Public versus Private Investment in Human Capital: Endogenous Growth and Income Inequality

Transcription

1 Public versus Private Investment in Human Capital: Endogenous Growth and Income Inequality Gerhard Glomm and B. Ravikumar JPE 1992 Presented by Prerna Dewan and Rajat Seth Gerhard Glomm and B. Ravikumar 1 / 37

2 Objectives To model human capital investment through formal schooling as the engine of growth To compare public education and private education regimes with respect to per capita income, growth and income inequality To endogenize the choice of educational system in an economy Gerhard Glomm and B. Ravikumar 2 / 37

3 Models such as Romer (1986) and Lucas (1988) study human capital as a factor of growth, however fail to account for the large involvement of the public sector in human capital investment Most models of long run growth are representative agent models, and do not address income distribution issues Gerhard Glomm and B. Ravikumar 3 / 37

4 ESSENCE OF THE MODEL Two period overlapping generations economy wherein parents have a bequest motive; they value the quality of education of their offspring Quality of education in the two regimes determined by 1 tax on the income of the old: public education 2 private choices of individuals: private education Gerhard Glomm and B. Ravikumar 4 / 37

5 THE BASIC FRAMEWORK OLG model with constant population Each generation consists of a continuum of agents who are differentiated by the stock of human capital of their parents Initial generation of old agents at t = 0; j th member endowed with knowledge h j0 Knowledge of initial generation distributed according to pdf G o (. ) which is lognormal with parameters µ 0 and σ 2 0. Gerhard Glomm and B. Ravikumar 5 / 37

6 THE BASIC FRAMEWORK Individuals have identical preferences given by a warm glow utility function: U t = ln n t + ln c t+1 + ln e t+1 (1) Individuals are endowed with one divisible unit of time in their youth which they allocate between leisure and human capital accumulation Human capital accumulation takes the form: h t+1 = θ(1 n t ) β e γ t h δ t (2) θ > 0 β, γ, δ (0, 1) At time t + 1, an individual s income is the same as his human capital h t+1 Gerhard Glomm and B. Ravikumar 6 / 37

7 PUBLIC EDUCATION REGIME Each individual s earning at time t + 1 is taxed at the rate τ t+1 Total tax revenues determine the quality of public schools at time t + 1: E t+1 = τ t+1 H t+1 (3) where H t+1 h t+1 dg t+1 (h t+1 ) Tax rate is determined endogenously by the old agents in each period through majority voting Gerhard Glomm and B. Ravikumar 7 / 37

8 PRIVATE EDUCATION REGIME Quality of education is chosen individually Each individual allocates her income h t+1 between own consumption, c t+1, and the quality of education for the offspring, e t+1 c t+1 + e t+1 = h t+1 (4) Gerhard Glomm and B. Ravikumar 8 / 37

9 EQUILIBRIUM UNDER PUBLIC EDUCATION The equilibrium for the public education economy is a set of sequences {n t } t=0,{h t+1} t=0, {c t} t=0, {G t+1(. )} t=0, {E t} t=0, and {τ t} t=0 such that 1 n t and c t+1 are the optimal choices of an agent born at time t whose parent s human capital is h t ; 2 the human capital of each agent is determined by h t+1 = θ(1 n t ) β e γ t h δ t ; 3 given the distribution G t (. ) at time t, the distribution of income G t+1 (. ) at time t + 1 is determined by the transformation of variables h t+1 = θ(1 n t ) β E γ t h δ t ; 4 the tax rate τ t is preferred by a majority of old agents at time t; and 5 the quality of schools at time t is E t = τ t ht dg t (h t ) Gerhard Glomm and B. Ravikumar 9 / 37

10 EQUILIBRIUM UNDER PUBLIC EDUCATION We solve individual j s optimization problem in two steps. 1 Solve for optimal effort, consumption and human capital investment in the first period subject to: 1 c t+1 = (1 τ t+1 )h t+1, 2 h t+1 = θ(1 n t ) β E γ t h δ t, given E t, h t, E t+1, and τ t+1. max n t,c t+1 ln n t + ln c t+1 + ln E t+1 Gerhard Glomm and B. Ravikumar 10 / 37

11 EQUILIBRIUM UNDER PUBLIC EDUCATION 2 Solve for the agent s preferred tax rate in the second period max τ t+1 ln[(1 τ t+1 )h t+1 ] + ln τ t+1 H t+1 where H t+1 is the mean income at time t + 1 Note that the old agent s choice of tax rate does not alter his income but affects the fraction of income he can consume. Gerhard Glomm and B. Ravikumar 11 / 37

12 EQUILIBRIUM UNDER PUBLIC EDUCATION The public education equilibrium is characterized by: 1 n t = β 1 + β (5) h t+1 = θet γ β ( 1 + β )β ht δ (6) τ t+1 = 1 2 (7) Note that time allocated to human capital investment by an individual born at time t is independent of the tax rate and the individual type because of log preferences and Cobb Douglas learning technology. Also, each individual s preferred tax rate is independent of individual income and constant over time because of log preferences over consumption and bequests. Gerhard Glomm and B. Ravikumar 12 / 37

13 EQUILBRIUM UNDER PUBLIC EDUCATION Since h t is lognormally distributed with mean µ t and variance σ 2 t, h t+1 is also lognormally distributed with mean µ t+1 and variance σ 2 t+1, where µ t+1 = ln[θet γ β ( 1 + β )β ] + δµ t (8) σ 2 t+1 = δ 2 σ 2 t (9) Substituting the values of E t from equation (3) and τ t+1 from equation (7) in equation (6), we get: h t+1 = θ( 1 2 )γ Ht γ β ( 1 + β )β ht δ AHt γ ht δ (10) For lognormal distribution, H t = e µt+( σ2 t 2 ). So: µ t+1 = ln A + γ ln(h t ) + δµ t = ln(a) + (γ + δ)µ t + γσ2 t 2 (11) Gerhard Glomm and B. Ravikumar 13 / 37

14 EQUILIBRIUM UNDER PRIVATE EDUCATION The equilibrium for the private education economy is a set of sequences {n t } t=0, {c t} t=0, {G t+1(. )} t=0, {e t} t=0, and {h t+1} t=0 such that 1 n t, c t+1 and e t+1 are the optimal choices of an agent born at time t whose parent s human capital is h t ; 2 the human capital of each agent is determined by h t+1 = θ(1 n t ) β e γ t h δ t ; 3 given the distribution G t (. ) at time t, the distribution of income G t+1 (. ) at time t + 1 is determined by the transformation of variables h t+1 = θ(1 n t ) β e γ t h δ t ; Gerhard Glomm and B. Ravikumar 14 / 37

15 EQUILIBRIUM UNDER PRIVATE EDUCATION The young individual s optimization problem at time t is as follows: subject to: 1 c t+1 = h t+1 e t+1 2 h t+1 = θ(1 n t ) β e γ t h δ t, given e t and h t. max n t,c t+1,e t+1 ln n t + ln c t+1 + ln e t+1 Gerhard Glomm and B. Ravikumar 15 / 37

16 EQUILIBRIUM UNDER PRIVATE EDUCATION The private education equilibrium is characterized by: c t+1 = e t+1 = 1 2 h t+1 (12) [Note that the quality of education is agent specific.] 1 n t = β β (13) [Note that the time devoted to human capital accumulation is different for the two economies. (Why?)] h t+1 = θ( 1 β 2 )γ [ β ]β ht γ+δ = Bht γ+δ (14) h t+1 is lognormally distributed with mean µ t+1 = ln B + (γ + δ)µ t and variance σ 2 t+1 = (γ + δ)2 σ 2 t Gerhard Glomm and B. Ravikumar 16 / 37

17 HOMOGENEOUS AGENTS Comparison of the two equilibrium paths with respect to levels and growth rates of income when initial generation is homogeneous Evolution of income in the two regimes is given by h u t+1 = A(h u t ) γ+δ (15) (since per capita income at time t coincides with the representative agent s income in the case of homogeneous agents economy) h r t+1 = B(h r t ) γ+δ (16) Gerhard Glomm and B. Ravikumar 17 / 37

18 HOMOGENEOUS AGENTS Conditions for the existence and uniqueness of the (nontrivial) steady-state income/human capital: PROPOSITION 1 (a) If γ + δ 1, then there exists a unique steady state given by 1 h u s > 0 such that h u t+1 = hu s whenever h u t = h u s, 2 h r s > 0 such that h r t+1 = hr s whenever h r t = h r s, and 3 for γ + δ < 1, h r s > h u s, and for γ + δ > 1, h r s < h u s. (b) If γ + δ = 1 and A 1, then there does not exist a steady state for the public education economy. (c) If γ + δ = 1 and B 1, then there does not exist a steady state for the private education economy. Gerhard Glomm and B. Ravikumar 18 / 37

19 HOMOGENEOUS AGENTS Figure: Human Capital accumulation in a homogeneous household economy : Decreasing Returns Gerhard Glomm and B. Ravikumar 19 / 37

20 HOMOGENEOUS AGENTS Figure: Human Capital accumulation in a homogeneous household economy : Constant Returns Gerhard Glomm and B. Ravikumar 20 / 37

21 HOMOGENEOUS AGENTS Figure: Human Capital accumulation in a homogeneous household economy : Increasing Returns Gerhard Glomm and B. Ravikumar 21 / 37

22 HOMOGENEOUS AGENTS Assume that both private and public education economies start off with the same (positive) level of initial income h 0. PROPOSITION 2 h r t > h u t for all t > 0 Along the equilibrium path, a private education economy yields higher income levels in all periods than a public education economy. Gerhard Glomm and B. Ravikumar 22 / 37

23 HOMOGENEOUS AGENTS PROPOSITION 3 Vt r > Vt u for all t 0, where Vt r is the equilibrium level of utility of an agent born at time t in the private education economy and Vt u is the corresponding utility in the public education economy. The representative agent in the private education economy is better off than his counterpart in the public education economy. Gerhard Glomm and B. Ravikumar 23 / 37

24 HOMOGENEOUS AGENTS PROPOSITION 4 (a) If γ + δ < 1, then lim t ( hr t+1 h r t (b) If γ + δ = 1, then hr t+1 ht r (c) For γ + δ > 1, 1 2 h r t+1 ht r h u t+1 ht u ) = lim t ( hu t+1 ht u ) = 1. = B > A = hu t+1 h u t for all t 0. is greater than one and increasing over time if h 0 > h r s and is greater than one and increasing over time if h 0 > h u s. Decreasing returns implies zero long run growth; Constant returns implies constant long run growth under certain parameter restrictions; Increasing returns are neither necessary nor sufficient for long-run growth in either economy. Gerhard Glomm and B. Ravikumar 24 / 37

25 HOMOGENEOUS AGENTS Private education economies with homogeneous population achieves higher incomes and growth rates than the public education economy whenever γ + δ 1. If a policy of mandatory schooling is enforced, then the allocations in the public education regime would be the same as in the private education regime. Gerhard Glomm and B. Ravikumar 25 / 37

26 HETEROGENEOUS AGENTS The initial income distribution is lognormal with parameters µ 0 and σ 2 0 ; income inequality at time t is described by the parameter σ t. PROPOSITION 5 (a) In the public education economy, income inequality declines over time. (b) In the private education economy, income inequality declines over time if γ + δ < 1, increases over time if γ + δ > 1, and remains constant over time if γ + δ = 1. Gerhard Glomm and B. Ravikumar 26 / 37

27 HETEROGENEOUS AGENTS Even if γ + δ < 1, income inequality in the private education economy does not decline as fast as in the public education economy. Conditions for long-run growth in per capita income with heterogeneous agents are identical to those in the homogeneous agent economy. Along the balanced growth path (γ + δ = 1) income inequality declines in the public education economy but stays the same in the private education economy. Gerhard Glomm and B. Ravikumar 27 / 37

28 HETEROGENOUS AGENTS PROPOSITION 6 (a) Consider two distinct public education economies with the same mean income at time t, that is, H t = H t. If σ t > σ t, then H t+1 > H t+1. (b) Similarly, consider two distinct private education economies with H t = H t and σ t > σ t. Then H t+1 > H t+1 if and only if γ + δ < 1. Although the proposition establishes an order relation between per capita income levels only for the next period, the relation holds for all future periods. Gerhard Glomm and B. Ravikumar 28 / 37

29 HETEROGENOUS AGENTS: AN EXAMPLE If the initial income inequality is sufficiently high, then the public education economy may yield higher per capita income for some future periods than the private education economy: Two economies with the same income distribution at time t; γ + 2δ < 1 (γ + δ) 2 < γ + δ 2. ln(h u t+1) = ln A + (γ + δ)µ t + (γ + δ2 )σ 2 t 2 ln(h r t+1) = ln B + (γ + δ)µ t + (γ + δ)2 σ 2 t 2 If σ 2 t is sufficiently large, H u t+1 > Hr t+1. Gerhard Glomm and B. Ravikumar 29 / 37

30 CHOICE OF EDUCATIONAL REGIME Parents decide by majority vote whether the educational system should be private or public and no one can opt out. PROPOSITION 7 A majority of old agents at time t would prefer public over private education if and only if σ 2 t > 0. Gerhard Glomm and B. Ravikumar 30 / 37

31 CONCLUSION Agents in the private education economy spend more time on human capital accumulation than their counterparts in a public education economy because they internalize the fact that more time spent on human capital accumulation increases consumption as well as bequests. For a homogeneous agent economy, per capita income, indirect utility and growth rate are higher in a private education regime than in a public education regime. Income inequality declines faster in public education regime than in pvt education regime. Gerhard Glomm and B. Ravikumar 31 / 37

32 CONCLUSION If two economies have the same per capita income but differing levels of inequality, the economy with higher inequality will have lower per capita income in all subsequent periods. If inequality is sufficiently high in an economy, then a public education regime may yield higher per capita income for some future periods than a pvt education economy. If majority voting decides the choice of education regime in an economy, public education is chosen over private if a majority of agents have income below the average. Gerhard Glomm and B. Ravikumar 32 / 37

33 Appendix:Proof of Proposition 3 Gerhard Glomm and B. Ravikumar 33 / 37

34 Appendix:Proof of Proposition 3 Gerhard Glomm and B. Ravikumar 34 / 37

35 Appendix:Proof of Proposition 6 Gerhard Glomm and B. Ravikumar 35 / 37

36 Appendix:Proof of Proposition 6 Gerhard Glomm and B. Ravikumar 36 / 37

37 The End Gerhard Glomm and B. Ravikumar 37 / 37