7520 problem set
TRANSCRIPT
1. Indicate whether each of the following statements is true, false, or uncertain. Provide a brief but complete explanation of your answer.
a) Estimation of a labor supply model when income is subject to a continuous (i.e., no brackets) regressive tax is a simple extension of estimation in the no-tax case, because the same necessary first order conditions for an optimum apply in both cases.
False. The continuously regressive tax means that the budget constraint is no longer convex. In the face of such a constraint, problems such as
Max(X,H): U(X,H) such that pX = wH + Y – T(wH)
where T’ > 0 and T” < 0
no longer have unique interior maximums. This is not a simple extension – there is no longer a labor supply function which is uniquely determined.
b) If labor supply surveys asked nonworkers what their wage rate would be if they were to work, then selection bias would not be a problem when estimating labor supply models.
False. The simple selection problem comes from the existence of unobserved heterogeneity between workers and nonworkers. A wage equation can easily be imputed for nonworkers, but this doesn’t change the heterogeneity, which is due to a different ε for different people. ε is supposed to represent unobserved characteristics such as preferences and tastes for work, and these are not the same for workers and non-workers. Thus, it is not the absence of wage rates for the non-working population that creates sample selection bias. Rather, the bias arises because samples used in labor supply estimation are not random draws from a population as a whole. The samples are comprised of individuals who have self-selected into the sample on the basis of unobserved characteristics. Because these unobserved characteristics are likely to be a function of observed characteristics such as age, there is non-zero covariance between the unobserved characteristics and observed characteristics. In any statistical estimation, this non-zero covariance yields biased estimates. This bias is not corrected by knowledge of wage rates for the non-workers because it was not created by them.
c) Models of labor demand are usually estimated with either aggregate time series data, where the unit of analysis is an entire country, or aggregate cross sectional data, where the unit of analysis might be a city or county. Labor demand models sometimes include a measure of output on the right hand side and sometimes omit it. If the goal is to obtain an estimate of the long-run elasticity of labor demand, then the correct specification excludes output and uses aggregate cross-sectional data.
False. At the aggregate level, the wage rate is likely to be endogenous to the system. Use of aggregate data, therefore, yields biased and inconsistent parameter estimates in the presence of endogenity. Further, a specification that omitted output as a regressor wold
fail to account for shifts in the labor demand curve, which are a result of macroeconomic conditions. Thus, omitting the output regressor would confound macro effects with other effects. Omitting the output regressor specifically would confound movements along the labor demand curve with shifts of the labor demand curve. A correct specification would use firm level data in longitudinal or time series form.
d) Given current knowledge about labor supply behavior, it is likely that a major decrease in the implicit tax rate on earnings in welfare programs (the welfare benefit reduction rate) would result in relatively small increases in hours worked in the welfare-eligible population.
Uncertain. Let G denote a government transfer. The budget constraint is pX = (1-t)(wH + Y) + G if G > t(wH + Y) and wH + Y otherwise.
For those with G > 0,
dH/dt (given G > 0) = (dH/dw)(dw/dt) + (dH/dY)(dY/dt)
= { dH/dw (holding utility constant) + H(dH/dY)}dw/dt + (dH/dY)(dY/dt)
= (dw/dt)(dH/dw) (holding utility constant) + {(dw/dt)H + dY/dt}dH/dY
dw/dt = -w and dY/dt = -Y.
G-tY
Y
Consumption
Hours of Work Time
β
(1-t2)wH
(1-t1)wH
dH/dt (given G > 0) = (-w)(dH/dw) (holding utility constant) + {-wH – Y}dH/dY
= -{w(dH/dw) (holding utility constant) + (wH + Y)(dH/dY)} is ambiguous.
dH/dt (given G > 0) is ambiguous.
e) It is impossible to produce reliable estimates of the intertemporal substitution elasticity of labor supply without panel data.
True. Panel data studies a fixed sample as it moves through time. Intertemporal substitution effect needs this. Cross sectional data lacks the time variation. In cross sectional data, time is fixed, though the agents vary. It measures the effects of differences among agents at a given moment, not a given agent through time. With time series data, time varies and the sample varies. It measures differences among agents over time, not a given agent over time. Panel data is time series where the sample is fixed.
In order to produce reliable estimates of the intertemporal substitution elasticity, at least two time periods are needed. Further, λ (the way to determine marginal utility of wealth), must be constant, implying that there are no parameters outside period t affecting the choice variables of they dynamic programming model.
f) Basic labor supply theory predicts that ‘a lot’ of sample members should be observed at the kink points in their budget constraints. Since almost no one is observed at a kink, the theory must be wrong.
False. There could simply be measurement error. For example, if H = 40 was a kink, due to measurement error, 80/2.01 would not be at that kink. So, there’s nothing wrong with the theory.
2. Write a comparative evaluation of the Berndt and Christiansen and Hamermesh and Grant articles. What are the goals of each paper, what models are specified, and what parameters are estimated in order to achieve these goals, what important assumptions are made in order to specify and estimate the models, and what type of data and econometric methods are used? Compare the articles in terms of the inherent interest of the issues addressed and the success of the article in achieving its goal.
Berndt and Christiansen:
Examine U.S. time series data for manufacturing. Should blue and white collar workers be added up?
Goal is to estimate a production function with labor disaggregated and then test to see whether aggregation is consistent. See if there is any substantial bias for aggregation between blue and white collar workers.
The model is a translog production function (a production function where the AES are not constrained a priori.)
lnF = lnα0 + αBlnB + αWlnW + αKlnK + 1/2γBB(lnB)2 + γBWlnBlnW + γBKlnBlnK + 1/2γWW(lnW)2 + γWKlnWlnK + 1/2 γKK(lnK)2.
Output = Y = Y(F(B,W,K),G(E,M)) – separability assumption – assume a consistent way exists to separate the marginal product of workers from energy and machines. F is like value added. They focus on F.
Assume (1) constant returns to scale, (2) perfect competition, (3) all deviations from cost chare are deviations from firm optimization, (4) translog is exact representation of F, (5) perfect competition means firms are price-takers.
F(B,W,K) = H[J(B,W),K] is roughly implies σBK = σWK. Test whether elasticity of substations are identical. Thus, they’re assuming B, W, and K are weakly separable from all other inputs.
MB γWK = MW γBK
γBW γWK = γWW γBK
αB γWK = αW γBK.
These are testable parameters and constitute the main hypothesis testing.
dF/dlnB = MB = cost share of B.
Estimate these:
MB = αB + γBBlnB + γBWlnW + γBKlnK + ε1
MW = αW + γWWlnW + γBWlnB + γWKlnK + ε2
MK = αK + γKKlnK + γKWlnW + γBKlnB + ε3
MB + MW + MK = 1
αW + αB + αK = 1.
Column restrictions:γBB + γBW + γBK = 0 γWB + γWW + γWK = 0γKB + γKW + γKK = 0
Symmetry restrictions:γBK = γKB, γWK = γKW, etc.
So, only 5 parameters remain to be estimated: αW, αK, γWK, γWW, and γKK.
Estimate the 3 equations jointly, testing for endogeneity. Use iterative 3SLS to allow for systems estimation. Also, if iterated, residual covariance matrix is estimated until its an identity.
Results: Allow elasticities of substitution to vary with inputs. σBK > σWK, so reject σBK = σWK. We can reject all symmetry restrictions. Standard errors are small, so you can reject the argument of symmetry.
Why might the data reject symmetry? Autocorrelation (biased estimates). Standard errors without correction for autocorrelation are too low. True standard errors are bigger, making us unable to reject symmetry. Since there was no way to correct for autocorrelation, at the time this was written, symmetry was merely imposed. Estimates from imposed symmetry:
BC and WC workers are good substitutes.
WC and K are good complements.
BC and K are good substitutes.
As PK increases, K decreases, W decreases, and B increases. So, it’s unlikely that BC and WC workers can be aggregated.
The marginal product of labor can change by changing the amount of BC and WC workers holding total workers constant.
Tests reveal separability restrictions should be rejected. Though BC and WC workers are good substitutes, they can’t be aggregated. Test statistics of separability restrictions show we can strongly reject separability for any pair of inputs. Thus, we conclude that no consistent aggregate index of BW, BK, or WK exists for U.S. manufacturing data. Also, see question 13 for more information on this topic.
Grant and Hamermesh:
They look at the separability of workers between demographics: age, gender, and race. So, disaggregated data by demographics is needed. They use Census data. For example, they aggregate total number of black workers in a given SMSA. They add them up. They find capital stock data considering SMSA areas in manufacturing in different geographical areas.
Framework is similar to Berndt and Christiansen. Instead of W, B,and K, they use categories of aggregation – micro data on those who hire in a certain SMSA area with capital data in manufacturing.
Y = youths aged 14 – 24OB = older blacks (over age 24)OWF = older white females (over age 24)OWM = older white males (over age 24)K = capital
They estimate partial elasticities of complementarity: Cij.
Cij = (γij + MiMj)/MiMj = FijY/FiFj where σij = Y|Fij|/XiXj|F|.
Goal: see how influx of women working has affected the labor supply of youths.
The model is the translog approximation:
lnQ = lnα0 + αilnXi + 1/2 γijlnXilnXj.
dlnQ/dXi = pi = αi + γijXj.
Assumptions: (1) constant returns to scale, (2) firms are price-takers, (3) price elasticities of substitution and price elasticities are inappropriate for considering the effects of exogenous changes in factor quantity on factor price, (4) impose symmetry, (5) input supply is exogenous, (6) use a production function rather than a cost function.
They go on and impose symmetry without testing it. They estimate cost share equations using seemingly unrelated iterated estimation: a Zellner process. Estimate a system of equations jointly when you do not believe there are endogeneity problems. 3SLSs allows
for endogeneity. Zellner’s process allows for correlated errors between equation errors, not between inputs. The estimates of 3SLS and Zellner’s will be the same if the variables are the same in each equation. So, the same equations between GH and BC have same parameter estimates. There will be no efficiency gains for GH from using correlated errors. 3SLS was used to impose symmetry. 2SLS wouldn’t impose cross-equation symmetry. So, a systems estimator is used to impose symmetry. This aids in testing for aggregation and separability. They are not using a systems estimator to gain efficiency from correlated errors because there are none when compared to BC. Like BC, they test for aggregation. They reject every form of aggregation except for Y and OWF.
Y = F(Y, OB, OWF, OWM, K) = F(G(Y, OWF), OB, OWM, F).
So, CY,K = COWF,K.
Cij interpretation: an exogenous increase in OWF will depress wage rate of youths because they’re substitutes.
OWF and Y as well as OB and OWM are substitutes. The rest are complements. That is, if the quantity of i increases, the price of j increases.
Elasticities of factor prices were small – elastic demand.
Inputs are good substitutes if as supply if i goes up, the price of j doesn’t go up.
Relying on aggregation within manufacturing, they find the impact of a 10% increase in the quantity of OWF on labor force wages: depress wages of OWF and Y because they’re substitutes. But, the wages of others actually increase – they’re complements.
3. An employer accused of discrimination against black workers argues that it is necessary to treat different groups of workers differently because they have different sets of skills, different productivity, and are not perfect substitutes for each other. It is argued that this applies to groups of workers defined not only by race but also by age, education, gender, and other characteristics. Suppose you had access to a time series of disaggregated data on this employer’s work force and its demographic characteristics and wages, other inputs, and their prices, output, and output price. Describe carefully how you would use this information to determine the plausibility of the firm’s argument.
Let LB and LNB denote black and non-black workers.
Assume a production function: Y = f(LB, LNB, K).
The firm maximizes profit:
Max(LB,LW,K): Π = pf(LB, LNB, K) – wBLB – wNBLNB – rK
First order conditions are
dΠ/dLB = 0 implies pfLB = wB
dΠ/dLNB = 0 implies pfLNB = wNB
or (df/dLB / df/dLNB) = wB/wNB.
Now, take the data and form the following set: for each period t = 1, … T and for each variable i = 1, …N, I would partition the data into the black measures and the non-black measures. For each period, find the average for the partitioned measures. Then, for each period, take the log of the ratio of each variable across the partition.
Two of the more important variables are given below as examples:
wt = ln(average black wage at period t/average non-black wage at period t)
Xt = ln(average number of black employees at time t/average number of non-black employees at time t).
We now have an equation:
wt = α0 + α1Xt + α2(average age of blacks/average age of non-blacks) + α3(average education of blacks/average education of non-blacks) + … etc.
H0: no discrimination, which suggests α1 = 1 and some other αi ≠ 0 for i not equal to 1.
H1: not H0.
α1 is the partial elasticity of substitution.
If α1 = 1, then this implies blacks and non-blacks are perfect substitutes once all other variables are controlled for (all other things that would affect productivity).
If α1 ≠ 1, then wB/wNB varies systematically with race, and, hence, there is discrimination.
So, we need to ask:
Does systematic variation in Xt affect variation in wt?
Does systematic variation in education, etc., affect wt?
4. Suppose you are interested in evaluating the effects of being disabled on the labor supply of prime age men. You have access to a typical cross-section data set with the usual variables plus a variable indicating whether each man considers himself disabled. Some men who consider themselves disabled work a positive number of hours, while others are not in the labor force. Some men receive disability insurance payments from government programs. Describe how you would estimate the effect of being disabled on labor force participation. Discuss both the economic and econometric issues involved, and be specific about what equation you would estimate, how you would specify it, and how you would interpret the results. Now, suppose you have reason to suspect that disabled and non-disabled men have different income and substitution effects of wage changed. Write down and justify the equations you would estimate to test this conjecture. How would you recover income and substitution effects, and how would you interpret the differences in these effects between the disabled and non-disabled?
To examine this, assume a static model, that being disabled is exogenous, and that disability pay is part of non-wage income.
Max U(X, H, D, A, ε)
where D = 1 if disabled and 0 otherwise,
such that pX = wH + Y
where Y = Y0 + disability insurance.
LFP = α0 + α1Di + α2wi + α3Diwi + α4Yi + α5DiYi + α6(f/F) + α7A + εi,
where f/F is the hazard rate or the hazard rate correction or Heckman correction for the probability you’ll work.
A is a vector of other characteristics.
Next, control for unobserved heterogeneity.
dLFP/dD = α1 + α3wi + α5Yi,
which is the effect on expected LFP of being disabled.
dLFP/dwidDi = α3.
dLFP/dYidDi = α5.
Model hours:
Hi = β0 + β1Di + β2wi + β3Diwi + β4Yi + β5DiYi + β6(f/F) + β7A + εi.
To determine the effect of disability on hours worked,
dH/dD = β1 + β3wi + β5Yi,
dH/dwi = β2 + β3Di,
dH/dYi = β4 + β5Di,
d2H/dDdw = β3,
and
d2H/dDdY = β5.
Now we can recover:
the income effect: (β4 + β5Di),
the uncompensated wage effect: β2 + β3Di,
and
the compensated wage effect: β2 + β3Di - (β4 + β5Di).
β2 + β3Di = β2 + β3Di - (β4 + β5Di) + (β4 + β5Di)
where the interaction terms with Di measure (β3Di and β5Di) the effect of being disabled.
5. The Earned Income Tax Credit (EITC) is a federal tax credit available to working poor families with children in the United States. The purpose of the credit is to supplement the earnings of workers with low wages in order to increase their income. Because the credit is intended to increase the reward to working, it is hoped that the EITC not only reduces poverty directly but also encourages increased labor supply among the poor. However, in order to avoid paying the credit to middle income workers, it is gradually phased out as earnings rise. For the purpose of this question, ignore other taxes and subsidies. A worker’s budget constraint under the EITC is X = wH + Y + C, where X is consumption, w is the wage, H is hours worked, Y is non-wage income, and C is the credit under the EITC, determined by the following schedule:
C = 0 if wH = 0 [No credit for non-workers]C = wHt1 if 0 < wH ≤ q1 [Credit rises with earnings until earnings reach q1]C = q1t1 if q1 < wH ≤ q2 [Credit is independent of earnings in the range q1 – q2]C = q1t1 – (wH-q2)t2 if q2 < wH ≤ q3 [Credit is reduced with earnings in the range q2 – q3]C = 0 if q3 < wH [No credit for workers with high earnings]
where the q’s and t’s are positive constants with 0 < ti < 1 for i = 1,2. The diagram illustrates the budget constraint for a typical worker with and without the EITC.
a) How would introduction of the EITC affect the labor supply of individuals whose hours worked in absence of the EITC was in each of the following ranges. Use diagrams (or equations, if you prefer) to illustrate your answers and explain each answer briefly.
$
Y
H3 H2 H1 0
slope = -w slope = -w(1-t2)
slope = -w
slope = -w(1+t1)
(i) H = 0(ii) 0 < H < H1
(iii) H1 < H < H2
(iv) H2 < H < H3
(v) H3 < H
b) Discuss the main issues that would have to be dealt with in order to estimate empirically the effect of the EITC on labor supply, given the non-linearity of the budget constraint induced by the EITC. Describe the advantages and disadvantages of alternative approaches to these issues.
a)
(i) H = 0 Ambiguous; dH ≥ 0; d(LFP)/dw ≥ 0
(ii) 0 < H < H1 There exists both income and substitution effects
(iii) H1 < H < H2 Pure income effect will decrease hours.
(iv) H2 < H < H3 Negative substitution effect, negative income effect, dH < 0, like Greenberg and Kosters.
(v) H3 < H May lower, cannot raise.
Within the model, the following must be dealt with: measurement error, kinked budget, non-convexity, endogeneity of non-participation.
Advantage/disadvantage of various methods:
See lectures 11, 12, 13 on Hausman versus MGP studies or complete budget constraint methods vs. simple regression, controlling for endogeneity.
For information on non-linear budget constraints of all kinks, see lectures 4 and 8.
6. The literature on labor supply behavior of married women contains a very wide range of estimates of the uncompensated wage effect on hours worked, and a correspondingly wide range of model specifications and estimation methods. Describe three specifications and/or estimation issues that could affect the magnitude of the estimated uncompensated wage effect on hours worked. For each issue, explain the main alternative approaches used in the literature and how those approaches may have influenced the estimated wage effect.
See notes for answers.
7. A common empirical finding in the literature on male labor supply is that the wage rate has a negative effect on hours worked. Is this finding consistent with the standard theory of labor supply? Why or why not? Another relatively common finding is that the substitution effect of the wage rate on hours of work is negative. Is this finding consistent with the standard theory of labor supply? Why or why not? If your answer is no in their case, discuss what might be wrong with the standard theory that could lead to such an inconsistency.
Negative effect of the wage rate? Negative substitution effect?
Slutsky decomposition of H = H(w, p, Y, A, ε) for the effect of w on H is:
dH/dw = dH/dw (holding utility constant) + (dH/dY)(dY/dw).
dH/dy < 0 by assumption that leisure is a normal good.
dH/dw (holding utility constant) > 0 is predicted by the theory. dH/dw = {+} + {-} is ambiguous. Note that if the income effect is larger in absolute value than the substitution effect, then dH/dw < 0.
Proof of positive substitution effect:
L: U(X, H, A, ε) + λ(wH + Y – pX)
First order conditions:
-UX = - λp
UH = λw
and
wH + Y – pX = 0.
Totally differentiate the first order conditions:
UXX UHX -p dX λdp
UX
H
UH
H W dH = - Λdw -p w 0 dλ H(dw)-X(dp)+dY
Let D be the bordered Hessian matrix and Dii be the ii-th cofactor.
By the second order conditions:
|D| > 0 and |Dii| < 0.
So, dH/dw (holding utility constant) = - λ |D22|/|D| > 0 by the second order conditions.
We assume the income effect is negative because leisure is a normal good.
If preferences don’t follow the theory, then there could be (i) measurement error or (ii) an error with our original assumptions about the utility maximization process. There could be (iii) a problem with assuming individual agent theory models when the family model is better.
Pencavel finds most income effects to be negative and most substitution effects to be positive.
(iv) Greenberg and Kosters: are the correct variables included in the regression?
(v) Hausman’s specification: are we dealing with taxes correctly? He gives an example of how we can impose restrictions based on theory. Hausman doesn’t believe the theory as much as he wants a likelihood function which works.
(vi) With negative substitution effects, do we through out the theory? It is a matter of judgment. Note that the simple static model is simple and it might just not work.
(vii) We could go back to the theoretical model and change it to fit the data. Maybe a dynamic model would do better.
(viii) Also, sometimes incorrectly signed effects are not really statistically significant (statistically different than zero).
(ix) Biased estimates: endogeneity or sample selection bias – if sample selection bias exists and you don’t correct for it, your estimates are biased (potentially downward).
(x) Cagan’s fixed costs and Moffitt’s minimum hours models show individuals may not be free to choose or change H when the wage changes.
8. Describe the sample selection bias problem in the context of a labor supply model. What are the sources of the problem and what methods are available to deal with it? Discuss how to interpret labor supply parameters that are estimated with and without a correction for sample selection bias.
See notes – lecture 7, for example.
9. Suppose you are interested in determining whether the leisure hours of husbands and wives are substitutes or complements. Describe how you would investigate this issue. What model would you specify, what type of data would you use, how would you estimate the question of interest?
Are leisure hours of husbands and wives substitutes or complements? Investigate the issue, specify a model, define data, and explain how to estimate parameters.
Use a family labor supply functional form allowing for the labor supply choices of others to be included in a joint utility function.
Max(HF,HM,X): U(X, HF, HM, A, ε)
subject to the budget constraint
wFHF + wMHM + Y = pX.
HMi = α0 + α1wMi + α2Yi + α3A + α4wFi + εi
and
HFi = β0 + β1wFi + β2Yi + β3A + β4wMi + εi.
Impose symmetry: dHMi/dwFi = dHFi/dwMi and then test.
Tests will show whether the model is specified correctly.
The coefficients α4 and β4 (held in equality) will show the effects of the male’s labor supply function on the female’s labor supply and vice versa.
If these are negative, then hubands’ and wives’ hours are substitutes. If positive, then they are complements.
Estimate these equations as a system of equations using SUR or 3SLS. Then, test for symmetry restrictions.
First test: H0: α4 + α2 = β4 + β2 and H1: not H0.
If we accept H0, then impose this constraint. Then estimate as a system with constraint imposed. Then test for symmetry. Finally, see whether α4 and β4 are positive or negative.
Determine whether the leisure hours of husbands and wives are substitutes or complements.
Labor supply equations:
Assume that the family acts like it has a twice continuously differentiable utility function:
U = U(LM, LF, X)
with the budget constraint
pX = R + wMHM + wFHF
where R is non-wage income.
The first order conditions are:
dU/dLi = λwi if Hi > 0 or dU/dLi > λwi if Hi = 0.
dU/dX = λp.
These first order conditions allow us to define labor supply functions for all family members.
HM = HM(wM, wF, p, R) and
HF = HF(wM,wF, p, R).
Slutsky decomposition imposes testable restrictions:
Slutsky equation:
dLi/dwj = Sij – Hj(dLi/dR) or
dHi/dwj = Sij + Hj(dHi/dwj)R
Restriction imposed:
Own-substitution effects are positive: Sij > 0.
Symmetry: SMF = SFM
Matrix of substitution effects is negative semi-definite.
SMM SMF SFM SFF
0.
Now, test to see whether the leisure time of husbands and wives are substitutes or complements:
If Sij > 0, then LM and LF are substitutes.
If Sij < 0, then LM and LF are complements.
10. During the 20th century, the labor force participation rate of married women in the United States has increased substantially while the average number of hours worked per year among working married women has decreased. Can these apparently contradictory trends be explained in the context of the standard theory of labor supply? If not, why not? If so, explain how the theory can account for the simultaneous occurrence of these two trends.
The labor force participation of married women has increased and average hours worked has decreased. Explain.
Labor force participation is defined as whether a person works or not. If labor force participation increases, hours worked need not. Average hours equal total hours divided by total women. If all women entering the workforce work fewer hours than the previous average, then the average must fall.
An increase in wage causes an unambiguous increase in labor force participation, but due to the substitution and income effects, increases in wages could have a negative effect on hours worked.
The Slutsky decomposition of H = H(w, p, Y, A, ε) for the effect of w on H:
dH/dw = dH/dw (holding utility constant) + (dH/dY)(dY/dw) is the total effect equals the substitution effect plus the income effect.
dH/dY < 0 because we assume leisure is a normal good. dH/dw (holding utility constant) > 0 is predicted by theory. So, dH/dw = {+} + {-}, which is ambiguous.
If the income effect is larger than the substitution effect, then dH/dw < 0.
See proof of increased labor force participation from increased wage in lecture 3.
So, if the change is a change in the wage, we could see hours decrease and labor force participation increase.
A decrease in fixed costs of work could cause an increase in the labor force participation rate but a decrease in average hours. Fixed costs are relevant with women because they have children. If they work, there are fixed costs they must incur like paying for day care for the children.
Suppose a decrease in fixed costs:
Now, reservation hours have decreased, which increases labor force participation, but only for those interested in working low hours. So, these new workers have brought the average hours worked down.
In Zabel’s 1993 article, he defines hours worked as
Hi* = β1lnwi* + β2NIi + β3Xi + u1i
and wages as
lnwi* = α1Y1i + u2i.
Cogan’s fixed cost reservation hours equation can be written as:
Hr = γ1NIi + γ2Fi + u3i
where F is a vector of individual characteristics.
The labor force participation decision is work if Hi* > Hir, and don’t if Hi* < Hi
r.
Or, Hi* ≤ Hir implies (u1i + β1u2i – u3i)/σ ≤ -( β1Y1α1 + (β2- γ1)NIi + Xiβ3 - Fiγ2)/σ
FC1
FC2
Y H1
r H2r H0
11. What are the effects of a typical welfare program on labor supply behavior among members of a population eligible for the program? That is, how would the labor supply of members of this population be predicted to differ in the presence of such a program compared to the absence of the program? Demonstrate the effects in the context of a standard labor supply model.
See notes – Greenberg and Kosters.
12. An important U.S. senator has a proposal for reforming the federal income tax by making it strictly proportional, i.e. eliminating progressivity at the upper end and eliminating the high rate of implicit taxation of welfare benefits at the lower end of the income distribution. The senator’s assistant has convinced him that empirical evidence from labor supply studies shows that this reform could have a large impact on the supply of labor of both men and women. However, the senator’s assistant received his Ph.D. in 1982 and has not read any labor supply studies that have appeared since then. Write a memo to the senator’s assistant in which you describe what has been learned about labor supply behavior from empirical studies conducted since 1982, how it differs from the conventional wisdom as of 1982, and the implications of this new knowledge for the effects of the senator’s tax reform proposal on labor supply.
See portions of notes.
13. Describe the methods typically used in the labor demand literature to estimate the output-constant own-wage elasticity of demand for labor. What problems do you see with these methods, and how may these problems have affected the empirical results obtained in this literature? Identify a policy issue for which knowledge of the magnitude of this parameter is important. Based on the magnitude of this parameter typically estimated in the literature, discuss how this parameter influences the outcome of policy for your policy issue.
Typical estimation of labor demand in the literature typically uses aggregate data (time series or cross-sectional). It assumes constant returns to scale, using flexible functional forms like the translog or generalized leontieff production/cost function. The literature needs output and wage data to estimate a series of demand equations. From these, we get share and elasticity of substitutions.
One possible functional form is the Cobb-Douglas form:
Y = AX1β1X2
β2X3β3…
with constant returns to scale where βi = 1.
However, a problem with this functional form is that the resulting partial elasticities of complementarity all equal one by functional form:
Cij = Yfij/fifj = 1
and partial elasticities of substitution all equal one:
σij = Y|fij|/XiXj|F| = 1.
So, if you’re interested in the relationship among factors, this is not the correct functional form.
The constant elastiticity of substitution model:
Y = [ βi Xiσ-1/σ]σ/σ-1 where
C = Y[ αiwiσ(1-σ)]1/1-σ.
With the constant elasticity of substitution model, partial elasticities of substitution
σij = Cgii/gigj where σ is the same for all inputs in the input space. The elasticity of substitution is the same everywhere for any two inputs. Instead of σij = 1, they are all the same. So, we need to go beyond simple forms to avoid imposing restrictions we don’t want. We need a flexible form with no restrictions.
The translog form:
lnC = lnY + α0 + αilnwi + 1/2 βijlnwilnwj
where αi = 1 and βij = 0 for all i.
Cost share si = derivative of the translog form with respect to wi.
si = αi + βilnwj > one cost share function for each input.
We’ll estimate all the cost share equations jointly to find the parameter estimates we’re interested in.
σij = Cgij/gigj
where there are no restrictions on σij, and σij varies depending on where you are in the income space.
One problem that may arise is the problem of endogeneity. Is the wage endogenous? Well, firms are assumed to be price takers. If firms are price takers, then there should be no endogeneity in the wage. But what about regional or time variations? There still could be endogeneity.
Aside: The generalized Leontief: C = Y αijwi1/2wj
1/2 is another flexible form.
In general, with production function
lnY = α0 + αilnXi + βijlnXilnXj + ε
where Xi are inputs. Are inputs exogenous? Firms choose their inputs. If the Xs are truly exogenous, then the error ε will only measure error or error from functional form.
Anther problem results from aggregation. These forms are usually not used with firms but with industry data. Aggregated data doesn’t lend itself to firms and firm’s ability to
make decisions. So, there could be aggregation bias. If underlying functional form for the firm is non-linear and we aggregate for the industry, then there is aggregation bias. Aggregation doesn’t do anything to get around the problem of endogeneity. Further, with aggregation, industries are not price takers, so w is now kept from being exogenous. Aggregation bias makes things worse.
With aggregated data, = αi + αij .
But, we want average log, not the log of the average.
There is also the issue of the error, εt, where Y = f(K, L, A) + ε.
If K* and L* are chosen before ε is realized, then K and L are uncorrelated with εi, and εi is optimization error. In other words, there are no endogeneity problems.
If we know the value of εt before we choose K* and L*, then K and L are correlated, and the production function is biased. If εi are measurement error, then the production function estimates will be biased. So, from where εi comes will affect bias of the production function.
Anther problem in estimation is measurement error. We probably can’t aggregate workers and hours worked.
Y = f(# of workers, # of hours worked per worker).
The marginal cost of an additional worker could be higher than the marginal cost of an additional hour of work because a new worker will have to be trained, incurring training costs. Hours and workers may be separate inputs. Further, workers could be heterogeneous – secretaries and hard-hat workers, etc. Its hard to measure the price of labor with hourly earnings, fringe benefits, and the fact that overtime hours pay time and a half.
How do we know workers can be aggregated? (See question #2). If f(K, L1, L2) = f(K,H(L1, L2)), then we can say the production function is weakly separable in labor inputs. Thus, σL1,K = σL2,K. We can test this. If true, then we have a consistent aggregate index of labor input.
We assume constant returns to scale because it is then easy to calculate elasticities. But what if constant returns to scale is incorrect?
I want to investigate the minimum wage. Most workers earn more than the minimum wage. It has been thought that the minimum wage has a small negative effect on teen working hours. Time series data gives 0.06. Then, in 1990, the minimum wage changed and we got new data. Deere, Murphy, and Welch: They find the minimum wage does matter. They find many people are earning right at the minimum wage. They find in
1990 when the minimum wage was rising, there was a strong drop in teenage employment. For ages 15-19, 97.8 (1990) to 86.6 (1992) with an index equal to 100 (1985). DMW find the percentage of different kinds of workers working at the minimum wage. (44% of teens aged 15-19, for example). Then, when the minimum wage was increased in 1990, they found the biggest drop in employment came from low wage earners. So, there are negative employment effects from increasing the minimum wage.
The minimum wage went from $3.35 to $4.25. Male teens about 7% lower, female teens, about 11% lower, black teens about 10% lower, all statistically significant.
Stronger results than previous studies. But here, there was an actual minimum wage change, not an implicit change from inflation.
Other factors were at work, too: measurement error, recession, trend for employment to grow in states with low wages which offset the change in employment from the minimum wage increase. So, variation across states.
14. Consider a firm that employs three factors of production: skilled labor (S), unskilled labor (U), and capital (K). Assume that the firm’s input and output markets are perfectly competitive. Derive the output-constant and total effects on the firm’s use of each factor of an investment tax credit that has the effect of reducing the price of capital. Discuss your interpretation of the signs of the effects you derive, and describe any additional assumptions you make and why you make them.
Min(S,U,K): w1S + w2U + (w3 – t)K + λ( - f(S,U,K))
The first order conditions are:
- f(S,U,K) = 0
w1 - λfS = 0
w2 – λfU = 0
(w3 – t) – λUK = 0.
The second order conditions are:
0 -fS -fU -fK -fS -λfSS -λfSU -λfSK -fU -λfUS -λfUU -λfUK -fK -λfKS -λfKU -λfKK
= [Hb].
At a regular interior optimum, det [Hb] = |Hb| < 0.
The output-constant effect is measured by the system:
0 -fS -fU -fK dλ/dt 0 -fS -λfSS -λfSU -λfSK dS/dt = - 0 -fU -λfUS -λfUU -λfUK dU/dt 0 -fK -λfKS -λfKU -λfKK dK/dt 1
So, dS/dt =
0 0 0 -fK
-fS 0 0 -λfSK
-fU 0 0 -λfUK
-fK -1 -1 -λfKK
0 -fS -fU -fK
-fS -λfSS -λfSU -λfSK
-fU -λfUS -λfUU -λfUK
-fK -λfKS -λfKU -λfKK
= fS(λfUfUK - λfUUfUK) + fU(λfUSfK - λfUfSK)/|Hb| which is ambiguous.
Assumptions: fSK > 0, fUK < 0, and fUS = 0.
B and C find B and W to be substitutes, B and K to be substitutes, and W and K to be complements. B and H find K and W to be complements, B and K to be complements, some W and B to be substitutes (OB and OWM only) and (Y and OWF).
fSdS + fUdU + fKdK = 0.
So, dU/dt =
0 -fS -fU -fK
-fS -λfSS -λfSU -λfSK
-fU -λfUS -λfUU -λfUK
-fK -λfKS -λfKU -λfKK
0 -fS -fU -fK
-fS -λfSS -λfSU -λfSK
-fU -λfUS -λfUU -λfUK
-fK -λfKS -λfKU -λfKK
= -[λfS(fSfUK - fUSfK) + λfU(fSSfK – fSfSK)/|Hb| = -[-]/[-] = +/- < 0.
So, dU/dt < 0.
Similarly, dK/dt > 0.
A fall in w3 causes a rise in optimal K and optimal U falls.
Normalize p to 1.
Max(S,U,K): f(S,U,K) – w1S – w2U – (w3-t)K
The first order conditions are:
fS – w1 = 0
fU – w2 = 0
fK – (w3-t) = 0.
The second order conditions are:
fSS fSU fSK fUS fUU fUK fKS fKU fKK
= [H] = negative semi-definite, so |H| < 0.
fSS fSU fSK dS/dt 0 fUS fUU fUK dU/dt = - 0 fKS fKU fKK dK/dt -1
dS/dt =
0 fSU fSK
0 fUU fUK
-1 fKU fKK
fSS fSU fSK
fUS fUU fUK
fKS fKU fKK
= -(fSUfUK - fUUfSK) / (-) > 0 because fSU = 0.
Similarly,
dU/dt =
fSS 0 fSK
fUS 0 fUK
fKS -1 fKK
fSS fSU fSK
fUS fUU fUK
fKS fKU fKK
= (fSSfUK – fUSfSK) / (-) < 0 because fSU = 0 if fUK < 0.
and
dK/dt =
fSS fSU 0fUS fUU 0fKS fKU -1
fSS fSU fSK
fUS fUU fUK
fKS fKU fKK
= (fUSfSU – fSSfUU) / (-) > 0 because fSU = 0.
15. Conventional economic wisdom suggests that an increase in the legal minimum hourly wage rate for labor will cause a decrease in employment among those workers affected by the increase. This conclusion follows from the assumptions that profit-maximizing firms take input prices as given, the minimum wage is binding for at least some of the firm’s workers, and that all firms are subject to the minimum wage law. Suppose that a sample of firms is observed before and after an increase in the minimum wage. Data are collected on the quantity of output produced and quantities of inputs used, and output and input prices. The data show no change in average employment per firm after the minimum wage increase compared to before.
a) Why could a simple before-after comparison of means be misleading in this case and in general? Describe how you would use the data to estimate econometrically the impact of the minimum wage increase. Justify your choice of specification and estimation method and discuss any problems of interpretation that might be important.
Suppose that your best econometric estimate still shows no impact of the minimum wage increase on employment. This might lead you to reconsider some of the basic assumptions of the conventional model alluded to above.
b) Discuss how changing each of the following assumptions would alter the prediction of the theory concerning the impact of the minimum wage on employment. Would a change in the assumption definitely alter the prediction, maybe alter the prediction, or definitely not alter the prediction. Explain.
(i) Firms take input prices as given.(ii) The minimum wage law applies to all firms.
The results could be misleading because there are other macroeconomic changes going on in the economy which confound the effect of the minimum wage. For example, Deer, Murphy, and Welch discovered that states with the highest percentage of their workforce being low wage workers saw smaller declines in employment when there was a minimum wage increase in 1990 than other states. This is the result of a trend of employment growth in low wage states which offset the effect of the minimum wage increase. Similarly, women are more likely to receive low wages near or at the minimum wage than men. Yet, female employment fell less than the decline men experienced from the minimum wage increase. Similarly, in this case, the effect of the change in minimum wage is being confounded by the rising secular trend in female employment during this period. Further, we could be dealing with a state where the minimum wage is higher by state law than the federal minimum wage level. Thus, a change in the federal minimum wage would have little effect on employment in such a state. (See also criticisms of Card and Krueger’s work described below – these criticisms apply here, too).
In general, results could be due to some kind of measurement error. In the 1980s, the effect of changes in the minimum wage were found to be small. But, these changes were changes due to inflation – not actual nominal changes. Then, in 1990, there was an actual
change in the minimum wage in nominal terms. A resulting effect was experienced which was stronger than predicted by 1980s estimation.
In estimation, I would assume the minimum wage is not binding on all groups. I would identify a legitimate control group, use difference-in-difference estimators, use data to estimate an own-price elasticity of labor demand in the time prior to the minimum wage law.
Estimate Li = α0 + α1wi + α2Yi + α3pi + δTai + εi
Where the variables are in logs.
Li = labor inputwi = wageYi = outputpi = price of outputai = vector containing other inputs and their prices.
We know from the data how much the minimum wage changed and how much labor demand for those workers on whom the minimum wage is binding changed. We can predict using estimates what the effect on those bounded would be for a small change in the minimum wage.
α1 is the expected change in labor input due to a change in wage (wi). Check to see if the change in minimum wage prediction equals the actual change.
H0: The change in labor demand for those workers bounded by the minimum wage is due wholly to the change in the minimum wage.
H1: Not H0.
Test to see whether the prediction is significantly different than the truth.
If firm input prices are no longer taken as given – This imposition might allow different behavior, possibly altering predictions. Then we have a monopsonist model: an increase in the minimum wage causes an increase in employment.
When firms hire more workers, they raise the wage. The cost for one additional worker equals the cost of one new worker plus the higher wage you must pay to all existing workers.
We go from a wage of w1 to because of a minimum wage law. Then employment rises from L1 to L2.
This might alter predictions; however this explanation is shaky because few situations exist where a firm is the only buyer of labor. Many firms, like those in the restaurant business, are in competitive markets where there are many of them (firms). So, it is hard, for example, to think of a firm in the restaurant business being a monopsonist.
If the minimum wage applies to only certain firms, then we can estimate with control groups like Card and Krueger did for New Jersey and Pennsylvania. NJ experienced a change in the minimum wage while PA did not. Find the differences in differences:
Lit = α0 + α1MWi + δTXit + μi + εit
Ljt = α0 + δTXjt + μj + εjt
ΔLi = α1(ΔMW) + δT(ΔXi) + Δεi
ΔLj = δT(ΔXj) + Δεj.
ΔLi – ΔLj = α1(ΔMW) + δT(ΔXi- ΔXj) + Δεi – Δεj.
δT(ΔXi- ΔXj) goes to zero due to control.
I think relaxing assumption ii would probably not change predictions about the effect of the minimum wage. However, we still might find changes in the minimum wage having
W
L1 L2 Labor
MLC
SL
w1
DL
little effect on employment. Why? Reasons that were also criticisms of Card and Krueger’s work.
Firms might anticipate the change in the minimum wage before the change actually occurs. So, you must be sure your survey occurs at the right time and not too late. Your control and other groups may be experiencing differences (and additional) shocks to labor demand. We don’t see the “to-be” firms who choose not to come into existence because of the minimum wage change. This is a change in employment we don’t detect. There would also simply be measurement error. For example, there may be terms in the survey not defined the same by all who were surveyed – measurement error due to the survey – bad survey design – which generates inaccurate data. Experiment might not really be a natural experiment.
16a. For the labor supply model characterized by the concave utility function U(X, H, A, ε) and the linear budget constraint pX = wH + Y, where U = utility, X = consumption, H = hours worked, A = observed exogenous taste variables, ε = unobserved tastes, p = price of X, w = wage rate, and Y = nonwage income, prove the following:
a) The labor supply function is homogeneous of degree zero in w, p, and Y.
b) The substitution effect of w on H is positive.
c) Symmetry of the cross-substitution effects of w on X and p on H.
d) Engel aggregation.
e) Roy’s identity and dV/dw > 0, where V is the indirect utility function for an interior solution.
f) dV/dY > 0, dV*/dY > 0, dV/dY<dV*/dY, where V* is the indirect utility function conditional on H=0 (i.e., an increase in nonwage income reduces the likelihood of working).
g) The probability that H>0 for a randomly selected individual is increasing in w and decreasing in Y.
The labor supply function is derived from the first order condition: -M=w/p. For 0<α<1, the first order condition is –M= αw/αp, which implies no change in the labor supply function.
Totally differentiate with first order conditions:
UXX UXH -p dX λdp UXH UHH W dH = - -λdw -p w 0 Dλ -Xdp + Hdw + dy
Let D = the bordered Hessian matrix and Dii be the iith cofactor. By the second order conditions, |D|>0, |Dii|<0.
dH/dw (holding utility constant) = -λ|D22|/|D| > 0 by the second order conditions.
dH/dp (holding utility constant) = λ|D21|/|D| = λ|D12|/|D| = dX/dw (holding utility constant by symmetry of the bordered Hessian.
pX – wH = Y, which implies p(dX/dY) – w(dH/dY) = dY/dY
= (pXY/XY)(dX/dY) – (wHY)/(HY)(dH/dY) = 1
= (pX/Y)ηXY – (wH/Y) ηHY = 1
= αXηXY - αHηHY = 1
where αX + αY = 1, αX = (pX/Y), αH = (wH/Y), ηXY = (dX/dY)(Y/X), and ηHY = (dH/dY)(Y/H).
The indirect utility function is V=V(w, p, Y, A, ε)
= U(H(w, p, Y, A, ε), X(w, p, Y, A, ε), A, ε).
dV/dw = (dU/dX)(dX/dw) + (dU/dH)(dH/dw)
but pX –wH – Y = 0, which implies p(dX/dw) – w(dH/dw) – H = 0,
which implies H = p(dX/dw) – w(dH/dw).
From the first order conditions, dU/dX = λp and dU/dH = -λw,
which implies (dU/dX)(dX/dw) + (dU/dH)(dH/dw) = λp(dX/dw) – λw(dH/dw)
= λ(p(dX/dw) – w(dH/dw) = λH = dV/dw > 0.
Similarly,
dV/dY = (dU/dX)(dX/dY) + (dU/dH)(dH/dY)
= λ(p(dX/dY) – w(dH/dY) = λ.
This implies (dV/dw)/(dV/dY) = λH/λ = H.
From the above part, dV/dY = λ = (dU/dX)(1/p) from the first order conditions evaluated at I/p = (wH* + Y)/p, where H* is the optimal H.
V* = U(X, 0, A, ε) = U(U/p, 0, A, ε) = V*(Y, p, A, ε).
dV*/dY = (dU/dX)(1/p) evaluated at I/p = Y/p.
Sice wH* + Y > Y, dU/dX (given wH*+Y) < dU/dX (given Y).
This implies dV*/dY > dV/dY.
16b. Consider the following specific form of the direct utility function, known as the Stone-Geary or Linear Expenditure System:
U(X, H, A, ε) = αln(X- ) + (1-α)ln(T-H)
where α, T, are positive constants, X is consumption, and H is hours of marketplace work.
a) Find the labor supply function.b) Find the substitution effect of w on H. Is it positive)?c) Calculate the indirect utility function.d) Find the expenditure function for labor (i.e. the function wH = f(w, p, Y, A, ε)e) How should A and ε be incorporated?
L = αln(X- ) + (1-α)ln(T-H) + λ(wH + Y – pX)
The first order conditions are:
∂L/∂X: α/(X- ) – λp = 0
∂L/∂H: -(1- α)/(T-H) + λw = 0
and
∂L/∂: wH + Y – pX = 0.
Dividing the first first order condition by the second yields
α(T-H) / (1-α)(X- ) = p/w
α(T-H) = (p/w)(1-α)(X- )
-αH = (p/w)(1-α)(X- ) - αT
H = T - (p/αw)(1-α)(X- )
Now, plug this into the third first order condition:
w[T - (p/αw)(1-α)(X- )] + Y – pX = 0
wT - (p/α)(1-α)(X- ) + Y – pX = 0
wT - (p/α)(1-α)X + (p/α)(1-α) + Y – pX = 0
wT + (p/α)(1-α) + Y = (p/α)(1-α)X + pX
wT + (p/α)(1-α) + Y = p[(1/α)(1-α)X + X]
wT + (p/α)(1-α) + Y = p[(1/α)(1-α)X + (α/α)X]
wT + (p/α)(1-α) + Y = (p/α)X
(α/p) [wT + (p/α)(1-α) + Y] = X
(α/p)[wT + Y] + (1-α) = X.
So, X* = α(Tw + Y)/p + (1- α) .
Recall that H = T - (p/αw)(1-α)(X- ).
So, H = T - (p/αw)(1-α)([α(Tw + Y)/p + (1- α) ]- )
H = T - (p/αw)(1-α)((α(Tw + Y)/p) + (1- α) - )
H = T - (p/αw)(1-α)((α(Tw + Y)/p) - α )
H = T - (p/αw)(1-α)(α(Tw + Y)/p) + (p/αw)(1-α)α
H = T - (1/w)(1-α)(Tw + Y) + (p/w)(1-α)
H = T - (1/w)(1-α)Tw - (1/w)(1-α)Y + (p/w)(1-α)
H = T - (1-α)T - (1/w)(1-α)Y + (p/w)(1-α)
H = αT - (1/w)(1-α)Y + (p/w)(1-α) .
So, H* = αT + (1- α)(p -Y)/w
∂H/∂w = -(1- α)(p -Y)/w2
Recall that H* + αT = (1- α)(p -Y)/w.
So, ∂H/∂w = (H* - αT)/w
∂H/∂Y = -(1- α)/w
∂H/∂w (holding utility constant) = ∂H/∂w – H*(∂H/∂Y)
= -[(H* – αT)/w] + [H*(1 – α)/w]
= α(T-H*)/w > 0.
V(w, Y, p) = αln(X*- ) + (1- α)ln(T-H*)
V(w, Y, p) = αln{[(α(Tw + Y)/p) + (1- α) ]- } + (1- α)ln{T-[αT + ((1- α)(p -Y)/w)]}
V(w, Y, p) = αln{(α(Tw + Y)/p) - α } + (1- α)ln{(1-α)T – ((1- α)(p -Y)/w)}
V(w, Y, p) = αln{(α/p)(Tw + Y - p )} + (1- α)ln{((1-α)/w)(wT - p + Y}
Let q = Tw + Y – p .
Then V(w, Y, p) = αln(αq/p) + (1- α)ln((1- α)q/w)
From above, H* = αT + (1- α)(p -Y)/w
So wH = αTw + (1- α)(p -Y). Notice that it is linear in the variables, while H is not.
A and ε can be put in α and/or .
16c. Consider the following linear labor supply function for an interior solution:
H = αw + βY + γA + ε
where w and Y have been divided by p already.
a) Derive the indirect utility function. [Hint: use the fact that along an indifference curve, dV = 0 to derive an exact differential equation in dw and dy].
b) Derive the substitution effect (is it positive?).c) Interpret α and β.d) Suppose α = -20, β = -0.02, mean H = 2000, and mean w = $10/hour.
Compute (i) the substitution effect at mean H, (ii) the total income elasticity or marginal propensity to earn out of nonwage income at mean H and w, (iii) the substitution elasticity at mean H and w, and (iv) the range of values of H for which the substitution effect is nonnegative.
Derive V(w, Y, A, ε)
Along an indifference curve in w-Y space, (dV/dw)dw + (dV/dY)dY = 0
= dY + [(dV/dw)/(dV/dY)]dw = 0
= dY + Hdw = 0 by Roy’s identity
= dY + (αw + βY + γA + ε)dw = 0
= eβwdY + eβw(αw + βY + γA + ε)dw = 0
= MdY + Ndw = 0
This is an exact differential equation since dM/dw = dN/dY = βeβw.
So, V(w, Y, A, ε) = Ndw + Ψ(Y)
= eβw(αw + βY + γA + ε)dw + Ψ(Y)
= [eβw/p][βY + γA + ε – α/β + αw] + Ψ(Y)
dV/dY = eβw + Ψ’(Y) = M = eβw = Ψ’(Y) = 0
So, Ψ(Y) = K, a constant.
V(w, Y, A, ε) = eβw[Y + (γA + ε)/β + (α/β)w – α/β2] + K
Verify by showing (dV/dw)/(dV/dY) = (αw + βY + γA + ε) = H.
S = dH/dw – h(dH/dY) = α – Hβ.
Not guaranteed to be positive since α could be positive or negative and β<0 if leisure is normal.
α is the uncompensated wage effect, hours.
Β is the income effect.
S = -20 – (-0.02)2000 = 20
β = -0.2
20(10/2000) = 0.1
-20 + 0.02H > 0 if H > 1000.
16d. Consider the standard life cycle labor supply model with intertemporal separability of preferences:
U = StUt, Ut = U(Xt, Ht, At, εt), S = 1/(1+ρ)
K0 + RtwtHt = RtptXt, R = 1/(1+r).
For convenience, you may assume r = ρ = 0. Assume leisure is a normal good.
a) Prove that dHt/dwt (holding λ constant) > 0.b) Prove that dHt/dwt (holding λ constant) > dHt/dwt not holding λ constant.
(Hint: use the assumption that leisure is normal to show that dλ/dwt < 0 and dHt/dλ > 0).
c) Prove that dHt/dwt (holding λ constant) = 0 for t not equal to s.d) Prove that dHt/dwt (not holding λ constant) < 0 for t not equal s.
The first order conditions are
UHt + λwt = 0, for t = 0,…N
UXt – λpt = 0, for t = 0,…N.
With λ constant, these two equations can be solved for dHt/dwt = λD-1 > 0 where D = UXXUHH – UXH
2 > 0 by the second order conditions.
With λ allowed to vary, the first order conditions include N+1 pairs of evaluations as above plus the budget constraint. Totally differentiate these 2(N+1) + 1 equations and solve:
-λdw0 UHH UXH 0 0 . . w0 dH0 λdp0 UXH UXX 0 0 . . -p0 dX0 -λdw1 0 0 UHH UXH . . w1 dH1 λdp1 0 0 UXH UXX . . p1 dX1 = - . . . . . . . . . . . . . . . w0 -p0 w1 -p1 . . 0 dλ Q
Where Q = - Htdwt + ptdXt – dK.
This matrix is block-diagonal, (N+2 X N+2).
dHt/dwt = (1/D*)(-λDtt - HtDt,N+2), where D* is the determinant of the bordered Hessian and D*>0 by the second order conditions; Dtt is the determinant of the tt-th cofactor; Dt,N+2 is the determinant of the t, N+2-th cofactor. Dtt<0 by the second order conditions, and Dt, N+2 > 0 if leisure is normal.
From above, dHt/dλ = -(1/D)[wUXX + pUHX] > 0 if leisure is normal. From the above matrix equation,
dλ/dwt = -(1/D*)[λ Dt, N+2 + H1DN+2, N+2] and DN+2, N+2 > 0, so dλ/dwt < 0.
So, dHt/dwt = dHt/dwt (holding λ constant) + (dHt/dλ)(dλ/dwt) < dHt/dwt (holding λ constant).
From above, dHt/dws (holding λ constant) = 0 follows immediately.
dHt/dws = dHt/dws (holding λ constant) + (dHt/dλ)(dλ/dws) = (dHt/dλ)(dλ/dws) < 0. The first term is positive and the second is negative.
Or,
dHt/dws = -(1/D*)[λDts + HsDs, N+2].
Dts > 0 can be shown, and Ds, N+2 > 0 if leisure is a normal good, so dHt/dws < 0.
17. Two alternative specifications of the labor supply function of married women have commonly been used in the literature: (A) H=H(w, p, Y, A, ε) and (B) H = H(w, p, wH, Y*, A, ε), where Y = nonwage income plus the husband’s earnings, Y* = nonwage income, and wH = the husband’s wage rate.
a) Describe the underlying optimization problems that yield specifications A and B. What are the main differences in assumptions between these two problems?
b) What biases or other problems would result from estimating specification A if model B is the true model? Can model A be explicitly tested against model B? If so, how?
c) Suppose that some husbands are not employed. What are the implications of this for estimation of model A, assuming it is the true model, and model B, assuming it is the true model?
A: H = H(w, p, Y, A, ε) from individual’s model.
Max(H,X): U(X, H, A, ε) + λ(wH + Y – pX)
At an interior optimum, dL/dX = 0 implies UX = λp,
dL/dH = 0 implies UH = -λw,
and dL/dλ = 0 implies wH + Y – pX = 0.
From these first order conditions, we can divide the first two to get
UH/UX = -w/p, which is the marginal rate of substitution.
From the first order conditions, we can find the Marshallian demand functions:
Hi = H(w, p, Y, A, ε) and Xi = X(w, p, Y, A, ε).
The main assumptions are that this takes the husband’s labor supply decisions as exogenous, earnings part of nonwage income. There isn’t a joint labor supply decision for the family and the husband’s income is part of the wife’s nonwage income.
B: H = H(w, p, wH, Y*, A, ε) from the family model.
L: Max(Li,C): U(L1, … Li, C, A, ε) subject to nonwage income, or,
L: Max(X,H,HH): U(X, H, HH, A, ε) + λ(wH + wHHH + Y* - pX)
The first order conditions are:
UH = λw
UHH = λwH
UX = - λp
Uλ = wH + wHHH + Y – pX.
From the first order conditions, assuming an interior optimum, we can derive Hi = H((w, p, wH, Y*, A, ε).
Here, we assume labor supply decisions are made jointly. The husband’s hours decision affects the wife’s hours decision. The wife doesn’t take the husband’s labor supply as given.
If we estimate A when B is true, then we have a standard problem of omission of relevant regressors. The results will be biased and inconsistent. Estimates will be biased downward.
In A, a change in wH has only an income effect.
In B, a change in wH has both income and substitution effects.
We can test:
H0: no substitution effectH1: Not H0.
A failure to reject H0 leads to a rejection of family model B.
If H0 is rejected, then family labor supply model is the better model.
Some husbands are not employed.
With model A, if the husband isn’t employed, the nonwage income would be less and the women’s labor supply function would be affected through Y. No estimation problems.
With model B, if the husband were unemployed, then Y*=Y and wH=0. Therefore, this model would be the same as A. Since no corner solutions are allowed in family model B, the system cannot be estimated.
18. Answer the following questions about dynamic models of labor demand:
a) What are the implications of strictly convex versus linear adjustment costs for the path of adjustment to equilibrium employment following an exogenous chock to a firm? Use diagrams or equations to illustrate your answer.
b) What important issues must be dealt with in order to specify an empirical model of labor demand dynamics that is consistent with the theory of adjustment costs? Describe how alternative assumptions about these issues affect the interpretation of empirical results, using examples from Nickell or Hamermesh to illustrate your points.
Convex: total adjustment is asymptotic.
Linear: if shock is sufficiently large and not too costly, instantaneous response and adjustment.
Convex:
(t) = X(t) – δN(t)
(t) = (1/C’’){w – RN + (1+δ)C’(X(t))
Linear: C’ = β if X>0 and α if X<0.
If pRN(N*) > w + (r+δ) β then jump immediately.
Data: aggregate versus disaggregate; time series.
Expectations: point expectations (risk neutrality) versus rational;
Nature of adjustment costs: convex versus linear.
See also lectures 23-26.
19. Consider an individual whose goal at each age is to maximize the expected present discounted value of remaining lifetime utility. When she reaches age T, the individual must retire. She can retire early (before age T), but once retired cannot reenter the labor force. When retired, she receives an annual retirement benefit of B dollars per year, independent of the age at which she retires.
All jobs include a health insurance plan with the premiums paid by the employer. Beginning at age T, the government provides health insurance. If an individual decides to take early retirement (i.e. retire at age T-1 or earlier), then she will not have any health insurance until she reaches age T.
An individual still in the labor force at age t (t < T) receives an annual wage offer for that year drawn from the probability density function f(w) with cumulative distribution function F(w). The wage offer distribution is constant over time, and each year she receives an independent draw from this distribution. After receiving the wage offer for a given year, the individual chooses between continued work and retirement.
The probability of becoming ill at a given age, p, does not depend upon the age of the individual and is independent of previous illnesses. Individuals must make their work/retirement decision before they know whether they will be ill during the current year.
An illness imposes a monetary cot on an individual. If she is in the labor force, this cost is CW. If she is retired, then the cost is CR with CW>CR. Health insurance (both employer and government) pays CR during any year when an individual is ill, provided she has health insurance. Illnesses last for only one year.
Assume workers cannot borrow or lend, do not discount future utility, and that death occurs with certainty at age T+M. Suppose that per period utility is an increasing, concave function of net annual income Y.
a) Write expressions for net annual income in each of the six states of the world: (1) working and not ill; (2) working and ill; (3) retired, younger than age T and not ill; (4) retired, younger than age T and ill; (5) retired, age T or older and not ill; (6) retired, age T or older, and ill.
b) Consider an individual aged T-1 who was working at age T-2. Suppose that the wage draw for this time period is w. Set up her optimization problem and state the conditions under which she will choose to retire at age T-1.
c) How does the retirement decision at age T-1 depend upon the probability of becoming ill? How does it depend upon the size of the retirement benefit?
d) Now, consider a women aged T-2 who had worked at age T-3. She receives a wage draw w at age T-2. Set up her lifetime optimization problem from age T-2 onwards and state conditions under which she would choose to retire rather than work at that age.
e) Describe how the retirement decision at age T-2 differs from that at age T-1. In particular, would the “reservation wage” (that wage that makes her indifferent between retiring and working) be higher at age T-1 or at age T-2?
f) Suppose that the government institutes an optimal health insurance program for people who retire before age T. Early retirees can pay the amount K as an insurance premium and receive CR if they become ill during the year. You should assume that this insurance premium is actuarially fair. How does the ability to purchase health insurance alter the retirement decision at age T-1? How does it later the decision at age T-2? Does the ability to purchase insurance have a larger impact on the reservation wage at age T-1 or at age T-2?
(1) Y = w
(2) Y = w – CW + CR
(3) Y = B
(4) Y = B – CR
(5) Y = B
(6) Y = B – CR + CR = B
Let the value from choosing to work in period T-1 be VW, T-1 = pU(wT-1 – CW + CR) + (1 –
p)U(wT-1) + VT where VT = U(B).
Let the value from choosing to retire in period T-1 be VR, T-1 = pU(B – CR) + (1-p)U(B) + VT.
The individual will maximize utility by selecting V such that VT-1 = Max[VW, T-1, VR, T-1].
That is, the individual will retire if VR, T-1 > VW, T-1 or VR, T-1 – VW, T-1 > 0,
or {pU(B – CR) + (1-p)U(B) + VT} – {pU(wT-1 – CW + CR) + (1 – p)U(wT-1) + VT},
which equals {pU(B – CR) + (1-p)U(B)} – {pU(wT-1 – CW + CR) + (1 – p)U(wT-1)}.
∂[VR,T-1–VW¸T-1]/∂p = U(B – CR) – U(B) - U(wT-1 – CW + CR) + U(wT-1), which could be greater or less than 0.
∂[VR,T-1–VW,T-1]/∂B = pU’(B – CR) + (1-p)U’(B) > 0. The probability of retiring is increasing in B.
Let the value from choosing to work in period T-2 be VW, T-2 = pU(wT-2 – CW + CR) + (1 – p)U(wT-2) + EMax[VW, T-1, VR, T-1].
Let the value from choosing to retire in period T-2 be VR, T-2 = pU(B – CR) + (1-p)U(B) + VR, T-1.
The individual will maximize utility by selecting V in period T-2 such that VT-2 = Max[VW, T-2, VR, T-2].
That is, the individual will retire if VR, T-2 > VW, T-2 or VR, T-2 – VW, T-2 > 0,
or
{pU(B – CR) + (1-p)U(B) + VR, T-1} – {pU(wT-2 – CW + CR) + (1 – p)U(wT-2) + EMax[VW, T-
1, VR, T-1]}.
Let wT-1* be the period T-1 reservation wage and let wT-2* be the period T-2 reservation wage. The difference in the period T-1 and T-2 retirement decisions is:
{pU(B – CR) + (1-p)U(B) + VR, T-1} – {pU(wT-2 – CW + CR) + (1 – p)U(wT-2) + EMax[VW, T-
1, VR, T-1]} – [{pU(B – CR) + (1-p)U(B)} – {pU(wT-1 – CW + CR) + (1 – p)U(wT-1)}].
The probability of working in T-2 is greater than the probability of working in T-1. To see this, subtract the probability of retiring in period T-1 from the probability of retiring in period T-2. It should be less likely that the individual will retire in period T-2 because the difference in these retirement decisions is VR, T-1 - EMax[VW, T-1, VR, T-1], which is less than 0.
For the insurance premium to be actuarially fair, K must equal pCR. Since the individual is risk averse (which is true since utility is increasing in income at a decreasing rate), the individual will purchase insurance. However, this will not change the value from working in period T-1 because the insurance can only be purchased if the individual retires early.
The value from choosing to work in period T-1 will remain VW, T-1 = pU(wT-1 – CW + CR) + (1 – p)U(wT-1) + VT.
Let the value from choosing to retire in period T-1 now be VR,T-1I = U(B – K) + VT.
As before, the individual will maximize utility by selecting V such that VT-1 = Max[VW, T-
1, VR, T-1I].
That is, the individual now will retire if VR, T-1I > VW, T-1 or VR, T-1
I – VW, T-1 > 0,
or {U(B – K) + VT} – {pU(wT-1 – CW + CR) + (1 – p)U(wT-1) + VT} > 0,
or
{U(B – K) + VT} – {pU(wT-1 – CW + CR) + (1 – p)U(wT-1)} > 0.
Essentially, the insurance increases the value of retiring, raising the reservation wage. To see this, compare the period T-1 retirement decisions with and without the insurance. Subtracting the probability of retiring with insurance from the probability of retiring without insurance is:
{pU(B – CR) + (1-p)U(B)} – {U(B – K)}. Since the second term is larger, the individual is more likely to retire with the insurance and the corresponding reservation wage will be higher.
Similarly, for period T-2, let the value from choosing to work in period T-2 now be VW, T-
2I = pU(wT-2 – CW + CR) + (1 – p)U(wT-2) + EMax[VW, T-1, VR, T-1
I]. This is different than before because VR, T-1 has changed.
Let the value from choosing to retire in period T-2 now be VR, T-2I = U(B – K) + VR, T-1
I.
The individual will maximize utility by selecting V in period T-2 such that VT-2 = Max[VW, T-2
I, VR, T-2I].
That is, the individual will retire if VR, T-2I > VW, T-2
I or VR, T-2I – VW, T-2
I > 0,
or
{U(B – K) + VR, T-1I} – {pU(wT-2 – CW + CR) + (1 – p)U(wT-2) + EMax[VW, T-1, VR, T-1
I]}.
As before, the insurance increases the value of retiring, raising the reservation wage. To see this, subtract the period T-2 probability of retirement without insurance from the period T-2 probability of retirement with the insurance.
The difference is:
{pU(B – CR) + (1-p)U(B) + VR, T-1} – {pU(wT-2 – CW + CR) + (1 – p)U(wT-2) + EMax[VW, T-
1, VR, T-1]} - [{U(B – K) + VR, T-1I} – {pU(wT-2 – CW + CR) + (1 – p)U(wT-2) + EMax[VW, T-1,
VR, T-1I]}],
or
{pU(B – CR) + (1-p)U(B) + VR, T-1} – {EMax[VW, T-1, VR, T-1]} - [{U(B – K) + VR, T-1I} –
{EMax[VW, T-1, VR, T-1I]}]
First, note that [pU(B – CR) + (1-p)U(B)] is smaller than [U(B – K)].
Second, EMax[VW, T-1, VR, T-1] is smaller than EMax[VW, T-1, VR, T-1I] because though the
VW, T-1 terms are the same, VR, T-1 is smaller than VR, T-1I.
Reason 2 also indicates that VR, T-1 is smaller than VR, T-1I.
Again, the reservation wage is higher in the insurance version.
20. A major component of proposed welfare reform legislation is to limit how long a single mother can receive welfare benefits. Policymakers would like to know the impact of this proposed reform on human capital acquisition and labor supply behavior. Consider the following very simple life-cycle model as a vehicle for addressing these issues. The population of interest is single mothers who have completed their schooling. Treat schooling and fertility as exogenous.
Assume a two-period planning horizon, and suppose that an interior solution is optimal in both periods. To further simplify the analysis, assume that the wage rate in period one is exogenous, there is no depreciation of human capital, there are no financial markets (i.e. no savings), and no time discounting. Assume that there are no taxes and that welfare benefits are not reduced as earnings increase. Also assume that welfare benefits are the only source of nonwage income.
Define as follows:
wt = the wage rate in period t, t = 1, 2.Bt = the welfare benefit in period t.Ht = hours of work in period t.Ct = consumption in period t.Zt = a random variable.Ut = utility in period t = Zt[1-Ht]δ + [Ct]β.
Let the wage rate in period 2 be determined by w2 = w1(1+H1)θ. Z1 is observed by the mother prior to the beginning of period 1, but she does not observe Z2 until after her period 1 decisions are made.
a) What restriction does economic theory imply for δ, β, and θ?b) Give an interpretation of the equation for the period 2 wage.c) Specify the mother’s optimization problem, derive the first-order conditions
for an optimum, and interpret them.d) Derive the effects of B1, B2, and w1 on H1 and H2. You may assume that the
second-order conditions for an optimum are satisfied.e) Explain how your results can be used to help understand the impact on labor
supply and wages of limiting the length of time welfare benefits can be received. Would you expect the effects you find to be valid in less restrictive models? Why or why not?
f) How would you propose to estimate the parameters needed to quantify these effects? Which restrictions of the simple model would you relax? What additional assumptions would you make? What sort of data would you use? What kind of estimation method would be appropriate?
Restrictions: δ, β, and θ must all be between 0 and 1. θ is a form of human capital accumulation. These are all positive and yet diminishing.
Period 2 wage is w2 = w1(1+H1)θ. Human capital accumulation occurs via working: that is, one gains on-the-job experience, which has a positive effect on marginal productivity, so the wage rises.
V2 = Max Z2[1-H2]δ + [C2]β such that w2H2 + B2 = C2 where w2 = w1(1+H1)θ.
V2 = Max(H2): Z2[1-H2]δ + [[w1(1+H1)θ]H2 + B2]β
V1 = Max(H1,C1,H2,C2): Z1[1-H1]δ + [C1]β + EV2 such that w1H1 + B1 = C1.
Let 2 = E[Z2]
V = Max(H1,H2): Z1[1-H1]δ + [w1H1 + B1]β + 2 [1-H2]δ + [[w1(1+H1)θ]H2 + B2]β.
The first order conditions are:
dV/dH1:
-δZ1[1-H1]δ-1 + βw1[w1H1 + B1]β-1 + βw1θ(1+H1)θ-1H2[w1(1+H1)θH2 + B2] β-1=0
The first term is the marginal disutility of work. The second is the marginal utility of consumptions, and the third is the marginal utility of future consumption.
dV/dH2: -δ 2[1-H2]δ-1 + βw1(1+H1)θ[w1(1+H1)θH2 + B2] β-1 = 0.
The first term is the marginal disutility of work and the second is the marginal utility of consumption.
The second order conditions are
H11 H12 H21 H22
where [H] is negative-definite. So, H11 < 0, H22 < 0, and |H| > 0.
So, H12 = βw1θ(1+H1)θ-1[w1(1+H1)θH2 + B2] β-1 + βw1(1+H1)θ(β-1)w1θ(1+H1)θ-
1H2[w1(1+H1)θH2 + B2] β-2
which equals
= βw1θ(1+H1)θ-1[w1(1+H1)θH2 + B2] β-2[w1(1+H1)θH2 + B2 + (1+H1)θ(β-1)w1H2]
= βw1θ(1+H1)θ-1[w1(1+H1)θH2 + B2] β-2[B2 + (1+H1)θβw1H2] > 0.
So, H12 = H21 > 0.
Comparative statics:
∂B1:
H11 H12 ∂H1/∂B1 = - β(β-1)w1[w1H1+B1] β-2 H21 H22 ∂H2/∂B1 0
=
H11 H12 ∂H1/∂B1 = - {-} H21 H22 ∂H2/∂B1 0
=
H11 H12 ∂H1/∂B1 = {+} H21 H22 ∂H2/∂B1 0
So, ∂H1/∂B1 =
+ +0 -
- ++ -
which is {-}/{+} < 0.
Similarly, ∂H2/∂B1 =
- ++ 0
- ++ -
which is {-}/{+} < 0.
∂V/∂H1∂B2 = βw1θ(1+H1)θ-1H2(β-1)[w1(1+H1)θH2 + B2] β-2 < 0
and
∂V/∂H1∂B2 = βw1(1+H1)θ(β-1)[w1(1+H1)θH2 + B2] β-2 < 0.
∂B2:
H11 H12 ∂H1/∂B2 = - {-} H21 H22 ∂H2/∂B2 {-}
=
H11 H12 ∂H1/∂B2 = {+} H21 H22 ∂H2/∂B2 {+}
So, ∂H1/∂B2 =
+ ++ -
- ++ -
which is {-}/{+} < 0.
Similarly, ∂H2/∂B2 =
- ++ +
- ++ -
which is {-}/{+} < 0.
∂w1:
H11 H12 ∂H1/∂w1 = - (equation 1) H21 H22 ∂H2/∂w1 (equation 2)
∂V/∂H1∂w1 = β[w1H1 + B1]β-1 + β(β-1)w1H1[w1H1 + B1]β-2
+ βθ(1+H1)θ-1H2[w1(1+H1)θH2 + B2] β-1
+ βw1θ(1+H1)θ-1H2(β-1)(1+H1)θH2[w1(1+H1)θH2 + B2] β-2
which equals
= β[w1H1 + B1]β-2[βw1H1 + βB1 + β(β-1)w1H1]
+ βθ(1+H1)θ-1H2[w1(1+H1)θH2 + B2] β-2[w1(1+H1)θH2 + B2 + w1(β-1)(1+H1)θH2]
which equals
= β2[w1H1 + B1]β-2[βw1H + B1] + βθ(1+H1)θ-1H2[w1(1+H1)θH2 + B2] β-2[B2 + w1β(1+H1)θH2]
∂V/∂H1∂w1 > 0.
Similarly,
∂V/∂H2∂w1 =
β(1+H1)θ[w1(1+H1)θH2 + B2] β-1 + βw1(1+H1)θ(β-1)[w1(1+H1)θH2 + B2] β-2(1+H1)θH2
= [β(1+H1)θ][w1(1+H1)θH2 + B2] β-2 [w1(1+H1)θH2 + B2] + w1(β-1)(1+H1)θH2]
= [β(1+H1)θ][w1(1+H1)θH2 + B2] β-2 [βw1(1+H1)θH2 + B2]
∂V/∂H2∂w1 > 0.
So, H11 H12 ∂H1/∂w1 = - {+} H21 H22 ∂H2/∂w1 {+}
=
H11 H12 ∂H1/∂w1 = {-} H21 H22 ∂H2/∂w1 {-}
Sign: ∂H1/∂w1 =
- +- -
- ++ -
which is {+}/{+} > 0.
Similarly, sign: ∂H2/∂w1 =
- -+ -
- ++ -
which is {+}/{+} > 0.
In summary, ∂H1/∂B1 < 0, ∂H2/∂B1 < 0, ∂H1/∂B2 < 0, ∂H2/∂B2 < 0, ∂H1/∂w1 > 0, and ∂H2/∂w1 < 0.
The comparative statics show that welfare benefits in both periods lower labor supply. This means lower human capital accumulation. Ultimately, welfare benefits lower human capital in two ways. The first way is a direct pure income effect each period. That is, assuming leisure is a normal good, you work less because welfare benefits raise your real income. The second way is an indirect effect. That is, welfare benefits, by indirectly reducing work, reduce human capital accumulation. This results in a lower wage in the future with offsetting substitution and income effects with the income effect dominating. In this setup, time limiting welfare benefits are sensible.
In a less restrictive model, with U=U(1-Ht,Ct) where utility is not additively separable, results might be different because additive separability gave H12 and H21 its sign. Allowing a perfect capital market would also change things.
21. Consider the following extension of the standard static one-person labor supply model:
U = U(X,H;A,ε)
X = wH + Y – F if H > 0
X = Y if H = 0
F = Zγ + v
where H = hours worked, X = consumption, w = hourly wage rate, Y = nonwage income, F = (unobserved) fixed costs of work, A and Z are vectors of observed variables (assume Z contains at least one variable excluded from A), and ε and v are disturbances known to the individual but unobserved by the researcher.
a) Give a complete characterization of labor supply behavior in this model: derive the hours of work function, the conditional under which it holds, and the function characterizing the labor force participation decision. Be as specific as possible.
b) Describe an estimation strategy that will yield consistent estimates of the parameters of interest. Specify the equations to be estimated, interpret the parameters, and discuss the estimation method and identification.
c) Describe how your answers to parts a and b would change if F=0 (no fixed costs).
Max(X,H): U(X,H;A,ε) + λ(wH + Y – F – pX)
Assuming an interior optimum,
-UH = λw from L/ H = 0
-UX = λp from L/ X = 0
and
wH + Y – F – pX = 0 from L/ λ = 0.
-UH/UX = w/p = -M(X,H;A,ε), which is the marginal rate of substitution.
Hi = H(p,w,Y, A, ε, F)
and
Xi = X(p,w,Y, A, ε, F)
are Marshallian demand function.
Hi = H(p,w,Y, A, ε, F) if H > Hr where Hr is the reservation wage.
Hi = 0 if H < Hr.
U = U(X,H;A,ε) = U(X(p,w,Y, A, ε, F), H(p,w,Y, A, ε, F), A, ε, F) is the direct utility function and V = V(p,w,Y, A, ε, F) is the indirect utility function.
V/ w > 0, V/ Y > 0, V/ p < 0, and V/ F < 0.
HR is reservation hours. If H > Hr, the person will work. If H < Hr, then H = 0.
Fixed costs introduced a discontinuity into the size of the labor force, the size of which depends on individual preferences, Y, and the money cost of work.
Wage
w
Hr
LS
Fixed costs can make some leave the labor force who were willing to work H where H < Hr with no fixed costs.
Estimation:
Let Hi* = β1lnwi + β2NI + β3X + μ1 be the hours of work function and
Hr = γ1NI + γ2F + μ2 be the reservation hours equation
with wage equation lnwi = αYi + μ3
where Xi and Yi are vectors of individual characteristics.
An individual is observed to be working only if H* > Hr. So, the probability that an individual in the sample is observed to be working is given by
Pr(H* > Hr) = Pr[β1lnwi + β2NI + β3X + μ1 > γ1NI + γ2F + μ2].
Now, substitute for lnwi from the wage equation:
= Pr[β1ln αYi + β1μ3 + β2NI + β3X + μ1 > γ1NI + γ2F + μ2].
Rearranging, we get
= Pr[β1μ2 + μ1 – μ3 > γ1NI + γ2F - β1ln αYi - β2NI - β3X].
C
Y/p
Hr T
Y/p – F/p
= Pr[μ > J].
The probability of non-participation is Pr[J > μ].
We can now form a likelihood function, where you work if H* > Hr and you do not work if H* ≤ Hr.
Pr[Hi* = 0] = Pr[J > μ] = Pr[J/σ > μ/σ] = 1-Φ(J/σ).
Pr[Hi* > Hr] = Φ(J/σ).
λ = σ(φ(J/σ)/Φ(J/σ)).
We can now use this to get consistent estimates.
Hi* = β1lnwi + β2NI + β3X + β4λ + ε.
This model is identified because the vector of unobserved characteristics from the fixed cost equation has a least one variable excluded from the hours of labor characteristics vector.
We must note that our estimates could change radically: a small change in the wage could lead to large discontinuous changes in labor supply. Small changes in the wage could shift our labor supply radically. Further, we might find situations where there are multiple optima. So, if we want to find the effect of a change in the wage on labor supply, we really can’t use parameter estimates because the labor supply function is not continuous. Instead, we will have to compare utility at every point. That is, we’ll have to make global comparisons.
V0 = V0(Y, p, A, ε, F) is the indirect utility function for H = 0.
V1 = V1( , , ε, A, p, F) is the indirect utility function for a segment (the maximum
utility attainable on a linear segment. The individual will choose H = if and only if Vi
> Vj for all i not equal to j and where is the optimum along segment i.
If there were no fixed costs, then we would have a continuous labor supply function. Parameter estimates would accurately predict the effects of changes in variables.
In part a, the labor force participation decision would be based on the reservation wage. That is, Hi = H(w, p, Y, A, ε) if w > w* where w* is the reservation wage and Hi = 0 if w < w*.
The indirect utility function for Hi = 0 is V*(Y, p, A, ε). If Hi = H(w, p, Y, A, ε), then V(w, p, Y, A, ε) > V*.
The probability of working is Pr[H > 0] = Pr[w > w*] = Pr[w > β0 + β1Y + β2A + ε] where w* = β0 + β1Y + β2A + ε.
H = α0 + α1w + α2Y + α3A + E(ε | w > β0 + β1Y + β2A + ε)
H = α0 + α1w + α2Y + α3A + E(ε | ε > -q) where Pr[H > 0] = Pr[ε/σ > -q/σ] = λi = σ(φ(
/σ)/Φ( /σ)).
= H = + α1w + α2Y + α3A + α4λi + ε.
We now have consistent estimates where an omitted variable used to be. We are now estimating a model with a linear budget constraint. Parameter estimates can predict and reflect the change in labor supply of a change in one of the exogenous variables.