Essays on endogenous growth

ESSAYS ON ENDOGENOUS GROWTH LI YANG A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF ECONOMICS NATIONAL UNIVERSITY OF SINGAPORE 2010 Acknowledgements I am deeply grateful to my supervisors Professor Basant K. Kapur, and Professor Jie Zhang, for their guidance, patience and constant support during this research. I would like to express my gratitude to Professor Aditya Goenka, Professor Liu Haoming, Professor Zeng Jinli, Dr Aamir Rafique Hashmi, Dr Jingfeng Lu, Dr Tomoo Kikuchi for their valuable suggestions and comments. I would like to thank my friends {everyone on my facebook} {everyone in PhD Room and 2} for sharing these wonderful years with me in Singapore. Finally, I am heartily thankful to my parents for their support and encouragement. Li Yang January, 2010 i Contents Acknowledgements i Summary iv List of Tables vi List of Figures vii List of Symbols x Compulsory Retirement Savings vs. Unfunded Social Security with Unintended Bequests and Public Schooling Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Compulsory Savings in Individual Accounts . . . . . . . . . . . . . . . 1.3 Unfunded Social Security with Annuity Payment . . . . . . . . . . . . 16 1.4 Numerical Comparison Among Different Programs . . . . . . . . . . . 23 1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ii Nonhomothetic Tastes and Structural Change in an OLG Model of a Small Open Economy 35 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.1 Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.2 Solve the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Nonhomothetic Tastes and Structural change of a Small Open Economy with Infinitely-Lived Agents 83 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.1 Infinite Horizon Model . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.2 Comparison between the OLG and IH model . . . . . . . . . . . . . . 112 3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Bibliography 127 iii Summary This thesis contains three chapters, each a separate essay on endogenous growth. In Chapter 1, we compare unfunded social security with reform-favored compulsory individual savings in an endogenous growth model with heterogeneous accidental bequests and public schooling and without annuity markets. We show that compulsory individual savings should be inframarginal and neutral. By contrast, the unfunded scheme improves welfare for workers receiving no bequests, unless the survival rate becomes too high or the taste for old-age consumption is too weak; and it may also promote growth. Because retired agents gain from receiving intergenerational transfers from workers, their joint voice with have-no workers makes it unlikely for compulsory individual savings to win majority support. Chapter studies the overlapping generation model of a small open economy, adopting an expanding-product-variety framework with two final good sectors, manufacture and service. The variety of intermediate goods increases and generates monopoly profits for R&D firms. We find that when the consumers’ tastes for manufacture and service goods are nonhomothetic, the economy is stable and will converge to the steady-state of iv the homothetic preferences case if the monopoly mark-up is high. When price of service grows slowly, our numerical simulations are consistent with the empirical facts of structural change, such that in the short run and long run, relative price for service goods will increase, the employment and nominal output share of service sector will also increase. In Chapter we discuss the nonhomothetic tastes of consumers and the structural change using infinite horizon agent model. We also compare our results with the overlapping generation model in Chapter 2. Unlike in Chapter 2, we find that there exists transitional dynamics in the homothetic system of the IH model in order to satisfy the transversality condition. The initial growth rate is determined by the initial endowment as well as the technological level. There are two steady-states in the homothetic system. Depending on the parameter values, one of them is stable and the other is unstable. The nonhomothetic system converges to the stable homothetic steady-state asymptotically. One of homothetic steady-states is empirically more likely to be stable. In such case, a higher initial endowment leads to a higher initial growth rate. When the initial growth rate is lower than the stable steady-state, the employment share and nominal output share of the service sector will increase at the early stages of the transition, which is consistent with Erich’s stylized facts for structural changes. v List of Tables Chapter • Table Results with different social security programs (σ = 1, low θ) 30 • Table Results with different social security programs (σ = 2, low θ) 31 • Table Results with different social security programs (σ = 1, high θ) 32 • Table Results with different social security programs (σ = 2, high θ) 33 Chapter • Table A1 Nonhomothetic Transversality Condition Case 120 • Table A2 Nonhomothetic Transversality Condition Case 121 vi List of Figures Chapter • Figure 1(a) Homothetic system and steady-state 52 • Figure 1(b) Homothetic system and steady-state 52 • Figure 2(a) Homothetic system and steady-state 53 • Figure 2(b) Homothetic system and steady-state 53 • Figure 3(a) Homothetic and Nonhomothetic system 55 • Figure 3(b) Homothetic and Nonhomothetic system 56 • Figure 3(c) Homothetic and Nonhomothetic system 57 • Figure 3(d) Homothetic and Nonhomothetic system 57 • Figure 4(a) Convergence of Nonhomothetic growth rate 59 • Figure 4(b) Convergence of Nonhomothetic growth rate 59 • Figure 5(a) Convergence of Nonhomothetic growth rate 61 vii • Figure 5(b) Convergence of Nonhomothetic growth rate • Figure Divergence of Nonhomothetic growth rate 62 64 • Figure 7(a) Convergence of Nonhomothetic growth rate 64 • Figure 7(b) Convergence of Nonhomothetic growth rate 65 • Figure 8(a) Convergence of Nonhomothetic growth rate 66 • Figure 8(b) Convergence of Nonhomothetic growth rate 66 • Figure 9(a) Convergence of Nonhomothetic growth rate 67 • Figure 9(b) Convergence of Nonhomothetic growth rate 68 • Figure 10(a) Convergence of Nonhomothetic growth rate 69 • Figure 10(b) Convergence of Nonhomothetic growth rate 70 • Figure A1 Transitional dynamics of Nonhomothetic system 75 • Figure A2 Transitional dynamics of Nonhomothetic system 76 • Figure A3 Transitional dynamics of Nonhomothetic system 77 • Figure A4 Transitional dynamics of Nonhomothetic system 78 • Figure A5 Transitional dynamics of Nonhomothetic system 79 • Figure A6 Transitional dynamics of Nonhomothetic system 80 viii Chapter • Figure 11 Homothetic steady-states 92 • Figure 12 Transitional dynamics of Homothetic system 97 • Figure 13 Transitional dynamics of Homothetic system 101 • Figure 14(a) Nonhomothetic asymptotic steady-states 103 • Figure 14(b) Nonhomothetic asymptotic steady-states 103 • Figure 15(a) Nonhomothetic transversality condition 104 • Figure 15(b) Nonhomothetic transversality condition 105 • Figure 16 Transitional dynamics of Nonhomothetic system • Figure 17(a) Nonhomothetic transversality condition 107 • Figure 17(b) Nonhomothetic transversality condition 107 106 • Figure 18 Transitional dynamics of Nonhomothetic system 109 • Figure A7 Transitional dynamics of Nonhomothetic system 122 ix 3.3 Conclusions In this chapter we have discussed the nonhomothetic tastes of consumers and the structural change using an infinite horizon agent model. We have also compared our results with the overlapping generation model in Chapter 2. As we have discussed in the previous section, in the IH model the convergence paths of the growth rates are much smoother than the convergence paths in the OLG model, and thus are more empirically reasonable. For the transitional dynamics of the OLG model to be consistent with Erich’s stylized facts, the monopoly mark-up of the R&D firms should be high enough and the gap between the labor intensities of the manufacture and the service should be small enough. Also, the initial growth rate of the convergence path always jumps up to be higher than the steady-state value. On the other hand, for the convergence paths of the IH model to be realistic, the monopoly mark-up should be high and the initial growth rate should be below the steady-state value. As a result, the empirically applications of the two models depend on the status (current growth rate, labor intensities of the manufacture and the service, technology level etc.) of the economy that we want to study. Another interesting result in this chapter is that there exists transitional dynamics in the homothetic system of the IH model in order to satisfy the transversality condition. The initial growth rate is determined by the initial endowment as well as the technology level. Similarly, in the nonhomothetic system, the numerical relationship of the initial growth rate and the initial endowment is unique for each technology level. Note this is a small open economy such that wealth accumulations are not bounded by the growth rate of productivity. By setting the initial growth rate, the economy will be able to consume 117 the extra savings or accumulate to pay the initial debts at the early stages of the convergence. Thus the existence of the transitional dynamics (especially under homotheticity) in the IH model is due to the excess or insufficient amount of asset at the beginning of the representative agent’s life. However, agents in the OLG model have no initial endowments but only labor incomes. Barro and Sala-i-Martin have discussed that when agents in the OLG model are altruistic, such that they care for the children’s utilities and have a bequest motive, there will be intergenerational transfers and the finite-horizon effect of the OLG model will vanish. Therefore we think allowing intended bequests in the OLG model can serve to bring the results of the OLG and the IH models closer. Appendices 3.1 Walras’ Law, Continuous Time Model Using (3.1.7), (3.1.9), (3.1.15) and (3.1.20)-(3.1.22), we have wt = S˙ t − rS t + P MtC Mt + PS tCS t = S˙ t − rS t + P Mt Y Mt + PS t YS t − Bt = (1 − β + αβ)P Mt Y Mt + (1 − θ + αθ)PS t YS t + n˙ t νt S˙ t − rS t = Bt + n˙ t νt − nt πt = Bt + n˙ t νt + nt ν˙ t − rnt νt V˙ t − rVt = n˙ t νt + nt ν˙ t − rnt νt 118 3.2 Analytical characterization of dynamics of the Continuous time model Define: − (1−α)β α zt = e(r−ρ)t nt When z˙t < 0, we have gt > have gt < α(r−ρ) ; (1−α)β when z˙t = 0, we have gt = α(r−ρ) (1−α)β and when z˙t > 0, we α(r−ρ) . (1−α)β From equation (3.1.35) we know gt = γa − γb zt . We then have d ln zt (1 − α)β = (r − ρ) − gt dt α (1 − α)β = (r − ρ) − (γa − γb zt ) α z˙t = we can also rewrite − (1−α)θ α nt θ θ = e− β (r−ρ)t ztβ Substituting back to the z˙t function, we can derive the following: θ θ z˙t = γc + γd zt − γe e− β (r−ρ)t ztβ γc = rαµ − (1 − α)2 β2 −ρ [α − (1 − α)β(1 − β + αβ)]µ (1−α)β γd γe (1−α)β (1 − α)2 βn0 α r(1 − β + αβ)µ (β − θ)C¯ S (1 − α)β α = [β − + ] − g n (1−α)θ [α − (1 − α)β(1 − β + αβ)]µ 1−α α α θ α n0 (1 − α)2 β(β − θ)C¯ S = αθ [α − (1 − α)β(1 − β + αβ)]µ where γc , γd and γe are constant. When the system is nonhomothetic, i.e. γe 0, 119 analytical solution does not exist. When the system is homothetic, γe = 0, we can solve zt as the following: zt = γc {( − (1−α)β α where z0 = n0 γc + γd )eγc t − γd }−1 z0 is the initial condition. Thus gt = γa − γb zt = γa + γb γc {γd − ( γc + γd )eγc t }−1 z0 which is identical to equation (3.1.37). 3.3 Derivation from (3.1.35) to (3.1.37) Solve the following equation for n(t) g(t) = (1−α)β n (t) = γa − γb e(r−ρ)t n(t)− α n(t) where γa and γb are time invariable in the homothetic system. We get n(t) = { (1−α)β α γb (1 − α)βe(r−ρ)t + eγa α t ξ} (1−α)β n0 α(ρ − r) + γa (1 − α)β where ξ is a constant and can be solved using initial conditions. substituting the n(t) equation back, we get equation (3.1.37). 120 3.4 Initial condition for the Transversality condition of continuous time homothetic system Let C Mt +PS tCS t = (1−ζt )wt , where ζt could be varying during the transitional dynamics. Thus S˙ t = rS t + ζt wt . If we define χt = e−rt S t , for the transversality condition to be satisfied, limt→∞ χt = 0. We have: χ˙ t = −rχt + e−rt S˙ t = −rχt + e−rt (rS t + ζt wt ) = e−rt ζt wt = e−rt (1 − 1+τ )wt by (3.1.10) (3.1.11) λ t wt (1−α)β α = e−rt αβ nt − e−ρ t (1 + τ)λ−1 (1−α)β γb (1 − α)βe(r−ρ)t + eγa α t ξ} − e−ρ t (1 + τ)λ−1 by Appendix 3.3 α(ρ − r) + γa (1 − α)β (1−α)β (1−α)β (1−α)β γb (1 − α)β 1+τ − ] + e[γa α −r] t ξαβ n0 α = e−ρ t [αβ n0 α α(ρ − r) + γa (1 − α)β λ0 (1−α)β α = e−rt αβ n0 { let (1−α)β α γc = [αβ n0 1+τ γb (1 − α)β − ] α(ρ − r) + γa (1 − α)β λ0 we have χ˙ t = e−ρt γc + e[γa (1−α)β α −r] t (1−α)β α ξαβ n0 we can solve (1−α)β ξαβ n0 α (1−α)β γc + χ0 χt = (1 − e−ρ t ) + (1 − e−[r−γa α ] t ) ρ r − γa (1−α)β α 121 where χ0 = S . It is easy to show when max(g(1), g(2)) < αr (1−α)β (1−α)β ξαβ n0 α γc lim χt = S + + t→∞ ρ r − γa (1−α)β α Thus, for limt→∞ χt = 0, we can solve for λ0 , (1 − α)2 β(β + θτ) − α(1 + τ)rµ λ0 = (1−α)β α {[(1 − α)2 β2 − αrµ]S − αβ n0 [α − (1 − α)β(1 − β + αβ)]µ}ρ Using equation (3.1.33) to solve for g0 , we have g0 = α[(1 − α)β − r(1 − β + αβ)µ − (1−α)(β−θ)τ ] λ0 w0 [α − (1 − α)β(1 − β + αβ)]µ Substituting λ0 into g0 , we get the g0 function of w0 and S in Proposition 3.1. 3.5 Propensity to consume at continuous time homothetic steadystate From equation (3.1.41) we let C Mt + PS tCS t = (1 − ζ)wt , where ≤ ζ ≤ is constant at g(1), or equals to one at g(2). Together with equation (3.1.9), we have: S˙ t = rS t + ζwt We can then solve for S t . Assuming initial conditions S , and g0 = g∗, we have: S t = ζ(ert w0 − wt )[r − (1 − α)β −1 g∗] + ert S α 122 Therefore propensity to consume at homothetic steady-state g(1) is C Mt + PS tCS t ρ(1 − ζ)wt = rt rS t + wt re (ζw0 + ρS ) + (ρ − rζ)wt However, since ert /wt is not constant at steady-state g(1), the propensity to consume is not constant. 3.6 Numerical solution of the initial growth rate in the Nonhomothetic system Since the analytical solution of g0 cannot be derived, we use numerical methods to find the relationship between g0 and S for different n0 levels. From Proposition 3.1 we know that at a given n0 level, g0 and S have a monotonic relationship in the homothetic system. Since the nonhomothetic system is converging to the homothetic system as nt grows, the relationship between g0 and S should also be converging to the equation in Proposition 3.1 as n0 increases. It is natural for us to assume that in the nonhomothetic system, the relationship between g0 and S is also monotonic, i.e. the solution for g0 is unique. Our numerical results has shown that for a given n0 and g0 , for a large range of S , the relationship between S and e−ρt λt S t for a large t is monotonic. This proves that for a given g0 , the solution for S is unique. After n0 is chosen, the first step of the simulation is to determine the economically meaningful range of gt by (3.1.48). Then for the chosen g0 values within the range, we use binary search to find the S value such that after 2500 recursive steps, |e−ρt λt S t | < 10−6 .Since Case is converging faster than Case 2, we use t = 50 for Case and t = 100 for Case 2. 123 Case 1: g(1) is asymptotically stable. The parameter values we use are the same as for Figure 16: α = 0.15, β = 0.3, θ = 0.1, r = 1.2, ρ = 1, τ = 5, µ = 0.1, C¯ S = 250. The two homothetic steady-states are g(1) = 0.117647 and g(2) = −6.21388. The range of gt (including g0 ) is [0, 0.446445). Table A1 below shows the numerical solutions plotted in Figure 15(a) and 15(b). Table A1 Nonhomothetic Transversality Condition Case n0 = 100 n0 = 1000 g0 S0 S0 0.00 6587.304 99230.9 0.05 6589.640 99350.3 0.10 6591.975 99469.8 0.15 6594.310 99589.3 0.20 6596.645 99708.7 0.25 6598.980 99828.2 0.30 6601.320 99947.6 0.35 6603.655 100067.1 0.40 6605.990 100186.6 0.446445 6608.160 100297.6 124 Case 2: g(2) is asymptotically stable. The parameter values are the same as for Figure 18: α = 0.4, β = 0.3, θ = 0.2, r = 1.2, ρ = 1, τ = 5, µ = 0.1, C¯ S = 250. g(1) = 0.444444, g(2) = 1.29319, and g(2) is asymptotically stable. The range of gt is (g(1), g(2)). Table A2 below shows the numerical solutions plotted in Figure 17(a) and 17(b). Table A2 Nonhomothetic Transversality Condition Case n0 = 1000 n0 = 107 g0 S0 S0 g(1)+ 123.9650 566.0490 0.5 -65.6335 368.1315 0.6 -119.6047 68.6645 0.7 -151.4932 -197.799 0.8 -175.4285 -445.413 0.9 -195.0774 -680.2295 1.0 -211.9962 -905.572 1.1 -227.0043 -1123.519 1.2 -240.5904 -1335.477 g(2)− -252.2512 -1528.492 125 3.7 Nonhomothetic Case 2, when n0 is small As shown in Figure A7 (same parameter values as in Figure 18), at the initial stages of the convergence, the consumption over wage and the production and employment share of the manufacture sector are negative, which is not economically meaningful. 126 Bibliography Chapter • Altonji, J.G., Hayashi, F., Kotlikoff, L.J., ’Parental altruism and inter vivos transfers: theory and evidence.’ Journal of Political Economy 1997, 105, 1121-1166. • Brown, J.R., Casey, M.D., Mitchell, O.S., ’Who values the social security annuity? 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[θp + (1 − θ)(1 − τ − η)] (1.3.9) Denote the solution by (τ∗ , η∗ ) The uniqueness of the solution follows the features of the model: The feasible sets under the constraints are convex and compact; and the log utility function and Cobb-Douglas functions in production and education are strictly increasing, strictly concave and differentiable (hence continuous too) The ratio of physical to human capital... low survival rates and a sufficiently strong taste for old-age consumption Q.E.D The reasons for Proposition 4 are as follows First, the public annuity return is adjusted above the wage growth rate by the reciprocal of the rate of survival When the survival rate is low enough and when the taste for old-age consumption is strong enough, the annuity rate of return on the unfunded social security scheme... sufficiently strong taste for old-age consumption, even when the wage growth rate is below the interest rate However, when the contribution rate becomes higher, the weight on the negative wage growth feedback effect, η/[w0 (1 − p)], becomes larger, while the weight on the positive interest rate feedback effect, s0 /w0 , becomes 0 smaller We thus expect that the optimal contribution rate is positive but bounded... variable that can affect children’s utility is the input on their education Thus we use the measurement of the school quality in the utility function for tractability And since we assume education is purely public, it will only be shown in the government’s budget constraint 6 the after-tax wage income and the bequest income on life-cycle consumption: j c1,t = (1 + rt )btj + (1 − τt )wt − stj , (1.1.4)... for unfunded social security and the last term on the right-hand side of (1.3.2) is the equal amount of pooled public annuity payment from a pay-as-you-go pension When the contribution rate is defined as a constant ηt = η for all t such that ηt = ηt+1 , the return on the contribution to this unfunded program is equal to (wt+1 /wt )/(1 − p), i.e the wage growth rate divided by the rate of survival The... together form a majority in a political equilibrium among different age groups with conflicting interests In Kaganovich and Zilcha (1999), unfunded social security, together with government funding for education, may improve welfare in the long run only when the desire for old-age consumption and the altruism toward child education are sufficiently strong In a dynastic family model, unfunded social security . Convergence of Nonhomothetic growth rate 66 • Figure 8(b) Convergence of Nonhomothetic growth rate 66 • Figure 9(a) Convergence of Nonhomothetic growth rate 67 • Figure 9(b) Convergence of Nonhomothetic. of Nonhomothetic growth rate 62 • Figure 6 Divergence of Nonhomothetic growth rate 64 • Figure 7(a) Convergence of Nonhomothetic growth rate 64 • Figure 7(b) Convergence of Nonhomothetic growth. 4(a) Convergence of Nonhomothetic growth rate 59 • Figure 4(b) Convergence of Nonhomothetic growth rate 59 • Figure 5(a) Convergence of Nonhomothetic growth rate 61 vii • Figure 5(b) Convergence