The name is absent

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) ≡ l^αk^1-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.,

m_lax 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.

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