results. All proofs are contained in the appendix.
2 The Individual Agents
2.1 The Firm
There is only one firm with Cobb-Douglas production function:
Yt = Ktα (AtLt)1-α ;
where 0 < α < 1, Yt is output in period t, Kt is the capital stock in period t, Lt
is employment in period t, At is ”knowlegde” or the ”productivity of labour” in
period t and t = 0, 1, 2, .... We assume that At changes according to the equation
At+1 = (1 + g)At where g is the rate of growth of productivity. Output and
capital are the same produced good, and capital in period t is the output saved
in period t-1. For simplicity we assume complete depreciation of the stock of
capital. The real wage in period t is denoted by wt and the rate of interest in
period t by Rt . The firm maximizes profits every period. The formal program
is the following:
Program Ft : Given wt , Rt , choose Kt and Lt in order to maximize:
Ktα(AtLt)1-α - (1 + Rt)Kt - wtLt .
As it is well known, the zero profits condition implies a relationship between
the rate of interest and the real wage given by:
w α-1
1 + Rt = R(wt) = α[(1 - α)A] α . (1)
If the zero profits condition holds then the relationship between the demand
of capital, Ktd , and the demand of labor, Ltd , is given by the optimal capi-
tal/effective labor ratio function kd = AKtd and, if the production function is
AtLt
Cobb-Douglas, ktd is given by:
kd = ^(wt) = [ ∩ w∖ A ]1 ; (2)
(1 - α)At
The function R(wt) is called the factor-price frontier (see Diamond [4]). The
function k(wt) gives the optimal capital/effective labor ratio for a given wage. It
is easy to check that R' < 0 and k' > 0, which means that if the wage increases
the rate of interest decreases and the capital effective labor ratio increases, i.e,
the technology becomes mores capital intensive. We call to the term WAt- the
wage per unit of effective labour and we denote it by ωt . The output supply in
period t is denoted by Yts .