2 Theory
Consider a sector with N>0 identical producers. The output by each pro-
ducer is characterized by the following constant-returns-to-scale production
function:
q(l,k) ≡ lαk1-α (1)
where l and k denote labor and capital, respectively, and α ∈ (0, 1).Inthe
short run, k>0 is fixed. With r denoting the unit price of capital, rk is
the fixed cost.
A perfectly competitive firm takes prices as given in maximizing its profit,
i.e.,
mlax π (q (l) ,l) ≡ pq (l) - wl - rk (2)
where p and w are the unit prices of output and labor, respectively.3 The
first-order condition is
q0 (l*) = w/p (3)
where l* denotes the optimal amount of labor. By rearranging this equation,
we get the following short-run factor demand:
l* (p,w) = k (αp∕w)1-α (4)
3To ensure that the producers are operating in the market, we assume that π (l0) ≥ -rk,
or equivalently p ≥ wl0 /q (l0), holds with l0 > 0 denoting the labor input per producer
without immigration.