derived from (4) is:
χ (г) = [(ι - β) p (г) lχ (г)]1/в А (г)l (г) ,
(5)
where χ (г) is the price of machine ʃ(i). Machines in each sector are produced by
a monopolist. The unit cost of producing any machine is normalized to (1 — β)2.
Together with isoelastic demand (2), this implies that the monopolist in each sector
charges a constant price, χ (г) = (1 — β). Substituting χ (г) and (5) into (4), yields
the quantity produced in sector г as a linear function of the level of technology А(г)
and employed labor l (г):
y (г) = p (г)(1-в)/в А (г) l (г) .
(6)
The linearity of y (г) in А (г) is crucial for endogenous growth, but it is not a sufficient
condition. As it will become clear later on, an expansion of y (г) can reduce its price
р(г) and this can effectively generate decreasing returns. Given the Cobb-Douglas
specification in (4), the wage bill in each sector is a fraction β of sectoral output.
Therefore, equation (6) can be used to find the relation between equilibrium prices
and the wage:
w = βp (г)1/в а (г).
(7)
Since there is perfect mobility of labor across sectors, the wage rate has to be equal-
ized in the economy. Dividing equation (7) by its counterpart in sector j delivers
the equilibrium relative price of any two varieties:
p (г)
P (j)
Г А (j) f
И (г)
(8)
Intuitively, sectors with higher productivity have lower prices. Using (7), integrating
over the interval [0,1] and making use of (3) shows that the equilibrium wage rate
is a CES function of sectoral productivity:
j А (г)в(е-1)
1∕β(e-1)
(9)