rate of growth of productivity. 2) Constant rate of unemployment with real
wage per unit of effective labour increasing or decreasing to a long run wage
and with the rate of growth of output increasing or decreasing to (1 + n)(1 + g)
(a case similar to the same OLG model with perfect competition in the labor
market). 3) Real wage rigidity with increasing rates of growth of output and
employment with time. These special cases show that the dynamics of this OLG
model with a non competitive labour market may be very different depending
on the assumptions about the specific wage setting process and the behaviour
of the government.
6 The Case 0 < γ < 1.
If γt = γ , βt = β for all t, 0 < γ < 1 and we assume that there is always
unemployment substituting ( 21) in ( 18) on the one hand and taking equation
( 22) on the other we have the equilibrium wage and the effective quantity
of labour employed are given by the following system of non linear first order
difference equations:
*
wt
γ * + 1 - γ (1 - β)wtLt
(29)
(1 - α) wt-1 ' 1 - α)β Nt - Lt
Lt
1-α
At α cwt*-iLt —1
[ iwl ]1
1-α
(30)
Where the endogenous variables are Lt and w*.
We can transform this system into the following non-linear first order system
of two difference equations in terms of the wage per unit of effective labour and
the rate of unemployment.
* γ * (1 -γ)(i -β) *r 1 -∣ι
ωt = (1 - α)(1+ g) ωt-1 + (1 - α)β ωt [Ut - 1]
(31)
(1 - ut)(1 + n)
(1 - ut-1)
c(1 - α)1-αω*
(1 + g)(ω* )1
(32)
It is easy to compute that the steady state wage per unit of effective la-
1 α
bor and rate of unemployment are: ω* = (1 - α) 1-α ( (1+nc(1+g) ) 1-α and u =
__________(1+g)(1—γ)(1-β)__________
(1+g)(1—γ)(1 - β)+[(1+g)(1-α)—γ]β.
Analyzing the linearized system around the
steady-state we obtain the following result:
Proposition 6.1 There exists γ' such that if γ < γ' then the linearized system
converges to ω* and u.
In this case it is easy to see that the long run unemployment rate is less than
one and that decreases with g and β. It increases with γ if and only if g < i—α.
15