Search Model

Labor Search Model

1 The Model

Basic Assumptions:

Time: Continuous, inÐ'...nite horizon;

Demography: Mass 1 of inÐ'...nite lived homogeneous workers and mass 1 of

Ð'...rms. Jobs are subject to destruction at arrival rate  ;

Preference: Both are risk neutral, with common discount rate r , and the

workersÐ''Ð'‡ow value of leisure is b ;

Technologies: Matched worker/job pair produces p, which follows a technologically-

determined distribution of G(p); Jobs cost a to advertise; Standard Poisson

matching process with arrival rates w and f ;

Information: Both know G(p);

Institutions: Nash bargaining.

1.1 Nash bargaining without minimum wage

We Ð'...rst recall the values functions in model of Pissarides (2000):

Unemployed Value for workers: rUw = b+w(Vw фЂЂЂUw) (1)

Employed Value for workers: rVw = w + (Uw фЂЂЂ Vw) (2)

Unemployed Value for Ð'...rms: rUf = фЂЂЂa+f (Vf фЂЂЂUf )фЂЂЂUf (3)

Employed Value for Ð'...rms: rVf = p фЂЂЂ w фЂЂЂ Vf (4)

and wage determination by Nash bargaining solution:

w = arg max

w

(Vf фЂЂЂ Uf )(Vw фЂЂЂ Uw)1фЂЂЂ (5)

where  is the bargaining power of the Ð'...rm.

We notice that for any given unemployed value of Uw there exists a corre-

sponding critical productivity p = rUw, which has the property that the match

pair produce at least as great as p will result in employment while others will

not. For any p  p, we can rewrite the Nash bargaining condition by using the

equations (1) to (4) and the free entry condition Uf = 0 such like:

w = arg max

w

( pфЂЂЂw

r+ )(wфЂЂЂrUw

r+ )1фЂЂЂ (6)

And Ð'...nd the F.O.C w.r.t , we get the wage equation:

w(p;Uw) = (1 фЂЂЂ )p + rUw (7)

From equation (7), we can see that the reservation wage w = p = rUw.

Moreover, the modiÐ'...ed unemployed value for workers will be

rUw = b + w Z rUw

[Vw(w(p;Uw)) фЂЂЂ Uw] dG (p)

Since

1

Vw(w(p;Uw)) = (1фЂЂЂ)p+rUw+Uw

r+

We have the Ð'...nal expression for the unemployed value is

rUw = b + (1фЂЂЂ)w

r+ Z rUw

[p фЂЂЂ rUw] dG (p)

Since

p(w;Uw) = wфЂЂЂrUw

(1фЂЂЂ)

Therefore, the density function of wages is given by

f(w) =

(1фЂЂЂ)фЂЂЂ1g(p(w;Uw))

1фЂЂЂG(p) w  p

0 w < p

(8)

where g(p) = G0(p)

1.2 Nash bargaining with minimum wage

Now we impose the minimum wage wm in the model. It is clear that any

wm  w = p = rUw has no eÐ'¤ect on the behavior of the workers or Ð'...rms and

thus we consider only the imposition of an wm > p. To determine the wage,

we need to solve the constrained Nash bargaining problem which is given by:

w = arg max

wwm

( pфЂЂЂw

r+ )(wфЂЂЂrUw

r+ )1фЂЂЂ (9)

Under the imposition of minimum wage, we deÐ'...ne the unemployed search

value as Uw (wm), which is not equal to Uw. Thus, the new reservation wage

shoule be equal to rUw (wm). Under the Nash bargaining condition (9), we

should have the wage equation:

w(p;Uw (wm)) = (1 фЂЂЂ )p + rUw (wm) (10)

we assume the worker would receive the minimum wage wm when p = bp,

where

bp(w;Uw (wm)) = wmфЂЂЂrUw(wm)

(1фЂЂЂ)

When matched pair produces p belongs to the set [wm; bp), the wage oÐ'¤er

according to (10) is less than wm. Therefore, the Ð'...rm pays the wage of wm for

all p 2 [wm; bp). And for any p  bp, the wage oÐ'¤ers are determined according

to (10). We can now consider the unemployed search value for workers which is

given by

rUw (wm) = b + wfZ bp(w;Uw(wm))

wm