1. Basic Model of Labor Supply

Transcription

1 Static Labor Supply. Basic Model of Labor Supply.. Basic Model In this model, the economic unit is a family. Each faimily maximizes U (L, L 2,.., L m, C, C 2,.., C n ) s.t. V + w i ( L i ) p j C j, C j 0 j, 0 L i i where L i is leisure of family member i, C j is family consumption of good j, w i is the market wage of family member i, p j is the price of consumption good j, and V is nonwage family income. There is no saving. Note: think of the constraint as V + w i w i L i + p j C j. (.) The left-hand side of equation (.) is potential income or full income. The right-hand side is expenditure including expenditure on leisure. First order conditions for the appropriate Lagrangian are U Li λw i = 0 if 0 < L i < 0 if L i = 0 0 if L i = U Cj λp j = 0 if C j > 0 0 if C j = 0 We can solve demand equations (in theory): j. i L i = L i (w, p, V ) i; C j = C j (w, p, V ) j; λ = λ (w, p, V ).

2 We define w r i, the reservation wage for family member i, as the highest wage where L i =. The reservation wage is w r i = w r i (w, p, V ). Now consider the case where there exists only one family member; i.e. m =. We can draw indifference curves in C-L space ((for n ) = ). See Figure.] wi r is equal to the slope of the indifference curve at, V : p w r i = p U L U C L=. In general, an interior optimum occurs where U L λw = 0, U C λp = 0 w p = U L U C. (.2) The left-hand side of equation (.2) is the slope of the budget line, and the righthand side is the slope of the indifference curve. What happens if there is more than one good (n > )? There is no change because U C j p j and C k. We can differentiate the demand curves to get dl i = dc j = k= k= dw k + C j dw k + k= and, using the Slutsky equation, we can get dl i = S Li w k + H k I i ] dw k + dc j = k= ] SCj w i + H i J j dwi + where S Li w k = U, S Li p j = p j (= hours worked), I i = = U C k p k p j dp j + V dv ; C j p k dp k + C j V dv, ] SLi p j C j I i dpj ; ] SCj p k C k J j dpk k= U, S Cj w i = C j U, S Cj p k = C j p k for all interior C j U, H i = L i, and J V j = C j.note that when w V i rises, the cost of leisure rises but so does income; thus the H i term. 2

3 = j.2. Review of Slutsky equations Let e (p, u) = min p i h i s.t. u (h) = u be the expenditure function. Then and e/ h i = p i λu i = 0. This implies that e (p, u) = p i h i + λ u u (h)], (.3) u i p i = u j p j i, j. Define h (p, u) as the solution to equation (.3). Then e p i = h j p j + h i p i = h i by the Envelope Theorem. One can show concavity of the expenditure function in the following way: Consider to price vectors, p and p 2, and let p = δp +( δ) p 2. Let h i = h (p i, u) for i =, 2, e (p i, u) = p i jh i j for i =, 2, and e ( p, u) = j p j h j ( p, u) ] δp j + ( δ) p 2 j hj ( p, u) = δ j p jh j ( p, u) + ( δ) j p 2 jh j ( p, u) δ j p jh j ( p, u ) + ( δ) j p 2 jh j ( p 2, u ) = δe ( p, u ) + ( δ) e ( p 2, u ). The Slutsky equation is End of review. x i p j u = x i p j + x j x i y. 3

4 .3. Restrictions There exist a number of restrictions on the Slutsky coeffi cients: ) Engel Aggregation Condition: µ Li η Li + µ Cj η Cj = where µ Li = w il i (share of total income spent on L M i), µ Cj = p jc j M income spent on C j ), η Li = M (elasticity of L L i M i wrt total income), η Cj (elasticity of C j wrt total income), and M = V + m condition comes from the budget constraint: 0 = M w i L i M = = = w i M w i L i M µ Li η Li ] p j C j p j C j M M L i M µ Cj η Cj. p j C j M (share of total = M C j C j M w i (total income). This M C j C j M 2) Cournot Aggregation Condition: µ Li η Li w k + µ Cj η Cj w k = w k ( L k ) M where η Li w k = w k L i and η Cj w k = w k C j C j. This also follows from the budget 4

5 constraint: 0 = M = L k = L k w i L i w i w i L i w k ] p j C j w k L i C j p j p j C j w k w k C j C j which implies that 0 = ( L k) w k M = ( L k) w k M w i L i M w k L i µ Li η Li w k p j C j M µ Cj η Cj w k. w k C j C j 3) S Li w i < 0, S Cj p j < 0. 4) S Li w k = S Lk w i, S Cj p k = S Ck p j, and S Li p j = S Cj w i. 5) The matrix of Slutsky substitution terms is negative semidefinite. Note: Conditions (3-5) come from differentiation of the (concave) expenditure function. 6) Homogeneity: dl i U = dc j U = S Li w k w k dw k + S Li p j p j dp j = 0; k= S Cj w i w i dw i + k= S Cj p k p k dp k = Example Let u = β log L + β 2 log L 2 + β 3 log C. 5

6 Then the maximization problem is s.t. The expenditure function problems is s.t. max U (L, L 2, C) V + w + w 2 = pc + w L + w 2 L 2. min pc + w L + w 2 L 2 U (L, L 2, C) = u. Continuing with the maximization problem, the Lagrangian is Υ = U (L, L 2, C) + λ V + w + w 2 pc w L w 2 L 2 ] with derivative conditions (for an interior solution), Υ L = β λw = 0; L Υ L 2 = β 2 λw 2 = 0; L 2 Υ C = β 3 λp = 0; C and Υ λ = V + w + w 2 pc w L w 2 L 2 = 0. Demand equations are solved for as follows: which implies w L = β λ ; w 2 L 2 = β 2 λ ; pc = β 3 λ and V + w + w 2 = β + β 2 + β 3. λ 6

7 Without loss of generality, let β + β 2 + β 3 =. Then So Note that λ = V + w + w 2 ] = M. L (w, w 2, p, V ) = β V + w + w 2 w ; L 2 (w, w 2, p, V ) = β 2 V + w + w 2 w 2 ; and V + w + w 2 C (w, w 2, p, V ) = β 3. p dl V + w 2 = β dw w < 0; dl dw 2 = β > 0; w dl dp = 0; dc dw i = β 3 p > 0; dl i dv = β i > 0; w i and dc dv = β 3 p > 0. The reservation wages are found by solving L i = β i V + w + w 2 w i = for w i : for j i. Note that dwr i dv solution for L j.] = dwr i dw j w r i = β i β i (V + w j ) > 0 and dwr i dβ i > 0. Note: this assumes an interior 7

8 Slutsky demands can be found as U = ( L i ) w i w i V = β i V + w j w 2 i ( L i ) β i w i = β i V + w wi 2 j + w i ( L i )] < 0; U = ( L j ) w j w j V p = β i w i ( L j ) β i w i = L j β i w i > 0; U = p + C V = C β i w i > 0; C U = C ( L i ) C w i w i V C p = β 3 p ( L i) β 3 p = L i β 3 p > 0; and U = C p + C C V V + w + w 2 = β 3 + C β 3 p 2 p = β 3 ( β 3 ) V + w p 2 + w 2 ] < 0. 8

9 2. Analysis with Kinked Budget Curves 2.. Fixed Costs of Working The fact that wi r and L i are continuous functions implies that we should find a continuous supply of labor curve. Yet, in the data, we find very few people working a small number of hours. This can be explained by adding fixed costs to the model. First, assume there exists a fixed money cost to working (maybe carfare, the cost of appropriate clothing, other supplies). If, in Figure.2, non-wage income is V (assume just one consumption good) and the fixed cost of working is F, then p p one doesn t work until the wage is wh r, and one works a discrete amount. Next, assume there exists a fixed time cost, F, as in Figure.3. Then again one doesn t work until the wage is wh r, and one works a discrete amount. Next, assume that the tax schedule causes kinks in the budget space. We consider this as an estimation problem later. 3. Problems ) We have assumed that the vector of wages does not depend on the vector of hours worked. This may not be so. 2) There is a question of how households make decisions and how they were formed. 3) There are dynamic issues. 4. Selection Bias 4.. Basic Selection Problem Consider the problem of two groups of married women with the same distribution of wage offers but different distributions of reservation wages. If we compared average wages for the two groups, then women with the higher reservation wage would be observed having higher wage rates. Now consider the case with all women having the same reservation wage but the two groups having different levels of schooling. Assume that schooling increases the mean of the distribution of wage offers. Then the effect of selection is to reduce the coeffi cient on schooling. 9

10 4.2. Heckman s Shadow Wages Approach Let U = U (L w, L h, X) where L w is the wife s leisure, L h is the husband s leisure, and X is family consumption. The family maximizes U subject to the budget constraint, A + w w h w + w h h h = px where A is nonwage income, w w is the wife s wage, w h is the husband s wage, h w = L w is the wife s hours working, h h = L h is the husband s hours working, and p is the price of the consumption good. First order conditions are U Lw λw w = 0; U Lh λw h = 0; and U X λp = 0. We can write U Lw U X and define w w as the shadow wage, = w w p w w = w w (h w, h h, X). If we substitute in demand equations, we get w w = w w h w, h h (w w, w h, p, A), X (w w, w h, p, A)] = w w (h w, w h, p, A). None of the derivatives of w w can be signed from theory even though we have strong a priori expectations of the signs. Heckman now argues that wage offers are a function of experience and schooling: w w = w w (E, S). If h w > 0, then ww = ww, and if h p w = 0, then ww ww p wage). Heckman now specifies two equations: (this case is a reservation l (w wi) = β 0 + β h wi + β 2 w hi + β 3 p i + β 4 A i + β 5 Z i + ε i, l (w wi ) = b 0 + b S i + b 2 E i + u i 0

11 where i indexes women, Z i are other characteristics, ε i and u i are distributed iid joint Normal, and l ( ) is a Box-Cox transformation, l (w) = wλ. λ Note that as λ, l (w) w, and that as λ 0, l (w) log w. The condition for h wi = 0 is b 0 + b S i + b 2 E i β 0 β 2 w hi β 3 p i β 4 A i β 5 Z i < ε i u i i.e., l (w wi ) l (w wi) hwi =0< 0. If h wi > 0, then h wi = β b 0 + b S i + b 2 E i β 0 β 2 w hi β 3 p i β 4 A i β 5 Z i ] + u i ε i β i.e., l (w wi ) = l (wwi). But, for the sample of working women, ] ui ε i E S i, E i, w hi, A i, Z i, h wi > 0 0, β and E u i S i, E i, h wi > 0] 0. Therefore, neither the hours equation nor the wage equation can be consistently estimated using OLS. Let the joint density of h wi and l (w wi ), given that l (w wi ) = l (w wi), be j h wi, l (w wi )] = N h wi, l (w wi )] Pr l (w wi ) > l (w wi ) h wi = 0] where N, ] is a joint normal density. Assume there are T married women, the first K of whom work. Then the log likelihood function is log L = K T log N h wi, l (w wi )] + log Pr l (w wi ) < l (wwi) h wi = 0]. i=k+ We can compute MLE s for the parameters of the log likelihood function.

12 4.3. Heckman s Specification Error Approach Let y i = X i β + u i, (4.) y 2i = X 2i β 2 + u 2i where u i and u 2i are distributed iid h (u i, u 2i ). Assume there exists missing data in the first equation. The crucial question is why is it missing? We can write for the whole population, E (y i X i ) = X i β, but for the observed sample, we can write only E (y i X i, SSR) = X i β + E (u i X i, SSR) where SSR is the sample selection rule. If people are randomly selected, then E (u i X i, SSR) = 0. But, consider the following example: We observe y i iff y 2i > 0 (consider the case where y i is wage and y 2i is wage minus reservation wage). Then E (u i X i, SSR) = E (u i X i, y 2i > 0) = E (u i X i, u 2i > X 2i β 2 ). If u i and u 2i are independent, then E (u i X i, u 2i > X 2i β 2 ) = 0 and there is no selection bias. But, if they are correlated, then there is selection bias. To measure the selection bias, let where Pr y 2i > 0] = = h 2 (u 2i ) = X 2i β 2 X 2i β 2 h 2 (u 2i ) du 2i h (u i, u 2i ) du i du 2i (4.2) h (u i, u 2i ) du i is the marginal density of u 2i. We can write equation (4.2) as The conditional density of u i is Pr y 2i > 0] = H 2 ( X 2i β 2 ). j (u i u 2i > X 2i β 2 ) = X 2iβ h (u i, u 2i ) du 2i 2. H 2 ( X 2i β 2 ) 2

13 So and E (u i u 2i > X 2i β 2 ) = u i j (u i u 2i > X 2i β 2 ) du i, E (y i u 2i > X 2i β 2 ) = X i β + E (u i u 2i > X 2i β 2 ). (4.3) So we can think of selection bias as a specification errror. Since E (u i u 2i > X 2i β 2 ) depends upon X 2i,E (y i u 2i > X 2i β 2 ) will also depend upon X 2i. For each element of X 2i that is also an element of X i, we will get a biased estimate of the corresponding element of β, and for each element of X 2i that is not an element of X i, we will measure a nonzero effect of that element on y i. Now, if h (u i, u 2i ) is joint normal, then E (u i u 2i > X 2i β 2 ) = σ 2 λ i where and λ i = φ (Z i) Φ (Z i ) Z i = X 2iβ 2 Show this]. λ i is called the hazard rate and is also called the inverse Mill s ratio. Also, E (u 2i u 2i > X 2i β 2 ) = σ 22 λ i ; E (y i u 2i > X 2i β 2 ) = X i β + σ 2 λ i ; and We can now write E (y 2i u 2i > X 2i β 2 ) = X 2i β 2 + σ 22 λ i. y i = E (y i u 2i > X 2i β 2 ) + v i (4.4) y 2i = E (y 2i u 2i > X 2i β 2 ) + v 2i where v i and v 2i are deviations (random variables) of y i and y 2i from their respective means. We can show that E (v i ) = E (v 2i ) = E (v i v 2j ) = 0, and that E ( v 2 i) = σ ( ρ 2 ) + ρ 2 ( + Z i λ i λ 2 i )] ; E ( v 2 2i) = ( + Zi λ i λ 2 i ) ; and E (v i v 2i ) = σ 2 ( + Zi λ i λ 2 i ). 3

14 Note that there is heteroskedasticity. We now estimate the model using a nonlinear regression technique. First we get a consistent estimate of λ i by estimating β 2 ; we can do this with probit estimation. Then we set Ẑi = X 2iˆβ2 / σ 22 and ˆλ ) )] i = φ (Ẑi / Φ (Ẑi. ˆλ i is a consistent (but biased) estimate of λ i. Now, using equation (4.4), we can write y i = X i β + σ 2 λ i + v i (4.5) = X i β + σ ( 2 λ i ˆλ i + ˆλ ) i + v i X i β + σ 2 ˆλi + η i where η i = σ ( 2 λ i ˆλ ) i + v i. Equation (4.5) can be estimated with OLS (ineffi ciently). Identification occurs either through zero restrictions or because of the nonlinearity of λ i. Unfortunately, this procedure can be sensitive to functional form assumptions Mroz s Empirical Results Mroz (Ecta, 987, p 765) is a sensitivity analysis. He considers the model h i = a 0 + a ln (w i ) + a 2 Y i + Z i a 3 + e i where h i is hours worked, w i is wage,y i is other household income, and Z i are other relevant characteristics. He measures the sensitivity of the parameter estimates to various exogeneity assumptions, selection bias corrections, and kinked budget set adjustments. ) Exogeneity: he looks at wage rates, nonwife income, children, and wife s labor market experience. Why might each of these be endogenous? He tests for exogeneity by comparing OLS and 2SLS estimates. Look at Table V (p 77) and discuss choice of instruments. He finds that children and nonwife income pass exogeneity tests, but wage rate and experience do not. 2) Sample selection bias: he looks at the joint normality assumption and at zero restrictions. He finds some sensitivity to model specification and significant sensitivity to zero restrictions. 4

15 4.5. Nonparametric Selection Bias Correction Now write equation (4.3) as Ê (y i X i β ) = y i = X i β + τ (X 2i β 2 ) + v i where τ (X 2i β 2 ) is an (unspecified) function of X 2i β 2 correcting for selection bias. Mroz and others suggest that parameter estimates may be sensitive to the joint normality of the errors assumption in equation (4.). So Ichimura suggests estimating the model semiparametrically. Consider the following estimate of y i X i β : ( ) n j i (y X2i β j X j β ) K 2 X 2j β 2 b n n j i K ( X2i β 2 X 2j β 2 b n ) (4.6) where K ( ) is a kernel function with the properties, K (x) 0 as x and K (x) 0 (the second property is not really necessary and is sometimes dominated by K (x) sometimes negative), and b n is a bandwidth with the properties that b n 0 as n and nb n as n. K ( ) is a weighting function that gives positive weight to observations j close to observation i in the sense that X 2j β 2 is close to X 2i β 2. As n, since b n 0 (and therefore X 2iβ 2 X 2j β 2 b n whenever X 2i β 2 X 2j β 2 > 0), only observations j very close to observation i recieve any weight. But, since nb n, the number of observations receiving positive weight. Equation (4.6) provides a consistent estimate of τ (X 2i β 2 ). One can not identify constants and a scale effect (why?). Now consider the objective function, n (y i X i β ) Ê (y i X i β )] 2. (4.7) Consider two different ways of estimating β : ) Minimize equation (4.7) over both β and β 2. This is Ichimura s method and provides consistent estimates of both β and β 2. 2) Write the equation for y 2i as (y 2i > 0) = ς (X 2i β 2 ) + e 2i where ς (X 2i β 2 ) = E (y 2i > 0 X 2i β 2 )] = Pr y 2i > 0 X 2i β 2 ]. If we assume that u 2i N (0, ), then estimation of β 2 involves probit. If we are not willing to make 5

16 a functional form assumption about the density of u 2i, then we can estimate β 2 by minimizing ] 2 (y 2i > 0) n Ê (y 2i > 0) where Ê (y 2i > 0) = n n ( ) X2i β 2 X 2j β 2 b n j i (y 2j > 0) K ( ). j i K X2i β 2 X 2j β 2 b n This provides a consistent estimate of β 2 which can be used in equation (4.7) so that one needs only minimize over β. By comparing estimates from the two procedures, one can test the functional form assumptions of the model (Stern 997 rejects them). The procedures discussed above also provide consistent estimates of τ ( ) and/or ς ( ). For example, and ˆτ (x) = ˆς (x) = n j i n n n (y j X j ˆβ ) ( ) x X2j ˆβ2 K b n j i K ( x X2j ˆβ2 b n ), (4.8) ( ) x X2j ˆβ2 b n j i (y 2j > 0) K ( ). (4.9) j i K x X2j ˆβ2 b n Equations (4.8) and (4.9) can be used to test the probit assumption (by comparing ˆς (x) to Φ (x) properly standardized) and the joint normality assumption leading to the inverse Mills Ratio form (by comparing ˆτ (x) to λ (x) properly standardized). Stern 997 accepts the probit assumption but rejects joint normality. 5. Kinked Budget Set Estimation 5.. Graphical analysis Consider a budget line like in Figures (.4) or (.5). In Figure (.4), there are progressive taxes. Explain each segment. It is characterized by kinks but by unique tangencies of the budget line and indifference curves. Assuming that indifference curves are continuously distributed, there should be a large number of people at the kinks. Why? 6

17 In Figure (.5), there is no progressivity. Why might there be nonprogressivity in the US? This figure has some points where no people should be, and there are possibly multiple tangencies of indifference curves and the budget line. For now, assume that the budget line in concave as in Figure (.4). One could characterize the solution of the individual s optimization problem as follows: ) For each segment, extend the segment in both directions. Let the y-intercept of the extended segment be called the virtual income of that segment. 2) Compute where a tangency occurs between the extended segment and the set of indifference curves. If the tangency is on the segment (rather than the extended segment), then that is the optimum. If it is to the left (right) of the segment, then the optimum is to the left (right) of the segment. If for two adjacent segments, let s say A and B with A to the left of B, the optimum for extended A is to the right of segment A and the optimum for extended B is to the left of segment B, then the optimum is at the kink point where A and B meet Econometrics In the traditional model, h = g (w, y, z, θ) + ε = h + ε (5.) where h is hours worked, w is net wage, y is nonlabor income (the y-intercept of the budget line), θ is a set of parameters to estimate, and ε is an error representing mistakes. The goal is to minimize h i g (w i, y i, z i, θ)] 2 i over θ. The problem is that ε i is correlated with w i and y i because ε i affects which budget segment one is on and therefore what the net wage and y-intercept are. In the Hausman approach, rewrite equation (5.) separately for each budget segment: h j = g (w j, y j, z, θ) for each segment j. Assume a concave budget line, and let the lower boundary of each segment be H j. If H j h j H j+, then h j is the optimum. When taxes are nonprogressive, then all segments must be checked to guard against the possibility of multiple tangencies. When a multiple tangency is found, then the utility function must be evaluated at both points and the dominant one is chosen. 7

18 The utility function can be evaluated using Roy s identity. See below.] If there exists a j such that h j < H j and h j > H j, the optimum is at the kink H j. If h < 0 (= H ), then the individual should not work. The utility function can be evaluated using Roy s identity: v (w, y) / w v (w, y) / y = h = g (w, y, z, θ) = αw + βy + zγ. This is a differential equation that can be integrated to get v (w, y) = exp {βw} y + α β w + α β 2 + zγ ]. (5.2) β The point is that as soon as you write down an hours equation, it implies a utility function. Note: equation (5.2) is problematic as a utility function in that, assuming α 0 and β 0, the utility function is poorly behaved wrt w. Hausman specifies β as a random parameter, β truncn ( β, ) σ 2 β where the upper truncation point is zero. Next, we derive a series of more and more complicated likelihood functions: ) Assume we know that h is in segment j and that σ 2 β = 0. Then L ( ] h i w ij, y ij, z i, α, β, ) γ, σ 2 hi h ε = φ (5.3) where h = αw ij + βy ij + z i γ. 2) If σ 2 β > 0, equation (5.3) becomes L ( ] h i w ij, y ij, z i, α, β, γ, σ 2 ε, σβ) 2 hi h (β) = φ φ σ β where h (β) = αw ij + βy ij + z i γ. 3) If we can not condition on the proper segment, then define D ij = min 0, H ] j αw ij z i γ ; y ij U ij = min 0, H ] j+ αw ij z i γ. y ij β β σ β ] dβ (5.4) 8

19 Equation (5.4) becomes ( L h i w i, y i, z i, α, β, ) γ, σ 2 ε, σ 2 β = Uij φ j D ij + Dij+ φ j U ij hi h (β) hi H j+ ] σ β φ β β ] σ β φ σ β ] dβ β β σ β ] dβ. The first right-hand term is the density for h i conditional on β, φ hi h (β) ], multiplied by the density of β for all values of β such that H j h (β) H j+, Uij D ij ] φ hi h (β) σ β φ β β σ β ] dβ, added up over all segments j. The second righthand term is analogous for kink points. Discuss estimation results in Tables 2, 3, and 4. 9