Question

4. A firm produces computers with two factors of production: labor L and capital K. It's pro- duction function is y 10 . Suppose the factor prices are wL = 10 and wk = 100. (a) Graph the isoquants for y equal to 1,2, and 3. Does this technology show increasing, constant, or decreasing returns to scale? Why? (b) Derive the conditional factor demands. (c) Derive the long-run cost function C(y). (d) If the firm wants to produce one computer, how many units of labor and how many units of capital should it use? How much will it cost? What if the firm wants to produce two computers? (c) Derive the firm's long-run average cost function AC(y) and long-run marginal cost func- tion MC(y). Graph AC(y) and MC(y). (1) Label the firm's long-run supply curve on the graph.

Answer 1

a. Plotting isoquants:

Increasing both inputs by $\dpi{80} \fn_phv \small \alpha$ times leads to an increase in output by more than $\dpi{80} \fn_phv \small \alpha$ times:
$\dpi{80} \fn_phv \small f(\alpha L,\alpha K)=\frac{(\alpha L)(\alpha K)}{10}=\alpha^2*\frac{LK}{10}=\alpha^2f(L,K)$

b. At equilibrium, the firm sets the marginal rate of technical substitution equal to the factor price ratio:
$\dpi{80} \fn_phv \small MRTS=\frac{MP_L}{MP_K}=\frac{\frac{K}{10}}{\frac{L}{10}}=\frac{K}{L}=\frac{10}{100}=\frac{1}{10}\rightarrow L=10K$
Substituting this into the production function:
$\dpi{80} \fn_phv \small y=\frac{10K^2}{10}\rightarrow K=\sqrt{y}, L=10\sqrt{y}$

c. The firm's cost function is:
$\dpi{80} \fn_phv \small Cost=10*10\sqrt{y}+100*\sqrt{y}=200\sqrt{y}$

d. For one unit of output, demand for inputs is:
$\dpi{80} \fn_phv \small K=\sqrt{1}=1, L=10\sqrt{1}=10$
The cost of this combination is:
$\dpi{80} \fn_phv \small C_1=10*10+100=200$
For two units of output, demand for inputs is:
$\dpi{80} \fn_phv \small K=\sqrt{2}, L=10\sqrt{2}$
Cost of this combination is:
$\dpi{80} \fn_phv \small C_2=100\sqrt{2}+100\sqrt{2}=200\sqrt{2}$