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Expenditure Function, Slutsky Equation, Consumption and Leisure, Instrument is a Dummy, Interpret the Parameter, Preferred Strategy, Measured Wages, Hourly Wages, Intertemporal Optimization, Bellman Equation are some points from questions of this exam paper.
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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
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).
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.
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.
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?