Risks and Human Capital Investment

Transcription

1 Risks and Human Capital Investment Human Capital Investment Risk and Human Capital Investment Parental Income Risk and Human Capital Investment

2 Levhari and Weiss (1974) Model Basic Question: What is the effect of labor market risks on an individual s choice of human capital investment? Two-period model In the first-period, an individual chooses amount of human capital investment, s, and saving, k. Gross rate of interest R is certain (risk-less capital investment). Human capital investment in the first period increases earnings/human capital in the second period. The human capital function is given by φh(s) with h s (s) > 0 and h ss (s) < 0. φ is a random variable with mean φ and variance, σ 2 φ.

3 Optimization Problem The risk-averse individual s problem is subject to max U(c 1) + βeu(c 2 ) c 1,c 2,s,k c 1 + s + k = y 1 & (1) c 2 = φh(s) + Rs + y 2 (2) where y 1 and y 2 are endowment incomes in period 1 and 2 respectively. β is the discount factor and E is the expectation operator.

4 Optimal Choices The first order conditions are: s : U c (c 1 ) = βeu c (c 2 )φh s (s) & (3) Combining (3) and (4), we have k : U c (c 1 ) = βreu c (c 2 ). (4) EU c (c 2 )[φh s (s) R] = 0. (5)

5 Certainty Case Suppose that there is no uncertainty and φ = φ. Then (5) implies that the optimal s will be given by φh s (s) = R. (6) An individual will equate the marginal return from human capital investment to the rate of interest. This also characterizes efficient level of human capital investment.

6 Risky Human Capital Investment Now (5) can be written as EU c (c 2 )[φh s (s) R] = [φh s (s) R]EU c (c 2 )+Cov(U c (c 2 ), φh s (s)) = 0. (7) Now we have [φh s (s) R]EU c (c 2 ) = Cov(U c (c 2 ), φh s (s)). (8) Since, individual is risk-averse Cov will be negative. Thus, with risk φh s (s) > R. (9) The human capital investment will be lower than the certainty case and also below the efficient level.

7 Kumar (2015) Model Basic Question: What is the effect of parental income risk on the parental investment in the human capital of their children, when the human capital investment is risky? Two-period model A family consists of a parent and a child. Parent is altruistic and cares about the utility of child. In the first-period, the parent chooses amount of its saving, k, and human capital investment for its child, s. In the second period, the parent chooses amount of bequest, b, for its child.

8 Kumar (2015) Model Let y 1 and y 2 be the parental endowment income in period 1 and 2 respectively. Gross rate of interest R is certain (risk-less capital investment). Human capital investment in the first period increases earnings/human capital of the child in the second period. The human capital function of child is given by φh(s) with h s (s) > 0 and h ss (s) < 0. Suppose both y 2 and φ are random variables with means, y 2 and φ and variances, σ 2 y 2 and σ 2 φ respectively. Denote the covariance by σ φ,y2.

9 Parental Optimization Problem The parental problem is subject to max U(c 1) + βe[u(c 2 ) + δu(c)] c 1,c 2,s,k,b c 1 + s + k = y 1 (10) c 2 = Rs + y 2 b & (11) c = φh(s) + b (12) where δ (0, 1) is the degree of parental altruism and c is consumption of child in the second period.

10 Optimal Choices The first order conditions are: s : U c (c 1 ) = βδeu c (c)φh s (s); (13) b : U c (c 2 ) = δeu c (c) & (14) k : U c (c 1 ) = βreu c (c 2 ). (15)

11 Bequest as risk sharing device From (14), we have db & db dy 2 dr > 0 & db < 0. (16) dφ A higher second period parental endowment income and the rate of return on saving increases and a higher productivity of the human capital investment reduces bequest from the parent for a given level of human capital investment. The reason is that a higher endowment income and return on saving reduces the marginal cost of bequest, while a higher productivity of the human capital investment reduces the marginal benefit of bequest.

12 Optimal Human Capital Investment Combining (13)-(15), we have EU c (c 2 )[φh s (s) R] = 0. (17) Certainty Case Then (17) implies that the optimal s will be given by φh s (s) = R. (18) The parent will equate the marginal return from human capital investment to the rate of interest as before.

13 Uncertainty Case Using the co-variance decomposition, (17) can be written as [φh s (s) R]EU c (c 2 ) = Cov(U c (c 2 ), φh s (s)). (19) Suppose that the period utility function is homothetic, then c = (1 M)(y 2 +Rk +φh(s)) & c 2 = M((y 2 +Rk +φh(s)). (20) Using second order Taylor series approximation around (y 2 & φ), we can derive Cov(U c (c 2 ), φh s (s)) MU cc (c p 2 )[h(s)σ2 φ + σ y 2,φ]. (21)

14 Uncertainty Case (19) and (21) imply that [φh s (s) R]EU c (c 2 ) MU cc (c p 2 )[h(s)σ2 φ + σ y 2,φ]. (22) Now suppose that only human capital investment is risky, σ 2 y 2 = 0. Then (22) implies that φh s (s) > R. (23) Now suppose that only parental income is risky, σφ 2 = 0. Then, φh s (s) = R. (24)

15 Both risks Suppose now that both the parental income and the human capital investment are risky, σy 2 2 & σφ 2 > 0. Then, the effect will depend on the covariance term, σ y2,φ. Now suppose that σ y2,φ 0. Then (22) implies that However, if σ y2,φ < 0, it is possible that φh s (s) > R. (25) φh s (s) < R. (26) The human capital investment can be inefficiently high.