Question

(1) A firm has the following production technology: F(LK)-4LIKİ. (a) Derive the conditional labor demand in the short run, IS (Q, w,r, R) (b) Derive unconditional labor demand in short run, 15 (p,w,r,R (c) In two sentences or less explain your answers in (a) and (b) differ

Answer 1

a)

Production function:

$F(K,L) = 4L^{1/4}K^{1/2}$

In the short run, K is fixed at $\overline{K}$

Thus, production function is only a function of L,

$F(K,L) = 4L^{1/4}\overline{K}^{1/2}$

A conditional factor demand is the cost-minimizing level of an input like labor or capital that is required to produce a given level of output for a given wage rate and rental rate of capital.

Total cost = wL + r$\overline{K}$

Objective problem:

Min wL + r$\overline{K}$ w.r.t L such that $Q = 4L^{1/4}\overline{K}^{1/2}$

$Q = 4L^{1/4}\overline{K}^{1/2}$$\Rightarrow \overline{K}^{1/2} = \frac{Q}{4L^{1/4} } \Rightarrow \overline{K} = \frac{Q^2}{16L^{1/2} }$

Substituting in TC function -

Min TC = $wL + \frac{rQ^2 L^{-1/2}}{16}$ wrt L

Using first order conditions, we get:

$\frac{\partial TC}{\partial L} = 0 \Rightarrow w - \frac{rQ^2L^{-3/2}}{32} = 0$

$\Rightarrow L^{-3/2} = \frac{32w}{rQ^2} \Rightarrow L = (\frac{rQ^2}{32w})^{2/3}$

b)

Unconditional factor demand can be obtained from profit maximization -

Profit = Revenue - Total cost

$max \pi = pF(L, \overline{K}) - wL - r\overline{K}$ w.r.t L

$\pi = 4pL^{1/4}\overline{K}^{1/2} - wL - r\overline{K}$

Maximizing w.r.t. L

Using first order conditions -

$\frac{\partial \pi}{\partial L } = 0 \Rightarrow pL^{-3/4}\overline{K}^{1/2} = w \Rightarrow L^{3/4} = \frac{p\overline{K}^{1/2}}{w}$

$\Rightarrow L = (\frac{p\overline{K}^{1/2}}{w})^{4/3}$

This is the unconditional factor demand function.

c)

Functions in (a) and (b) are different because L in (a) is conditional factor demand obtained by minimizing the cost function. On the other hand, L in (b) is unconditional factor demand obtained by maximizing the profit function. Thus their functional form is different in both the cases.