Macro theory

Description

Question 1. (70 points) Consider the matching model of the labor market

that we studied in class. We are going to use the following ingredients.

E is employment, U is unemployment, L is the (exogenous or inelastic)

labor supply, and L = E + U. Matches occur according to the function

M (U, V ), where V are vacancies. Let   V/U be the tightness ratio,

f = M (U, V ) /U = M (1, ) be the job finding rate and q = M (U, V ) /V =

M (1/, 1) be the vacancy filling rate. The law of motion of employment is

˙E

= M (U, V )  sE,

where s is the exogenous separation rate.

Let r denote the interest rate, Awc the profit to the employer from a

filled job (w is the wage and c the non-wage cost of maintaining the job) and

c the profit from an unfilled job. Finally, let E and U denote, respectively,

the value for the worker of being employed and of being unemployed, and

let J and V denote, respectively, the value for the employer of a filled job

and a vacant job.

Answer the following questions.

1. Agents’ behavior is characterized by the following equations:

r =

w

E  s

E  U

E

;

r =

A  w  c

J  s

J  V

J

;

r = f

E  U

U

;

r = 

c

V

+ q

J  V

V

.

Interpret carefully these relations.

2. Assume that whenever a match occurs, employer and worker set the

wage so that

wi = argmax (Ei  U) (Ji  V )1 .

Interpret this characterization.

3. Assume that the cost of creating a vacancy is zero and that there is

free entry in vacancy creation so that V = 0. Interpret this condition.

(Careful here: recall the abuse of notation where V at the numerator

of  is vacancies, not the value of a vacancy, which we are setting at

zero.) Show that the five conditions just discussed yield a symmetric

equilibrium where the wage curve is

w =  (A  c + c) .

Interpret this equation.

4. Show that there exists an instantaneous equilibrium in (,w) space

given by the intersection of a job creation curve an the wage curve

derived in 3 above. Interpret this equilibrium.

5. Now show that the model’s dynamics reduces to a di§erential equation

in the unemployment rate u  U/L.

6. Use the di§erential equation to discuss the e§ects of a permanent increase

in each one of A, c, s, r.