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University of California Department of Economics Field Exam August, 2011 Labor Economics
Answer question I and any two of questions II, III, or IV. Note there is some choice in question III. You should plan to spend about 1 hour per question. Be sure to use graphs and/or equations whenever they would be helpful in clarifying your reasoning.
WRITE YOUR ANSWERS FOR EACH QUESTION IN A SEPARATE BOOK
Question I (required of all candidates). Answer all parts of this question
- Making use of the expenditure function, derive the Slutsky equation for hours of work in a two-good model with consumption and leisure. Carefully define your notation and state the assumptions you are using.
- Ashenfelter et al. estimate a model for the miles driven by cab-driver i in month t ( mit ) of the form: log mit = αx (^) it + βθit + εit
where θit is defined as revenues per mile received by the cab driver in month t (i.e., θit=Rit /mit , where R are measured revenues in period t), and the set of controls x includes a fixed effect for each cab. They estimate this model from a sample period that includes data before and after a fare increase, using OLS and IV. In the IV procedure the instrument is a dummy for months after the fare increase.
a) How would you interpret the parameter β? Would you expect a positive or negative estimate for β?
b) Explain why IV is a preferred strategy in this analysis. How would you expect the IV and OLS estimates to compare? (Hint: think about division bias).
- An economist has access to data on measured wages ( w ) and hours of work ( h ) of a sample of individuals who are observed in multiple periods. She fits a model of the form
Δlog hit = α + η Δlog wit + εit
where Δlog hit is the change in log hours from period t-1 to t, and Δlog wit is the change in log hourly wages from period t-1 to t.
a) Explain how an equation of this form can be derived from the first order conditions of an intertemporal optimization problem of the individual (use 3-4 equations and at most 1 paragraph, starting from a statement of the Bellman equation). What is the interpretation of the parameter η?
b) Discuss the different factors that are implicitly contained in the “error term” εit. Discuss at least 2 different approaches that have been used in the literature to obtain consistent estimates of η , while recognizing that some component of the observed wage change is unanticipated in period t-1.
QUESTIONS II-IV: ANSWER ALL PARTS OF ANY TWO OF THESE QUESTIONS
Question II. Answer all parts.
In the “double auction" game of labor markets studied by Hall and Lazear (1984), an employer knows the marginal product of a worker, say V , which is drawn from a uniform distribution on [0, 1], and a worker knows her reservation wage, say R, also drawn from a uniform distribution on [0, 1]. All the above is common knowledge. The game then proceeds as follows: The firm names a wage it is willing to pay, wF , and, simultaneously, the worker names a wage she is willing to accept, w (^) W. If wW ≤ wF trade takes place at w = 1/2(wF + wW). Otherwise there is no trade, and each party gets payoff of 0.
(a) Show that there exists an equilibrium with linear strategies, i.e., in which the firm posts:
(1) wF (V ) = a 1 + a 2 V ,
and the worker posts
(2) wW(R) = b 1 + b2 R.
Solve for the values for a 1 , a 2 , b1 , b.
(b) Using your solutions from part (a) find the range of values of V and R over which trade occurs. Explain intuitively why we can have V > R and yet there is no trade.
(c) Show that the linear-strategies equilibrium is not the only Bayesian Nash equilibrium. If you have a little extra time, show that the linear-strategies equilibrium yields higher expected gains to the players than any other Bayesian Nash equilibrium.
Question III, continued.
NOTE: answer #1 or #2 of this question.
- Consider a simple search model applied to an experiment on the labor supply of undergraduates. You take a random sample of 100 undergraduates. For each such undergraduate, you explain that they will receive a series of wage offers that will vary from day to day. Each offer can be accepted or not and acceptance or rejection of a day’s offer does not affect subsequent wage offers. Each such task takes 10 minutes to accomplish and can be done over the internet. Wage offers will be given daily at 6am over the internet by e-mail; the task can be accomplished any time prior to 5:59am the next day. You do not specify a final period to the experiment and describe it as being an open-ended arrangement. Students are indexed by s , and let their wage offers be distributed exponential with parameter λs. You randomly assign the parameters from a list of possible parameters. Let Bst denote the random variable associated with the wage offer to student s on day t. You inform students of the range of offers they are likely to get by giving them 100 examples of draws from their wage distribution. Assume that the students are able to infer the parameters from the list. Assume that undergraduates have utility of zero associated with not working for you on the menial task. Assume linear utility.
a) Write down the Bellman equation for this problem.
b) Give a formula for the expected value of the value function associated with a random draw Bst. Your formula may depend on the reservation wage.
c) Give a formula for the reservation wage. Your formula may depend on the expected value of the value function associated with a random wage offer.
d) Derive the log-likelihood function for the set of binary labor supply decisions recorded by the experiment, as a function of the discount factor.
e) Would the model be identified if you only gave one wage offer to the undergraduates?
Question IV. Answer all parts of this question. A popular program evaluation strategy is to match program participants to controls with similar covariates. Dehijia and Wahba (1999) evaluated the National Supported Work program using observational controls via propensity score methods. The key covariate in their analysis of the earnings e§ects of this program is lagged (pre- program) earnings.
Suppose in their sample, individual iís earnings at time t are generated according to the model:
Yit = (^) i + (^) t + Dit + "it (1)
where Dit is an indicator for program participation at time t, (^) i is a time invariant unobserved e§ect, and "it are unobserved shocks to earnings. For simplicity, assume that only two time periods are available so that t 2 f 1 ; 2 g and that program participation only occurs in period 2 (e.g. Di 1 = 0 8 i).
A (Di§erence in Di§erences) A simple evaluation strategy is to eliminate the unobserved e§ect (^) i via a Örst di§erence transformation as follows:
Yi 2 Yi 1 = 2 1 + (Di 2 Di 1 ) + "i 2 "i 1 = 2 1 + Di 2 + "i 2 "i 1 (2)
Estimation proceeds via OLS applied to (1) treating ( 2 1 ) and as unknown parameters.
a) What restriction(s) on the errors "i 2 "i 1 are necessary to ensure the OLS estimator identiÖes?
B (Matching)
b) Derive an expression for Yi 2 in terms of Yi 1 ; ( 2 1 ) ; Di 2 ; and "i 2 "i 1
c) Provide a set of restrictions on the errors ("i 1 ; "i 2 ; (^) i) such that E [Yi 2 jYi 1 ; Di 2 = 1] E [Yi 2 jYi 1 ; Di 2 = 0] identiÖes.
d) Discuss the plausibility of this assumption vis-a-vis those provided in your answer to a).
e) (BONUS) Smith and Todd (2005) advocate use of a matched di§erences in di§erences estimator. Are there conditions any weaker than those used in your answer to c) under which E [Yi 2 Yi 1 jYi 1 ; Di 2 = 1] E [Yi 2 Yi 1 jYi 1 ; Di 2 = 0] identiÖes?
