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Problem Set 1: Due February 18 at the beginning of class Instructions: Please type your answers, although it.s OK to write in any equations and draw any .gures by hand. Your answers can be submitted in class on paper or by e-mail as a .pdf .le. You are to work on these problems individually.
Note: Problems 1 and 2 are intended to be relatively straightforward. How- ever, you will likely .nd Problem 3 to be more challenging, so you should start to think about it well in advance of the due date.
1. Consider a worker who lives forever, discounting the future at rate r: While unemployed, this worker enjoys .ow utility b and receives job o¤ers, which are iid draws from a known exogenous wage o¤er distribution F(w); at Poisson rate _: Jobs end at exogenous Poisson rate.
(i) Derive the comparative statics of the reservation wage with respect to the job destruction rate, i.e., .nd dR=d_:
(ii) Let u denote the fraction of time that this worker is unemployed. The unemployment rate u can be derived from the steady-state condition,
_(1 F(R))u = _(1 u):
Derive the comparative statics of the unemployment rate with respect to the job destruction rate, i.e., .nd du=d_: You should .nd that the sign of du=d_ is ambiguous.
2. Sketch the comparative statics of an increase in _ in the basic Diamond- Mortensen-Pissarides (DMP) model. That is, how does an increase in _ a¤ect the equilibrium level of labor market tightness, the wage rate, and the unemployment rate? To answer this question, you should draw the graphs that represent the equilibrium conditions of the model and then explain how the increase in _ shifts these curves.
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3. Consider the following extension of the basic DMP model. Unemployed workers meet vacant jobs according to the constant-returns-to-scale contact function, M(u; v); just as in the basic model. However, not all contacts lead to a match. When a worker and .rm meet, a match-special productivity, y; is drawn from an exogenous distribution function, F(y): These match- special productivity draws are independently and identically distributed
across all worker-.rm meetings. If y _ R; the match goes forward, and the worker receives a wage w(y) = _y + (1 _)rU; while the .rm receives yw(y); so long as the match lasts. Otherwise, the worker and .rm continue to search. Matches end at exogenous Poisson rate _; again just as in the basic model.
In this extension, the key endogenous variables are (i) labor market tight- ness, _; (ii) the reservation productivity, R; and (iii) the unemployment rate, u: The corresponding equilibrium conditions are (i) free entry of vacancies,
(ii) zero net surplus at y = R; and (iii) the Beveridge curve. The free-entry condition is more complicated than it is in the basic model because a .rm doesn.t know in advance what value of y will be realized when it meets a worker; that is, ex ante, a .rm with a vacancy can only compute the ex- pected value of meeting a job applicant. The zero net surplus condition is J(R) = 0 or, equivalently, N(R)U = 0: Here J(R) is the value to a .rm of .lling a job when the productivity of the match is y = R; and similarly for N(R): (In general, the value to a .rm of a .lled job is J(y); i.e., depends on the productivity of the match; similarly, the value to the worker is N(y):)
Finally, the Beveridge curve condition needs to account for the fact that not all contacts lead to a match. Write down (and explain) the equations that describe the equilibrium of this extended version of the basic DMP model.
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