Three essays on macroeconomic dynamics

THREE ESSAYS ON MACROECONOMIC DYNAMICS WAN JING (B.A. 2003, TianJin University M.A. 2006, Nankai University) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF ECONOMICS NATIONAL UNIVERSITY OF SINGAPORE 2012 ACKNOWLEDGEMENTS I have benefited greatly from the guidance and support of many people over the past four years. In the first place, I owe an enormous debt of gratitude to my main supervisor, Professor Zhang Jie, for his supervision from the very early stage of this research. I believe his passion, perseverance and wisdom in pursuit of the truth in science as well as his integrity, extraordinary patience and unflinching encouragement in guiding students will leave me a life-long influence. I am always feeling lucky and honorable to be supervised by him. I would also like to sincerely thank my co-supervisor, Professor Zhu Shenghao, for his supervision and support in various ways. In particular, the second chapter of this thesis was under the guidance of him. I also gratefully acknowledge him for his constructive comments on this thesis. Along with these professors, I also wish to thank my friends and colleagues at the department of Economics for their thoughtful suggestions and comments, especially to Zhang Shen and Li Bei. Finally, to my parents, my husband and my son, all I can say is that it is your unconditional love that gives me the courage and strength to face the challenges and difficulties in pursuing my dreams. Thanks for your acceptance and endless support to the choices I make all the time. i TABLE OF CONTENTS Acknowledgements i Table of Contents ii Summary v List of Tables vii List of Figures ix Chapter 1: Inflation, Taxation, Welfare and Growth Through Cycles with Money in the Utility Function 1.1 Introduction 1.2 The model with money in the utility function 1.2.1 The consumer 1.2.2 Production and innovation 1.3 Equilibrium and results 10 1.3.1 The steady state 15 1.3.2 The dynamics 17 1.4 Calibration and simulation results 23 1.4.1 Calibration 23 1.4.2 Simulations 25 1.5 Conclusion 28 ii Chapter 2: Social Optimality, Inflation and Taxation in an Endogenous Business Cycles Model with Innovation, Investment and Cash-in-advance Constraints 39 2.1 Introduction 39 2.2 The model 42 2.2.1 The consumer 43 2.2.2 Production and innovation 44 2.2.3 Government 47 2.2.4 Equilibrium 47 2.3 A tractable equilibrium with money and investment subsidization 50 2.3.1 The steady state 53 2.3.2 The dynamics 55 2.4 The socially optimal path and government policies 56 2.4.1 The socially optimal path 57 2.4.2 Optimal policy 61 2.5 Calibration and simulation results 64 2.6 Conclusion 68 Chapter 3: Intergenerational Links, Taxation, and Wealth Distribution 73 3.1 Introduction 73 3.2 The model 76 3.2.1 Agent’s problem 76 3.2.2 Firm’s problem 79 iii 3.2.3 Government 80 3.2.4 General equilibrium 80 3.3 Wealth distribution 82 3.4 Inequality measures 84 3.4.1 Lorenz dominance 84 3.4.2 The convex order 86 3.5 Bequest motives and wealth inequality 87 3.6 Ability inheritance and wealth inequality 89 3.7 Estate taxes and wealth inequality 93 3.8 Conclusion 96 3.9 Appendices 97 3.9.1 Proof of proposition 97 3.9.2 Proof of proposition 99 2.9.3 Proof of proposition 100 3.9.4 Proof of theorem 100 3.9.5 Proof of proposition 102 3.9.6 Proof of proposition 103 3.9.7 Proof of theorem 11 104 3.9.8 Proof of lemma 12 107 3.9.9 Proof of theorem 13 108 iv SUMMARY This thesis is composed of three essays on macroeconomic dynamics. The first chapter is a joint work with my supervisor, and it explores whether inflation taxation, a substitute for income taxation given fixed government spending, can mitigate business fluctuations, promote growth and enhance welfare by extending the Matsuyama model with endogenous growth through endogenous cycles to incorporate money in the utility function. Here, faster money growth promotes capital accumulation, innovation and growth by reducing income taxes. At low money growth rates, faster money growth enlarges fluctuations of periodtwo cycles. However, sufficiently high money growth rates can eliminate endogenous cycles and accelerate oscillatory convergence under plausible conditions. Numerically, optimal money growth enhances welfare based on calibration. The second chapter is a joint work with Assistant Professor Zhu Shenghao, and it determines the social optimal path in the innovation-cycle model of Matsuyama (1999, 2001) and explore whether inflation and taxation can be used to obtain the social optimum under a cash-in-advance constraint. The socially optimal path allows innovation to occur at a lower level of the capital-variety ratio than the equilibrium path. Also, starting from a binding capital constraint on innovation, the socially optimal path can move from the neoclassical regime without innovation towards the balanced path with innovation through a temporary transition. v The third chapter again is a joint work with my supervisor, which extends one of the main findings in Bossmann et al. (2007). ("Bequests, taxation and the distribution of wealth in a general equilibrium model", Journal of Public Economics, 91, 1247-1271.) Bequest motives per se reduce wealth inequality. We show that the result holds for a stronger criterion of inequality comparison between distributions. Bossmann et al. (2007) use the coefficient of variation as the inequality measure. Our Lorenz dominance result implies their result. We also strengthen two other conclusions in Bossmann et al. (2007). Earnings ability inheritance could increase wealth inequality and estate taxes could decrease wealth inequality. vi LIST OF TABLES Tables for chapter Table 1.1.Total tax revenue as percentage of GDP and per capita GDP levels in G7 countries 31 Table 1.2. Standard deviation, autocorrelation, and correlation with output: US data 1929-2011 at 10 year frequency 32 Table 1.3. Regression results for GDP growth rate and Average income tax / GDP 33 Table 1.4. Parameters from related literature 34 Table 1.5. Parameters calibrated to US data 35 Table 1.6. Simulation result I: Targeting inverse velocity of M1 36 Table 1.7. Simulation result II: Targeting inverse velocity of M2 37 Tables for chapter Table 2.1. Parameters come from related literature 70 Table 2.2. Parameters calibrated according to US observations 70 Table 2.3. Benchmark equilibrium with 71 and vii Table 2.4. Socially optimal policies and investment return with 71 Table 2.5. Comparison between the benchmark and the social optimum 71 viii Note that > > > > > > < vs v Thus v s cs4 v c4 cs4 = > c4 > > > > > : cs4 for s if s=0 if s=1 if s + c4s v + cs4 v + 0. For t + c4 v s + vs 1, at is an increasing function of a1 , l1 , and "2 , "3 , , "t . Also lt is an increasing function of a1 , l1 , and "2 , "3 , "2 , "3 , , "t . And a1 , l1 , and , "t are independent. By Proposition 20.I.13 of Marshall and Olkin (2007), we know that for t 1, at and lt are positively associated. 3.9.7 Proof of theorem 11 Proof: In economy H, v = 0. Thus H aH t+1 = c3 l + "t + c4 at + c5 : Note that l + "t and aH t are independent. In economy I, < v < 1. From proposition we know that aIt and lt are positively associated. Let aI1 = and aH = 1. Thus H aH = c l + " + c a1 + c 104 cx c3 l1 + c4 aI1 + c5 = a2 since l + "1 cx l1 by proposition 8.18 Now suppose that aH t aIt . Find two independent random variables U and V cx such that U =st lt and V =st aIt Thus l + "t cx U and aH t cx V . Thus c l + " t + c aH t cx c3 U + c4 V by the property of the convex order in footnote 17 and part (d) of Theorem 3.A.12 of Shaked and Shanthikumar (2010). By lemma 10 we have c3 U + c4 V cx c3 lt + c4 aIt since aIt and lt are positively associated. 18 X cx Y implies bX + c cx bY + c for any b; c R. Note that (bx + c) is a convex function of x R if (x) is a convex function of x R. 105 By the transitivity of the convex order we have c l + " t + c aH t cx c3 lt + c4 aIt : H aH t+1 = c3 l + "t + c4 at + c5 cx c3 lt + c4 aIt + c5 = aIt+1 Thus by the property of the convex order in footnote 18. By mathematical induction we have aH t cx aIt ; 8t 1: H I I Since aH t !st a1 and at !st a1 , we have aH cx aI1 by part (c) of Theorem 3.A.12 of Shaked and Shanthikumar (2010). By lemma we have aH I since E aH = E a1 = c5 +c3 . c4 106 L aI1 3.9.8 Proof of lemma 12 Proof: Let g(x) = (1 )x + E (X) , x [0; +1) h(x) = (1 ^)x + ^E (X) , x [0; +1) and Note that g( ) and h( ) are non-negative increasing functions de…ned on [0; +1), since ^ < 1. Also g(x) > and h(x) > for x > 0. Note that h(x) g(x) is increasing in x (0; +1), since (1 h(x) = g(x) (1 = 1 ^)x + ^E (X) ^ )x + E (X) 41 ^ ^ (1 E(X) 5: E(X) x + ) By Theorem 3.A.26 of Shaked and Shanthikumar (2010) we have g(X) (1 (1 L h(X), i.e. (1 ^)X + ^E (X). By lemma we have (1 )X + E (X) cx h i ^)X + ^E (X) since E [(1 )X + E (X)] = E(X) = E (1 ^)X + ^E (X) . )X + E (X) L 107 3.9.9 Proof of theorem 13 Proof: Note that a1 is the stationary distribution of the stochastic process fat g which is generated by h at+1 = c6 lt + c7 (1 )at + K i ^ ^ and a given a1 . And a1 is the stationary distribution of the stochastic process fat g which is generated by h ^ at+1 = c6 lt + c7 (1 ^)at^ + ^K i ^ and a given a1 . ^ Let a1 =st a1 . Thus a1 a1 by the de…nition of the convex order. ^ Now suppose that at cx (1 since E at ^ cx at . By lemma 12 we have )at + K cx (1 ^)at + ^K = K. By Corollary 3.A.22 of Shaked and Shanthikumar (2010) we have (1 (1 ^)at^ since ^)at cx ^ is independent of at and at^ . By Part (d) of Theorem 3.A.12 108 of Shaked and Shanthikumar (2010) we have ^)at + ^K (1 since ^K is independent of (1 cx ^)at and (1 (1 ^)at^ + ^K ^)at^ . By the transitivity of the convex order we have (1 )at + K cx (1 ^)at^ + ^K: h i By Corollary 3.A.22 of Shaked and Shanthikumar (2010) we have c7 (1 )at + K i h ^ ^ ^ ^ )at + K since c7 is independent of (1 )at + K and (1 ^)at + ^K. Note c7 (1 h i h i ^ ^ ^ that c6 lt and c7 (1 )at + K are independent. And c6 lt and c7 (1 )at + K are independent. Thus by part (d) of Theorem 3.A.12 of Shaked and Shanthikumar (2010), we have h c6 lt + c7 (1 i cx h c6 lt + c7 (1 at+1 cx at+1 : )at + K Thus we have ^ By mathematical induction we have at ^ cx 109 at ; 8t 1: i ^)at^ + ^K : cx ^ ^ Since at !st a1 and at !st a1 thus a1 ^ cx a1 by part (c) of Theorem 3.A.12 of Shaked and Shanthikumar (2010). By lemma we have a1 ^ since E a1 = E a1 = K. 110 ^ L a1 BIBLIOGRAPHY Adam, K., Billi, R.M., 2008. Monetary conservatism and fiscal policy. Journal of Monetary Economics 55, 1376-1388. Aiyagari, S. R., 1994. Uninsured idiosyncratic risk and aggregate saving. Quarterly Journal of Economics, 109, 659-684. 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Equation (9) follows from the consumer budget constraint and the government budget constraint with Equation (10) emerges from the optimal conditions (4), (5) and (6) According to (10), the inflation rate between period and has a direct negative effect on money demand in period since it drives up the cost of holding money A full characterization of the equilibrium solution will be given after considering... requirement constraint on innovation and intermediate goods production, initially magnifying period-two cycles but eventually inducing oscillatory convergence Also, money contraction in our model may lead to chaotic dynamics The remainder of the paper proceeds as follows Section 2 introduces the model Section 3 focuses on the equilibrium and derives the analytical results Section 4 provides the calibration and... on education and health are included in private consumption instead of in capital, then the ratio of consumption to output would be above 60% All the values of parameters in the second group are pinned down by the calibration simultaneously We view the calibration as the benchmark It is worth clarifying how we choose the measure of money balance for the calibration On one hand, for the determination... money-in-the-utility-function into the Matsuyama (1999, 2001) model It sheds some new light on monetary policy First, faster money growth promotes capital accumulation, innovation, and output growth by reducing income tax rates and making money holding more costly Second, faster money growth magnifies fluctuations of period-two cycles at low money growth rates but eliminates cyclical fluctuations asymptotically at high money... money growth rate on capital accumulation is based on equation (21) in both regimes The absence of an immediate effect of a change in the money growth rate on innovation is based on equation (16) As a result, it has a positive effect on the capital-variety ratio certain , which is transparent in equation (22) With periods of lag, the increased capital variety ratio will eventually increase innovation... monopoly pricing, a welfare improving combination of money growth and income taxation is better than any combination of money growth and fiscal policy (such as lump-sum or consumption taxation) that maintains neutrality The consumer's problem can be formulated as , subject to the solvency condition, given initial stocks in period 0 The optimal conditions for this problem are provided below for , :... function, the consumer's choice of a sequence 6 in equilibrium is determined by the consumer budget constraint (2), the government budget constraint (3), , , and the optimal conditions (4)-(7) The consumer’s choice is a function of market prices , per capita output , the tax rate , and the initial asset for , the rate of inflation : , (8) , (9) (10) Here, equation (8) follows from the optimal conditions... the consumer’s taste parameter for real money balances, it is appropriate to consider M1 and M2, which are closer to the money holdings of consumers than the money base M0 On the other hand, M0 may seem more suitable for seignorage revenue collected from inflation taxation However, there is no banking sector in this model to link the money base M0 to consumers’ money balances M1 an M2 via a money multiplier... externalities in their AK model and pushes the economy from one balanced growth 14 path to another immediately In the present model, the positive effect of faster money growth has a delayed positive effect on innovation by relaxing the capital requirement constraint on innovation and intermediate goods production The consequences of faster money growth on the entire equilibrium path are more complicated . The dynamics 17 1.4 Calibration and simulation results 23 1.4.1 Calibration 23 1.4.2 Simulations 25 1.5 Conclusion 28 iii Chapter 2: Social Optimality, Inflation and Taxation in. Optimal policy 61 2.5 Calibration and simulation results 64 2.6 Conclusion 68 Chapter 3: Intergenerational Links, Taxation, and Wealth Distribution 73 3.1 Introduction 73 3.2 The model 76 3.2.1. composed of three essays on macroeconomic dynamics. The first chapter is a joint work with my supervisor, and it explores whether inflation taxation, a substitute for income taxation given fixed