Labor Economics: Problem Set 1

Transcription

1 Labor Economics: Problem Set 1 Paul Schrimpf October 3, Problem a The budget constraint is: U = U(c f + c m, l f, l m ) (1) p(c f + c m ) + w f l f + w m l m w f T f + w l T l + y (2) There is no point in the distinction between c f and c m. Any c f and c m such that c f +c m = c will yield the same utility and satisfy the same budget constraint. Henceforth, I will let c = c f + c m and no longer consider them separately. 1.2 b The Lagrangean is: max U(c, l f, l m ) λ(pc + w f l f + w m l m w f T f w m T m y) (3) l f,l m,c The first order conditions are: U =λp c (4) U =λw m l m (5) U =λw f l f (6) pc + w f l f + w m l m =w f T f + w m T m + y (7) The solution to these equations gives the uncompensated demand for leisure and consumption as a function of prices and endowment, l f (p, w m, w f, y), l m (p, w m, w f, y), and c(p, w m, w f, y). 1.3 c If the utility function is concave, then the Lagrangean will have an optimum, so the first order conditions will determine labor supply. 1

2 1.4 d Let e(p, w m, w f, ū) be the expenditure function i.e.: e(p, w m, w f, ū) = min l m,l f,c pc + w ml m + w f l f w f T f w m T m U(c, l f, l m ) ū (8) The solution to this minimization problem gives the compensated demands for leisure and consumption as functions of prices and utility, l c f (p, w m, w f, y), l c m (p, w m, w f, y), and c c (p, w m, w f, y). By the usual duality argument: l c f(p, w m, w f, ū) = l f (p, w m, w f, e(p, w m, w f, ū)) (9) Differentiating both sides with respect w f yields: lf c = l f + l f e (10) y e By the envelope theorem, = lf c T f. Labor supply, h f = T f l f, so equation 13 can be rewritten as: hc f = h f + h f y h f (11) Thus, the uncompensated labor supply reponse of a female with respect to her wage is: h f = hc f + h f y h f (12) h c f h = S ff is the female own wage substitution effect. It is positive. h is the income y effect. It will be negative assuming that leisure is a normal good. Similarly, we can derive the uncompensated labor supply response of a female with respect to her husband s wage by differentiating equation 9 with respect to w m : lf c = l f + l f e (13) w m w m y w m Applying the envelope theorem and rewriting in terms of labor supply yields: hc f w m = h f w m + h f y h m (14) Thus, the uncompensated labor supply reponse of a female with respect to her husband s wage is: h f = hc f + h f w m w m y h m?? (15) As above the second part of this equation, h y m is the income effect, which will be h negative if leisure is a normal good. The first part, c f w m = S fm is the female substitution effect in reponse to a change in male wages. Its sign is not clear. h f 2

3 The uncompensated labor supply responses of males the same as for females with the m and f subscripts reversed. The uncompensated response in male labor supply to a change in the male wage is: h m = hc m + h m w m w m y h m (16) The uncompensated response in male labor suppply to a change in the female wage is: h m = hc m + h m y h f (17) The same remarks about the signs ( of the substitution ) effects apply as were made above. Sff S The substitution matrix, S = fm, is equal to the second derivative of the S mf S mm expenditure function by the envelope theorem. By the usual argument, the expenditure function is concave, so its matrix of second derivatives with respect to leisure is negative semi-definite. Since the derivative with respect to labor is the opposite of the derivative with respect to leisure, the substitution matrix is positive semi-definite. This means that S is symmetric and has positive diagonal elements. Therefore, S mf = S fm, S mm 0, and S ff e Let i be m or f. The total differential of h i is: [ hi dh i = h i w m h i p h i y ] dw f dw m dp dy ) dh i =(S if + B i h f )dw f + (S im + B i h m )dw m + B i dy =S if dw f + S im dw m + B i (h f dw f + h m dw m + dy) (18) where B i = h, and the results from part d and the fact that dp = 0 were substituted in y to get the second equality. A negative income tax with a subsidy of G and a tax rate of t will change the budget constraint from equation 2 to: p(c f + c m ) + w f (1 t)l f + w m (1 t)l m w f (1 t)t f + w l (1 t)t l + y + G (19) Where, we have assumed that h m w m + h f w f + y G/t, i.e. that the family is eligible for the negative income tax. If the family is not eligible, (and does not choose to become eligible), there is no change in the budget constraint. The wage rate has changed by wi = w i t and income has changed by y = G. 1 For small G and t the change in gross earnings for the husband or wife will be approximately given by the total differential of labor supply times evaluated using wi and y instead of d wi and d yi times the wage. That is, hi w i w i ( Sif wf + S im wm + B i (h f wf + h m wm + y ) ) (20) 1 If the family was not eligible for the negative income, these changes will be different, but in any case, wi 0 and y 0. 3

4 The last term in this equation, h f wf + h m wm + y is the change in the family s total income. This term must be greater than or equal to zero since the budget constraint has shifted outward. B i is negative. Now, let i = m. S mm > 0, so S mm wm 0. However, the sign of S mf ambiguous and there is no necessary relationship among the magnitudes of the variables in the equation. Analogous remarks apply to i = f. Thus, hi w i can be either positive or negative. The change in family income is just the sum of the changes in male and female income. hmw m+h f w f ( w i Sif wf + S im wm + B i (h f wf + h m wm + y ) ) (21) i=m,f It is useful to write this sum in matrix notation using the substitution matrix from above. Let w = [ ] w f w m. Note that wi = tw i, so i=m,f w i(s if wf +S im wm ) = tw Sw. Therefore, we have hmw m+h f w f tw Sw w i B i (h f wf + h m wm + y ) (22) i=m,f The second term in this equation is the sum of the income effects, both of which are negative as explained above. Also, from the above, S is positive semi-definite, so by definition w Sw 0. Consequently family earnings must decrease. 1.6 f With some source of data on changes in husbands and wives wages and labor supplies, income and substitution effects can be estimated from equation 23, replacing differentials with discrete changes. That is, one could estimate income and substitution effects from: hi = S if wf + S im wm + B i (h f wf + h m wm + y ) + ǫ i (23) These changes in wages could be generated from a change in the tax code, time series variation in a panel dataset, or if panal data is unavaliable, as in Ashenfelter and Heckman (1974), as cross-sectional deviations from means. Also, if accurate information on unearned income is unavailable, as in Ashenfelter and Heckman (1974), then h f wf + h m wm + y can be replaced by changes in total family income, F = wf h f + wmh m + y. 1.7 g OLS is inappropriate because by construction, F depends on hi and so is correlated with ǫ m and ǫ f. To test the classical restrictions on the Slutsky hypothesis, I would jointly test the null hypthesis that: H 0 : S mm = 0, S ff = 0, det(s) = 0 (24) Against the alternative: H a : S mm > 0, S ff > 0, det(s) > 0 (25) 4

5 And hope to reject. This would test whether the own wage substitution effects are positive and the substitution matrix is positive definite. The estimates of S have an asymptotic normal distribution. Let f S denote the density of S the test statistic is then: S R 4 S mm> ˆ S mm,s ff > ˆ S ff,det(s)> det(s) To test whether S mf = S fm I would test the null hypothesis, f S (S)dS (26) H 0 : S mf = S fm (27) against the alternative that they re different and hope to accept. 1.8 h Without standard errors, these estimatates don t necessarily imply anything. However, assuming that they are significantly different from 0, then yes, they do imply that leisure is a normal good. 1.9 i The uncompensated cross-effects are given by equation??. Plugging in the numbers, for females gives = For males it is = These effects mean that a man s labor supply is insensitive to his wife s wage, but a woman s labor supply responds to changes in her husband s wage. The higher a woman s husband s wage, the less she will work j If husbands and wives each maximize their own utility function, they each solve: max c i,l i U i (c m, l m, c f, l f ) s.t. p(c m + c f ) + w m l m + w f l f (w m + w f )T + y (28) I have assumed that each person takes their spouse s leisure and consumption choices as given when maximizing utility. Alternatively, one could suppose that each person takes into account their spouse s responses when choosing consumption and leisure. In either case, an allocation is efficient if one spouse cannot be made better off without making the other worse off. 2 Problem a Suppose that mothers must provide childcare for their children at all times. While working, a mother must pay for childcare at price p c. Assume that while consuming leisure a mother can care for her children without changing her utility of leisure. Then as shown in the graph below, children lower a mother s effective wage from w to w p c. 5

6 2.2 b Assume that mothers receive the lumpsum allowance regardless of whether they work. Let A be the amount of the allowance, and τ be the phase out rate. Then for wh < A/τ, the effective wage rate decreases to w(1 τ) p c and unearned income increases by A. Thus, labor supply will further decrease. When child care is publicly provided, mothers effective wage is kept at w as though they don t have children. Therefore the labor supply of mothers will increase. 2.3 c No, increasing the labor supply of mothers is not unambiguously desirable. If mothers and child care providers produce child care of equal quality, then increasing the labor supply of mothers is only desirable in terms of maximizing production if their wages are greater than the cost of childcare. 3 Problem 3 The Macurdy (1981) utility function is: U i (C i,t, N i,t ) = Γ 1,i,t C ω 1 i,t Γ 2,i,tN ω 2 i,t (29) Henceforth, I will leave out the i subscript except for when needed for emphasis. Consumers maximize their lifetime utility, subject to the constraint that they leave no debt, ( ) t 1 (Γ 1,tC ω 1 t Γ 2,t N ω 2 t ) (30) 1 + ρ A 0 + N t W t (1 + r) = C t (31) t (1 + r) t The Lagrangean is: ( ) ( t 1 max (Γ C 1,tC ω 1 t Γ 2,t N ω 2 t ) + λ A 0 + t,n t 1 + ρ The first order conditions are: ) N t W t (1 + r) C t t (1 + r) t (32) A 0 + Γ 1,t ω 1 C ω 1 1 t (1 + ρ) t = λ (1 + r) t (33) Γ 2,t ω 2 N ω 2 1 t = λw t (34) (1 + ρ) t (1 + r) t N t W t (1 + r) = C t (35) t (1 + r) t 6

7 Rearranging equation 33 yields the λ-constant commodity demand function. ( λ(1 + ρ) t C t λ (r, λ) = (1 + r) t Γ 1,t ω 1 ) 1 ω 1 1 (36) Similarly, equation 34 can be manipulated to get the λ-constant labor supply. ( λwt (1 + ρ) t N t λ (r, w t, λ) = (1 + r) t Γ 2,t ω 2 ) 1 ω 2 1 (37) i Taking logs of equation 37 yields: log(n t λ ) = 1 ω 2 1 (log(λ) + log(w t) + t(log(1 + ρ) log(1 + r)) log(γ 2,t ω 2 )) (38) Thus, the intertemporal substitution elasticity is log(nt λ) log(w t) = 1 ω 2. Since the utility 1 function is concave, 2 U = Γ 2,tω 2 (ω 2 1)N ω2 2 L 2 t < 0, which is only possible if ω (1+ρ) t 2 > 1. Thus, log(n t λ ) log(w t) > 0. The intertemporal substitution elasticity of labor supply is positive. Holding the marginal utility of wealth constant, a change in the wage during one period does not affect consumption. Since consumers can freely smooth consumption across periods they will simply choose their consumption each period such that its marginal utility is equal to the marginal utility of wealth properly discounted. If the marginal utility of wealth is not held constant, an increase in wages will affect both the marginal utility of wealth and labor supply. From equation 37 we see that if wages increase, λ must fall or N t must increase. The change in consumption will depend on the change in λ. Assuming that the marginal utility of wealth is decreasing as a function of W t, an increase in wages will decrease λ. Consumers must then decrease their marginal utility of consumption in all periods by increasing consumption ii For this section, it will be helpful to rewrite equation 38 with the i subscripts included. log(n i,t λ ) = 1 ω 2 1 (log(λ i) + log(w i,t ) + t(log(1 + ρ) log(1 + r)) log(γ 2,i,t ω 2 )) (39) Permanent differences in human capital can be thought of as permanent differences lifetime wealth. People with different levels of human capital face different schedules of wages, so they have different lifetime wealth. In this model, differences in lifetime wealth appear as differences in the the marginal utility of wealth, λ i. Thus, this model says that people with different levels of human capital will have log-wage versus log-hours profiles that differ by a constant, but will display the same intertemporal substitution elasticity. Wage increases generated by changes in labor market experience are generally expected. Consequently, such changes do not represent changes in a person s lifetime wealth 7

8 and should not change the marginal utility of wealth. This model predicts that the elasticity of labor supply in response to changes in wages associated with experience to be 1 ω 2 1. If a transitory shock in wages is anticipated, it s effect is the same as a change in wages associated with experience. However, in the more plausible case that the shock is not anticipated, it represents a change in both wages and lifetime wealth. If the change in wealth is small, so that we may think of it as holding λ constant, the labor supply response will just be the intertemporal substitution effect. However, if the shock is large, it will shift λ. If the shock reduced wages, wealth has decreased, so its marginal utility increase, causing labor supply to rise at all subsequent time periods. At the same time wages at the time of the shock have decreased, which will decrease labor supply. Conversely, if the shock increased wages, λ will decrease and labor supply will fall at the time of the shock and afterward. Also, the higher wages at the time of shock will cause labor supply to temporarily rise. In summary, an unanticipated wage change has an ambiguous effect on contemporaneous labor supply, but causes future labor supply to move in the opposite directioon of wage change. A temporary negative income tax and job loss at age fifty can be thought of in the same way as a transitory shock in wages. The temporary negative income tax increases wages today relative to the future, so labor supply will increase due to intertemporal substitution. However, the program also increases lifetime wealth, decreasing λ, causing labor supply to fall at all times. Job loss at age fifty likely causes wages to fall for the rest of a person s life due to their loss of job-specific human capital. This would increase the marginal utility of wealth, which increases labor supply. However, wages have fallen, so intertemporal substitution decreases labor supply. 3.1 b Data for this section was extracted by Matt Notowidigo i MaCurdy s specification is: ln h it = A 0 + A 1 t + A 2 t 2 + θ θ 3 ln(w it h it ) + θ θ 3 ln λ i + u it (40) The grouped-data, or efficient Wald estimator, is a two-staged least squares estimator using year dummies as instruments. The ANCOVA estimator is the same as a fixed effects estimator. The OLS estimates are straightforward. Each of set of estimates are shown in the tables below. Interestingly, the OLS and ANCOVA estimates are very similar to those in Angrist (1991), but the grouped-data estimates are significantly larger. Although not shown, this result holds if a linear instead of quadratic time trend is included. 8

9 Table 1: Grouped-Data Variable Coefficient (Std. Err.) θ 3 1+θ (0.103) A (0.006) A (0.000) Intercept (0.903) Significance levels : : 10% : 5% : 1% Table 2: ANCOVA Variable Coefficient (Std. Err.) θ 3 1+θ (0.006) A (0.002) A (0.000) Intercept (0.049) Significance levels : : 10% : 5% : 1% Table 3: OLS Variable Coefficient (Std. Err.) θ 3 1+θ (0.004) A (0.003) A (0.000) Intercept (0.035) Significance levels : : 10% : 5% : 1% 9

10 3.1.2 ii For MaCurdy s specification, Angrist finds a smaller, but not statistically significantly θ different, estimate of 3 1+θ 3 using the efficient Wald estimator than using the fixed effects estimator. Angrist notes that measurement error would bias the fixed effects estimator upward. In the other two specifications estimated by Angrist, he finds that the fixed effects estimates are significantly different from the efficient Wald estimates. Angrist argues that this is likely due to the fact that wages are much more poorly measured than annual earnings in the PSID. Card is skeptical of the large positive intertemporal substitution elasticities because he believes that the assumption of perfect foresight, required for λ i to be constant, is likely to be false. Card points out that from wages grew at about a constant rate, but following 1975 wage growth stopped. He argues that this change was not anticipated, and so individuals suffered an unexpected fall in their lifetime wealth. Therefore, to estimate lifecycle labor supply, changes in the marginal utility of wealth should be explicitly modelled. 3.2 Do file used for estimates /* */ Do file for Labor Economics Problem Set 1 // load the dataset created by Matt Notowidigdo do loadfile // v1 is a 1..nobs identifier rename v1 id // we want logs, not levels gen lnh=ln(hours) gen lny=ln(earnings) gen lnw=ln(wages) // going to outtex my results, make nice labels for the tables label var lny "$\frac{\theta_3}{1+\theta_3}$" label var t "$ A_1 $" label var tsquared "$ A_2 $" /* The grouped data, aka efficient Wald, estimator is simply iv using year dummies as instruments */ xi i.t ivreg lnh (lny = _It_* ) t tsquared outtex, legend file("grouped.tex") title("grouped-data") /// replace labels nocheck /* The OLS estimate */ 10

11 reg lnh lny t tsquared outtex, legend file("ols.tex") title("ols") /// replace labels nocheck /* The ANCOVA estimator is a fixed effects estimator */ xtreg lnh lny t tsquared, i(id) fe outtex, legend file("ancova.tex") title("ancova") /// replace labels nocheck 11