Essay About Job Pair Produces P And Minimum Wage

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Labor Search Model
1 The Model
Basic Assumptions:
Time: Continuous, inÐnite 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
workersÐЇow 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 Ѕrms: rUf = фЂЂЂa+f (Vf фЂЂЂUf )фЂЂЂUf (3)
Employed Value for Ѕrms: rVf = p фЂЂЂ w фЂЂЂ Vf (4)
and wage determination by Nash bargaining solution:
w = arg max
(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
( 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
Vw(w(p;Uw)) = (1фЂЂЂ)p+rUw+Uw
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 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 eФect 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
( 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 oФer
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 oФers 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))