Foreword by Guido Cozzi (University of St. Gallen, Switzerland)Advanced Macroeconomics covers selected topics in advance

*204*
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*11MB*

*English*
*Pages 172
[170]*
*Year 2020*

- Author / Uploaded
- Angus Chi Ho Chu

*Table of contents : ContentsForewordPrefaceAbout the AuthorAcknowledgementsChapter 1. A Static General-Equilibrium Model 1.1. The Model 1.2. Equilibrium 1.3. Elastic Labour Supply 1.4. Neutrality of Money 1.5. Summary 1.6. ExercisesChapter 2. The Neoclassical Growth Model 2.1. Household 2.2. Hamiltonian 2.3. Firm 2.4. Steady-State Equilibrium 2.5. Summary 2.6. ExercisesChapter 3. Dynamics in the Neoclassical Growth Model 3.1. Short-Run Effects of Technology 3.2. Long-Run Effects of Technology 3.3. Dynamics 3.4. Summary 3.5. ExercisesChapter 4. The Neoclassical Growth Model with Elastic Labour Supply 4.1. Household 4.2. Hamiltonian 4.3. Firm 4.4. Short-Run Effects of Technology 4.5. Long-Run Effects of Technology 4.6. Dynamics 4.7. Summary 4.8. ExercisesChapter 5. Fiscal Policy: Government Spending 5.1. Household 5.2. Government 5.3. Hamiltonian 5.4. Firm 5.5. Long-Run Effects of Government Spending 5.6. Short-Run Effects of Government Spending 5.7. Summary 5.8. ExercisesChapter 6. Fiscal Policy: Labour Income Tax 6.1. Household 6.2. Government 6.3. Hamiltonian 6.4. Firm 6.5. Long-Run Effects of Labour Income Tax 6.6. Short-Run Effects of Labour Income Tax 6.7. Government Spending Financed by Labour Income Tax 6.8. Summary 6.9. ExercisesChapter 7. Fiscal Policy: Capital Income Tax 7.1. Household 7.2. Government 7.3. Hamiltonian 7.4. Firm 7.5. Long-Run Effects of Capital Income Tax 7.6. Government Spending Financed by Capital Income Tax 7.7. Summary 7.8. ExercisesChapter 8. Monetary Policy in the New Keynesian Model 8.1. A Simple New Keynesian Model 8.2. Final Output 8.3. Intermediate Goods 8.4. Short-Run Effects of Monetary Policy 8.5. Long-Run Effects of Monetary Policy 8.6. Summary 8.7. ExercisesChapter 9. The Solow Growth Model 9.1. The Solow Model Without Technological Progress 9.2. The Solow Model with Technological Progress 9.3. Summary 9.4. ExercisesChapter 10. The Ramsey Model 10.1. Household 10.2. Hamiltonian 10.3. Steady State 10.4. Summary 10.5. ExercisesChapter 11. The Ramsey Model with a Perfectly Competitive Market 11.1. Household 11.2. Firm 11.3. Equilibrium 11.4. Summary 11.5. ExercisesChapter 12. The Ramsey Model with a Monopolistically Competitive Market 12.1. Household 12.2. Final Good 12.3. Intermediate Goods 12.4. Aggregation 12.5. Monopolistic Competition vs. Perfect Competition 12.6. Market Power of Monopolistic Firms 12.7. Summary 12.8. ExercisesChapter 13. The Romer Model of Endogenous Technological Change 13.1. Household 13.2. Final Good 13.3. Intermediate Goods 13.4. R&D 13.5. Aggregation 13.6. Solving the Model 13.6.1. R&D productivity 13.6.2. Discount rate 13.6.3. Capital and labour intensity in production 13.6.4. Labour force 13.7. Summary 13.8. ExercisesChapter 14. Scale Effect in the Romer Model 14.1. The Jones Model 14.2. Solving the Model 14.3. Summary 14.4. ExercisesChapter 15. R&D Underinvestment and Subsi 15.1. Socially Optimal Level of R&D 15.2. R&D Underinvestment 15.3. R&D Subsidies 15.4. Summary 15.5. ExercisesChapter 16. The Schumpeterian Growth Model 16.1. Household 16.2. Final Good 16.3. Intermediate Goods 16.4. R&D 16.5. Aggregation 16.6. Market Equilibrium Level of R&D 16.7. Socially Optimal Level of R&D 16.8. Capital 16.9. Summary 16.10. ExercisesChapter 17. Appendix on Dynamic OptimisationReferencesIndex*

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“Angus Chu’s book superbly manages to teach undergraduate students the intertemporal macroeconomic methods needed to fully understand the research going on in this ﬁeld. Hence it ﬁlls an important gap in the current didactic literature, which badly misses it and leaves Bachelor students unable to write, for example, a theoretical thesis. Without dwelling on abstract math, this book introduces intertemporal methods directly into the most important macroeconomics topics, spanning from the neoclassical workhorse growth and business cycle framework to the basic New Keynesian model to recent R&D-driven growth theory. I strongly recommend this book in any good economics curriculum.” Professor Guido Cozzi University of St. Gallen, Switzerland

“Angus Chu’s text is an excellent exposition of modern macroeconomics for advanced undergraduates, ﬁlling a gap between intermediate and advanced graduate material. It is distinctive as it systematically develops models from ﬁrst principles using dynamic optimisation techniques. Macroeconomics involves studying variables over time and this concise and lucid book introduces the framework of modern macroeconomics. It shows the development of ideas and to go from studying growth, to business cycles, to monetary and ﬁscal policy with the same framework and tools.” Professor Aditya Goenka University of Birmingham, UK

“The book introduces the tools of modern dynamic macroeconomics to an advanced undergraduate audience in a user-friendly way. It ﬁlls the gap between the standard undergraduate textbook with very little or no calculus and the graduate textbook with too much mathematics for an undergraduate course. The presentation of modern growth theory is excellent, allowing readers to access frontier tools and research topics at a fairly low technical cost. I envision this book being a useful tool in the classroom.” Professor Giammario Impullitti University of Nottingham, UK

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“This book provides an excellent overview of the key macroeconomic models with relevance for government policy. In doing so, it addresses not only the traditional models, but it also gives insights from recent research in this ﬁeld. The book is a useful toolkit for those who wish to go beyond the standard macroeconomic models; it provides various roadmaps to tackle challenging modern policy questions.” Professor Jozef Konings The University of Liverpool, UK

“This is a highly readable and easily accessible account of the modern tools and practices in macroeconomic theory and policy in the context of both developed and emerging market economies. It is essential reading for those with an interest in the issues of alternative approaches in policy modelling. This advanced textbook oﬀers an interesting blend of analytical, quantitative and policy insights about the best means to model and assess diﬀerent mechanisms of economic growth and macroeconomic policy questions.” Professor Sushanta Mallick Queen Mary University of London, UK

“Angus Chu covers the dynamic general-equilibrium approach to macroeconomics in a way that is accessible to advanced undergraduate students. This approach has the potential to change the way undergraduate students think about modern macroeconomics. This book covers a range of policy issues, such as ﬁscal policy, monetary policy, and innovation policy, in a coherent analytical framework, which has the advantage of helping students understand how diﬀerent topics in macroeconomics relate to each other.” Professor Pietro Peretto Duke University, USA

“An excellent textbook that covers a substantial number of advanced macroeconomic topics. This is a wonderful learning and reference guide designed for undergraduate students who want to explore advanced macroeconomics and can be easily adopted at postgraduate level. The textbook presents modern macroeconomic views via a selection of neoclassical and New Keynesian

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models. Concepts such as proﬁt maximisation, general equilibrium, consumption, labour, capital and money supply are carefully explained. The theoretical insights are accompanied by ﬁscal and monetary policies analysis. The book also covers a wealth of economic growth models including diﬀerent market structures and endogenous technological change. This stimulating book provides students with solid grounding and rigorous analysis of mathematical modelling. I particularly like the easy-to-follow structure and comprehensive language which are unique features possessed by this musthave book.” Professor Oleksandr Talavera University of Birmingham, UK

“Filling the gap between undergraduate and graduate textbooks has always been a challenge. Using Advanced Macroeconomics: An Introduction for Undergraduates, instructors can meet it. This well written textbook expertly bridges the gap between the two levels of study. The author starts with the one sector growth model, the workhorse model of modern macroeconomics. He extends this model in several directions in one step at the time to make the material easily accessible for students. In this way, students get well acquainted with the basic toolkit of modern macroeconomics, and the rigorous policy analysis shows them how and when the government can improve upon market outcomes.” Professor Akos Valentinyi University of Manchester, UK

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Foreword by Guido Cozzi

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Published by World Scientific Publishing Europe Ltd. 57 Shelton Street, Covent Garden, London WC2H 9HE Head office: 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601

Library of Congress Cataloging-in-Publication Data Names: Chu, Angus C., 1978– author. Title: Advanced macroeconomics : an introduction for undergraduates / Angus C. Chu, University of Liverpool, UK ; with contributions from Xilin Wang. Description: USA : World Scientific, 2020. | Includes bibliographical references and index. Identifiers: LCCN 2020037076 | ISBN 9781786349125 (hardcover) | ISBN 9781786349132 (ebook) | ISBN 9781786349149 (ebook other) Subjects: LCSH: Macroeconomics. Classification: LCC HB172.5 .C478 2020 | DDC 339--dc23 LC record available at https://lccn.loc.gov/2020037076

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Copyright © 2021 by Angus C. Chu All rights reserved.

For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/Q0267#t=suppl

Desk Editors: Aanand Jayaraman/Michael Beale/Shi Ying Koe

Typeset by Stallion Press Email: [email protected] Printed in Singapore

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In memory of Elsie

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Foreword

Undergraduate students often ﬁnd macroeconomics frustrating. Many books they encounter in their career explain the main theories in good detail, while surgically avoiding teaching the underlying intertemporal methods. This widespread approach leaves the students of macroeconomics in a disturbingly unclear state of mind. They suspect that there is something more rigorous behind the simple graphs they are shown but also realise that they cannot fully understand what it really is. Angus Chu’s Advanced Macroeconomics does quite the opposite: it follows a rigorous and crystal clear path from start to ﬁnish. During this process, students will gradually build up the analytical skills to eventually help them ﬁnd their way through the highly complex frontier of research in macroeconomics. I am sure that they will appreciate this eﬀort and be grateful to its author. Personally, I would not have expected anything less rigorous from a macroeconomic theory master like Angus Chu. Since I met him, many years ago, in a dinner at Shanghai, where half an hour of creative discussion about a potential new theory ended up in some twenty hand-written pages of perfectly consistent mathematical formulas, I know Angus would never deliver any half-explained or halfanalysed theory. In a book for undergraduate students, he would rather simplify the ornaments of some theories while keeping the focus at the highest analytical standard. This is exactly what is needed to build a sound bridge between bachelor and post-graduate studies. Equipped with a careful study of this book, students will be xi

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able to successfully start their theses without too much fear of reading the frontier literature. It will also avoid the scenario of students graduating and then becoming traumatised by the mathematics they suddenly see in their ﬁrst months of their Master or PhD classes. In studying this book, the reader will receive a complete training in macroeconomics, highlighting all the most important concepts and accompanying the student through the build-up of necessary methodologies. At the same time, this book, in the style of the author, will not be a nasty trainer but rather a gentle, gradual reinforcement of the reader’s skills. Concepts are introduced in a very digestible sequence, without putting too much burden on the learning process. Hence I think that students with one or two years of standard economics studies will relatively easily manage to digest and understand this book. With all that in mind, I wish the reader enjoys reading this book as much as I have. Professor Guido Cozzi University of St. Gallen, Switzerland

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Preface

This book covers selected topics in advanced macroeconomics at the undergraduate level and bridges the gap between intermediate macroeconomics for undergraduates and advanced macroeconomics for postgraduates. By building on materials in intermediate macroeconomics textbooks (e.g., Barro et al., 2017) and covering the mathematics of some classic dynamic general-equilibrium models, this book will give undergraduate students a ﬁrm appreciation of modern developments in macroeconomics. Dynamic general equilibrium is the foundation of modern macroeconomics. Chapter 1 begins with a simple static model to demonstrate the concept of general equilibrium. Chapters 2–4 cover the neoclassical growth model to explore the eﬀects of exogenous changes in the level of technology, which are an important source of business cycle ﬂuctuations. Chapters 5–7 use the neoclassical growth model to explore the eﬀects of ﬁscal policy instruments, such as government spending, labour income tax and capital income tax. Chapter 8 develops a simple New Keynesian model to analyse the eﬀects of monetary policy. Chapter 9 begins the analysis of economic growth by reviewing the Solow growth model. Chapters 10–12 present the Ramsey model and introduce diﬀerent market structures to the model to lay down the foundation of the Romer model. Chapter 13 incorporates an R&D sector into the Ramsey model with a monopolistically

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competitive market structure to develop the Romer model of endogenous technological change. Chapters 14 and 15 examine the implications of the Romer model. Chapter 16 concludes this book by presenting the Schumpeterian growth model and examining its diﬀerent implications from the Romer model.

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About the Author

Angus C. Chu was born in Hong Kong and studied at Simon Fraser University and the University of British Columbia in Canada. He obtained his PhD degree in Economics from the University of Michigan in the US. He is currently Chair in Macroeconomics at the University of Liverpool in the UK and was previously Professor of Economics at Fudan University in China. Angus’s research focuses on macroeconomics, monetary economics, economic growth, innovation and intellectual property rights. His research appears in journals, such as the European Economic Review, International Economic Review, Journal of Development Economics, Journal of Economic Growth, Journal of International Economics, Journal of Money, Credit and Banking, and the Review of Economic Dynamics. His textbook Intermediate Macroeconomics with Robert J. Barro and Guido Cozzi was published by Cengage Learning in 2017. According to IDEAS/RePEc Rankings, he is among the top 10% of economists in the world. He also serves as a Co-Editor-inChief for Economic Modelling.

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Acknowledgements

My biggest debt is to Amanda, Alvin and Alice for their loving support. I am also very grateful for the helpful comments and feedback from my students at Fudan University and the University of Liverpool.

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Contents

Foreword

xi

Preface

xiii

About the Author

xv

Acknowledgements

xvii

Chapter 1.

A Static General-Equilibrium Model

1

Chapter 2.

The Neoclassical Growth Model

9

Chapter 3.

Dynamics in the Neoclassical Growth Model

17

Chapter 4.

The Neoclassical Growth Model with Elastic Labour Supply

25

Chapter 5.

Fiscal Policy: Government Spending

35

Chapter 6.

Fiscal Policy: Labour Income Tax

45

Chapter 7.

Fiscal Policy: Capital Income Tax

55

Chapter 8.

Monetary Policy in the New Keynesian Model

63

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Contents

Chapter 9.

The Solow Growth Model

71

Chapter 10.

The Ramsey Model

79

Chapter 11.

The Ramsey Model with a Perfectly Competitive Market

87

The Ramsey Model with a Monopolistically Competitive Market

93

Chapter 12.

Chapter 13.

The Romer Model of Endogenous Technological Change

103

Chapter 14.

Scale Eﬀect in the Romer Model

113

Chapter 15.

R&D Underinvestment and Subsidies

121

Chapter 16.

The Schumpeterian Growth Model

129

Chapter 17.

Appendix on Dynamic Optimisation

139

References

143

Index

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Chapter 1

A Static General-Equilibrium Model

Dynamic general equilibrium is the foundation of modern macroeconomic models. In this chapter, we ﬁrst explore the concept of general equilibrium using a simple static model. Our simple economy involves two groups of economic agents: consumers and ﬁrms. We consider a representative household, which determines the behaviour of consumers. We also consider a representative ﬁrm, which determines the behaviour of ﬁrms. The representative household supplies labour and capital to the representative ﬁrm, which then uses these factor inputs to produce output and sells the output back to the household. We use this model to explore how changes in the level of technology aﬀect the labour market and the capital market and how the two markets interact with each other. 1.1.

The Model

In general, the representative household maximises utility. For now, we simply assume that the household supplies labour Ls and capital K s inelastically to earn a wage income W and a capital rental income R. Perfectly inelastic supply implies a vertical labour supply curve (i.e., Ls = L) and a vertical capital supply curve (K s = K), where L and K are exogenous parameters. We now consider the ﬁrm’s optimisation problem. The representative ﬁrm hires labour L and rents capital K from the household 1

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to produce output Y using the following Cobb–Douglas production function: Y = AK α L1−α ,

(1.1)

where the parameter α ∈ (0, 1) is the degree of capital intensity in production and A is the exogenous level of technology. The proﬁt function Π is Π = P Y − RK − W L.

(1.2)

We consider a perfectly competitive product market, in which the ﬁrm takes the market price P as given. To maximise proﬁt, we diﬀerentiate (1.2) with respect to K and L: ∂Y ∂Π =P − R = 0, ∂K ∂K

(1.3)

∂Y ∂Π =P − W = 0. ∂L ∂L

(1.4)

Rewriting (1.3) and (1.4) yields R ∂Y = αAK α−1 L1−α = , ∂K P

(1.5)

W ∂Y = (1 − α)AK α L−α = , ∂L P

(1.6)

which states that a proﬁt-maximising ﬁrm would equate the marginal product of capital to the real rental price R/P and the marginal product of labour to the real wage rate W/P . Equations (1.5) and (1.6) are the demand curves for capital and labour. To see this, they can be re-expressed as αA 1/(1−α) d L, (1.7) K = R/P (1 − α)A 1/α d K, (1.8) L = W/P where capital demand K d is decreasing in the real rental price R/P and labour demand Ld is decreasing in the real wage rate W/P . In other words, the demand for capital and labour is determined by

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the proﬁt-maximising behaviour of ﬁrms. We also have the following implications from (1.7) and (1.8). First, an increase in the level of technology increases the demand for capital and labour. Second, an increase in the level of labour increases the demand for capital. Third, an increase in the level of capital increases the demand for labour. 1.2.

Equilibrium

Combining the demand and supply curves of capital (labour) yields the equilibrium level of the real rental price (real wage rate) as follows: 1−α L R = αA , (1.9) P K α K W = (1 − α)A . (1.10) P L Together with the production function α

Y = AK L

1−α

,

(1.11)

we have the following results. First, an increase in the level of technology A raises the level of output Y , the real rental price R/P (by increasing capital demand) and the real wage rate W/P (by increasing labour demand); see Figures 1.1 and 1.2.

Figure 1.1.

Labour market: Inelastic labour supply.

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Figure 1.2.

Capital market: Inelastic capital supply.

Second, an increase in the level of capital K increases the level of output Y and the real wage rate W/P but decreases the real rental price R/P . Third, an increase in the level of labour L increases the level of output Y and the real rental price R/P but decreases the real wage rate W/P . As we can see, changes in the supply of one factor input (e.g., labour) not only aﬀect its own market (e.g., labour market) but also aﬀect the other market (e.g., capital market). This interaction between the two markets represents a general-equilibrium eﬀect.

1.3.

Elastic Labour Supply

Suppose we generalise the model by allowing for elastic labour supply. Then we have an upward-sloping labour supply curve in the labour market. In this case, an increase in the level of technology A raises the real wage rate and the equilibrium level of labour by shifting the labour demand curve to the right; see Figure 1.3. The resulting increase in the equilibrium level of labour in turn has a general-equilibrium eﬀect on the capital market by shifting the capital demand curve further to the right and causing an additional positive eﬀect on the real rental price; see Figure 1.4.

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1.4.

Figure 1.3.

Labour market: Elastic labour supply.

Figure 1.4.

Capital market: Elastic labour supply.

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5

Neutrality of Money

So far, we have only determined the rea1 wage rate W/P and the real rental price R/P . However, we haven’t determined the nominal wage rate W and the nominal rental price R, which in turn are determined by the price level P . To determine the price level, we introduce the quantity theory of money: M V = P Y,

(1.12)

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where M is the level of money supply and V is the velocity of money in the economy. For simplicity, we set V = 1.1 Substituting (1.11) into (1.12), we have P =

M M = α 1−α , Y AK L

(1.13)

which shows that the level of money supply in the economy determines the price level. When the central bank increases the level of money supply M , the price level P increases by the same proportion without aﬀecting the level of output Y . The rea1 wage rate W/P and the real rental price R/P also remain unchanged, whereas the nominal wage rate W and the nominal rental price R increase by the same proportion as the level of money supply M . To see this, we substitute (1.13) into (1.9) and (1.10) to derive α M K (1.14) P = (1 − α) , W = (1 − α)A L L 1−α M L (1.15) P =α . R = αA K K This neutrality of money would also hold when the supply of capital and/or labour is elastic. The neutrality of money arises because the price level in the economy changes immediately to oﬀset any change in the money supply.2 1.5.

Summary

In this chapter, we use a simple static general-equilibrium model to explore the eﬀects of technology on the economy. We ﬁnd that an increase in the level of technology raises the level of output, the real rental price and the real wage rate. In the case of elastic labour supply, the equilibrium level of labour also increases, which in turn has a general-equilibrium eﬀect on the capital market by shifting the capital demand curve further to the right and causing a larger 1

Our analysis would hold so long as the money velocity V is exogenous. In Chapter 8, we will consider a New Keynesian model in which the neutrality of money does not hold because prices do not adjust immediately. 2

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increase in the real rental price. We also explore the eﬀects of changes in the level of money supply and ﬁnd that money supply only aﬀects nominal variables (i.e., the price level, the nominal rental price and the nominal wage rate) without aﬀecting any of the real variables (i.e., the level of output, the real rental price, the real wage rate, the level of labour and the level of capital). This result is known as the classical dichotomy, according to which real and nominal variables can be determined separately. 1.6.

Exercises

1. How do changes in the level of technology aﬀect the labour and capital markets when both labour supply and capital supply are elastic? 2. Show that the neutrality of money holds when the supply of capital and/or labour is elastic. 3. Suppose the supply of capital is elastic. How does an increase in the supply of labour aﬀect the nominal variables? 4. Suppose the supply of labour is chosen by a utility-maximising household with the following utility function: βL2 , 2 where β determines the importance of leisure relative to consumption C. The household’s budget constraint is given by U =C−

P C ≤ W L + RK. Show that the labour supply function Ls is given by Ls = which is increasing in 5. How do changes in β markets when capital 6. How do changes in β markets when capital

1W , β P

the real wage rate W/P . in Exercise 4 aﬀect the labour and capital supply is perfectly inelastic (i.e., K = K)? in Exercise 4 aﬀect the labour and capital supply is elastic?

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Chapter 2

The Neoclassical Growth Model

In this chapter, we convert the static general-equilibrium model into a dynamic general-equilibrium model, which is the foundation of modern macroeconomics. Speciﬁcally, we consider the neoclassical growth model, in which the representative household chooses consumption and saving to maximise lifetime utility. To solve this dynamic optimisation problem, we use a mathematical tool known as the Hamiltonian.1 This analysis enables us to endogenise the equilibrium levels of macroeconomic variables, such as capital and output, in order to explore their determinants, such as the level of technology and the preference of the representative household. 2.1.

Household

In the neoclassical growth model, there is a representative household, which has a utility function ut at time t. For simplicity, we consider a log utility function ut = ln Ct that depends on consumption Ct . In other words, increasing consumption makes the household better oﬀ. Furthermore, the log utility function has a number of nice properties. For example, it features diminishing marginal utility, and the log

1

In intermediate microeconomics, students often use the Lagrangian for solving static constrained optimisation problems. In the case of dynamic optimisation problems, we use the Hamiltonian instead. 9

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of zero is negative inﬁnity so that the household would avoid zero consumption. A forward-looking household should not only care about utility at time t but also lifetime utility, which is given by U = u 0 + u1 + u2 + · · · =

T

ut ,

(2.1)

t=0

where T is the length of a lifetime. Equation (2.1) assumes that current utility and future utility carry the same weight, which is unrealistic because future consumption is often discounted. To capture discounting, we introduce a discount rate ρ > 0 so that (2.1) becomes ∞

U = u0 +

ut u2 u1 + + ··· = , 2 1 + ρ (1 + ρ) (1 + ρ)t

(2.2)

t=0

where we assume that a lifetime is long enough to be approximated by inﬁnity.2 In the rest of the analysis, we will use a mathematical tool known as Hamiltonian that solves dynamic optimisation problems in continuous time. Therefore, we need to rewrite (2.2) in continuous time using the integral as ∞ ∞ −ρt e ut dt = e−ρt ln Ct dt, (2.3) U= 0

0

where the continuous-time discount factor e−ρt replaces the discretetime discount factor (1 + ρ)−t . The household inelastically supplies L units of labour to earn a wage income Wt . Furthermore, it accumulates capital Kt and rents it to the representative ﬁrm to earn a capital-rental income Rt . We assume that capital is the only productive asset in the economy.3 If we 2

As t becomes very large, the discounting would make ut /(1 + ρ)t not to matter too much in the utility function U . 3 Introducing a bond that is in zero net supply would allow us to determine the real interest rate but would not aﬀect the rest of our analysis.

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11

normalise the price of output to unity,4 then the asset-accumulation equation is5 K˙ t = Rt Kt + Wt L − Ct ,

(2.4)

where K˙ t ≡ ∂Kt /∂t is the change in the level of capital with respect to time t. Here we have assumed a zero depreciation rate of capital (i.e., δ = 0).6 2.2.

Hamiltonian

The household maximises (2.3) subject to (2.4). To solve this dynamic optimisation problem, we use the Hamiltonian.7 The Hamiltonian function Ht is given by Ht = ln Ct + λt (Rt Kt + Wt L − Ct ).

(2.5)

In other words, the Hamiltonian function at time t consists of (a) the utility function ln Ct , (b) the right-hand side of the assetaccumulation equation Rt Kt + Wt L − Ct , and (c) a multiplier λt for the asset-accumulation equation. To maximise the household’s utility, we derive the ﬁrst-order conditions, which include 1 ∂Ht = − λt = 0, ∂Ct Ct ∂Ht = λt Rt = λt ρ − λ˙ t . ∂Kt

(2.6) (2.7)

Note that Kt is a state variable (i.e., a variable that accumulates over time), so we have to treat its ﬁrst-order condition diﬀerently. 4 Recall from Chapter 1 that money supply determines the price level without aﬀecting any of the real variables in the economy. 5 Here we assume that one unit of output can be converted into one unit of consumption or one unit of capital. 6 In general, the asset-accumulation equation is given by K˙ t = (Rt − δ)Kt + W t L − Ct . 7 See the appendix on dynamic optimisation for further details.

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12

Instead of equating ∂Ht /∂Kt to zero, we set ∂Ht /∂Kt = λt ρ − λ˙ t . Taking the log of (2.6) yields ln Ct = − ln λt .

(2.8)

Diﬀerentiating both sides of (2.8) with respect to t yields8 λ˙ t C˙ t =− . Ct λt

(2.9)

Substituting this equation into (2.7) yields λ˙ t C˙ t = − = Rt − ρ, Ct λt

(2.10)

which is the optimal path of consumption chosen by the household. The optimal consumption path in (2.10) states that when the rental price Rt is greater than the discount rate ρ, the household’s consumption should be increasing over time (i.e., C˙ t > 0). Intuitively, when the return to capital is high relative to the household’s discount rate, the household should decrease current consumption and increase saving in order to invest in capital. As a result, consumption is increasing over time. Conversely, when the rental price Rt is less than the discount rate ρ, the household’s consumption should be decreasing over time (i.e., C˙ t < 0). Intuitively, when the return to capital is low relative to the household’s discount rate, the household should increase current consumption and decrease investment in capital. As a result, consumption is decreasing over time. 2.3.

Firm

To derive the equilibrium of the economy, we also need to consider the ﬁrm’s optimisation problem, which is quite simple because of its static setting. There is a representative ﬁrm in the economy, and this ﬁrm hires labour Lt and rents capital Kt from the household to produce output Yt using the following Cobb–Douglas production 8

Note that

∂ ln Ct ∂t

=

1 ∂Ct Ct ∂t

=

C˙ t . Ct

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13

function: Yt = AKtα L1−α , t

(2.11)

where the parameter α ∈ (0, 1) is the degree of capital intensity in production and A is the exogenous level of technology. The proﬁt function Πt is Πt = Yt − Rt Kt − Wt Lt .

(2.12)

Recall that we have chosen Yt as the numeraire (i.e., the price of Yt is normalised to unity). Diﬀerentiating (2.12) with respect to Kt and Lt yields ∂Πt ∂Yt = − Rt = αAKtα−1 L1−α − Rt = 0, t ∂Kt ∂Kt ∂Yt ∂Πt = − Wt = (1 − α)AKtα L−α t − Wt = 0. ∂Lt ∂Lt

(2.13) (2.14)

These two equations are the demand functions for Kt and Lt . 2.4.

Steady-State Equilibrium

from (2.13) into (2.10) yields Substituting Rt = αAKtα−1 L1−α t C˙ t = αAKtα−1 L1−α − ρ, Ct

(2.15)

=M P Kt

where we have set Lt = L. Equation (2.15) shows that the optimal path of consumption is determined by the return to capital, which in turn is determined by the marginal product of capital M P Kt . Substituting Rt = αAKtα−1 L1−α and Wt = (1 − α)AKtα L−α from (2.13) and (2.14) into (2.4) yields the capital-accumulation equation: K˙ t = αAKtα L1−α + (1 − α)AKtα L1−α − Ct = AKtα L1−α − Ct . (2.16) This equation shows that the accumulation of capital is determined by capital investment, which is the diﬀerence between output and

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consumption. Equations (2.15) and (2.16) are two diﬀerential equations in Ct and Kt , and these two equations determine the behaviour of the economy. Now we solve for the steady-state equilibrium.9 In the steady state, all variables are constant, such that C˙ t = 0 and K˙ t = 0. Imposing C˙ t = 0 on the optimal consumption path in (2.15) yields the steady-state equilibrium level of capital: ∗

K =

αA ρ

1/(1−α)

L,

(2.17)

which is increasing in the level of technology A and decreasing in the discount rate ρ. Intuitively, a higher level of technology A increases the return to capital and encourages the household to accumulate more capital. In contrast, a higher discount rate makes future consumption less attractive to the household, which prefers current consumption and accumulates less capital. Equation (2.17) also allows us to quantify the eﬀects on K ∗ . For example, the elasticity of K ∗ with respect to A is 1/(1 − α). To see this, ∂ ln K ∗ /∂ ln A = 1/(1 − α).10 In other words, increasing technology A by 1% would lead to an increase in capital K ∗ by 1/(1 − α) percent. If α = 1/3, then 1/(1 − α) is 1.5. If α = 1/2, then 1/(1 − α) is 2. Using the production function in (2.11), we can derive the steadystate equilibrium level of output: ∗

∗ α

Y = A(K ) L

1−α

=

αA ρ

α/(1−α)

AL,

(2.18)

which is increasing in A via a direct eﬀect from A on Y and an indirect eﬀect from A on K ∗ . The elasticity of Y ∗ with respect to A is also 1/(1 − α).11 Steady-state output Y ∗ is decreasing in ρ because a smaller capital stock K ∗ reduces the level of output Y ∗ .

9

One can use the phase diagram of (2.15) and (2.16) to show that the economy converges to this steady state. ∗ ∗ 10 Note that ddlnlnKA = dK / dA , which is the elasticity of K ∗ with respect to A. K∗ A 11 Note that 1 + α/(1 − α) = 1/(1 − α).

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15

Imposing K˙ t = 0 on the capital-accumulation equation in (2.16) yields the steady-state equilibrium level of consumption: αA α/(1−α) ∗ ∗ α 1−α = AL, (2.19) C = A(K ) L ρ which is also increasing in A and decreasing in ρ because C ∗ = Y ∗ . In this special case of δ = 0, the steady-state equilibrium level of investment is I ∗ = Y ∗ − C ∗ = 0. However, in the more general case of δ > 0, the steady-state equilibrium level of investment would be positive; see Exercise 1 at the end of this chapter. 2.5.

Summary

In this chapter, we explore the concept of dynamic general equilibrium by developing the neoclassical growth model. The model features a utility-maximising representative household, which chooses consumption and saving optimally. We use the Hamiltonian to solve this dynamic optimisation problem and derive the household’s optimal consumption path, in which the growth rate of consumption is increasing in the rental price of capital and decreasing in the household’s discount rate. A proﬁt-maximising representative ﬁrm interacts with the utility-maximising household in the market economy that determines the allocation of resources in equilibrium. Then, we derive the steady-state equilibrium levels of capital and output, which are both increasing in the level of technology but decreasing in the household’s discount rate and also the depreciation rate of capital (see Exercise 1). 2.6.

Exercises

1. Consider a positive capital depreciation rate δ > 0. In this case, the asset-accumulation equation becomes K˙ t = (Rt − δ)Kt + Wt L − Ct .

(2.20)

Show that the optimal consumption path is given by C˙ t = Rt − δ − ρ Ct

(2.21)

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and that the steady-state equilibrium levels of {K ∗ , Y ∗ , I ∗ , C ∗ } are given by αA 1/(1−α) ∗ K = L, (2.22) ρ+δ αA α/(1−α) ∗ ∗ α 1−α = AL, (2.23) Y = A(K ) L ρ+δ αA 1/(1−α) ∗ ∗ δL, (2.24) I = δK = ρ+δ αA α/(1−α) ρ + (1 − α)δ ∗ ∗ ∗ AL. (2.25) C =Y −I = ρ+δ ρ+δ 2. Consider a positive capital depreciation rate δ > 0. Compute the elasticity of {K ∗ , Y ∗ , I ∗ , C ∗ } with respect to technology A. 3. How does an increase in the capital depreciation rate δ aﬀect the steady-state consumption rate C ∗ /Y ∗ ? What happens to this eﬀect as α → 0? 4. Suppose the household has the following instantaneous utility function: ut =

Ct1−σ − 1 , 1−σ

where σ > 0 determines the elasticity of intertemporal substitution as 1/σ. This functional form is known as the isoelastic utility function. When σ → 1, ut becomes ln Ct . Show that the optimal consumption path is given by 1 C˙ t = (Rt − δ − ρ), Ct σ in which the change in the growth rate of consumption with respect to a change in the return to capital is determined by 1/σ.

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Chapter 3

Dynamics in the Neoclassical Growth Model

In Chapter 2, we focused on the steady state of the neoclassical growth model. In other words, we only looked at the long-run eﬀects of changes in the level of technology. In this chapter, we use a graphical approach to demonstrate the short-run eﬀects of technology, which is a potential source of economic ﬂuctuations in business cycles. We also graphically demonstrate the long-run eﬀects of technology for a comparison with its short-run eﬀects. This analysis enables us to compare the eﬀects of changes in the level of technology on the macroeconomy at diﬀerent time horizons. In summary, we ﬁnd that the short-run and long-run supply curves of capital are drastically diﬀerent, which in turn have interesting implications on the macroeconomic eﬀects of technology. 3.1.

Short-Run Eﬀects of Technology

We deﬁne the short run as the moment when a parameter (e.g., the level of technology A) changes. At this moment, the level of capital Kt in the economy is predetermined and cannot be changed immediately. In other words, the short-run supply curve of capital is vertical. The assumption of perfectly inelastic supply of labour (i.e., Lt = L for all t) implies that the labour supply curve is also vertical. As for the

17

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demand curves of capital and labour, they are given by1 1−α Lt Rt = αA , Kt α Kt , Wt = (1 − α)A Lt

(3.1) (3.2)

which equate the rental price Rt to the marginal product of capital and the wage rate Wt to the marginal product of labour. An increase in the level of technology A shifts the demand curves of capital and labour to the right. As a result, the wage rate Wt and the rental price Rt go up. However, given the vertical supply curves of capital and labour, the levels of capital and labour do not change in the short run; see Figures 3.1 and 3.2. Then, the production function , Yt = AKtα L1−α t

(3.3)

implies that an increase in technology A gives rise to an increase in the level of output Yt despite the fact that capital and labour do not

Figure 3.1. 1

Labour market in the short run.

Recall that we have normalised Pt to unity.

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Figure 3.2.

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19

Capital market in the short run.

change in the short run. The short-run eﬀects of technology A can be summarised as follows: Short-run eﬀects of an increase in

3.2.

A

Y

K

R

L

W

increase

no change

increase

no change

increase

Long-Run Eﬀects of Technology

In the long run, the level of capital fully adjusts to its steady-state equilibrium level. So, what does the long-run supply curve of capital look like? Recall that the optimal consumption path derived from the household’s utility maximisation is given by C˙ t = Rt − ρ, Ct

(3.4)

where the parameter ρ > 0 is the household’s discount rate. In the steady state, we have C˙ t = 0. Therefore, the steady-state version of the optimal consumption path is given by Rt = ρ,

(3.5)

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which gives us a horizontal long-run supply curve of capital. In other words, the long-run supply curve of capital is perfectly elastic. The increase in the level of technology A has shifted the demand curves of capital and labour to the right. The horizontal long-run supply curve of capital implies that the rental price returns to the initial level whereas the equilibrium level of capital increases in the long run; see Figure 3.4. The increase in the equilibrium level of capital has an additional positive eﬀect on the wage rate by shifting the labour demand curve further to the right in the long run; see Figure 3.3.

Figure 3.3.

Labour market in the long run.

Figure 3.4.

Capital market in the long run.

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21

Then, the production function Yt = AKtα L1−α implies that the t increases in technology A and capital K both give rise to an increase in the steady-state equilibrium level of output Y ∗ . Furthermore, the increase in the steady-state equilibrium level of capital K ∗ implies that the long-run increases in the level of output and the wage rate are larger than their short-run increases. The long-run eﬀects of technology A can be summarised as follows: Long-run eﬀects of an increase in

3.3.

A

Y

K

R

L

W

increase

increase

no change

no change

increase

Dynamics

We can also predict what happens as the economy moves from the short run to the long run.2 After the level of technology A increases, the level of output Yt increases immediately whereas the level of capital Kt increases gradually, which leads to a further gradual increase in output Yt ; see Figures 3.5 and 3.6. As we move along the capital demand curve, the rental price Rt gradually decreases towards the initial level; see Figure 3.7. The increase in the level of capital gradually shifts the labour demand curve further to the right. As we move along the labour supply curve, the wage rate Wt increases further and gradually converges towards the new steady-state equilibrium level; see Figure 3.8. 3.4.

Summary

In this chapter, we use a graphical approach to examine the eﬀects of technology in the neoclassical growth model. We ﬁnd that the eﬀects of permanent changes in the level of technology vary across time because the short-run and long-run supply curves of capital are very diﬀerent. In the short run, the capital supply curve is perfectly inelastic, so that changes in the level of technology do not aﬀect the 2

For a more precise analysis, one could use the phase diagram of (2.15) and (2.16) to explore the dynamics of the economy.

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Figure 3.5.

Time path of capital.

Figure 3.6.

Time path of output.

Figure 3.7.

Time path of rental price.

equilibrium level of capital. In this case, an increase in the level of technology increases the level of output, the wage rate and the rental price without aﬀecting the equilibrium levels of labour and capital. In the long run, the capital supply curve becomes perfectly elastic, so that an increase in the level of technology raises the equilibrium level of capital, which in turn has a general-equilibrium eﬀect on the labour market by shifting the labour demand curve further to the right and

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Dynamics in the Neoclassical Growth Model

Figure 3.8.

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23

Time path of wage rate.

causing a larger increase in the wage rate. In this case, an increase in the level of technology increases the level of output, the wage rate and the equilibrium level of capital without aﬀecting the rental price and the equilibrium level of labour (due to the assumption of perfectly inelastic labour supply). Finally, an unrealistic implication of the neoclassical growth model is that the level of labour (i.e., employment) never changes. In the next chapter, we will consider elastic labour supply. 3.5.

Exercises

1. What are the short-run and long-run eﬀects of an increase in the level of labour supply L on {Yt , Kt , Rt , Wt }? 2. What are the short-run and long-run eﬀects of an increase in the household’s discount rate ρ on {Yt , Kt , Rt , Wt }? 3. Consider a positive capital depreciation rate δ > 0. Derive the long-run capital supply curve. How does the capital depreciation rate δ aﬀect the long-run capital supply curve?

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Chapter 4

The Neoclassical Growth Model with Elastic Labour Supply

In Chapter 3, we considered perfectly inelastic labour supply in the neoclassical growth model, in which case the level of labour (i.e., employment) never changes. In this chapter, we generalise the neoclassical growth model to allow for elastic labour supply chosen by the utility-maximising household.1 In summary, the supply of labour is determined by a substitution eﬀect and an income eﬀect, which are both inﬂuenced by changes in technology. Therefore, this modiﬁcation of elastic labour supply allows for ﬂuctuations in employment, which are an important feature of business cycles. The neoclassical growth model with elastic labour supply is essentially a special case of the real business cycle (RBC) model.2 4.1.

Household

We introduce the choice of leisure into the household’s utility function: ∞ e−ρt [ln Ct + β ln(L − lt )]dt, (4.1) U= 0

1

See Romer (2018, chapter 11) for other approaches of modelling unemployment. The RBC model was developed by Kydland and Prescott (1982), who received the Nobel Memorial Prize in Economics in 2004 partly for this contribution. See McCandless (2008) and Romer (2018, chapter 5) for a textbook treatment of the RBC model. 2

25

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where the parameter ρ > 0 is the household’s discount rate and the parameter β > 0 determines the importance of leisure L − lt relative to consumption Ct in the utility function. L is the household’s total labour endowment, and lt is the level of employment chosen by the household. The household elastically supplies lt units of labour to earn a wage income Wt . Furthermore, it accumulates capital Kt and rents it to the representative ﬁrm to earn a capital-rental income Rt . The asset-accumulation equation is modiﬁed as follows: K˙ t = Rt Kt + Wt lt − Ct ,

(4.2)

where we have assumed a zero depreciation rate of capital (i.e., δ = 0).3 4.2.

Hamiltonian

The Hamiltonian function is given by Ht = ln Ct + β ln(L − lt ) + λt (Rt Kt + Wt lt − Ct ).

(4.3)

The ﬁrst-order conditions include β ∂Ht =− + λt Wt = 0, ∂lt L − lt 1 ∂Ht = − λt = 0, ∂Ct Ct ∂Ht = λt Rt = λt ρ − λ˙ t . ∂Kt

(4.4) (4.5) (4.6)

Recall that Kt is a state variable (i.e., a variable that accumulates over time), so we have to set ∂Ht /∂Kt = λt ρ − λ˙ t . Combining (4.4) and (4.5) yields the labour supply curve lts given by lts = L −

βCt , Wt

(4.7)

which is increasing in the wage rate Wt (i.e., a substitution eﬀect) and decreasing in consumption Ct (i.e., an income eﬀect). Intuitively, a 3 In general, the asset-accumulation equation is given by K˙ t = (Rt − δ)Kt + W t lt − C t .

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27

higher wage rate increases the opportunity cost of leisure and causes the household to supply more labour, which captures a substitution eﬀect, whereas a higher level of consumption decreases the marginal utility of consumption from wage income and causes the household to enjoy more leisure, which captures an income eﬀect. If leisure is not important to the household (i.e., β = 0), then we have lts = L, in which case we are back to the case of perfectly inelastic labour supply in the previous chapter. If leisure matters to the household (i.e., β > 0), then unemployment L − lts is positive. Taking the log of (4.5) and substituting it into (4.6) yields the optimal consumption path: C˙ t = Rt − ρ, Ct

(4.8)

which is the same as in the previous chapter. 4.3.

Firm

The ﬁrm’s optimisation problem is the same as before. There is a representative ﬁrm in the economy, and this ﬁrm hires labour and rents capital from the household to produce output using the following production function: Yt = AKtα lt1−α ,

(4.9)

where the parameter α ∈ (0, 1) is the degree of capital intensity in production and A is the exogenous level of technology. The proﬁt function Πt is Πt = Yt − Rt Kt − Wt lt ,

(4.10)

where we have chosen Yt as the numeraire (i.e., the price of Yt is normalised to unity). Diﬀerentiating (4.10) with respect to Kt and lt yields 1−α ∂Yt ∂Πt lt = − Rt = αA − Rt = 0, (4.11) ∂Kt ∂Kt Kt α ∂Yt ∂Πt Kt = − Wt = (1 − α)A − Wt = 0. (4.12) ∂lt ∂lt lt These two equations are the demand functions for Kt and lt .

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Short-Run Eﬀects of Technology

Once again, we deﬁne the short run as the moment when a parameter (e.g., the level of technology A) changes. At this moment, the level of capital in the economy is predetermined. In other words, the shortrun supply curve of capital is vertical as before. However, we now have an upward-sloping labour supply curve given by Wt =

βCt . L − lt

(4.13)

As for the demand curves of labour and capital, they are given by 1−α Yt lt =α , (4.14) Rt = αA Kt Kt α Yt Kt = (1 − α) . (4.15) Wt = (1 − α)A lt lt An increase in the level of technology A shifts the demand curves of labour and capital to the right. In the labour market, the wage rate Wt and the equilibrium level of labour lt increase.4 Therefore, elastic labour supply gives rise to a positive eﬀect of technology A on employment lt , which in turn has a general-equilibrium eﬀect on the capital market. In the capital market, the rental price Rt increases, whereas the equilibrium level of capital does not change in the short run; see Figures 4.1 and 4.2. Then, the production function Yt = AKtα lt1−α

(4.16)

implies that the increases in technology A and labour lt both give rise to an increase in the level of output Yt . The short-run eﬀects of technology A can be summarised as follows: Short-run eﬀects of an increase in A

4

Y

K

R

l

W

increase

no change

increase

increase

increase

Technically, the labour supply curve may shift due to changes in consumption Ct . We ignore this eﬀect in the short run.

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The Neoclassical Growth Model with Elastic Labour Supply

4.5.

Figure 4.1.

Labour market in the short run.

Figure 4.2.

Capital market in the short run.

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29

Long-Run Eﬀects of Technology

In the long run, the level of capital fully adjusts to its steady-state equilibrium level K ∗ . Recall that the optimal consumption path is given by C˙ t = Rt − ρ. Ct

(4.17)

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In the steady state, we have C˙ t = 0. Therefore, the long-run supply curve of capital is perfectly elastic and given by Rt = ρ.

(4.18)

The increase in the level of technology A shifts the demand curves of capital and labour to the right. In the capital market, the rental price Rt returns to the initial level whereas the equilibrium level of capital increases in the long run; see Figure 4.4. In the labour market, we now allow the labour supply curve to shift. As can be seen in Figure 4.3, the labour supply curve shifts to the left (due to an increase in the level of consumption) and completely oﬀsets the shift in the labour demand curve. Therefore, the equilibrium level of labour lt returns to the initial level. However, there continues to be a positive eﬀect on the wage rate Wt in the long run; see Figure 4.3. To see that the steady-state equilibrium level of labour returns to the initial level, we substitute (4.15) into (4.7) to derive lt = L −

βCt βCt =L− lt . Wt (1 − α)Yt

(4.19)

Given the assumption of a zero capital depreciation rate (i.e., δ = 0), the steady-state equilibrium level of investment I ∗ is zero. Therefore, the steady-state equilibrium levels of consumption and output are the

Figure 4.3.

Labour market in the long run.

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Figure 4.4.

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31

Capital market in the long run.

same such that C ∗ = Y ∗ . Therefore, (4.19) implies that the steadystate equilibrium level of labour l∗ is given by l∗ =

L , 1 + β/(1 − α)

(4.20)

which is independent of technology A. In other words, changes in the level of technology A do not aﬀect the steady-state equilibrium level of labour l∗ . Furthermore, the steady-state equilibrium level of labour l∗ is decreasing in leisure preference β and capital intensity α. Intuitively, if leisure becomes more important (i.e., a larger β) to the household, it would supply less labour to the labour market. Similarly, if labour becomes less important in production (i.e., a larger α), the ﬁrm would demand less labour in the labour market. Finally, the production function Yt = AKtα lt1−α implies that the increases in technology A and capital K both give rise to an increase in the steady-state equilibrium level of output Y ∗ . Furthermore, the increase in the steady-state equilibrium level of capital K ∗ and the leftward shift in the labour supply curve imply that the long-run increase in the wage rate is larger than its short-run increase. The long-run eﬀects of technology A can be summarised as follows: Long-run eﬀects of an increase in A

Y

K

R

l

W

increase

increase

no change

no change

increase

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4.6.

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Dynamics

Given the additional adjustment in labour, the dynamics becomes more complicated in the neoclassical growth model with elastic labour supply, but we can still conjecture what happens as the economy moves from the short run to the long run. After the level of technology A increases, the level of output Yt increases immediately whereas the level of capital Kt increases gradually. As we move along the capital demand curve, the rental price Rt gradually decreases towards the initial level. The increase in the level of capital gradually shifts the labour demand curve further to the right. Simultaneously, the labour supply curve gradually shifts to the left. These shifts in the labour demand and supply curves give rise to a gradual increase in the wage rate Wt towards the new steady-state equilibrium level and a gradual decrease in the level of labour lt towards the initial level.

4.7.

Summary

In this chapter, we extend the neoclassical growth model by allowing for elastic labour supply. In the model, the representative household chooses leisure in addition to consumption and saving. Maximising the household’s utility, we derive the labour supply curve. The household’s supply of labour is increasing in the wage rate, which captures a substitution eﬀect, and decreasing in the level of consumption, which captures an income eﬀect. Given the upward-sloping labour supply curve, an increase in the level of technology shifts the labour demand curve to the right and increases the equilibrium level of labour in the short run, which in turn has a general-equilibrium eﬀect on the capital market by shifting the capital demand curve further to the right and causing a larger increase in the rental price. In this case, an increase in the level of technology increases the level of output, the level of labour, the wage rate and the rental price without aﬀecting the level of capital in the short run. In the long run, the higher level of consumption gives rise to an income eﬀect on labour supply and shifts the labour supply curve to the left. As a result, the equilibrium level of labour returns to the initial level. However, the long-run capital supply curve becomes perfectly elastic, and the

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equilibrium level of capital increases in the long run. In this case, an increase in the level of technology increases the level of output, the level of capital and the wage rate without aﬀecting the rental price and the level of labour in the long run. 4.8.

Exercises

1. Consider a positive capital depreciation rate δ > 0. In this case, the asset-accumulation equation becomes K˙ t = (Rt − δ)Kt + Wt lt − Ct .

(4.21)

Show that the steady-state equilibrium level of labour l∗ is given by5 l∗ = 1+

β 1−α

L 1−

αδ ρ+δ

,

(4.22)

which continues to be independent of technology A. In other words, changes in the level of technology A do not aﬀect the steady-state equilibrium level of labour l∗ even in the presence of capital depreciation. Furthermore, (4.22) shows that the steadystate equilibrium level of labour l∗ is increasing in the capital depreciation rate δ and decreasing in the discount rate ρ. Intuitively, an increase in the capital depreciation rate δ or a decrease in the discount rate ρ reduces the consumption rate C/Y , which in turn decreases leisure and increases labour. 2. Derive the steady-state equilibrium levels of capital K ∗ and output Y ∗ . 3. Consider an alternative utility function given by ∞ e−ρt (ln Ct − βlt ) dt. (4.23) U= 0

Show that the labour supply curve becomes perfectly elastic. How does an increase in consumption Ct shift the labour supply curve? 5

Hint: Note that C ∗ /Y ∗ = 1 − I ∗ /Y ∗ = 1 − δK ∗ /Y ∗ .

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4. Consider the following household’s lifetime utility function: ∞ 1+ϕ l dt, U= e−ρt ln Ct − t 1+ϕ 0 where ϕ > 0. Derive the labour supply curve.

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Chapter 5

Fiscal Policy: Government Spending

The neoclassical growth model not only allows us to analyse the eﬀects of technology but also allows us to perform policy analysis. We now begin our analysis of government policies in the neoclassical growth model. We will consider a number of ﬁscal policy instruments in this chapter and the following chapters. The policy instrument that we consider in this chapter is government spending.1 Speciﬁcally, we analyse the macroeconomic eﬀects of changes in government spending in the neoclassical growth model with elastic labour supply, which is a crucial feature because the expansionary eﬀects of government spending operate through an income eﬀect on labour supply. 5.1.

Household

The representative household’s utility function U is given by2 ∞ U= e−ρt [ln Ct + β ln(L − lt )]dt, (5.1) 0

1

Here we only consider permanent changes in government spending; see Baxter and King (1993) for a comparison between permanent versus temporary changes in government spending in the neoclassical growth model. 2 We can generalise the utility function to allow government spending Gt to ∞ improve the household’s utility; e.g., U = 0 e−ρt [ln Ct + θ ln(L − lt ) + V (Gt )]dt. All our results would hold so long as Gt does not aﬀect the marginal utility of consumption and leisure. 35

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36

where the parameter ρ > 0 is the household’s discount rate and the parameter β > 0 determines the importance of leisure L − lt relative to consumption Ct in the utility function. L is the household’s total labour endowment, and lt is the level of employment chosen by the household. The household elastically supplies lt units of labour to earn a wage income Wt . Furthermore, it accumulates capital Kt and rents it to the representative ﬁrm to earn a capital-rental income Rt . The asset-accumulation equation is K˙ t = Rt Kt + Wt lt − Ct − Tt ,

(5.2)

where the capital depreciation rate is zero and Tt is a lump-sum tax.3 5.2.

Government

The government collects tax revenue Tt to pay for government spending Gt . The balanced budget condition is Gt = Tt . We deﬁne the ratio of government spending to output as γ ≡ Gt /Yt . We are interested in the eﬀects of changes in γ on other macroeconomic variables. 5.3.

Hamiltonian

The Hamiltonian function of the household is given by Ht = ln Ct + β ln(L − lt ) + λt (Rt Kt + Wt lt − Ct − Tt ).

(5.3)

The ﬁrst-order conditions include β ∂Ht =− + λt Wt = 0, ∂lt L − lt 1 ∂Ht = − λt = 0, ∂Ct Ct ∂Ht = λt Rt = λt ρ − λ˙ t . ∂Kt

(5.4) (5.5) (5.6)

Recall that Kt is a state variable (i.e., a variable that accumulates over time), so we have to set ∂Ht /∂Kt = λt ρ − λ˙ t . 3

We will consider other tax instruments in the next two chapters.

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37

Combining (5.4) and (5.5) yields the labour supply curve lts given by lts = L −

βCt , Wt

(5.7)

which is increasing in the wage rate Wt (i.e., a substitution eﬀect) and decreasing in consumption Ct (i.e., an income eﬀect). Unless β = 0, unemployment L − lts is positive. Taking the log of (5.5) and substituting it into (5.6) yields the optimal consumption path: C˙ t = Rt − ρ. Ct

(5.8)

In summary, the labour supply curve and the optimal consumption path are the same as before. 5.4.

Firm

The ﬁrm’s optimisation problem is also the same as before. There is a representative ﬁrm in the economy, and this ﬁrm hires labour lt and rents capital Kt from the household to produce output Yt using the following Cobb–Douglas production function: Yt = AKtα lt1−α ,

(5.9)

where the parameter α ∈ (0, 1) is the degree of capital intensity in production and A is the exogenous level of technology. The proﬁt function Πt is Πt = Yt − Rt Kt − Wt lt ,

(5.10)

where we have implicitly chosen Yt as the numeraire (i.e., the price of Yt is normalised to unity). Diﬀerentiating (5.10) with respect to Kt and lt yields 1−α ∂Πt ∂Yt lt = − Rt = αA − Rt = 0, (5.11) ∂Kt ∂Kt Kt α ∂Yt ∂Πt Kt = − Wt = (1 − α)A − Wt = 0. (5.12) ∂lt ∂lt lt

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These two equations are the demand functions for Kt and lt . In summary, the demand functions for Kt and lt are also the same as before. 5.5.

Long-Run Eﬀects of Government Spending

We start with the long-run eﬀects of permanent changes in government spending. In the long run, the level of capital fully adjusts to its steady-state equilibrium level. Recall that the optimal consumption path is given by C˙ t = Rt − ρ. Ct

(5.13)

In the steady state, we have C˙ t = 0. Therefore, the long-run supply curve of capital is perfectly elastic and given by Rt = ρ,

(5.14)

whereas the labour supply curve is Wt =

βCt . L − lt

(5.15)

As for the demand curves of capital and labour, they are given by Rt = αA

lt Kt

1−α

Wt = (1 − α)A

Kt lt

=α α

Yt , Kt

= (1 − α)

(5.16) Yt . lt

(5.17)

Combining labour supply in (5.15) and labour demand in (5.17) yields lt = L −

βCt βCt =L− lt . Wt (1 − α)Yt

(5.18)

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39

Given the assumption of a zero capital depreciation rate (i.e., δ = 0), the steady-state equilibrium level of investment I ∗ is zero. Therefore, the steady-state equilibrium level of consumption is given by C ∗ = Y ∗ − G∗ = (1 − γ)Y ∗ ,

(5.19)

which is proportional to the steady-state equilibrium level of output. Substituting (5.19) into (5.18) yields the steady-state equilibrium level of labour l∗ given by l∗ =

L , 1 + β(1 − γ)/(1 − α)

(5.20)

which is increasing in the government-spending ratio γ. Intuitively, an increase in government spending γ raises tax T and reduces the after-tax income to cause a negative income eﬀect on the household, which then consumes less leisure and supplies more labour l. Graphically, it shifts the labour supply curve to the right; as a result, the equilibrium level of labour l increases and the wage rate W decreases; see Figure 5.1. In the capital market, the increase in the level of labour shifts the capital demand curve to the right. Given the horizontal long-run capital supply curve, the rental price R remains at the initial level whereas the equilibrium level of capital increases in the long run; see Figure 5.2. The increase in capital shifts the labour demand curve to the right. As a result, the wage rate W

Figure 5.1.

Labour market in the long run.

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Figure 5.2.

Capital market in the long run.

increases and returns to the initial level because the capital–labour ratio K/l is independent of γ.4 Finally, the production function Yt = AKtα lt1−α implies that the increases in labour and capital both give rise to an increase in the steady-state equilibrium level of output Y ∗ . The long-run eﬀects of government spending γ can be summarised as follows: Long-run eﬀects of an increase in γ

5.6.

Y

K

R

l

W

increase

increase

no change

increase

no change

Short-Run Eﬀects of Government Spending

To complete our analysis, we now look at the short-run eﬀects of permanent changes in government spending. We deﬁne the short run as the moment when the parameter γ changes. At this moment, the level of capital in the economy is predetermined. In other words, the short-run supply curve of capital is vertical. As before, an increase in government spending γ raises tax and reduces the after-tax income to cause a negative income eﬀect on the household, which then consumes 4

Note from (5.17) that W is increasing in K/l. Then, note from (5.16) that R is decreasing in K/l and R = ρ does not change in the steady state.

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41

less leisure and supplies more labour. Graphically, it shifts the labour supply curve to the right; as a result, the equilibrium level of labour increases and the wage rate decreases; see Figure 5.3. In the capital market, the increase in the level of labour shifts the capital demand curve to the right. Given the vertical short-run capital supply curve, the rental price increases whereas the equilibrium level of capital remains unchanged in the short run; see Figure 5.4. Finally, the production function Yt = AKtα lt1−α implies that the increase in labour

Figure 5.3.

Labour market in the short run.

Figure 5.4.

Capital market in the short run.

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gives rise to an increase in the level of output. The short-run eﬀects of government spending γ can be summarised as follows: Short-run eﬀects of an increase in γ

5.7.

Y

K

R

l

W

increase

no change

increase

increase

decrease

Summary

In this chapter, we explore the eﬀects of permanent changes in the level of government spending in the neoclassical growth model with elastic labour supply. We ﬁnd that an increase in the level of government spending is accompanied by an increase in taxation, which in turn gives rise to an income eﬀect on labour supply and shifts the labour supply curve to the right. In the short run, the equilibrium level of labour increases and the wage rate decreases while the level of output increases. The increase in labour causes a general-equilibrium eﬀect on the capital market by shifting the capital demand curve to the right. As a result, the rental price of capital increases because the short-run capital supply curve is perfectly inelastic. Given that the capital supply curve becomes perfectly elastic in the long run, the equilibrium level of capital increases and in turn aﬀects the labour market by shifting the labour demand curve to the right. At the end, the equilibrium level of labour increases by an even larger amount whereas the wage rate returns to the initial level. In summary, an increase in the level of government spending has an expansionary eﬀect on the macroeconomy and increases the levels of output, capital and labour without aﬀecting the rental price and the wage rate in the long run. 5.8.

Exercises

1. Consider a positive capital depreciation rate δ > 0. In this case, the asset-accumulation equation becomes K˙ t = (Rt − δ)Kt + Wt lt − Ct − Tt .

(5.21)

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43

Show that the steady-state equilibrium level of labour l∗ is given by l∗ =

L β 1 + 1−α 1−γ −

αδ ρ+δ

,

(5.22)

which continues to be increasing in the government-spending ratio γ. 2. Derive the steady-state equilibrium levels of capital K ∗ and output Y ∗ . 3. What are the long-run eﬀects of government spending γ on consumption and investment? 4. Consider the following government’s budget constraint: Tt = Gt + T Rt = γYt + T Rt , where T Rt is the transfer payment from the government to the household. In this case, the household’s asset-accumulation equation is given by K˙ t = Rt Kt + Wt lt − Ct − Tt + T Rt . Suppose the lump-sum tax Tt is ﬁxed, and the transfer payment of government T Rt adjusts endogenously to oﬀset the change in γ and balance the government’s budget constraint. Does the presence of T Rt cause any diﬀerence to the eﬀects of γ on the labour and capital markets in the short run and long run?

b2530 International Strategic Relations and China’s National Security: World at the Crossroads

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Chapter 6

Fiscal Policy: Labour Income Tax

In this chapter, we continue our analysis of ﬁscal policies in the neoclassical growth model. In Chapter 5, we considered a lump-sum tax, which is a very rare ﬁscal policy instrument in reality. Therefore, we now consider a labour income tax, which is a more realistic tax instrument. Once again, we use in this policy analysis the neoclassical growth model with elastic labour supply, which is a crucial feature because the contractionary eﬀects of labour income tax operate through a substitution eﬀect on labour supply. 6.1.

Household

As before, the household’s utility function is given by ∞ U= e−ρt [ln Ct + β ln(L − lt )]dt,

(6.1)

0

where the parameter ρ > 0 is the household’s discount rate and the parameter β > 0 determines the importance of leisure L − lt relative to consumption Ct in the utility function. L is the household’s total labour endowment, and lt is the level of employment chosen by the household. The household elastically supplies lt units of labour to earn an after-tax wage income (1 − τW )Wt , where τW > 0 is the tax rate on labour income. Furthermore, the household accumulates capital Kt and rents it to the representative ﬁrm to earn a capital-rental

45

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income Rt . The asset-accumulation equation is K˙ t = Rt Kt + (1 − τW )Wt lt − Ct − Tt ,

(6.2)

where the capital depreciation rate is zero and Tt is a lump-sum tax. 6.2.

Government

The government collects tax revenue to pay for government spending Gt . The balanced budget condition is Gt = Tt + τW Wt lt . We deﬁne the ratio of government spending to output as γ ≡ Gt /Yt . We are interested in the eﬀects of changes in the labour income tax rate τW on other macroeconomic variables. In the previous chapter, we saw that changes in Gt cause an income eﬀect on the household. To separate this income eﬀect from our analysis, we assume that changes in the labour income tax rate τW are balanced by changes in the lump-sum tax Tt while the government-spending ratio γ does not change. Therefore, changes in the labour income tax rate τW only give rise to a substitution eﬀect on the household’s labour supply. 6.3.

Hamiltonian

The Hamiltonian function of the household is given by Ht = ln Ct + β ln(L − lt ) + λt [Rt Kt + (1 − τW )Wt lt − Ct − Tt ]. (6.3) The ﬁrst-order conditions include ∂Ht β =− + λt (1 − τW )Wt = 0, ∂lt L − lt 1 ∂Ht = − λt = 0, ∂Ct Ct ∂Ht = λt Rt = λt ρ − λ˙ t . ∂Kt

(6.4) (6.5) (6.6)

Recall that Kt is a state variable (i.e., a variable that accumulates over time), so we have to set ∂Ht /∂Kt = λt ρ − λ˙ t . Combining (6.4)

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and (6.5) yields the labour supply curve lts given by lts = L −

βCt , (1 − τW )Wt

(6.7)

which is increasing in the wage rate Wt but decreasing in the labour income tax rate τW . Taking the log of (6.5) and substituting it into (6.6) yields the optimal consumption path: C˙ t = Rt − ρ. Ct

(6.8)

In summary, the optimal consumption path is the same as before, whereas the labour supply curve now depends on the after-tax wage rate (1 − τW )Wt . 6.4.

Firm

The ﬁrm’s optimisation problem is the same as before. There is a representative ﬁrm in the economy, and this ﬁrm hires labour and rents capital from the household to produce output using the following production function: Yt = AKtα lt1−α ,

(6.9)

where the parameter α ∈ (0, 1) is the degree of capital intensity in production and A is the exogenous level of technology. The proﬁt function Πt is Πt = Yt − Rt Kt − Wt lt ,

(6.10)

where we have implicitly chosen Yt as the numeraire (i.e., the price of Yt is normalised to unity). Diﬀerentiating (6.10) with respect to Kt and lt yields 1−α ∂Yt ∂Πt lt = − Rt = αA − Rt = 0, (6.11) ∂Kt ∂Kt Kt α ∂Yt ∂Πt Kt = − Wt = (1 − α)A − Wt = 0. (6.12) ∂lt ∂lt lt

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These two equations are the demand functions for Kt and lt . In summary, the demand functions for Kt and lt are also the same as before. 6.5.

Long-Run Eﬀects of Labour Income Tax

We start with the long-run eﬀects of labour income tax. In the long run, the level of capital fully adjusts to its steady-state equilibrium level. Recall that the optimal consumption path is given by C˙ t = Rt − ρ. Ct

(6.13)

In the steady state, we have C˙ t = 0. Therefore, the long-run supply curve of capital is perfectly elastic and given by Rt = ρ,

(6.14)

whereas the labour supply curve is Wt =

1 βCt . 1 − τ W L − lt

(6.15)

As for the demand curves of capital and labour, they are given by Rt = αA

lt Kt

1−α

Wt = (1 − α)A

Kt lt

=α α

Yt , Kt

= (1 − α)

(6.16) Yt . lt

(6.17)

Combining labour supply in (6.15) and labour demand in (6.17) yields lt = L −

βCt βCt =L− lt . (1 − τW )Wt (1 − τW )(1 − α)Yt

(6.18)

Given the assumption of a zero capital depreciation rate (i.e., δ = 0), the steady-state equilibrium level of investment I ∗ is zero. Therefore,

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49

the steady-state equilibrium level of consumption is given by C ∗ = Y ∗ − G∗ = (1 − γ)Y ∗ ,

(6.19)

which is proportional to the steady-state equilibrium level of output. Substituting (6.19) into (6.18) yields the steady-state equilibrium level of labour l∗ given by l∗ =

L

1+

β 1−α

1−γ 1−τW

,

(6.20)

which is decreasing in the labour income tax rate τW . Intuitively, an increase in the labour income tax rate τW reduces the after-tax wage rate, which in turn causes the household to substitute leisure for consumption (i.e., a substitution eﬀect) and supply less labour l. Graphically, it shifts the labour supply curve to the left; as a result, the equilibrium level of labour l decreases and the pretax wage rate W increases; see Figure 6.1. In the capital market, the decrease in the level of labour shifts the capital demand curve to the left. Given the horizontal long-run capital supply curve, the rental price Rt remains at the initial level whereas the equilibrium level of capital decreases in the long run; see Figure 6.2. The decrease in capital shifts the labour demand curve to the left; see Figure 6.1. As a result, the pre-tax wage rate W decreases and returns to the initial

Figure 6.1.

Labour market in the long run.

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Figure 6.2.

Capital market in the long run.

level because the capital–labour ratio K/l is independent of τW .1 Although the pre-tax wage rate W does not change in the long run, the after-tax wage rate (1 − τW )W decreases. Finally, the production function Yt = AKtα lt1−α implies that the decreases in labour and capital both give rise to a decrease in the steady-state equilibrium level of output Y ∗ . The long-run eﬀects of labour income tax τW can be summarised as follows: Long-run eﬀects of an increase in τW

6.6.

Y

K

R

l

W

decrease

decrease

no change

decrease

no change

Short-Run Eﬀects of Labour Income Tax

To complete our analysis, we now look at the short-run eﬀects of labour income tax. We deﬁne the short run as the moment when the parameter τW changes. At this moment, the level of capital in the economy is predetermined. In other words, the short-run supply curve of capital is vertical. As before, an increase in the labour income tax rate τW reduces the after-tax wage rate, which in turn causes the 1

Note from (6.17) that W is increasing in K/l. Then, note from (6.16) that R is decreasing in K/l and R = ρ does not change in the steady state.

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Figure 6.3.

Labour market in the short run.

Figure 6.4.

Capital market in the short run.

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51

household to substitute leisure for consumption (i.e., a substitution eﬀect) and supply less labour. Graphically, it shifts the labour supply curve to the left; as a result, the equilibrium level of labour decreases and the pre-tax wage rate W increases; see Figure 6.3. In the capital market, the decrease in the level of labour shifts the capital demand curve to the left. Given the vertical short-run capital supply curve, the rental price decreases whereas the equilibrium level of capital remains unchanged in the short run; see Figure 6.4. Finally, the production function Yt = AKtα lt1−α implies that the decrease in labour gives rise to a decrease in the level of output. The short-run

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eﬀects of labour income tax rate τW can be summarised as follows: Short-run eﬀects of an increase in τW

6.7.

Y

K

R

l

W

decrease

no change

decrease

decrease

increase

Government Spending Financed by Labour Income Tax

What happens when the government uses labour income tax to ﬁnance its spending? In this case, we eliminate lump-sum tax Tt and modify the balanced budget condition as Gt = τW Wt lt , which can be re-expressed as γ = (1 − α)τW

(6.21)

using Gt = γYt and Wt lt = (1 − α)Yt . In other words, whenever the government increases spending γ, it has to raise the labour income tax rate according to τW = γ/(1−α). Substituting this equation into (6.20) yields l∗ =

1+β

L

1−γ 1−α−γ

,

(6.22)

which is decreasing in γ unless α → 0. In other words, the substitution eﬀect of government spending γ via labour income tax τW dominates the income eﬀect of government spending (unless the degree of capital intensity α approaches zero, in which case they cancel each other). As a result, the steady-state equilibrium levels of capital and output are decreasing in γ.2 6.8.

Summary

In this chapter, we explore the eﬀects of permanent changes in the labour income tax rate in the neoclassical growth model with elastic 2

Barro and Redlick (2011) provide empirical evidence that the estimated tax multiplier is larger than the spending multiplier in absolute terms such that the overall balanced-budget multiplier for government spending is negative.

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labour supply. We ﬁnd that by decreasing the after-tax wage rate, an increase in labour income tax gives rise to a substitution eﬀect on labour supply and shifts the labour supply curve to the left. In the short run, the equilibrium level of labour decreases and the pre-tax wage rate increases while the level of output decreases. The decrease in labour causes a general-equilibrium eﬀect on the capital market by shifting the capital demand curve to the left. As a result, the rental price of capital decreases because the short-run capital supply curve is perfectly inelastic. Given that the capital supply curve becomes perfectly elastic in the long run, the equilibrium level of capital decreases and in turn aﬀects the labour market by shifting the labour demand curve to the left. At the end, the equilibrium level of labour decreases by an even larger amount whereas the pre-tax wage rate returns to the initial level. In summary, an increase in the labour income tax rate has a contractionary eﬀect on the macroeconomy and decreases the levels of output, capital and labour without aﬀecting the rental price and the pre-tax wage rate in the long run. 6.9.

Exercises

1. Consider a positive capital depreciation rate δ > 0. In this case, the asset-accumulation equation becomes K˙ t = (Rt − δ)Kt + (1 − τW )Wt lt − Ct − Tt .

(6.23)

Show that the steady-state equilibrium level of labour l∗ is given by l∗ =

L β 1 + (1−α)(1−τ 1−γ− W)

αδ ρ+δ

,

(6.24)

which is decreasing in the labour income tax rate τW for a given γ.3 2. Derive the steady-state equilibrium levels of capital K ∗ and output Y ∗ . 3

Substituting τW = γ/(1 − α) into (6.24), one can show that l∗ continues to be decreasing in γ as in Section 6.7 unless α → 0 or ρ → 0.

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3. What are the long-run eﬀects of labour income tax τW on consumption and investment? 4. Suppose the government uses labour income tax to ﬁnance its spending and the capital depreciation rate is positive. Derive the steady-state equilibrium levels of {l∗ , K ∗ , I ∗ , Y ∗ , C ∗ }.

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Chapter 7

Fiscal Policy: Capital Income Tax

In this chapter, we conclude our analysis of ﬁscal policy in the neoclassical growth model. In Chapter 6, we considered a labour income tax. However, labour income is not the only source of income that is taxed by the government. In this chapter, we consider instead a capital income tax, which is another tax instrument that we commonly observe in reality.1 In this case, we ﬁnd that capital income tax is also contractionary by decreasing the accumulation of capital. 7.1.

Household

The household’s utility function is given by ∞ e−ρt [ln Ct + β ln(L − lt )]dt, U=

(7.1)

0

where the parameter ρ > 0 is the household’s discount rate and the parameter β > 0 determines the importance of leisure L − lt relative to consumption Ct in the utility function. lt is the level of employment chosen by the household. The household elastically supplies lt units of labour to earn a wage income Wt . Furthermore, the household

1 We do not consider government bonds here. Barro (1974) provides a theoretical foundation for the Ricardian equivalence proposition and shows that “shifts between debt and tax ﬁnance for a given amount of public expenditure would have no ﬁrst-order eﬀect”. See also Romer (2018, Chapter 13).

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accumulates capital Kt and rents it to the representative ﬁrm to earn an after-tax capital-rental income (1 − τR )Rt , where τR > 0 is the tax rate on capital income. The asset-accumulation equation is K˙ t = (1 − τR )Rt Kt + Wt lt − Ct − Tt ,

(7.2)

where the capital depreciation rate is zero and Tt is a lump-sum tax. 7.2.

Government

The government collects tax revenue to pay for government spending Gt . The balanced budget condition is Gt = Tt + τR Rt Kt . We deﬁne the ratio of government spending to output as γ ≡ Gt /Yt . We are interested in the eﬀects of changes in the capital tax rate τR on other macroeconomic variables. In Chapter 5, we saw that changes in Gt cause an income eﬀect on the household. To separate this income eﬀect from our analysis, we assume that changes in the capital tax rate τR are balanced by changes in the lump-sum tax Tt while the government-spending ratio γ does not change. 7.3.

Hamiltonian

The Hamiltonian function of the household is given by Ht = ln Ct + β ln(L − lt ) + λt [(1 − τR )Rt Kt + Wt lt − Ct − Tt ]. (7.3) The ﬁrst-order conditions include β ∂Ht =− + λt Wt = 0, ∂lt L − lt 1 ∂Ht = − λt = 0, ∂Ct Ct ∂Ht = λt (1 − τR )Rt = λt ρ − λ˙ t . ∂Kt

(7.4) (7.5) (7.6)

Recall that Kt is a state variable (i.e., a variable that accumulates over time), so we have to set ∂Ht /∂Kt = λt ρ − λ˙ t . Combining (7.4)

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and (7.5) yields the labour supply curve lts given by lts = L −

βCt , Wt

(7.7)

which is increasing in the wage rate Wt (i.e., a substitution eﬀect) and decreasing in consumption Ct (i.e., an income eﬀect). Taking the log of (7.5) and substituting it into (7.6) yields the optimal consumption path: C˙ t = (1 − τR )Rt − ρ, Ct

(7.8)

which now depends on the after-tax rental price (1 − τR )Rt . 7.4.

Firm

The ﬁrm’s optimisation problem is the same as before. There is a representative ﬁrm in the economy, and this ﬁrm hires labour and rents capital from the household to produce output using the following production function: Yt = AKtα lt1−α ,

(7.9)

where the parameter α ∈ (0, 1) is the degree of capital intensity in production and A is the exogenous level of technology. The proﬁt function Πt is Πt = Yt − Rt Kt − Wt lt ,

(7.10)

where we have implicitly chosen Yt as the numeraire (i.e., the price of Yt is normalised to unity). Diﬀerentiating (7.10) with respect to Kt and lt yields 1−α ∂Yt ∂Πt lt = − Rt = αA − Rt = 0, (7.11) ∂Kt ∂Kt Kt α ∂Yt ∂Πt Kt = − Wt = (1 − α)A − Wt = 0. (7.12) ∂lt ∂lt lt These two equations are the demand functions for Kt and lt . In summary, the demand functions for Kt and lt are also the same as before.

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Long-Run Eﬀects of Capital Income Tax

Equation (7.8) shows that capital income tax aﬀects the household’s optimal consumption path, which in turn determines the supply of capital. However, because the short-run capital supply curve is perfectly inelastic, capital income tax has no eﬀect on the equilibrium level of capital at the moment when the tax rate τR changes. Therefore, we focus on the long-run eﬀects of capital income tax in our analysis.2 In the long run, the level of capital fully adjusts to its steady-state equilibrium level. Recall that the optimal consumption path is given by C˙ t = (1 − τR )Rt − ρ. Ct

(7.13)

In the steady state, we have C˙ t = 0. Therefore, the long-run supply curve of capital is perfectly elastic and given by Rt =

ρ , 1 − τR

(7.14)

whereas the labour supply curve is Wt =

βCt . L − lt

(7.15)

As for the demand curves of capital and labour, they are given by 1−α Yt lt =α , (7.16) Rt = αA Kt Kt α Yt Kt = (1 − α) . (7.17) Wt = (1 − α)A lt lt

2

It can be shown that consumption jumps up at the moment when the tax rate τR increases. As a result, the labour supply curve shifts to the left, reducing the equilibrium levels of labour and output in the short run. Then, consumption gradually decreases towards a lower steady-state equilibrium level, shifting the labour supply curve to the right in the long run as we show in Figure 7.1.

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Figure 7.1.

Labour market in the long run.

Figure 7.2.

Capital market in the long run.

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An increase in the capital income tax rate τR reduces the aftertax capital rental price and causes the household to accumulate less capital. Graphically, it shifts the capital supply upwards; as a result, the equilibrium level of capital K decreases in the capital market; see Figure 7.2. In the labour market, the decrease in the level of capital shifts the labour demand curve to the left. However, as can be seen from Figure 7.1, the labour supply curve shifts to the right

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(due to a negative income eﬀect from the reduction in output Y and consumption C) to completely oﬀset the shift in the labour demand curve such that the equilibrium level of labour remains at the initial level.3 To derive the steady-state equilibrium level of labour, we combine labour supply in (7.15) and labour demand in (7.17) to obtain lt = L −

βCt βCt =L− lt . Wt (1 − α)Yt

(7.18)

Given the assumption of a zero capital depreciation rate (i.e., δ = 0), the steady-state equilibrium level of investment I ∗ is zero. Therefore, the steady-state equilibrium level of consumption is given by C ∗ = Y ∗ − G∗ = (1 − γ)Y ∗ ,

(7.19)

which is proportional to the steady-state equilibrium level of output. Substituting (7.19) into (7.18) yields the steady-state equilibrium level of labour l∗ given by l∗ =

L , 1 + β(1 − γ)/(1 − α)

(7.20)

which is independent of the capital income tax rate τR . In other words, changes in the capital tax rate τR do not aﬀect the steadystate equilibrium level of labour l∗ (in the absence of capital depreciation). Finally, the production function Yt = AKtα lt1−α implies that the decrease in capital gives rise to a decrease in the steady-state equilibrium level of output Y ∗ . Therefore, the long-run eﬀects of capital income tax τR can be summarised as follows: Long-run eﬀects of an increase in τR

3

Y

K

R

l

W

decrease

decrease

increase

no change

decrease

This result is due to the zero capital depreciation rate; see Exercise 1 at the end of this chapter.

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Government Spending Financed by Capital Income Tax

What happens when the government uses capital income tax to ﬁnance its spending? In this case, we eliminate lump-sum tax Tt and modify the balanced budget condition as Gt = τR Rt Kt , which can be re-expressed as γ = ατR ,

(7.21)

using Gt = γYt and Rt Kt = αYt . In other words, whenever the government increases spending γ, it has to raise the capital income tax rate according to τR = γ/α. Given that the capital income tax rate τR does not appear in (7.20), an increase in government spending γ increases the steady-state equilibrium level of labour as in Chapter 5. However, the resulting increase in the capital income tax rate τR leads to a decrease in the steady-state equilibrium level of capital as in Figure 7.2. Therefore, when government spending is ﬁnanced by capital income tax, its overall eﬀect on the steady-state equilibrium level of output is ambiguous. 7.7.

Summary

In this chapter, we explore the long-run eﬀects of permanent changes in the capital income tax rate in the neoclassical growth model with elastic labour supply. We ﬁnd that an increase in the capital income tax rate causes the household to accumulate less capital and reduces the steady-state equilibrium level of capital, which in turn has a general-equilibrium eﬀect on the labour market by shifting the labour demand curve to the left and by depressing the wage rate and the equilibrium level of labour. The decrease in the levels of capital and output reduces the level of consumption and gives rise to an income eﬀect, which shifts the labour supply curve to the right. As a result, the wage rate decreases by a larger amount and the equilibrium level of labour may return to the initial level (depending on the capital depreciation rate).4 In summary, an increase in the capital income tax 4

See Exercise 1 at the end of this chapter.

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rate has a contractionary eﬀect on the macroeconomy by decreasing the wage rate and the levels of output, capital and generally labour but raises the pre-tax rental price of capital such that the after-tax rental price remains unchanged in the long run. 7.8.

Exercises

1. Consider a positive capital depreciation rate δ > 0. In this case, the asset-accumulation equation becomes K˙ t = [(1 − τR )Rt − δ]Kt + Wt lt − Ct − Tt .

(7.22)

Show that the optimal consumption path is given by C˙ t = (1 − τR )Rt − δ − ρ, Ct

(7.23)

and that the steady-state equilibrium level of labour l∗ is given by l∗ = 1+

β 1−α

L 1−γ−

αδ ρ+δ (1

. − τR )

(7.24)

Under a positive capital depreciation rate δ > 0, although the labour supply curve shifts to the right as in Figure 7.1, it only partly oﬀsets the shift in the labour demand curve such that the equilibrium level of labour l∗ becomes decreasing in τR for a given γ.5 2. Derive the steady-state equilibrium levels of capital K ∗ and output Y ∗ . 3. What are the long-run eﬀects of capital income tax τR on consumption and investment? 4. Suppose the government uses capital income tax to ﬁnance its spending and the capital depreciation rate is positive. Derive the steady-state equilibrium levels of {l∗ , K ∗ , I ∗ , Y ∗ , C ∗ }.

5

Substituting τR = γ/α into (7.24), one can show that l∗ continues to be increasing in γ as in Section 7.6 unless ρ → 0.

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Chapter 8

Monetary Policy in the New Keynesian Model

In this chapter, we consider monetary policy. Recall that in the neoclassical growth model, prices are fully ﬂexible. In this case, changes in the level of money supply do not have any eﬀect on real variables (i.e., the neutrality of money). Therefore, we need to introduce sticky prices into our model.1 However, before we can consider sticky prices, we need to ﬁrst develop a model in which ﬁrms have price-setting power. In other words, ﬁrms need to have the power to set their prices before they can decide whether or not to change their prices. Consequently, we need to convert the market structure from perfect competition to monopolistic competition.2 In summary, we develop a New Keynesian model and ﬁnd that increasing the money supply has an expansionary eﬀect on the macroeconomy in the short run by increasing the demand for goods.3

1 In the 1990s, economists started to introduce sticky prices into the RBC model. This combination of neoclassical economics and Keynesian macroeconomics is known as the new neoclassical synthesis; see e.g., Kimball (1995) for an early study. 2 The model of monopolistic competition was developed by Dixit and Stiglitz (1977). 3 See Gali (2015) and Romer (2018, Chapters 6 and 7) for a textbook treatment of the New Keynesian model.

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A Simple New Keynesian Model

Given the complexity of the ﬁrm side, we keep the household side as simple as possible. Speciﬁcally, the household has an upward-sloping labour supply curve and a perfectly inelastic capital supply curve in the short run. We focus our analysis on the short run because sticky prices are a short-run phenomenon and monetary policy only has short-run eﬀects. In the long run, prices become fully ﬂexible, and the eﬀects of monetary policy become neutral. On the ﬁrm side, we need to distinguish between competitive ﬁrms that produce a ﬁnal good and monopolistic ﬁrms that produce intermediate goods. There are N monopolistic ﬁrms that are indexed by i ∈ [1, N ] and sell diﬀerentiated intermediate goods yi . The production function of monopolistic ﬁrm i is given by yi = AKiα li1−α ,

(8.1)

where α ∈ (0, 1) and {Ki , li } are capital and labour employed by ﬁrm i. There is also a representative ﬁrm that produces ﬁnal output Y by combining the diﬀerentiated intermediate goods using the following production function: N 1/ε Y = yiε , (8.2) i=1

which is known as a constant elasticity of substitution (CES) aggregator in which the parameter ε ∈ (0, 1) determines the substitution elasticity 1/(1 − ε) between the diﬀerentiated intermediate goods. As ε approaches unity, the substitution elasticity 1/(1 − ε) goes to inﬁnity, in which case the intermediate goods become perfect substitutes. In other words, the degree of substitutability between products is increasing in ε. The less substitutable the products are (i.e., a smaller ε), the more market power the monopolistic ﬁrms have. 8.2.

Final Output

A representative ﬁrm produces ﬁnal good, and the proﬁt function Π is N 1/ε N N ε Π = PY − pi y i = P yi − pi y i , (8.3) i=1

i=1

i=1

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where P is the price of ﬁnal good Y and pi is the price of intermediate good yi . The market structure in this sector is perfectly competitive, and the ﬁrm takes the prices as given. The ﬁrst-order condition with respect to yi is 1 ∂Π = P ∂yi ε

N

1−ε ε yiε

εyiε−1 − pi = 0,

(8.4)

i=1

which can be expressed as pi = P Y

1−ε ε−1 yi

⇔

yid

=

P pi

1/(1−ε)

Y.

(8.5)

This is the demand function yid , in which the demand elasticity is 1/(1− ε). As ε approaches unity, the demand elasticity 1/(1− ε) goes to inﬁnity, in which case the demand curve for yi becomes perfectly elastic. 8.3.

Intermediate Goods

A monopolistic ﬁrm produces intermediate product i. The proﬁt function πi is given by πi = pi yi − W li − RKi .

(8.6)

Here, we make the following assumption to simplify our analysis: the level of capital supplied to each ﬁrm is ﬁxed in the short run. Under this assumption, ﬁrm i can only change its labour input whenever yi changes in the short run.4 The market structure in this sector is monopolistically competitive, so that the ﬁrm has a price-setting power. In other words, the ﬁrm sets its own price pi , instead of taking it as given.

4

If both capital and labour inputs can adjust, then we need to ﬁrst perform cost minimisation to derive the marginal cost function before deriving the proﬁtmaximising price; see the exercises at the end of this chapter.

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Substituting the demand function in (8.5) into the proﬁt function in (8.6) yields πi = P Y 1−ε yiε − W li − RKi ,

(8.7)

where {W, R} are the wage rate and the capital rental price as before. Diﬀerentiating (8.7) with respect to li yields ∂yi ∂πi = εP Y 1−ε yiε−1 − W = 0, ∂li

∂li

(8.8)

=pi

which can be re-expressed as ﬁrm i’s labour demand: W = εpi M P Li ⇔ lid =

ε(1 − α)pi yi , W

(8.9)

where ε < 1.5 In other words, a proﬁt-maximising monopolistic ﬁrm would set its value of marginal product of labour above the wage rate (i.e., W < pi M P Li ). Equation (8.9) can also be re-expressed as pi =

1 W 1 = M Ci , ε M P Li ε

(8.10)

where W/M P Li is the marginal cost M Ci of production. Given that ε < 1, the monopolistic price is above the marginal cost of production (i.e., pi > M Ci ). The markup ratio 1/ε > 1 implies that the monopolistic ﬁrm makes a positive proﬁt. The positive monopolistic proﬁt enables the ﬁrm to allow its price to temporarily deviate from its proﬁt-maximising level without making a loss. For example, (8.10) shows that when the wage rate W increases, the ﬁrm would want to raise its price pi to maximise proﬁt. However, there may be some frictions (e.g., a menu cost) that prevent an immediate price adjustment. We summarise these frictions as sticky prices. When prices are sticky, monopolistic ﬁrms may not be able to maximise proﬁt, but they would continue production so long as pi > M Ci . 5

Therefore, aggregate labour demand in the real wage rate W/P .

N

d i=1 li

= ε(1 − α)P Y /W is decreasing

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Short-Run Eﬀects of Monetary Policy

In this section, we explore the short-run eﬀects of monetary policy. We deﬁne the short run as the duration in which the prices of ﬁrms do not change. Recall that the demand function yid is given by 1/(1−ε) P d yi = Y, (8.11) pi which is decreasing in the relative price pi /P and increasing in aggregate output Y . To relate aggregate output to the level of money supply in the economy, we introduce the quantity theory of money given by M V = P Y,

(8.12)

where M is the level of money supply and V is the velocity of money in the economy. For simplicity, we set V = 1. Substituting (8.12) into (8.11) yields 1/(1−ε) M P d , (8.13) yi = pi P which relates the demand function yid to the level of money supply M . Given the assumption of sticky prices pi , the aggregate price level P is also ﬁxed in the short run.6 Therefore, an increase in the level of money supply M would increase the demand yid for product i ∈ [1, N ] by increasing aggregate output Y . Suppose the level of money supply M increases by a small amount given by ΔM > 0. Then, the increase in the demand yid is 1/(1−ε) ΔM P d . (8.14) Δyi = pi P To satisfy the increased demand for its product, ﬁrm i ∈ [1, N ] needs to employ more labour, and the increase in the labour demand lid is 1/(1−ε) Δyid 1 ΔM P d , (8.15) = Δli = α M P Li (1 − α)A(Ki /li ) pi P 6

ε/(ε−1) (ε−1)/ε It can be shown that P = [ N ] . i=1 pi

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Figure 8.1.

Labour market in the short run.

where we have used the deﬁnition of the marginal product of labour M P Li ≡ Δyi /Δli . Given that all ﬁrms i ∈ [1, N ] demand more d labour, aggregate labour demand increases by N i=1 Δli . Graphically, the labour demand curve shifts to the right. In the labour market, the equilibrium level of labour l and the real wage rate W/P increase; see Figure 8.1. Therefore, the short-run eﬀects of an increase in money supply M can be summarised as follows: Short-run eﬀects of an increase in M

8.5.

Y

l

W/P

W

increase

increase

increase

increase

Long-Run Eﬀects of Monetary Policy

An increase in the level of money supply has an expansionary eﬀect on the economy, but only in the short run. When prices become fully ﬂexible in the long run, the price level P increases by the same proportion as the level of money supply M . Then, the expansionary eﬀect disappears, and the real variables {Y, l, W/P } return to their initial levels. Therefore, in the long run, an increase in the level of money supply M only causes the nominal variables {P, W, R} to increase by the same proportion without aﬀecting the real variables. This result is known as the neutrality of money and the classical dichotomy.

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69

Summary

In this chapter, we explore the eﬀects of monetary policy in a New Keynesian model. To allow for sticky prices, we consider a monopolistically competitive product market in which monopolistic ﬁrms have price-setting power. This price-setting power enables each ﬁrm to price its diﬀerentiated product above the marginal cost of production. The presence of this markup allows the ﬁrm to let its price temporarily deviate from the proﬁt-maximising level while still making a positive monopolistic proﬁt. When prices are sticky in the short run, an increase in the level of money supply increases the demand for goods, which in turn increases the demand for factor inputs (e.g., labour). In this case, an increase in the level of money supply has an expansionary eﬀect on the economy by increasing the level of output, the level of labour, the real wage rate and the nominal wage rate. However, this expansionary eﬀect disappears when prices fully adjust in the long run, in which case the higher level of money supply increases the price level and the nominal wage rate without aﬀecting the level of output, the level of labour and the real wage rate. 8.7.

Exercises

1. Suppose all the monopolistic ﬁrms i ∈ [1, N ] can now adjust their capital input Ki in addition to labour input li when yi changes in the short run. Show that the marginal cost of producing yi is given by7 1−α W 1 R α M Ci = . (8.16) A α 1−α 2. Given (8.16), show that the proﬁt-maximising price pi = M Ci /ε continues to hold. Also, derive the amount of monopolistic proﬁt πi and show that πi → 0 as ε → 1. Explain the intuition. 3. Suppose monopolistic ﬁrms can adjust their capital input Ki in addition to labour input li when yi changes in the short run. How does an increase in the level of money supply aﬀect the capital market in the short run? Hint: Minimise T Ci ≡ W li + RKi subject to yi = AKiα li1−α ≥ Q, where Q is an arbitrary number. Then, M Ci = ∂T Ci /∂Q. 7

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Chapter 9

The Solow Growth Model

The level of income varies drastically across countries with the richest country being more than 100 times richer than the poorest country in the world. According to the International Monetary Fund, GDP (at purchasing power parity) per capita across countries ranges from about $700 in the Central African Republic to about $130,000 in Qatar in 2018. Suppose we consider two hypothetical countries that have a real GDP per capita of $1,000 in the early 19th century. If one country grows at 1% per year, then after two centuries this country would have a real GDP per capita of about $7,300, which is roughly the income level of India and Vietnam. If the other country grows at 2% per year, then after two centuries this country would have a real GDP per capita of about $52,500, which is roughly the income level of Australia and Germany. Therefore, a small diﬀerence in the growth rates accumulated over a long period of time can lead to very large income diﬀerences. However, what determines the rate of economic growth? In this chapter, we begin our analysis of economic growth.1 A seminal model of economic growth is the Solow growth model.2 We will review the useful insights that one can obtain from this model 1

Other textbook treatments of economic growth include Aghion and Howitt (2009) and Jones and Vollrath (2013) at the advanced undergraduate level and Acemoglu (2009), Barro and Sala-i-Martin (2003), Galor (2011) and Romer (2018, chapters 1–4) at the postgraduate level. 2 The Solow growth model was developed by Solow (1956), who received the Nobel Memorial Prize in Economics in 1987 partly for this contribution. Swan 71

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and also discuss its limitations. In summary, the Solow model shows that capital accumulation cannot sustain long-run economic growth, which requires technological progress; however, the model treats technological progress as exogenous and does not inform us on its determinants. The Solow model consists of the following components: an aggregate production function; an accumulation equation for capital; and an exogenous saving rate.

9.1.

The Solow Model Without Technological Progress

Output Yt at time t is produced by the Cobb–Douglas production function: , Yt = AKtα L1−α t

(9.1)

where the parameter α ∈ (0, 1) is the degree of capital intensity in production and A is the exogenous level of technology. Kt is the stock of capital that has been accumulated as of time t. Lt is the size of the labour force in the economy at time t. For simplicity, we assume that the size of the labour force is constant, and we normalise Lt to unity so that other variables can be interpreted as per capita variables (e.g., output per capita). The second key equation in this model is the accumulation equation for capital given by K˙ t = It − δKt ,

(9.2)

where the parameter δ > 0 is the depreciation rate of capital, K˙ t ≡ ∂Kt /∂t denotes the change in the stock of capital with respect to time t, and It is capital investment. In this closed economy without

(1956) also developed a similar growth model; therefore, the Solow model is often referred to as the Solow–Swan model.

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a government sector, the national income account is simply Y t = Ct + I t ,

(9.3)

where Ct is consumption in the economy at time t. The Solow growth model is quite simple because it assumes an exogenous saving (or investment) rate denoted by s. In other words, s≡

Ct It =1− . Yt Yt

(9.4)

To solve this model, we substitute (9.1) and (9.4) into (9.2) to obtain K˙ t = sAKtα − δKt ,

(9.5)

where we have used Lt = 1. Equation (9.5) is a one-dimensional diﬀerential equation in Kt . Imposing K˙ t = 0 on (9.5) yields the steady-state level of capital given by ∗

K =

sA δ

1/(1−α)

.

(9.6)

Equation (9.5) implies that whenever Kt < K ∗ , Kt would increase over time until it reaches K ∗ ; see Figure 9.1. Similarly, whenever, Kt > K ∗ , Kt would decrease over time until it reaches K ∗ . Equation (9.6) shows that K ∗ is increasing in the saving rate s and the level of technology A but decreasing in the depreciation rate δ.

Figure 9.1.

Phase diagram.

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When the level of capital converges to its steady state K ∗ , the level of output also reaches its steady-state level given by Y ∗ = A(K ∗ )α . Although Y ∗ is increasing in the saving rate s, the level of output is stationary in the long run. Why doesn’t the output keep growing in the long run? To answer this question, we rewrite (9.5) to derive an expression for the growth rate of capital. sA K˙ t = 1−α − δ. Kt Kt

(9.7)

This equation shows that as Kt increases, the growth rate of capital K˙ t /Kt decreases and eventually converges to a long-run value of zero. An increase in the saving rate s would increase the growth rate of capital in the short run, but the growth rate of capital always converges to zero in the long run; see Figure 9.1. The reason behind this convergence process is decreasing returns to scale (i.e., α < 1) with respect to capital in the production function. As capital increases, output increases; however, the additional output that the additional capital produces is decreasing. This diminishing marginal product of capital implies that the additional investment created by the additional output is also decreasing. Given that capital accumulation requires capital investment, the growth rate of capital decreases and converges to zero. If the production function instead features constant returns to scale (i.e., α = 1), then the long-run growth rate of capital would be K˙ t /Kt = sA − δ, which remains positive so long as sA > δ. 9.2.

The Solow Model with Technological Progress

The previous section shows that in the more plausible case of decreasing returns to scale (i.e., α < 1), the stock of capital would converge to a steady state without economic growth in the long run. This result arises because there is no technological progress (i.e., A is assumed to be a constant parameter). In the rest of this section, we analyse the more interesting case in which At is a variable that grows over time according to an exogenous growth rate gA ≡ A˙ t /At > 0. Taking the natural log of the production function Yt = At Ktα yields ln Yt = ln At + α ln Kt .

(9.8)

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Diﬀerentiating this equation with respect to t yields3 Y˙ t A˙ t K˙ t = +α , Yt At Kt

(9.9)

where A˙ t /At = gA is an exogenous parameter. Substituting (9.4) into (9.2) and dividing by Kt yields Yt K˙ t =s − δ. Kt Kt

(9.10)

In the long run, the economy is on a balanced growth path (BGP) along which each variable grows at a constant rate.4 Because K˙ t /Kt is constant on the BGP, Yt /Kt is also constant implying that output Yt and capital Kt grow at the same rate on the BGP. Using this information and (9.9), we can now derive the long-run growth rate of output Yt and capital Kt on the BGP given by K˙ t gA Y˙ t . = = Yt Kt 1−α

(9.11)

This equation reveals a key insight of the Solow model: economic growth in the long run is driven by technological progress. Where does technological progress come from? Unfortunately, the Solow model cannot be used to analyse this question because technological progress is exogenous in the model. In other words, although the Solow model allows us to explore the relative importance of technological progress and capital accumulation, which Acemoglu (2009) refers to as proximate causes of economic growth, it does not allow us to explore the fundamental causes of economic growth. In Chapter 13, we study the Romer model in which the technology growth rate gA is endogenously determined by R&D and innovation in the market economy, which allows us to explore how fundamental causes, such as culture (the household’s preference) and institutions (intellectual property rights), aﬀect economic growth via technological progress. Similarly, the Solow model features an exogenous saving rate, which determines capital accumulation as a proximate cause of growth; 3

˙

Yt t Recall that ∂ ln = Y1t ∂Y = YYtt . ∂t ∂t Because this constant growth rate can be zero, a steady state is a special case of a balanced growth path. 4

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however, it cannot be used to analyse the question on why the saving rate diﬀers across countries. In Chapter 10, we introduce a utilitymaximising household, which chooses consumption and saving, to explore the fundamental determinants of the saving rate.

9.3.

Summary

In this chapter, we review the Solow growth model. In the absence of technological progress, the economy always converges to a steady state, in which the long-run levels of capital and output are stationary and increasing in the saving rate and the level of technology but decreasing in the depreciation rate of capital. The absence of long-run economic growth is due to the decreasing returns to scale with respect to capital in the production function. Therefore, unless capital exhibits constant returns to scale in production, capital accumulation alone cannot sustain economic growth in the long run without technological progress. In the presence of technological progress, the long-run growth rate of output and capital is determined by the growth rate of technology, which however is exogenous in the Solow growth model. Therefore, one cannot use the Solow growth model to explore the determinants of technological progress.

9.4.

Exercises

1. Use (9.7) and the phase diagram in Figure 9.1 to show the eﬀects of (a) a one-time increase in the level of technology A and (b) a continuous increase in the level of technology A. 2. Derive the steady-state level of consumption. Show that setting the saving rate to s = α maximises the steady-state level of consumption. 3. Suppose Lt = L. Derive the steady-state levels of capital per capita and output per capita. 4. Suppose the growth rate of labour is L˙ t /Lt = n. Derive the steadystate levels of capital per capita and output per capita in the Solow model.

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5. Suppose the growth rate of labour is L˙ t /Lt = n and the growth rate of technology is A˙ t /At = gA . What is the long-run growth rate of output per capita in the Solow model? 6. Consider an extension of the Solow growth model with human capital accumulation. The production function is Yt = AKtα Ht1−α , where Ht is human capital. The level of technology A is constant. The accumulation equation of physical capital Kt is K˙ t = sK Yt − δKt , where sK ∈ (0, 1) is the investment rate of Kt and δ > 0 is the depreciation rate of Kt . The accumulation equation of human capital Ht is H˙ t = sH Yt −δHt , where sK ∈ (0, 1) is the investment rate of Ht and δ > 0 is also the depreciation rate of Ht . What is the long-run growth rate of output Yt ?

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Chapter 10

The Ramsey Model

In this chapter, we continue our analysis of economic growth by developing the Ramsey model,1 which can be viewed as a generalisation of the Solow growth model. The Solow model assumes an exogenous saving rate, whereas the Ramsey model features a representative household which chooses the saving rate optimally. As we saw in the Solow model, although the saving rate does not aﬀect the longrun growth rate, it aﬀects the levels of capital and output. Therefore, the Ramsey model allows us to explore the question on why the saving rate diﬀers across countries, which explains some of the variation in the level of income across countries. In summary, we ﬁnd that the saving rate is determined by the household’s discount rate, the degree of capital intensity in production and the depreciation rate of capital but independent of the level of technology. Aside from the endogenous saving rate, the rest of the Ramsey model is the same as the Solow model. 10.1.

Household

In the Ramsey model, there is a representative household, which has a utility function ut at time t. As in the neoclassical growth model,

1

The Ramsey model was developed by Ramsey (1928) and further extended by Cass (1965) and Koopmans (1965), so it is often referred to as the Ramsey–Cass– Koopmans model. 79

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we consider a log utility function: ut = ln Ct ,

(10.1)

which depends on consumption Ct at time t. Given the discount rate ρ > 0, the lifetime utility function is given by ∞

U = u0 +

ut u2 u1 + + · · · = , 1 + ρ (1 + ρ)2 (1 + ρ)t

(10.2)

t=0

where we assume that a lifetime is long enough to be approximated by inﬁnity.2 In this analysis, we will once again use the Hamiltonian to solve the household’s dynamic optimisation problem in continuous time. Therefore, we need to rewrite (10.2) in continuous time using the integral as ∞ ∞ e−ρt ut dt = e−ρt ln Ct dt, (10.3) U= 0

0

where the continuous-time discount factor e−ρt replaces the discretetime discount factor (1 + ρ)−t . As in the Solow model, output Yt at time t is produced by the Cobb-Douglas production function: , Yt = AKtα L1−α t

(10.4)

where the parameter α ∈ (0, 1) is the degree of capital intensity in production and A is the exogenous level of technology. Kt is the stock of capital that has been accumulated as of time t. Lt is the size of the labour force, which we normalise to unity. The accumulation equation for capital is K˙ t = It − δKt ,

(10.5)

where the parameter δ > 0 is the depreciation rate of capital, and It is capital investment. In this closed economy without a government 2

As t becomes very large, the discounting would make ut /(1 + ρ)t not to matter too much in the utility function U .

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sector, the national income account is simply Y t = Ct + I t .

(10.6)

Substituting (10.4) and (10.6) into (10.5) yields K˙ t = Yt − Ct − δKt = AKtα − Ct − δKt ,

(10.7)

where we have set Lt = 1. With the above information, we can now set up the dynamic optimisation problem faced by the representative household. The household chooses consumption Ct and accumulates capital Kt in order to maximise lifetime utility. Formally, the optimisation problem is ∞ e−ρt ln Ct dt, (10.8) max U = Ct

0

subject to K˙ t = AKtα − Ct − δKt . 10.2.

(10.9)

Hamiltonian

To solve the dynamic optimisation problem in (10.8) and (10.9), we use the Hamiltonian. We proceed as follows. First, we set up the Hamiltonian function. Then, we derive the ﬁrst-order conditions. Finally, we use the ﬁrst-order conditions to derive the steady-state levels of Ct and Kt . The Hamiltonian function is given by Ht = ln Ct + λt (AKtα − Ct − δKt ).

(10.10)

The Hamiltonian consists of (a) the utility function ln Ct at time t, (b) the right-hand side of the capital-accumulation equation AKtα − Ct − δKt , and (c) a multiplier λt for the capital-accumulation equation. Now, we derive the ﬁrst-order conditions with respect to Ct and Kt . The ﬁrst-order conditions include 1 ∂Ht = − λt = 0, ∂Ct Ct

(10.11)

∂Ht = λt (αAKtα−1 − δ) = λt ρ − λ˙ t . ∂Kt

(10.12)

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Note that Kt is a state variable (i.e., a variable that accumulates over time), so we have to set ∂Ht /∂Kt = λt ρ − λ˙ t . From (10.11), we have Ct = λ−1 t . Taking the log of this equation yields ln Ct = − ln λt .

(10.13)

Diﬀerentiating it with respect to t yields λ˙ t C˙ t =− . Ct λt

(10.14)

Substituting this equation into (10.12) yields λ˙ t C˙ t = − = αAKtα−1 − δ − ρ. Ct λt

(10.15)

Equation (10.15) is the optimal path of consumption chosen by the household. The optimal path of consumption states that if the net return to capital (i.e., the marginal product of capital αAKtα−1 net of depreciation δ) is greater than the discount rate ρ, then the household should save more and consume less today. Because current consumption is relatively low, consumption must be increasing over time so that C˙ t > 0. On the other hand, if the net return to capital is less than the discount rate, then the household should save less and consume more today. Because current consumption is relatively high, consumption must be decreasing over time so that C˙ t < 0. 10.3.

Steady State

In summary, the Ramsey model provides us with two diﬀerential equations: C˙ t = αAKtα−1 − δ − ρ, Ct K˙ t = AKtα − Ct − δKt .

(10.16) (10.17)

Now, we solve for the steady state.3 In the steady state, C˙ t = 0 and K˙ t = 0. Imposing C˙ t = 0 on the optimal consumption path 3 One can use the phase diagram of (10.16) and (10.17) to show that the economy converges to this steady state.

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in (10.16) yields the steady-state level of capital: αA 1/(1−α) ∗ , K = ρ+δ

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(10.18)

which is increasing in the level of technology A and decreasing in the discount rate ρ and the depreciation rate δ. Intuitively, a higher level of technology A increases the return to capital and encourages the household to accumulate more capital. In contrast, a higher discount rate ρ makes future consumption less attractive to the household, which then accumulates less capital. Finally, a higher depreciation rate δ also makes capital depreciate more rapidly, so that the level of accumulated capital becomes lower. Imposing K˙ t = 0 on the capital-accumulation equation in (10.17) yields the steady-state level of consumption: αA α/(1−α) ρ + δ(1 − α) C ∗ = A(K ∗ )α − δK ∗ = A. (10.19) ρ+δ ρ+δ Using the production function, we can also derive the steady-state level of output given by αA α/(1−α) ∗ ∗ α Y = A(K ) = A. (10.20) ρ+δ As for the steady-state level of investment, it is given by αA 1/(1−α) ∗ ∗ ∗ ∗ I = Y − C = δK = δ. ρ+δ

(10.21)

To derive the steady-state saving rate s∗ , we use s∗ ≡

I∗ δK ∗ αδ , = = ∗ ∗ α Y A(K ) ρ+δ

(10.22)

which is increasing in the degree of capital intensity α and the depreciation rate δ but decreasing in the discount rate ρ. The intuition of these results can be explained as follows. If capital becomes more important in production (i.e., a larger α), then the household would save more to accumulate capital. A higher discount rate ρ makes future consumption less attractive to the household, which then saves less. As for the capital depreciation rate δ, it has two eﬀects on the household’s saving rate. First, it reduces the net return to capital

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and makes saving less attractive, which is captured by the δ in the denominator of s∗ . Second, capital depreciation requires investment to replace the depreciated capital. This replacement eﬀect, which is captured by the δ in the numerator of s∗ , implies that a larger δ requires a higher saving rate s∗ . Overall, the positive replacement eﬀect of δ dominates unless ρ → 0, in which case the two eﬀects exactly oﬀset each other. Finally, we substitute s∗ /δ = α/(ρ + δ) into (10.18) to obtain ∗

K =

s∗ A δ

1/(1−α)

,

(10.23)

which is the same as K ∗ in (9.6) except that s is exogenous in the Solow model whereas s∗ is endogenous in the Ramsey model. 10.4.

Summary

In this chapter, we explore the Ramsey model, which generalises the Solow growth model by featuring an endogenous saving rate that is optimally chosen by a utility-maximising household. The Solow growth model shows that an increase in the exogenous saving rate gives rise to a higher growth rate in the short run and a higher level of output in the long run, which demonstrates the importance of capital accumulation as a proximate cause of economic growth. However, one cannot use the Solow growth model to explore the fundamental determinants of capital accumulation, which the Ramsey model allows us to do. In summary, we ﬁnd that the steady-state saving rate is increasing in the degree of capital intensity and the depreciation rate of capital but decreasing in the discount rate, which reﬂects the preference of the representative household and captures a cultural trait. Therefore, cross-country variation in these determinants helps to explain some of the variation in income level across countries.

10.5.

Exercises

1. Compare the steady-state saving rate from the Ramsey model to the consumption-maximising saving rate s in the Solow model.

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Which one is higher? Under what condition would they be the same? 2. Suppose Lt = L. Derive the steady-state saving rate in the Ramsey model. 3. Consider the introduction of leisure to the household’s utility function: ∞ e−ρt [ln Ct + β ln(1 − lt )]dt, (10.24) U= 0

where the parameter β > 0 determines the importance of leisure 1 − lt in the utility function. In this case, the capital-accumulation equation becomes K˙ t = AKtα lt1−α − Ct − δKt .

(10.25)

Show that the steady-state level of labour l∗ is given by l∗ = 1+

β 1−α

1 1−

αδ ρ+δ

.

(10.26)

4. Suppose the growth rate of labour is L˙ t /Lt = n. Derive the steadystate saving rate in the Ramsey model. 5. Suppose the growth rate of labour is L˙ t /Lt = n and the growth rate of technology is A˙ t /At = gA . Derive the steady-state saving rate and the long-run growth rate of output in the Ramsey model.

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Chapter 11

The Ramsey Model with a Perfectly Competitive Market

In the previous chapter, we cover the Ramsey model, in which the representative household carries out the production of goods. However, this setting is unrealistic because goods are often produced by ﬁrms in the real world. In this chapter, we introduce a market economy to the Ramsey model, in which the representative household supplies labour and capital to a representative ﬁrm, which then uses these factor inputs to produce output and sells the output back to the household. As you can see, this familiar setting is basically the neoclassical growth model of Chapter 2. After deriving the equilibrium allocation of resources in the decentralised market economy, we can then compare it to the allocation in the centralised economy that is optimally chosen by the representative household as in Chapter 10. In summary, we ﬁnd that the two sets of allocations are the same, implying that the market economy is eﬃcient. In other words, the ﬁrst fundamental theorem of welfare economics holds in this setting due to the absence of distortion in the market economy. 11.1.

Household

In the Ramsey model, there is a representative household, which has the following lifetime utility function: ∞ e−ρt ln Ct dt, (11.1) U= 0

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where the parameter ρ > 0 is the household’s discount rate and Ct is the level of consumption at time t. The household inelastically supplies one unit of labour to earn a wage income Wt . Furthermore, it accumulates capital Kt and rents it to the representative ﬁrm to earn a capital-rental income Rt . If we normalise the price of output to unity, then the asset-accumulation equation is1 K˙ t = Rt Kt + Wt − Ct − δKt ,

(11.2)

where the parameter δ > 0 is the depreciation rate of capital. To solve this dynamic optimisation problem, we use the Hamiltonian. The Hamiltonian function is given by Ht = ln Ct + λt (Rt Kt + Wt − Ct − δKt ).

(11.3)

The ﬁrst-order conditions include 1 ∂Ht = − λt = 0, ∂Ct Ct

(11.4)

∂Ht = λt (Rt − δ) = λt ρ − λ˙ t . ∂Kt

(11.5)

Recall once again that Kt is a state variable (i.e., a variable that accumulates over time). Taking the log of (11.4) yields ln Ct = − ln λt .

(11.6)

Diﬀerentiating it with respect to t yields C˙ t λ˙ t =− . Ct λt

(11.7)

Substituting this equation into (11.5) yields C˙ t = Rt − δ − ρ, Ct

(11.8)

which is the optimal path of consumption chosen by the household. 1

Here, we assume that one unit of output can be converted into one unit of consumption or one unit of capital.

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11.2.

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Firm

We now consider the ﬁrm’s optimisation problem. There is a representative ﬁrm in the economy, and this ﬁrm hires labour Lt and rents capital Kt from the household to produce output Yt using the following production function: Yt = AKtα L1−α , t

(11.9)

where the parameter α ∈ (0, 1) is the degree of capital intensity in production and A is the exogenous level of technology. The proﬁt function Πt is Πt = Yt − Rt Kt − Wt Lt ,

(11.10)

where we have chosen Yt as the numeraire (i.e., the price of Yt is normalised to unity). Diﬀerentiating (11.10) with respect to Kt and Lt yields ∂Yt ∂Πt = − Rt = αAKtα−1 L1−α − Rt = 0, t ∂Kt ∂Kt ∂Yt ∂Πt = − Wt = (1 − α)AKtα L−α t − Wt = 0. ∂Lt ∂Lt

(11.11) (11.12)

These two equations are the demand functions for Kt and Lt . 11.3.

Equilibrium

Substituting (11.11) into the consumption path in (11.8) yields C˙ t = αAKtα−1 − δ − ρ, Ct

(11.13)

where we have set Lt = 1. Substituting (11.11) and (11.12) into (11.2) yields the capital-accumulation equation: K˙ t = αAKtα + (1 − α)AKtα − Ct − δKt = AKtα − Ct − δKt .

(11.14)

Equations (11.13) and (11.14) are two diﬀerential equations in Ct and Kt , and these two equations completely characterise the behaviour of the economy.

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It is important to note that (11.13) and (11.14) are exactly the same as (10.16) and (10.17) in the Ramsey model in the previous chapter. Therefore, the decentralised market economy has the same allocation of resources as the centralised economy that is optimally chosen by the representative household. The reason is that the rental price Rt in the market economy is equal to the marginal product of capital αAKtα−1 . This also implies that the decentralised economy has the same steady-state equilibrium allocations as the centralised economy. For example, the steady-state equilibrium level of capital is the same as before such that αA 1/(1−α) ∗ , (11.15) K = ρ+δ and the household’s saving rate s∗ is also the same as before such that s∗ ≡

I∗ δK ∗ αδ . = = ∗ Y A(K ∗ )α ρ+δ

(11.16)

This result is an example of the ﬁrst fundamental theorem of welfare economics, which states that any competitive equilibrium leads to a Pareto eﬃcient allocation of resources. This fundamental theorem holds because the decentralised economy is characterised by perfect competition in the production sector, so that Rt = M P Kt = αAKtα−1 . In Chapter 12, we will consider another version of the Ramsey model with monopolistic competition under which the decentralised economy diﬀers from the centralised economy (i.e., the decentralised economy exhibits market failure).

11.4.

Summary

In this chapter, we introduce a market economy into the Ramsey model. Speciﬁcally, we consider a perfectly competitive product market. In this case, a representative ﬁrm demands factor inputs from the representative household and supplies output to the household. Therefore, the household and the ﬁrm interact in the labour market, the capital market and the product market. We ﬁnd that the equilibrium allocation of resources in the market economy is

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the same as the socially optimal allocation in the centralised version of the Ramsey model in Chapter 10. In other words, due to the absence of distortion in the economy, the market equilibrium is eﬃcient. Therefore, the ﬁrst fundamental theorem of welfare economics holds in the Ramsey model with a perfectly competitive product market. 11.5.

Exercises

1. Compare the steady-state saving rate from the competitive version of the Ramsey model to the consumption-maximising saving rate s in the Solow model. Which one is higher? Under what condition would they be the same? 2. Suppose Lt = L. Derive the steady-state saving rate in the competitive version of Ramsey model. 3. Consider the introduction of leisure to the household’s utility function: ∞ e−ρt [ln Ct + β ln(1 − lt )]dt, (11.17) U= 0

where the parameter β > 0 determines the importance of leisure 1 − lt in the utility function. In this case, the asset-accumulation equation becomes K˙ t = Rt Kt + Wt lt − Ct − δKt .

(11.18)

Show that the steady-state equilibrium level of labour l∗ is given by l∗ =

1 β 1 + 1−α 1−

αδ ρ+δ

,

(11.19)

which is the same as (10.26) in the centralised economy of the Ramsey model. Once again, due to the absence of market failure, the level of employment in the decentralised market economy is the same as the centralised allocation of labour that is optimally chosen by the representative household.

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4. Suppose the growth rate of labour is L˙ t /Lt = n. Derive the steadystate saving rate in the competitive version of the Ramsey model. 5. Suppose the growth rate of labour is L˙ t /Lt = n and the growth rate of technology is A˙ t /At = gA . Derive the steady-state saving rate and the long-run growth rate of output in the competitive version of the Ramsey model.

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Chapter 12

The Ramsey Model with a Monopolistically Competitive Market

In this chapter, we will consider another form of market structure in the Ramsey model. Speciﬁcally, we consider monopolistic competition. In the model, there is a representative ﬁrm that produces the ﬁnal good. Also, there are a number of ﬁrms that produce diﬀerentiated intermediate goods. Because each of these intermediate-good ﬁrms sells a diﬀerentiated product, it has market power and charges a markup over the marginal cost. This monopolistic distortion in turn causes the decentralised economy to have a diﬀerent allocation of resources from the centralised economy. In other words, the ﬁrst fundamental theorem of welfare economics does not hold in this setting due to the distortion of monopolistic competition, which in turn gives rise to market failure. The model has the following components: (a) a representative household, (b) a representative ﬁrm that produces the ﬁnal good, and (c) N monopolistic ﬁrms that produce diﬀerentiated intermediate goods.

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12.1.

Household

The representative household has the following lifetime utility function: ∞ e−ρt ln Ct dt, (12.1) U= 0

where the parameter ρ > 0 is the household’s discount rate and Ct is the level of consumption at time t. As usual, the household inelastically supplies one unit of labour to earn a wage income Wt . Furthermore, it accumulates capital Kt and rents it to ﬁrms to earn a capital-rental income Rt . The asset-accumulation equation is K˙ t = Rt Kt + Wt +

N

πt (i) − Ct − δKt ,

(12.2)

i=1

where the parameter δ > 0 is the depreciation rate of capital. It is useful to note that each intermediate-good ﬁrm i generates monopolistic proﬁt πt (i), which is transferred back to the household as it owns the ﬁrm. We have dealt with this dynamic optimisation problem many times now, so we will skip the steps and simply write down the result. The familiar consumption path is given by C˙ t = Rt − δ − ρ, Ct

(12.3)

which determines the path of consumption chosen by the household.

12.2.

Final Good

There is a representative ﬁrm that produces the ﬁnal good Yt . This ﬁrm hires labour Lt and uses intermediate goods as inputs. The production function is diﬀerent from before and given by Yt = L1−α t

N

Xtα (i) = L1−α [Xtα (1) + Xtα (2) + · · · + Xtα (N )] , t

i=1

(12.4)

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where the parameter α ∈ (0, 1) determines the degree of labour intensity 1−α in production. In other words, there are N diﬀerent intermediate goods Xt (i) that are indexed by i ∈ [1, N ]. The proﬁt function for this ﬁnal-good ﬁrm is Πt = Yt − Wt Lt −

N

Pt (i)Xt (i),

(12.5)

i=1

where we have implicitly chosen Yt as the numeraire and Pt (i) is the price of Xt (i) for i ∈ [1, N ]. Diﬀerentiating (12.5) with respect to Lt and Xt (i) yields Yt ∂Πt = (1 − α)L−α Xtα (i) − Wt = (1 − α) − Wt = 0, t ∂Lt Lt N

(12.6)

i=1

∂Πt Xtα−1 (i) − Pt (i) = 0. = αL1−α t ∂Xt (i)

(12.7)

These two sets of equations are the demand functions for Lt and Xt (i) for i ∈ [1, N ].

12.3.

Intermediate Goods

Each variety of intermediate goods is produced by a ﬁrm that has monopolistic power over this variety. We refer to each variety of intermediate goods as an industry, and let’s consider an arbitrary industry i. In industry i, the monopolistic ﬁrm rents capital from the household to produce intermediate good i. We consider a simple production function given by Xt (i) = Kt (i).

(12.8)

In other words, one unit of capital produces one unit of intermediate good i. The proﬁt function for intermediate-good ﬁrm i is πt (i) = Pt (i)Xt (i) − Rt Kt (i).

(12.9)

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Given that this ﬁrm acts as a monopolist, it chooses its price to maximise proﬁt (rather than taking the price as given). Substituting (12.7) and (12.8) into (12.9) yields Xtα (i) − Rt Xt (i). πt (i) = αL1−α t

(12.10)

Diﬀerentiating πt (i) with respect to Xt (i) yields ∂πt (i) = α αL1−α X α−1 (i) − Rt = 0. t t ∂Xt (i)

(12.11)

=Pt (i)

Using (12.7), we can re-express (12.11) as Pt (i) = Rt /α > Rt , where 1/α > 1 is the markup ratio. Due to this markup pricing, the intermediate-good sector generates positive monopolistic proﬁts. As a result of these monopolistic proﬁts, the decentralised economy has a diﬀerent allocation of resources compared with the centralised economy as we show below.

12.4.

Aggregation

Equation (12.11) shows that Xt (i) = (α2 /Rt )1/(1−α) Lt is the same across all i ∈ [1, N ]. Using this information, we impose symmetry on (12.4) to obtain N Xtα (i). Yt = L1−α t

(12.12)

Then, we substitute the resource constraint on capital given by Xt (i) = Kt (i) = Kt /N into (12.12) to derive the aggregate production function: α Kt 1−α = N 1−α Ktα L1−α = AKtα L1−α , (12.13) Y t = Lt N t t N where we have relabelled N 1−α as A. This aggregate production function is the same as before. In this economy, it is the number of diﬀerentiated products that determines the level of technology; in other words, a higher level of technology is driven by a larger number of diﬀerentiated products.

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Setting Lt = 1 in (12.7) yields Pt (i) = αXtα−1 (i).

(12.14)

From (12.11), we know that Pt (i) = Rt /α. Substituting this equation and the resource constraint Xt (i) = Kt /N into (12.14) yields the market demand curve for capital as follows: α−1 Kt = α2 AKtα−1 , (12.15) Rt = α2 N where the second equality follows from N 1−α = A. We impose steady state on the household’s consumption path to derive the long-run capital supply curve as C˙ t = 0 ⇔ Rt = ρ + δ; Ct

(12.16)

see Figure 12.1. Finally, combining capital demand in (12.15) and capital supply in (12.16) yields the steady-state equilibrium level of capital under monopolistic competition as ∗ Km

Figure 12.1.

=

α2 A ρ+δ

1/(1−α) ,

(12.17)

Monopolistic competition vs. perfect competition.

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which is increasing in the level of technology A and decreasing in the discount rate ρ and the depreciation rate δ. 12.5.

Monopolistic Competition vs. Perfect Competition

It may seem that the Ramsey model with a monopolistically competitive market in this chapter is very diﬀerent from the Ramsey model with a perfectly competitive market in the previous chapter. However, the two models have the same aggregate structure: the same aggregate production function; the same capital-accumulation equation;1 and the same consumption path (except that the equilibrium rental price of capital diﬀers across the two models). We will use M P Kt to denote the marginal product of capital. Regardless of the underlying market structure, the consumption path in the Ramsey model is C˙ t = Rt − δ − ρ. Ct

(12.18)

In a perfectly competitive market, the capital demand curve is Rt = M P Kt = αAKtα−1 , which yields the following steady-state equilibrium level of capital: αA 1/(1−α) ∗ ; (12.19) Kp = ρ+δ see Figure 12.1. In a monopolistically competitive market, the capital demand curve is Rt = αM P Kt = α2 AKtα−1 , which yields the following steady-state equilibrium level of capital: ∗ Km

=

α2 A ρ+δ

1/(1−α)

< Kp∗ .

(12.20)

In other words, the steady-state equilibrium level of capital diﬀers in the two economies. The reason is that under monopolistic competition, the rental price of capital is less than the marginal product 1

One can show that K˙ t = Rt Kt + Wt + N πt − Ct − δKt = Yt − Ct − δKt .

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of capital (i.e., Rt = αM P Kt < M P Kt given α < 1). Intuitively, due to monopolistic distortion, the market return to capital is lower under monopolistic competition than under perfect competition; as a result, the household accumulates less capital. Finally, we derive the steady-state saving rate given by s∗ ≡

I∗ δK ∗ δ(K ∗ )1−α = = . Y∗ A(K ∗ )α A

(12.21)

In the Ramsey model with a monopolistically competitive market, the household’s saving rate s∗m is given by s∗m =

αδ α2 δ < = s∗p . ρ+δ ρ+δ

(12.22)

Given an empirically relevant range of capital intensity α ∈ [1/3, 1/2], the distortion of monopolistic competition can cause the household’s saving rate to be less than half of the optimal level. This distortion on the saving rate in turn reduces the level of output signiﬁcantly; to see this, ∗ )α A(Km Ym∗ = = αα/(1−α) . Yp∗ A(Kp∗ )α

(12.23)

For example, if α = 1/2, then the monopolistic level of output Ym∗ is only half of the competitive level of output Yp∗ . 12.6.

Market Power of Monopolistic Firms

Suppose the government imposes an upper bound on the monopolistic price given by Pt (i) = μRt < Rt /α,

(12.24)

where the policy parameter μ ∈ (1, 1/α) determines the markup ratio, capturing the market power of monopolistic ﬁrms. Substituting (12.24) and the resource constraint Xt (i) = Kt /N into (12.14) yields M P Kt α Kt α−1 αAKtα−1 = , (12.25) = Rt = μ N μ μ

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where the second equality follows from N 1−α = A. Equation (12.25) shows that the diﬀerence between the marginal product of capital and its rental price is increasing in μ, which in turn determines the degree of distortion in the market economy. Substituting the consumption path in (12.16) into (12.25) yields the steady-state equilibrium level of capital given by ∗ Km

αA = μ(ρ + δ)

1/(1−α)

0 determines the importance of leisure 1 − lt in the utility function. In this case, the asset-accumulation equation becomes K˙ t = Rt Kt + Wt lt +

N

πt (i) − Ct − δKt .

(12.29)

i=1

Suppose we assume μ ∈ (1, 1/α). Show that the steady-state equilibrium level of labour l∗ is given by l∗ =

1+

β 1−α

1 1−

αδ μ(ρ+δ)

,

(12.30)

which is less than l∗ in (11.19) in the competitive market of the Ramsey model unless δ = 0.2 4. Suppose the growth rate of labour is L˙ t /Lt = n. Derive the steady-state saving rate in the monopolistic version of the Ramsey model.

2 Although there is no distortion in the labour market (i.e., Wt = (1 − α)Yt /lt ), the monopolistic distortion in the capital market leads to a suboptimally low investment rate s∗ and a suboptimally high consumption rate C ∗ /Y ∗ = 1 − s∗ , which in turn leads to a suboptimally high level of leisure 1−l∗ unless s∗ = δ = 0.

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Chapter 13

The Romer Model of Endogenous Technological Change

Paul Romer greatly enhances economists’ understanding of endogenous technological change by developing a growth model in which technological progress is driven by the invention of new products, which in turn is due to research and development (R&D) by proﬁtseeking entrepreneurs. In this chapter, we cover the Romer model.1 In the previous chapter, we introduced a monopolistically competitive market structure to the Ramsey model. If we further introduce an R&D sector into the model to allow for endogenous growth in the number of products, then we have the Romer model. In other words, we have added several layers of structure to gradually extend the familiar Solow model into the less familiar Romer model.2 Once we derive the endogenous growth rate of technology in the Romer model, we can then perform comparative statics to explore the determinants of technological progress. As before, there is a representative ﬁrm that produces the ﬁnal good, and there are a number of ﬁrms that produce diﬀerentiated intermediate goods. The novel element here is that the number of these diﬀerentiated goods increases over time due to innovation. Because each of these intermediate-good ﬁrms sells a diﬀerentiated 1 The Romer model was developed by Romer (1990), who received the Nobel Memorial Prize in Economics in 2018 partly for this contribution. 2 See Chu (2018) for this approach of teaching the Romer model. Some discussion in this chapter follows from Chu (2018).

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product, it generates monopolistic proﬁts, which serve as the incentives for R&D. In summary, the model has the following components: (a) a representative household, (b) a representative ﬁrm that produces the ﬁnal good, (c) a number of monopolistic ﬁrms that produce diﬀerentiated intermediate goods, and (d) competitive R&D entrepreneurs who invest in R&D and create new varieties of intermediate goods. 13.1.

Household

The representative household has the following lifetime utility function: ∞ e−ρt ln Ct dt, U= 0

where the parameter ρ > 0 is the household’s discount rate and Ct is the level of consumption at time t. The household inelastically supplies L units3 of labour to earn a wage income Wt . Furthermore, it accumulates capital Kt and rents it to ﬁrms to earn a capitalrental income Rt . Here, we also introduce ﬁnancial asset Ft (i.e., the shares of monopolistic ﬁrms) and its rate of return given by rt , which is also the real interest rate. Then, the asset-accumulation equation becomes4 F˙t + K˙ t = rt Ft + Rt Kt + Wt L − Ct − δKt , where the parameter δ > 0 is the depreciation rate of capital. The Hamiltonian function is given by Ht = ln Ct + λt (rt Ft + Rt Kt + Wt L − Ct − δKt ). The ﬁrst-order conditions include 1 ∂Ht = − λt = 0, ∂Ct Ct 3 Here, we consider L units of labour supply instead of 1 unit for reasons to be explained. 4 It can be shown that the asset-accumulation equation can be re-expressed as ˙ Kt = Yt − Ct − δKt in equilibrium.

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∂Ht = λt (Rt − δ) = λt ρ − λ˙ t , ∂Kt ∂Ht = λt rt = λt ρ − λ˙ t , ∂Ft

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(13.1) (13.2)

where Ft is also a state variable. Combining (13.1) and (13.2) yields a no-arbitrage condition given by rt = Rt − δ. In other words, the two types of assets (Ft and Kt ) must yield the same rate of return. We have the consumption path given by C˙ t = rt − ρ, Ct

(13.3)

where rt = Rt − δ from the no-arbitrage condition. Usually, the consumption path is expressed in rt instead of Rt − δ. 13.2.

Final Good

There is a representative ﬁrm that produces the ﬁnal good Yt . This ﬁrm hires labour and uses intermediate goods Xt (i) as inputs. The production function is almost the same as in Chapter 12 and given by Yt =

L1−α Y,t

Nt 0

Xtα (i)di,

(13.4)

where the parameter α ∈ (0, 1) determines the intensity 1 − α of production labour. LY,t is the number of production workers (and we will use LR,t to denote R&D workers such that the resource constraint on labour is given by LY,t + LR,t = L). It is useful to note that we are treating the number of varieties Nt as a continuous number (instead of a discrete number) for modelling simplicity. The proﬁt function for the ﬁnal-good ﬁrm is Πt = Yt − Wt LY,t −

Nt 0

Pt (i)Xt (i)di,

(13.5)

where we have implicitly chosen Yt as the numeraire and Pt (i) is the price of Xt (i) for i ∈ [1, N ]. Diﬀerentiating (13.5) with respect to

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LY,t and Xt (i) yields ∂Πt = (1 − α)L−α Y,t ∂LY,t = (1 − α)

0

Nt

Xtα (i)di − Wt

Yt − Wt = 0, LY,t

∂Πt α−1 = αL1−α (i) − Pt (i) = 0, Y,t Xt ∂Xt (i)

(13.6) (13.7)

for i ∈ [0, Nt ]. These two sets of equations are the demand functions for LY,t and Xt (i) for i ∈ [0, Nt ]. 13.3.

Intermediate Goods

This sector is exactly the same as in Chapter 12. Each variety of intermediate goods is produced by a ﬁrm that has monopolistic power over this variety. We will refer to each variety of intermediate goods as an industry, and let’s consider an arbitrary industry i. In industry i, the monopolistic ﬁrm rents capital from the household to produce the intermediate good. We consider a simple production function as before. Xt (i) = Kt (i).

(13.8)

In other words, one unit of capital produces one unit of intermediate good. The proﬁt function for this intermediate-good ﬁrm is πt (i) = Pt (i)Xt (i) − Rt Kt (i).

(13.9)

Given that this ﬁrm acts as a monopolist, it chooses its price to maximise proﬁt (rather than taking the price as given). Substituting (13.7) and (13.8) into (13.9) yields α πt (i) = αL1−α Y,t Xt (i) − Rt Xt (i).

(13.10)

Diﬀerentiating πt (i) with respect to Xt (i) yields ∂πt (i) α−1 = α αL1−α (i) − Rt = 0. Y,t Xt ∂Xt (i) =Pt (i)

(13.11)

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Using (13.7), we can re-express (13.11) as Pt (i) = Rt /α > Rt , where 1/α > 1 is the markup ratio. Due to this markup pricing, the intermediate-good sector generates positive monopolistic proﬁts. 13.4.

R&D

The novel element in the Romer model is the R&D sector. The law of motion for the number of varieties is given by N˙ t = θNt LR,t ,

(13.12)

where LR,t is the number of R&D workers and θ > 0 is an R&D productivity parameter. Rewriting (13.12) yields the growth rate of Nt given by gN ≡ N˙ t /Nt = θLR,t ; therefore, increasing R&D labour stimulates economic growth. However, what determines the equilibrium allocation of R&D labour LR,t ? Let vt denote the market value of a new invention (i.e., a new variety of diﬀerentiated products). The market value of creating N˙ t new inventions is N˙ t vt . The cost of R&D is Wt LR,t . Given that there is free entry in the R&D sector, this sector generates zero proﬁt such that N˙ t vt = Wt LR,t .

(13.13)

Substituting (13.12) into (13.13), we obtain θNtvt = Wt ,

(13.14)

where we have cancelled LR,t from both sides. We will see that the R&D condition in (13.14) determines the equilibrium allocation of R&D labour. To show this result, we will need to ﬁnd out the value of vt . The value of an invention is the present value of all the future monopolistic proﬁts. Formally, πt , (13.15) vt = r − gπ where gπ is the steady-state equilibrium growth rate of πt . Substituting (13.15) into (13.14) yields θNt πt = Wt . r − gπ

(13.16)

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We already know from (13.6) that Wt = (1−α)Yt /LY,t . We also know from (13.3) that r = ρ + gC , where gC is the steady-state equilibrium growth rate of Ct . Therefore, what we need to do next is to derive an expression for πt . 13.5.

Aggregation

Equation (13.11) implies that Xt (i) = (α2 /Rt )1/(1−α) LY,t is the same across all i ∈ [0, Nt ]. Using this information, we impose symmetry α on (13.4) to obtain Yt = L1−α Y,t Nt Xt (i). Then, we substitute into this equation the resource constraint on capital given by Xt (i) = Kt (i) = Kt /Nt to derive the aggregate production function: α Kt 1−α α 1−α = Nt1−α Ktα L1−α (13.17) Yt = LY,t Nt Y,t = At Kt LY,t , Nt where we have relabelled Nt1−α as At . Therefore, the growth rate of At in the Solow model becomes gA = (1 − α)gN , which is endogenous in the Romer model. Also, it should be noted that the aggregate production function is the same as the one in the Solow model except that Lt is replaced by LY,t . Recall that the proﬁt function for the intermediate-good ﬁrm i is πt (i) = Pt (i)Xt (i) − Rt Kt (i). Substituting (13.8) and Pt (i) = Rt /α into the proﬁt function yields πt =

1−α 1 − α Rt Kt Rt Kt (i) = , α α Nt

(13.18)

which uses the resource constraint Xt (i) = Kt (i) = Kt /Nt . Recall from (13.6) that the production labour income share of output is Wt LY,t = (1−α)Yt , which implies that the remaining share of output αYt goes to capital income and monopolistic proﬁt5 : πt Nt + Rt Kt = Nt Pt Xt = Yt . 5

(13.19)

Note that GDP in this economy is given by GDPt = Yt +R&Dt = Yt +Wt LR,t .

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Combining (13.18) and (13.19) yields the proﬁt share of output given by πt = α(1 − α)

Yt , Nt

(13.20)

which implies that gπ = gY − gN . 13.6.

Solving the Model

Using (13.20) along with Wt = (1 − α)Yt /LY,t and r = ρ + gC , we can simplify (13.16) to 1 αθ = . ρ + gC − gY + gN LY

(13.21)

On the balanced growth path, the growth rates of output and consumption are the same such that gC = gY . Using gN = θLR from (13.12), we can further simplify (13.21) to 1 αθ = . ρ + θLR LY

(13.22)

Combining (13.22) with the resource constraint on labour LY +LR = L yields ρ

α L− . (13.23) LR = 1+α αθ Therefore, the equilibrium allocation of R&D labour has the following comparative statics: LR (ρ , θ , α, L). − + + +

The intuition of these comparative statics results can be explained as follows. 13.6.1.

R&D productivity

An improvement in R&D productivity θ increases the growth rate gN of technology for a given level of R&D labour LR and also makes R&D labour more productive, which in turn increases R&D labour in the economy. Therefore, gN is increasing in θ. To see this, recall that

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the growth rate of technology is gN = θLR (θ). Therefore, an increase in R&D productivity θ has a direct positive eﬀect on gN by increasing R&D productivity and also an indirect positive eﬀect by increasing R&D labour LR . The parameter θ captures the importance of human capital on the innovation capacity of an economy. To stimulate economic growth, policymakers could consider devoting more resources to education that improves the innovative capacity of entrepreneurs, scientists and engineers. 13.6.2.

Discount rate

A higher discount rate ρ increases the real interest rate and decreases the present value of monopolistic proﬁts as well as the value of inventions, which in turn decreases R&D labour in the economy. Therefore, gN is decreasing in ρ. To see this, recall that the growth rate of technology is gN = θLR (ρ). Therefore, an increase in the discount rate ρ decreases the growth rate of technology by decreasing R&D labour LR . The parameter ρ captures the eﬀects of household preference and also frictions in the ﬁnancial market, such as credit constraints, on innovation. To stimulate economic growth, policymakers could consider policies that improve the eﬃciency of the ﬁnancial market. 13.6.3.

Capital and labour intensity in production

An increase in α increases capital intensity and reduces labour intensity in the production process, allowing more labour to be devoted to R&D. Therefore, gN is increasing in α. To see this, recall that the growth rate of the number of diﬀerentiated products is gN = θLR (α). Therefore, an increase in capital intensity α increases the growth rate of technology by increasing R&D labour LR . The parameter α captures the eﬀect of structural transformation of an economy from a labour-intensive production process to a capital-intensive production process. However, the positive eﬀect of α on R&D is based on the assumption that R&D does not require the use of capital. If R&D also uses capital, then the eﬀect of α on growth may be reversed depending on the degree of capital intensity in the R&D process. 13.6.4.

Labour force

Finally, a larger labour force L increases the supply of labour in the economy, which in turn increases R&D labour and the growth rate

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of technology. Therefore, gN is increasing in L. To see this, recall that the growth rate of technology is gN = θLR (L). Therefore, an increase in the labour force L increases the growth rate of technology by increasing R&D labour LR . In the literature, this is known as the scale eﬀect, which is often viewed as a counterfactual implication of the Romer model. We will discuss the scale eﬀect in more detail in Chapter 14. 13.7.

Summary

In this chapter, we develop the Romer model by introducing an R&D sector into the Ramsey model with a monopolistically competitive product market. In this case, the positive monopolistic proﬁt in the economy serves as an incentive for entrepreneurs to do R&D, which in turn gives rise to technological progress. We derive the market equilibrium level of R&D labour, which determines the equilibrium growth rate of technology. In summary, the market equilibrium level of R&D labour is increasing in the monopolistic power of ﬁrms (see Exercise 1), the productivity of R&D, the degree of capital intensity in production and the size of the labour force but decreasing in the discount rate of the representative household. Therefore, crosscountry variation in these determinants helps to explain some of the variation in economic growth across countries. 13.8.

Exercises

1. Suppose the government imposes an upper bound on the monopolistic price given by Pt (i) = μRt < Rt /α,

(13.24)

where the parameter μ ∈ (1, 1/α) is the markup ratio, capturing the monopolistic power of ﬁrms. Show that the equilibrium allocation of R&D labour becomes μ(1 − α) ρ μ−1 LR = α , (13.25) L− μ−α μ − α αθ which is increasing in the monopolistic power μ of ﬁrms.

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2. Show that on the balanced growth path, the growth rate of capital, output and consumption is given by gN . 3. Show that on the balanced growth path, the saving rate s∗ in the Romer model is given by s∗ =

α δ + gN . μ ρ + δ + gN

(13.26)

4. Suppose the production function in the Romer model is given by Nt 1−α−β β Lt Xtα (i) di, Yt = HY,t 0

where HY,t is the number of high-skill production workers, and Lt is the number of low-skill workers. The law of motion for the number of varieties is given by N˙ t = θNt HR,t , where HR,t is the number of high-skill R&D workers. The household inelastically supplies L units of low-skill labour to earn a low-skill wage income Wt , and inelastically supplies H units of high-skill labour to earn a high-skill wage income ωt . In this case, the asset-accumulation equation becomes F˙t + K˙ t = rt Ft + Rt Kt + Wt L + ωt H − Ct − δKt . Derive the equilibrium allocation of high-skill R&D labour HR and the growth rate of output on the BGP.

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Chapter 14

Scale Eﬀect in the Romer Model

In this chapter, we will explore an important but counterfactual property of the Romer model. Recall from the Romer model the following key equation: gN ≡ N˙ t /Nt = θLR,t , where LR,t is the number of R&D workers and θ > 0 is an R&D productivity parameter. This equation implies that the growth rate gN of technology is increasing in the number of R&D workers LR,t . This property is known as the scale eﬀect in the literature. Jones (1995) points out that this prediction from the Romer model is inconsistent with empirical evidence. Jones (1995) documents that the number of R&D scientists in the US steadily increased from 1950 to the late 1980s; however, the growth rate of technology did not exhibit any positive trend during this period; see Figures 14.1 and 14.2 in which we look at more recent data from 1990 to 2013.1 Motivated by this empirical fact, Jones developed a modiﬁed version of the Romer model, which is free from the scale eﬀect.

1

The trend of total factor productivity (TFP) growth rates is computed using the HP filter.

113

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14,00,000 13,00,000 12,00,000 11,00,000 10,00,000 9,00,000 8,00,000 7,00,000 6,00,000 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012

Figure 14.1.

Researchers in the US.

3.0% 2.5% 2.0% 1.5% 1.0% 0.5% 0.0% −0.5% −1.0% −1.5% 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 data

Figure 14.2.

trend

TFP growth in the US.

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The Jones Model

The basic structure of the Jones model is the same as the Romer model. The only major diﬀerence is in the law of motion for the number of varieties. Romer (1990) assumes the following law of motion: N˙ t = θNt LR,t .

(14.1)

Jones (1995) generalises this speciﬁcation to N˙ t = θNtφ LR,t ,

(14.2)

where the parameter φ < 1 measures the degree of intertemporal knowledge spillovers. In other words, Romer (1990) implicitly assumes that φ = 1. In contrast, Jones (1995) relaxes this assumption and considers the general case of φ < 1. To see the diﬀerent implications, we divide (14.2) by Nt to obtain LR,t N˙ t = θ 1−φ . Nt Nt

(14.3)

On the balanced growth path, the growth rate N˙ t /Nt is constant, which in turn implies that LR,t and Nt1−φ grow at the same rate; formally, (1 − φ)

L˙ R,t N˙ t 1 L˙ R,t N˙ t = ⇔ = . Nt LR,t Nt 1 − φ LR,t

(14.4)

Therefore, the long-run growth rate of technology is now determined by the growth rate of LR,t instead of the level of LR,t , eliminating the scale eﬀect. Figure 14.3 presents the growth rate of researchers and shows that its trend exhibits a similar pattern as the trend of TFP growth rates in Figure 14.2.

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116 10.0% 8.0% 6.0% 4.0% 2.0% 0.0% −2.0% −4.0% −6.0%

1991 1993 1995 1997 1999 2001 2003 2005 2007 2009 2011 2013 data

Figure 14.3.

14.2.

trend

Growth of US researchers.

Solving the Model

As mentioned before, the basic structure of the Jones model is the same as in the Romer model, so we do not repeat all the equations here. The key diﬀerence is in Equation (14.2). Furthermore, the Jones model allows for growth in the labour force Lt , which increases over time at a constant exogenous growth rate n. On the balanced growth path, the resource constraint on labour given by LR,t + LY,t = Lt ⇔

LR,t LY,t + =1 Lt Lt

(14.5)

implies that LR,t /Lt must be constant. Therefore, LR,t and Lt grow at the same rate on the balanced growth path. Using this information along with (14.4), we ﬁnd that the steady-state equilibrium growth rate of Nt is n , (14.6) gN = 1−φ which is determined by exogenous parameters n and φ. Therefore, the Jones model is also known as the semi-endogenous growth model.2 2

See Cozzi (2017a, b) for a hybrid growth model that captures both semiendogenous growth and fully endogenous growth.

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Scale Eﬀect in the Romer Model

Figure 14.4.

Romer model.

Figure 14.5.

Jones model.

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117

Increasing R&D labour LR,t only leads to a higher growth rate in the short run. From the following equation, we see that for a given Nt , an increase in R&D labour LR,t raises the growth rate N˙ t /Nt temporarily. LR,t N˙ t = θ 1−φ . Nt Nt However, Nt1−φ increases over time to oﬀset the eﬀect of LR,t . Eventually, the growth rate N˙ t /Nt returns to gN = n/(1 − φ) in the long run; see Figures 14.4 and 14.5 for a comparison between the Romer model and the Jones model. The steady-state equilibrium growth rate gN is easy to determine in the Jones model. However, we will have to go through the same derivations as in the Romer model to derive the steady-state equilibrium allocation of R&D labour, which determines the growth rate of technology in the short run and the level of technology in the long run.

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As in the Romer model, the free-entry condition in the R&D sector is N˙ t vt = Wt LR,t ,

(14.7)

where N˙ t is now given by (14.2) instead of (14.1). The value vt of an invention is the same as before and given by vt =

πt , r − gπ

(14.8)

where monopolistic proﬁt πt is also the same as before and given by πt = α(1 − α)

Yt . Nt

(14.9)

Therefore, the steady-state equilibrium growth rate of πt is the same as before and given by gπ = gY − gN , where gY is the growth rate of output Yt . Similarly, the wage rate Wt is equal to the marginal product of production labour LY,t as before and given by Wt = (1 − α)

Yt . LY,t

(14.10)

Substituting (14.8), (14.9) and (14.10) into (14.7) yields LR,t α N˙ t = , r − g π Nt LY,t

(14.11)

where N˙ t /Nt = gN = n/(1 − φ). Combining the real interest rate r from the consumption path r = ρ + gC with gπ = gY − gN yields r − gπ = ρ + gN = ρ + n/(1 − φ),

(14.12)

where we have used the steady-state condition gC = gY , equating the growth rate of consumption to the growth rate of output. Finally, if we deﬁne SR ≡ LR /LY as the steady-state equilibrium ratio of R&D labour LR to production labour LY , then Equation (14.11) becomes SR =

αn αgN αn/(1 − φ) = . = ρ + gN ρ + n/(1 − φ) (1 − φ)ρ + n

(14.13)

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Therefore, the steady-state equilibrium ratio of R&D labour to production labour has the following comparative statics: SR (ρ , φ, α, n). − + + +

Intuitively, a higher discount rate ρ increases the real interest rate and decreases the present value of monopolistic proﬁts. A larger α increases capital intensity and reduces labour intensity in the production process, allowing more labour to be devoted to R&D. An increase in either knowledge spillover φ or population growth n raises the technology growth rate gN , which in turn increases the beneﬁt of R&D. 14.3.

Summary

In this chapter, we explore the scale eﬀect in the Romer model, in which the steady-state growth rate of technology is increasing in the level of R&D labour. This implication is inconsistent with the evidence documented in Jones (1995), who then generalises the Romer model by parameterising the degree of intertemporal knowledge spillovers and restricting this parameter value to be less than unity. In this case, the steady-state growth rate of technology depends on the growth rate of R&D labour (instead of its level), removing the scale eﬀect in the Romer model. Then, we derive the market equilibrium allocation of R&D labour and explore its determinants. In this case, we ﬁnd that the growth rate of the labour force and the degree of intertemporal knowledge spillovers also aﬀect the market equilibrium allocation of R&D labour. 14.4.

Exercises

1. Suppose the government imposes an upper bound on the monopolistic price given by Pt (i) = μRt < Rt /α,

(14.14)

where the parameter μ ∈ (1, 1/α) is the markup ratio, capturing the monopolistic power of ﬁrms. Show that the equilibrium ratio

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of R&D labour to production labour becomes SR =

αn μ − 1 αgN μ−1 , = μ(1 − α) ρ + gN μ(1 − α) (1 − φ)ρ + n

(14.15)

which is increasing in the monopolistic power μ of ﬁrms. 2. Show that on the balanced growth path, the growth rate of capital, output and consumption is given by gN + n. 3. Show that on the balanced growth path, the saving rate s∗ in the Jones model is given by s∗ =

α δ + gN + n . μ ρ + δ + gN + n

(14.16)

4. Consider the introduction of leisure to the household’s utility function in the Jones model: ∞ e−ρt [ln Ct + β ln (1 − lt )] dt, U= 0

where the parameter β > 0 determines the importance of leisure 1 − lt in the utility function. In this case, the asset-accumulation equation becomes F˙t + K˙ t = rt Ft + Rt Kt + Wt lt − Ct − δKt . Suppose we assume μ ∈ (1, 1/α). Derive the equilibrium allocation of R&D labour LR and the growth rate of output on the BGP.

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Chapter 15

R&D Underinvestment and Subsidies

In this chapter, we will compare the decentralised equilibrium allocation in the Jones model to its socially optimal allocation chosen by a social planner. As we will show, due to the presence of R&D externality, the market equilibrium level of R&D labour is lower than its socially optimal level. In other words, as a result of market failure, the market economy exhibits a social problem of R&D underinvestment. In this case, it is socially optimal for the government to provide policy intervention in the form of R&D subsidy. Recall that the law of motion for the number of varieties in the Jones model is given by N˙ t = θNtφ LR,t , where LR,t is the number of R&D workers and θ > 0 is an R&D productivity parameter. For simplicity, we will consider a special case of φ = 0 (i.e., zero knowledge spillovers), so that the law of motion for the number of varieties simplifies to N˙ t = θLR,t . In this case, the steady-state equilibrium ratio of R&D labour to production labour in the Jones model simplifies to αgN αn , (15.1) = SR = ρ + gN ρ+n where we have set φ = 0 and gN = n.

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15.1.

Socially Optimal Level of R&D

To derive the socially optimal allocation in the Jones model, we first write down its key equations. The representative household has the following lifetime utility function: U=

0

∞

e−ρt ln Ct dt,

(15.2)

where the parameter ρ > 0 is the household’s discount rate and Ct is the level of consumption at time t. The aggregate production function for output Yt is given by Yt = Nt1−α Ktα L1−α Y,t ,

(15.3)

where the parameter α ∈ (0, 1) determines the intensity 1 − α of production labour LY,t . The accumulation equation of capital Kt is K˙ t = Yt − Ct − δKt ,

(15.4)

where the parameter δ > 0 is the depreciation rate of capital. The law of motion for the number of varieties is N˙ t = θLR,t .

(15.5)

Finally, the resource constraint on labour is LR,t + LY,t = Lt ,

(15.6)

where the labour force Lt grows at a constant exogenous growth rate n. Given that we are deriving the socially optimal allocation, we are solving the centralised version of the model. In other words, we are not dealing with the market equilibrium, but instead, we directly maximise the household’s utility in (15.2) subject to (15.3)–(15.6). For convenience, we will drop all the time subscripts. The Hamiltonian function is − C − δKt ) + λN θ(L − LY ), H = ln C + λK (N 1−α K α L1−α Y

(15.7)

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123

where λK is the multiplier for K˙ and λN is the multiplier for N˙ . The first-order conditions are 1 ∂H = − λK = 0, ∂C C Y ∂H = λK (1 − α) − λN θ = 0, ∂LY LY Y ∂H = λK α − δ = λK ρ − λ˙ K , ∂K K Y ∂H = λK (1 − α) = λN ρ − λ˙ N . ∂N N

(15.8) (15.9) (15.10) (15.11)

Equation (15.9) can be re-expressed as (1 − α)λK Y = θLY . λN

(15.12)

On the balanced growth path (BGP), the growth rate of Y and C is1 gY = gC = gN + n = 2n,

(15.13)

where the second equality holds due to φ = 0 and gN = n. Rewriting (15.11) yields (1 − α)λK Y λ˙ N =ρ− , λN N λN

(15.14)

where both sides are stationary in the long run. From (15.8), we have λK C = 1, which is stationary and implies that λK Y is also stationary. Therefore, λN N must be stationary, implying that λ˙ N /λN = −gN = −n. Now we can substitute (15.12) into (15.14) to obtain θLY = ρ + n, N 1

(15.15)

One can use the production function in (15.3) and the long-run implication that the growth rate of output and capital is the same to derive this growth rate.

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which also uses −λ˙ N /λN = n. We use the law of motion for N˙ to derive LR LR N˙ =θ =n⇔N =θ . N N n

(15.16)

∗ Substituting (15.16) into (15.15) yields the socially optimal ratio SR of R&D labour to production labour given by ∗ SR =

15.2.

n . ρ+n

(15.17)

R&D Underinvestment

Comparing (15.1) and (15.17) shows R&D underinvestment in the market economy: SR =

αn ∗ = αSR , ρ+n

(15.18)

where α < 1. For example, if α = 1/2, then the market equi∗. librium R&D ratio SR is only half of the optimal R&D ratio SR The decentralised equilibrium allocates too little labour to the R&D sector because the market economy does not provide enough incentives. To see this, it is useful to note that the marginal product of Nt is Yt ∂Yt = (1 − α) , ∂Nt Nt

(15.19)

but the market allocates less than this amount to monopolistic profit: πt = α(1 − α)

Yt ∂Yt =α , Nt ∂Nt

(15.20)

where α < 1. Given that the social benefit of R&D (i.e., the marginal product of Nt ) is larger than the private benefit to entrepreneurs (i.e., the monopolistic profit), this is a positive R&D externality, which is known as the surplus appropriability problem. To solve this R&D underinvestment problem, the government could make use of R&D subsidies, which we will analyse in the following section.

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125

R&D Subsidies

We now determine the equilibrium allocation of R&D labour in the Jones model with R&D subsidies. We continue to focus on the special case with φ = 0. The cost of R&D is (1 − ς)Wt LR,t , where ς ∈ [0, 1] is the rate of R&D subsidies provided by the government. Therefore, the zero-profit condition in the R&D sector is modified to N˙ t vt = (1 − ς)Wt LR,t .

(15.21)

The value vt of an invention is vt =

πt , r − gπ

(15.22)

where the amount of monopolistic profit πt is πt = α(1 − α)

Yt . Nt

(15.23)

Therefore, the steady-state equilibrium growth rate of πt is gπ = gY − gN . The wage rate is Wt = (1 − α)

Yt . LY,t

(15.24)

Substituting (15.22)–(15.24) into (15.21) yields LR,t α N˙ t = (1 − ς) , r − g π Nt LY,t

(15.25)

where N˙ t /Nt = gN = n (due to φ = 0). Combining the consumption path r = ρ + gC with gπ = gY − gN yields r − gπ = ρ + gN = ρ + n,

(15.26)

which also uses gY = gC . If we define SR ≡ LR /LY , then Equation (15.25) becomes SR =

αgN αn 1 1 . = 1 − ς ρ + gN 1−ς ρ+n

(15.27)

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Therefore, the steady-state equilibrium ratio of R&D labour to production labour is increasing in the rate of R&D subsidies as expected. ∗ in (15.18), we need to set the To equate SR in (15.27) with SR rate of R&D subsidies to the following value: ς ∗ = 1 − α.

(15.28)

Given that the capital share is α ∈ (1/3, 1/2) in the data, our analysis implies that the optimal rate of R&D subsidies can be as high as 50–67%!

15.4.

Summary

In this chapter, we derive the socially optimal level of R&D labour in the Jones specification of the Romer model and compare it with the market equilibrium level. We find that the market underinvests in R&D due to the presence of positive R&D externality in the economy. In other words, the model features market failure in the form of R&D underinvestment. In this case, policy intervention by the government subsidising R&D can eliminate the social problem of R&D underinvestment and restore the optimal level of R&D labour in the market equilibrium. However, our model predicts that the required rate of R&D subsidies may be very high.

15.5.

Exercises

1. Consider the general law of motion for the number of varieties given by N˙ t = θNtφ LR,t .

(15.29)

Show that the socially optimal rate of R&D subsidies is given by ς∗ = 1 − α

ρ+n , ρ + n/(1 − φ)

(15.30)

which is increasing in φ. Intuitively, φ is an additional force of R&D externality. When φ > 0, current R&D improves the

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productivity of future R&D. The productivity of R&D is determined by ∂ N˙ t = θNtφ . ∂LR,t

(15.31)

Therefore, when an entrepreneur invests in R&D today, it increases Nt and R&D productivity θNtφ in the future. However, the entrepreneur does not take into account this social benefit of R&D when deciding how much R&D to do. This positive R&D externality is known as intertemporal knowledge spillover.2 2. Show that the socially optimal allocation of R&D labour in the Romer model is given by ρ (15.32) L∗R = L − . θ In the Romer model, intertemporal knowledge spillovers are also present because N˙ = θN LR . Therefore, a larger N leads to a higher R&D productivity θN . 3. Derive the socially optimal rate of R&D subsidies in the Romer model.

2

Jones and Williams (2000) provide a discussion and formalisation of other R&D externalities in the Romer model.

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Chapter 16

The Schumpeterian Growth Model

In this chapter, we cover the Schumpeterian quality-ladder model developed by Aghion and Howitt (1992) and follow the treatment of the model in Grossman and Helpman (1991).1 In the qualityladder model, economic growth is driven by the development of higher-quality products that replace lower-quality products. Joseph Schumpeter (1942) argues that this process of creative destruction is the main engine of long-run economic growth. As we will show, innovation in the form of quality improvement has a diﬀerent implication from innovation in the form of new product development.2 Speciﬁcally, the replacement of lower-quality products by higher-quality products gives rise to a business-stealing eﬀect, which is a negative R&D externality. Therefore, in the Schumpeterian growth model, the market economy may exhibit R&D underinvestment or overinvestment, which is diﬀerent from the Romer model that always exhibits R&D underinvestment. The Schumpeterian model features the following four components: a representative household; ﬁnal good; intermediate goods; and R&D. The household supplies labour to the R&D sector and the intermediate-good sector. Then, monopolistic ﬁrms that have

1

The materials covered in this chapter are relatively advanced. Peretto (1994) combines the two dimensions of innovation in quality improvement and new product development to develop the second-generation Schumpeterian growth model that removes the scale eﬀect and endogenises the market structure of the economy. 2

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the most advanced technologies produce diﬀerentiated intermediate goods. Finally, a representative ﬁnal-good ﬁrm aggregates intermediate goods into the ﬁnal good to be consumed by the household. 16.1.

Household

There is a representative household which has a log utility function: ∞ e−ρt ln Ct dt, (16.1) U= 0

where the parameter ρ > 0 is the household’s discount rate and Ct is the level of consumption at time t. The household maximises utility subject to3 F˙ t = rt Ft + Wt L − Ct .

(16.2)

rt is the real interest rate. Ft is the value of ﬁnancial assets owned by the household. Wt is the wage rate. The household supplies L units of labour. Using Hamiltonian, we can derive the familiar consumption path given by C˙ t = rt − ρ. Ct 16.2.

(16.3)

Final Good

This sector is perfectly competitive, and ﬁrms take the output and input prices as given. Final good Yt is produced by aggregating a unit continuum of diﬀerentiated intermediate goods Xt (i) indexed by i ∈ [0, 1]. We consider a Cobb–Douglas aggregator given by4 1 Yt = exp ln Xt (i)di . (16.4) 0

3

We do not consider capital in the baseline model and will add capital in an extension. 4 The aggregator can be re-expressed in a multiplicative form: Yt = Π10 Xt (i)di.

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From proﬁt maximisation, the demand function for Xt (i) is ∂Yt Yt = = Pt (i), ∂Xt (i) Xt (i)

(16.5)

where Pt (i) is the price of Xt (i) for i ∈ [0, 1]. 16.3.

Intermediate Goods

There is a unit continuum of intermediate goods indexed by i ∈ [0, 1]. Each industry i is dominated by a temporary monopolistic leader, who holds a patent on the latest innovation and dominates the market until the next innovation arrives. The production function for the industry leader is Xt (i) = z qt (i) Lx,t (i).

(16.6)

The parameter z > 1 is the step size of a quality improvement. qt (i) is the number of quality improvements that have occurred in industry i as of time t. In other words, z qt (i) is the highest level of quality in industry i. Lx,t (i) is production labour in industry i. The industry leader’s marginal cost of producing Xt (i) is Wt . (16.7) z qt (i) To derive the equilibrium price, we assume that the current and former industry leaders engage in Bertrand competition. Because the current industry leader’s product has a higher quality by a factor of z, the proﬁt-maximising price for the current leader is a constant markup over the marginal cost. M Ct (i) =

Pt (i) = μM Ct (i),

(16.8)

where the markup ratio μ ∈ (1, z] is less than or equal to the quality step size, depending on the level of patent protection, which determines the market power of the industry leader.5 Monopolistic proﬁt 5 Aghion and Howitt (1992) and Grossman and Helpman (1991) assume that the markup ratio is determined by the quality step size z. Li (2001) generalises the Schumpeterian model by introducing a patent policy parameter µ ≤ z that determines the markup ratio.

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in industry i is πt (i) = [Pt (i) − M Ct (i)] Xt (i) μ−1 μ−1 Pt (i)Xt (i) = Yt , = μ μ

(16.9)

where the last equality uses (16.5) and (16.8). Finally, wage income to production labour is Wt Lx,t (i) = M Ct (i)Xt (i) =

16.4.

Yt Pt (i)Xt (i) = . μ μ

(16.10)

R&D

Denote the value of an invention in industry i as vt (i). Due to the Cobb–Douglas speciﬁcation in (16.4), the amount of proﬁt is the same across industries (i.e., πt (i) = πt for i ∈ [0, 1]). As a result, vt (i) = vt in a symmetric equilibrium in which the arrival rate of innovation is equal across industries.6 The value of an invention is given by vt =

πt . r − gπ + σ

(16.11)

The present value of monopolistic proﬁts is discounted by r − gπ + σ, where σ is the arrival rate of innovation, because when an innovation arrives, the current industry leader loses the market to the next industry leader. Competitive R&D entrepreneurs hire R&D labour Lr,t to create inventions. The expected beneﬁt of R&D is vt σt whereas the cost of R&D is Wt Lr,t . Zero expected proﬁt in R&D implies that vt σt = Wt Lr,t . 6

(16.12)

Cozzi et al. (2007) provide a theoretical foundation for the symmetric equilibrium to be the unique rational-expectation equilibrium in the Schumpeterian model.

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The arrival rate of innovation is σt = θLr,t , where the parameter θ > 0 determines R&D productivity. Therefore, (16.12) can be reexpressed as θvt = Wt .

(16.13)

This R&D condition determines the allocation of labour between production and R&D. 16.5.

Aggregation

Substituting (16.6) into (16.4) yields the aggregate production function: Yt = At Lx,t ,

(16.14)

where the (log) level of aggregate technology At is deﬁned as 1 t qt (i)di ln z = σs ds ln z. (16.15) ln At ≡ 0

0

The last equality uses the law of large numbers, 1 which equates the average number of quality improvements 0 qt (i)di that have occurred as of time t to the total number of innovation arrivals t σ ds up to time t. Diﬀerentiating ln At with respect to time yields 0 s the law of motion for At given by7 A˙ t = σt ln z. At

(16.16)

Therefore, the steady-state growth rate of technology is gA = σ ln z = (θ ln z)Lr , which is increasing in R&D labour Lr . 16.6.

Market Equilibrium Level of R&D

In this model without capital, all the ﬁnal good is consumed by the household such that Y t = Ct . 7

(16.17)

Here, we apply the Leibniz rule for diﬀerentiation under the integral sign.

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Also, the labour market must clear such that Lx,t + Lr,t = L.

(16.18)

To solve for the steady-state equilibrium labour allocation, we use gπ = gY = gC ,

(16.19)

where gπ denotes the growth rate of the proﬁt in (16.9). Substituting (16.9), (16.10), (16.11) and (16.19) into the R&D condition in (16.13) yields θ(μ − 1)Lx = θLr + ρ,

(16.20)

where r−gπ = ρ because gπ = gY = gC . Combining (16.18) and (16.20) yields the market equilibrium level of R&D labour given by ρ μ−1 L− . (16.21) Lr (ρ , θ , μ, L) = μ μθ − + + + The comparative statics are as follows. Lr is decreasing in ρ because a higher discount rate decreases the present value of future monopolistic proﬁts. Lr is increasing in θ because higher R&D productivity increases the incentives for R&D. Lr is increasing in μ because a larger markup increases the amount of monopolistic proﬁt, which determines the private return to R&D. Finally, Lr is increasing in L implying the presence of the scale eﬀect.8 16.7.

Socially Optimal Level of R&D

In this section, we derive the socially optimal allocation of the model. Suppose a benevolent social planner maximises the house∞ hold’s utility U = 0 e−ρt ln Ct dt subject to the aggregate production Ct = At Lx,t , the law of motion for technology A˙ t = At (θ ln z)Lr,t and the labour resource constraint Lx,t + Lr,t = L. Then, setting up 8

See Segerstrom (1998) for a semi-endogenous-growth version of the Schumpeterian model.

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the Hamiltonian function Ht = ln At + ln(L − Lr,t ) + λt At (θ ln z)Lr,t , we can derive the optimal allocation of R&D labour L∗r given by9 ρ . (16.22) L∗r (ρ , θ , z , L) = L − + + + θ ln z − Comparing (16.21) and (16.22), we see that the market equilibrium allocation Lr is not necessarily equal to and can be larger or smaller than the socially optimal allocation L∗r . Speciﬁcally, there exists a threshold value of the markup ratio μ above (below) which the market equilibrium allocation Lr is above (below) the socially optimal allocation L∗r . The threshold μ is θL ln z, μ≡ 1+ ρ which is increasing in the quality step size z. The reason is that the Schumpeterian growth model features both positive and negative externalities of R&D. Intertemporal knowledge spillover in A˙ t = At (θ ln z)Lr,t is a positive externality of R&D, whereas the business-stealing eﬀect (i.e., new industry leaders taking over the market from current industry leaders) is a negative externality of R&D. It is useful to note that this business-stealing eﬀect is absent in the Romer model, which features only positive externality. When the positive externality dominates the negative externality, the market underinvests in R&D such that Lr < L∗r . R&D underinvestment happens when the quality step size z, which captures the social beneﬁt of R&D, is suﬃciently large. When the negative externality dominates the positive externality, the market overinvests in R&D such that Lr > L∗r . R&D overinvestment happens when the markup μ, which captures the private beneﬁt of R&D, is suﬃciently large. 16.8.

Capital

The canonical quality-ladder model does not feature physical capital Kt . In this section, we extend the quality-ladder model by allowing 9

The derivations of L∗r are left as an exercise.

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for capital accumulation. We assume for simplicity that only the intermediate-good sector employs capital. We modify the production function (16.6) to Xt (i) = z qt (i) [Kt (i)]α [Lx,t (i)]1−α ,

(16.23)

where the parameter α ∈ (0, 1) determines the intensity of capital Kt (i) in the production of Xt (i). From cost minimisation, the marginal cost of production is M Ct (i) =

1

z qt (i)

Rt α

α

Wt 1−α

1−α

,

(16.24)

where Rt denotes the rental price of capital. The proﬁt income is still given by (16.9) whereas capital income and production-labour income are, respectively, Rt Kt = Wt Lx,t =

αYt , μ

(1 − α)Yt . μ

(16.25) (16.26)

Substituting (16.9), (16.19) and (16.26) into (16.13) yields θ(μ − 1)Lx = (1 − α)(θLr + ρ).

(16.27)

Combining (16.18) and (16.27) yields the market equilibrium level of R&D labour given by μ−1 1−α ρ . (16.28) Lr (ρ , θ , μ, L, α) = L− μ−α μ−α θ − + + + + The comparative statics for ρ, θ, μ and L are the same as before. As for the capital intensity, Lr is increasing in α because a decrease in labour intensity in production reduces the usage of labour for production and leads to more labour available for R&D. To complete the analysis in this section, we substitute (16.23) into (16.4) to derive the aggregate production given by Yt = At Ktα L1−α x,t .

(16.29)

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Also, the resource constraint on the ﬁnal good is Y t = Ct + I t ,

(16.30)

and the law of motion for capital accumulation is .

K t = It − δKt .

(16.31)

Along the balanced-growth path, the growth rate of output equals the growth rate of capital; therefore, the long-run growth rate of output is 1 θ ln z gA = Lr , (16.32) gY = 1−α 1−α where the market equilibrium level of Lr has been derived in (16.28). 16.9.

Summary

In this chapter, we develop the Schumpeterian growth model in which innovation is driven by the quality improvement of existing products, instead of the development of new products as in the Romer model. In addition to the usual positive R&D externality, this alternative engine of innovation gives rise to negative R&D externality in the form of a business-stealing eﬀect, in which entrants take away the market share of incumbents when inventing a higher-quality product. As a result of this negative R&D externality, the market may overinvest or underinvest in R&D. R&D overinvestment occurs when ﬁrms have too much market power which determines the private beneﬁt of R&D, whereas R&D underinvestment occurs given a suﬃciently large quality step size which determines the social beneﬁt of R&D. 16.10.

Exercises

1. Consider the Schumpeterian growth model without capital. Use the Hamiltonian function Ht = ln At + ln(L − Lr,t ) + λt At (θ ln z)Lr,t ,

(16.33)

to show that the optimal allocation of R&D labour L∗r is given by ρ L∗r = L − . (16.34) θ ln z

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2. Suppose the government provides R&D subsidies. Derive the equilibrium level of R&D labour in the Schumpeterian growth model without capital. 3. Consider the introduction of leisure to the household’s utility function: ∞ e−ρt [ln Ct + β ln(L − lt )]dt, (16.35) U= 0

where the parameter β > 0 determines the importance of leisure L − lt in the utility function. In this case, the asset-accumulation becomes F˙t = rt Ft + Wt lt − Ct .

(16.36)

Show that the equilibrium level of R&D labour in the Schumpeterian growth model without capital is given by ρ μ−1 1 L − (1 + βμ) , (16.37) Lr = 1+β μ μθ which is the same as (16.21) when β = 0. 4. Suppose the price regulation policy is not binding, i.e., μ ≥ z. In this case, the Bertrand competition between the current and former industry leaders implies Pt (i) = zM Ct (i).

(16.38)

Derive the equilibrium level of R&D labour Lr and the growth rate of output.

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Chapter 17

Appendix on Dynamic Optimisation

In this appendix, we demonstrate how to use a mathematical tool known as the Hamiltonian (named after William Rowan Hamilton) to solve dynamic optimisation problems in continuous time. We use the Ramsey model as an example. The household’s lifetime utility function is given by ∞ e−ρt u(Ct )dt, (17.1) U= 0

where u(Ct ) is a diﬀerentiable, strictly increasing and concave function in Ct ,1 which denotes consumption. The parameter ρ > 0 is the subjective discount rate. The household maximises utility in (17.1) subject to the following capital-accumulation equation: K˙ t = F (Kt ) − Ct ,

(17.2)

where K˙ t ≡ ∂Kt /∂t denotes the change in the level of capital with respect to time t, and F (Kt ) is a diﬀerentiable and concave production function in Kt net of capital depreciation.2

1

In the main text, we consider a log utility function u(Ct ) = ln Ct for simplicity. In the main text, we consider a Cobb–Douglas production function for simplicity. 2

139

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The household maximises (17.1) subject to (17.2). To solve this dynamic optimisation problem, we use the Hamiltonian. The currentvalue Hamiltonian function Ht is given by3 Ht = u(Ct ) + λt [F (Kt ) − Ct ].

(17.3)

In other words, the Hamiltonian function at time t consists of (a) the utility function u(Ct ), (b) the right-hand side of the capitalaccumulation equation F (Kt ) − Ct , and (c) a multiplier λt (known as the co-state variable) for the capital-accumulation equation. To maximise the household’s utility, we derive the ﬁrst-order conditions with respect to {Ct , Kt , λt }: ∂u(Ct ) ∂Ht = − λt = 0, ∂Ct ∂Ct ∂F (Kt ) ∂Ht = λt = λt ρ − λ˙ t , ∂Kt ∂Kt ∂Ht = F (Kt ) − Ct = K˙ t . ∂λt

(17.4) (17.5) (17.6)

It is useful to note that Kt is a state variable (i.e., a variable that accumulates over time),4 so we have to treat its ﬁrst-order condition diﬀerently. Instead of equating ∂Ht /∂Kt to zero, we set ∂Ht /∂Kt = λt ρ − λ˙ t . As for the ﬁrst-order condition with respect to the co-state variable λt , it simply yields the capital-accumulation in (17.2). If the household is maximising utility in (17.1), then its consumption behaviour and its accumulation of capital must satisfy (17.4)–(17.6). Suppose u(Ct ) = ln Ct . Then, (17.4) can be expressed as λt =

∂u(Ct ) 1 = , ∂Ct Ct

(17.7)

where 1/Ct is the marginal utility of consumption. Taking the log of (17.7) yields ln λt = − ln Ct . 3

(17.8)

There is an alternative formulation, known as the present-value Hamiltonian. In contrast, Ct is a control variable that can jump to a diﬀerent value at any time t. 4

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Diﬀerentiating both sides of (17.8) with respect to t yields5 C˙ t λ˙ t =− . λt Ct

(17.9)

Substituting (17.9) into (17.5) yields λ˙ t ∂F (Kt ) C˙ t =− = − ρ, Ct λt ∂Kt

(17.10)

where ∂F (Kt )/∂Kt is the marginal product of capital net of depreciation. C˙ t /Ct in (17.10) is the optimal consumption path of the household. If the household is maximising utility, then its evolution of consumption must satisfy the optimal consumption path in (17.10). Also, its accumulation of capital must satisfy the capital-accumulation equation in (17.2). Finally, one can use (17.2) and (17.10) to draw a two-dimension phase diagram to show that Ct and Kt converge to a steady state with C˙ t = K˙ t = 0.6

5 6

˙

λt t Note that ∂ ln = λ1t ∂λ = λλtt . ∂t ∂t See Romer (2018, Chapter 2).

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References

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

11. 12. 13.

14.

Acemoglu, D., 2009. Introduction to Modern Economic Growth. Princeton, NJ: Princeton University Press. Aghion, P., and Howitt, P., 1992. A model of growth through creative destruction. Econometrica, 60, 323–351. Aghion, P., and Howitt, P., 2009. The Economics of Growth. Cambridge, MA: MIT Press. Barro, R., 1974. Are government bonds net wealth? Journal of Political Economy, 82, 1095–1117. Barro, R., Chu, A., and Cozzi., G., 2017. Intermediate Macroeconomics. Cengage Learning. Barro, R., and Redlick, C., 2011. Macroeconomic eﬀects from government purchases and taxes. Quarterly Journal of Economics, 126, 51–102. Barro, R., and Sala-i-Martin, X., 2003. Economic Growth. Cambridge, MA: MIT Press. Baxter, M., and King, R., 1993. Fiscal policy in general equilibrium. American Economic Review, 83, 315–334. Cass, D., 1965. Optimum growth in an aggregative model of capital accumulation. Review of Economic Studies, 32, 233–240. Chu, A., 2018. From Solow to Romer: Teaching endogenous technological change in undergraduate economics. International Review of Economics Education, 27, 10–15. Cozzi, G., 2017a. Endogenous growth, semi-endogenous growth. . . or both? A simple hybrid model. Economics Letters, 154, 28–30. Cozzi, G., 2017b. Combining semi-endogenous and fully endogenous growth: A generalization. Economics Letters, 155, 89–91. Cozzi, G., Giordani, P., and Zamparelli, L., 2007. The refoundation of the symmetric equilibrium in Schumpeterian growth models. Journal of Economic Theory, 136, 788–797. Dixit, A., and Stiglitz, J., 1977. Monopolistic competition and optimum product diversity. American Economic Review, 67, 297–308. 143

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16. 17. 18. 19. 20. 21. 22.

23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33.

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References Gali, J., 2015. Monetary Policy, Inflation, and the Business Cycle: An Introduction to the New Keynesian Framework and its Applications. Princeton, NJ: Princeton University Press. Galor, O., 2011. Unified Growth Theory. Princeton, NJ: Princeton University Press. Grossman, G., and Helpman, E., 1991. Quality ladders in the theory of growth. Review of Economic Studies, 58, 43–61. Jones, C., 1995. R&D-based models of economic growth. Journal of Political Economy, 103, 759–784. Jones, C., and Vollrath, D., 2013. Introduction to Economic Growth. New York: W. W. Norton & Company. Jones, C., and Williams, J., 2000. Too much of a good thing? The economics of investment in R&D. Journal of Economic Growth, 5, 65–85. Kimball, M., 1995. The quantitative analytics of the basic neomonetarist model. Journal of Money, Credit and Banking, 27, 1241–1277. Koopmans, T., 1965. On the concept of optimal economic growth. In: The Econometric Approach to Development Planning, Amsterdam: NorthHolland Publishing Company. Kydland, F. and Prescott, E., 1982. Time to build and aggregate ﬂuctuations. Econometrica, 50, 1345–1370. Li, C.-W., 2001. On the policy implications of endogenous technological progress. Economic Journal, 111, C164–C179. McCandless, G., 2008. ABCs of RBCs: An Introduction to Dynamic Macroeconomic Models. Cambridge, MA: Harvard University Press. Peretto, P., 1994. Essays on Market Structure and Economic Growth. Ph.D. dissertation, Yale University. Ramsey, F., 1928. A mathematical theory of saving. Economic Journal, 38, 543–559. Romer, D., 2018. Advanced Macroeconomics. New York: McGraw-Hill Education. Romer, P., 1990. Endogenous technological change. Journal of Political Economy, 98, S71–S102. Schumpeter, J., 1942. Capitalism, Socialism and Democracy. New York: Harper and Brothers. Segerstrom, P., 1998. Endogenous growth without scale eﬀects. American Economic Review, 88, 1290–1310. Solow, R., 1956. A contribution to the theory of economic growth. Quarterly Journal of Economics, 70, 65–94. Swan, T., 1956. Economic growth and capital accumulation. Economic Record, 32, 334–361.

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Index A Acemoglu, D., 71, 75 aggregate labour demand, 66, 68 aggregate production function, 72, 96, 98, 108, 122, 133 aggregation Ramsey model with a monopolistically competitive market, 96–98 Romer model, 108–109 Schumpeterian growth model, 133 Aghion, P., 71, 129, 131 asset-accumulation equation, 11, 15, 26, 33, 36, 42–43, 46, 53, 56, 62, 88, 91, 94, 101, 104, 112, 120

Hamiltonian, 56–57 household, 55–56 long-run eﬀects of, 58–60 Cass, D., 79 Chu, A., 103 classical dichotomy, 7, 68 Cobb–Douglas aggregator, 130 Cobb–Douglas production function (see also production function), 2, 12–13, 37, 72, 80, 130, 139 Cobb–Douglas speciﬁcation, 132 constant elasticity of substitution, 64 Cozzi, G., 116, 132 creative destruction, 129 D demand functions, 13, 27, 38, 48, 57, 89, 95, 106 discount rate, 10, 12, 14–15, 19, 23, 26, 33, 36, 45, 55, 79–80, 82–84, 88, 94, 98, 104, 110–111, 119, 122, 130, 134, 139 Dixit, A., 63 dynamic general equilibrium, 1, 9, 15 dynamic optimisation, 9–11, 15, 80–81, 88, 94, 139–141

B balanced growth path (BGP), 75, 109, 112, 115–116, 120, 123 Barro, R., 52, 55, 71 Baxter, M., 35 C capital-accumulation equation, 13, 15, 81, 83, 85, 89, 139, 140–141 capital demand curve, 2, 6, 18, 20–21, 28, 30, 32, 38, 39, 41–42, 48–49, 51, 53, 58, 97–98 capital income tax ﬁrm, 57 government spending, 56, 61

E economic growth, 71–72, 74–76, 79, 84, 107, 110–111, 129 equilibrium, 3–4 capital market, 4 145

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Index

146 labour market, 3 real rental price, 3 real wage rate, 3 F ﬁrm ﬁscal policy: capital income tax, 57 ﬁscal policy: government spending, 37–38 ﬁscal policy: labour income tax, 47–48 neoclassical growth model, 12–13 neoclassical growth model with elastic labour supply, 27 Ramsey model with a perfectly competitive market, 89 ﬁrst fundamental theorem of welfare economics, 87, 90–91, 93, 100 G Gali, J., 63 general equilibrium (see also equilibrium), 1 general-equilibrium eﬀect, 4, 6, 22, 28, 32, 42, 53, 61 government spending, 35–36 capital income tax, 56, 61 ﬁrm, 37–38 Hamiltonian, 36–37 household, 35–36 labour income tax, 46, 52 long-run eﬀects, 38–40 short-run eﬀects, 40–42 Grossman, G., 129, 131 H Hamilton, William Rowan, 139 Hamiltonian, 9–10, 139 ﬁscal policy: capital income tax, 56–57 ﬁscal policy: government spending, 36–37 ﬁscal policy: labour income tax, 46–47 neoclassical growth model, 11–12

neoclassical growth model with elastic labour supply, 26–27 Ramsey model, 81–82 Hamiltonian function, 11, 26, 36, 46, 56, 81, 88, 104, 122, 135, 140 Helpman, E., 129, 131 household ﬁscal policy: capital income tax, 55–56 ﬁscal policy: government spending, 35–36 ﬁscal policy: labour income tax, 45–46 neoclassical growth model, 9–11 neoclassical growth model with elastic labour supply, 25–26 Ramsey model, 79–81 Ramsey model with a perfectly competitive market, 87–88 Ramsey model with a monopolistically competitive market, 94 Romer model, 104–105 Schumpeterian growth model, 130 static general-equilibrium model, 1 Howitt, P., 71, 129, 131 I income eﬀect, 25–27, 35, 37, 39–40, 46, 52, 56–57, 60–61 intermediate microeconomics, 9 intertemporal knowledge spillover, 115, 127, 135 International Monetary Fund, 71 isoelastic utility function, 16 J Jones, C., 71, 113, 115, 119, 127 Jones model, 115–119 K Kimball, M., 63 King, R., 35

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Index Koopmans, T., 79 Kydland, F., 25 L labour demand curve, 2, 4, 6, 18, 20, 21–22, 28, 30, 32, 38–39, 42, 48–49, 53, 58–60, 68 labour income tax (see also capital income tax) government spending, 46, 52 Hamiltonian, 46–47 household, 45–46 long-run eﬀects of, 48–50 short-run eﬀects of, 50–52 labour supply curve, 1, 4, 17, 21, 26, 28, 30–32, 37–39, 41, 47–48, 51, 57–59 Lagrangian, 9 Leibniz rule, 133 log utility function, 9, 80, 130, 139 long-run eﬀects capital income tax of, 58–60 capital market, 20, 31, 40, 50, 59 government spending of, 38–40 labour income tax of, 48–50 labour market, 20, 30, 39, 49, 59 monetary policy of, 68 of technology, 19–21, 29–31 M McCandless, G., 25 monopolistic competition, 63 versus perfect competition, 98–99 monopolistic distortion, 93, 99 N neoclassical growth model, 9 dynamics, 21 ﬁrm, 12–13 Hamiltonian, 11–12 household, 9–11 long-run eﬀects of technology, 19–21

b3947-index

147 short-run eﬀects of technology, 17–19 steady-state equilibrium, 13–15 neoclassical growth model with elastic labour supply, 25 dynamics, 32 ﬁrm, 27 Hamiltonian, 26–27 household, 25–26 long-run eﬀects of technology, 29–31 short-run eﬀects of technology, 28–29 neutrality of money, 5–6, 63, 68 nominal rental price, 5–6 nominal wage rate, 5–6 price level, 5–6 New Keynesian Model, 63 ﬁnal output, 64–65 intermediate goods, 65–66 long-run eﬀects of monetary policy, 68 short-run eﬀects of monetary policy, 67–68 simple model, 64 new neoclassical synthesis, 63 O optimal consumption path, 12, 19, 27, 29, 37–38, 47–48, 57–58, 94, 141 P Peretto, P., 129 Prescott, E., 25 present-value Hamiltonian, 140 production function, 3, 14, 18, 21, 27–28, 31, 37, 40, 47, 50–51, 57, 60, 64, 74, 83, 89, 94–95, 105–106, 131, 136, 139 proﬁt function, 2, 13, 27, 37, 47, 57, 64–66, 89, 95, 105–106, 108 R Ramsey, F., 79 Ramsey–Cass–Koopmans model, 79

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148 Ramsey model (see also Solow growth model) Hamiltonian, 81–82 household, 79–81 steady state, 82–84 Ramsey model with perfectly competitive market equilibrium, 89–90 ﬁrm, 89 household, 87–88 Ramsey model with monopolistically competitive market aggregation, 96–98 ﬁnal good, 94–95 household, 94 intermediate goods, 95–96 market power of monopolistic ﬁrms, 99–100 monopolistic competition versus perfect competition, 98–99 R&D productivity, 107, 109–110, 113, 121, 127, 133–134 real business cycle model, 25 Redlick, C., 52 research and development (see also R&D), 103–104, 107–108 externality, 121, 124, 129 labour, 107, 109–111, 117–119, 121, 124–126, 132–136 socially optimal level of, 122–124, 134–135 Schumpeterian growth model, 132–133 subsidies, 125–126 underinvestment, 124 Romer, D., 25, 55, 63, 71, 141 Romer, P., 103, 115 Romer model, 103 Romer model, scale eﬀect in Jones model, 115–116 R&D labour, 117–119 solving model, 116–119 Romer model of endogenous technological change aggregation, 108–109

b3947-index

Index capital and labour intensity in production, 110 discount rate, 110 ﬁnal good, 105–106 household, 104–105 intermediate goods, 106–107 labour force, 110–111 R&D (see also research and development), 107–108 R&D productivity, 109–110 solving the model, 109 S scale eﬀect, 111, 113 Schumpeter, J., 129 Schumpeterian growth model aggregation, 133 capital, 135–137 ﬁnal good, 130–131 household, 130 intermediate goods, 131–132 market equilibrium level of R&D, 133–134 negative R&D externality, 135 R&D, 132–133 R&D underinvestment or overinvestment, 135 socially optimal level of R&D, 134–135 Segerstrom, P., 134 semi-endogenous growth model (see also Jones model), 116 short-run eﬀects capital market, 19, 29, 41, 51 government spending of, 40–42 labour income tax of, 50–52 labour market, 18, 29, 41, 51, 68 monetary policy of, 67–68 technology (neoclassical growth model) of, 17–19, 28–29 Solow, R., 71 Solow growth model exogenous saving rate, 73 phase diagram, 73

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b3947-index

Index with technological progress, 74–76 without technological progress, 72–74 Solow–Swan model, (see also Solow model) static general-equilibrium model (see also general equilibrium), 1–3, 9 elastic labour supply, 4–5 equilibrium, 3–4 neutrality of money, 5–6 steady-state equilibrium, 13–15 growth rate, 117–118, 125, 133–134 level of consumption, 15, 39, 49, 60, 83 level of labour, 31, 39, 49, 52, 60 levels of capital and output, 14, 83, 90, 97–98, 100 saving rate, 83, 99–100

149 Stiglitz, J., 63 substitution eﬀect, 25–27, 37, 45–46, 49, 51–52, 57 surplus appropriability problem, 124 T total factor productivity, 113–114 U utility function (see also log utility function), 9, 25, 35, 45, 55, 80, 87, 94, 104, 122, 130, 139 W Williams, J., 127 Z zero capital depreciation rate, 30, 39, 48, 60

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