Proposition 2.3.1 Given wt-1, τt-1, st-1, Le-1, Nt-1 and wt; 1 + R* =
R(wt). Moreover there exists wc = (1 - α)(At)1-α[SN-1 ]α such that: If wt > wc
then:
Kd = KCt(.) = St-1 (11)
and
Lte = Ltd
If wt = wtc then:
Ktd
and
If wt < wtc then:
1-α
Kd = At ~ St-1
Atk(wt) [ 1-α ]α
KS(wt-ι,Let-ι) = St-1
Lte = Ltd =Nt.
Lte = Ltd = Nt
and
Kd = AtLdk(wt) = N [ w' ]α < Kt(.).
At α 1 — α
(12)
(13)
(14)
(15)
(16)
Proposition 2.3.1 says that the equilibrium interest rate is given by the zero
profits condition 1 + R* = R(wt). Moreover, if the wage is greater than the
competitive wage, the capital market is in equilibrium, the effective quantity of
labor employed is given by the labor demand ( 12) and there is always unem-
ployment. In this case, if the wage in period t increases then the labor demand
in period t decreases. This may seem strange because we are in the particular
case in which saving do not depend on the interest rate. The explanation is that
an increase in the wage switches to a more capital intensive technique and labor
is reduced. This fact can be seen in equation ( 12) where Ktd does not depend on
Rt but if wt increases then k(wt) decreases. Note also that the elasticity labor
demand is —1 < —1, and, thus, the wage bill before taxes wtLd(.) decreases
when wt increases.
If the capital market is in equilibrium one can show that the output market
is in equilibrium and then aggregate output in period t, Yt , is given by:
Yt = Ytt
t
St-1
1-α
At — St—1
[___wt___] 1αα
[ At(1-α)] α
[ — ] 1αα
1-α
(17)
In this case the elasticity of output with respect to the wage is
is, in order to increase aggregate employment by the percentage of ɪ
output must increase by the percentage of 1-α (Okun’s Law).
1—α, that
α,
aggregate
Finally, if the wage is less than the competitive wage, the labor market is
in equilibrium and there is an excess supply of capital. In this last case, we
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