Consider a firm that produces using capital (K) and labor (N) given a production technology represented with the following production function:
zF (K, N) = zK^αN^(1âα)
The firm has a fixed amount of capital K and can hire labor at a market (given) wage (w) (per unit of labor hired). Then the profits of the firm are:
Ï = max N zF (K, N) â wN
= max zK^αN^(1âα) â wN
(a) Find the FOC of the firm and interpret it.
(b) Solve for the optimal labor demand of the firm.
(c) Is the labor demand increasing in K? Why?
(d) The elasticity of the labor demand to wages is defined as: ξw =â ln N/â ln w
Find it and interpret it.
(e) The elasticity with respect to capital and productivity measures how much the demand for labor varies when the amount capital or the productivity vary. We can use our labor demand function to find these elasticities:
ξK =â ln N/â ln K, ξz =â ln N/â ln z
Find them and interpret them.