Question

9. Suppose the firm's production function is given by f(K,L) min (K",L" (a) For what values of a will the firm exhibit decreasing returns to scale? Constant returns to scale? Increasing returns to scale? (b) Derive the long-run cost function and the optimal input choices. (c) Suppose the capital is fixed at R = 10,000 and a =. Assuming that the firm wants to produce less than 100 units, derive 10. Consider the production function: f(K, L) = KLi. Let w and r denote the price of labor and capital, and let p denote the price of the output good. (a) Find the cost minimizing input bundle and the cost function as a function of w, r, and q (b) Find the profit maximizing output level and the profit as a function of w, r, and p

Answer 1

9. a. For the production function to exhibit decreasing returns to scale, increasing inputs by $\dpi{80} \fn_phv \small \beta$ times increases the output by less than $\dpi{80} \fn_phv \small \beta$ times. $\dpi{80} \fn_phv \small f(\alpha L, \alpha K)=min((\beta K^\alpha),(\beta L^\alpha))=\beta^{\alpha}min(K^\alpha,L^\alpha)\rightarrow \alpha<1$
For the production function to exhibit increasing returns to scale, increasing inputs by $\dpi{80} \fn_phv \small \beta$ times increases the output by greater than $\dpi{80} \fn_phv \small \beta$ times. $\dpi{80} \fn_phv \small f(\alpha L, \alpha K)=min((\beta K^\alpha),(\beta L^\alpha))=\beta^{\alpha}min(K^\alpha,L^\alpha)\rightarrow \alpha>1$
For the production function to exhibit constant returns to scale, increasing inputs by $\dpi{80} \fn_phv \small \beta$ times increases the output by $\dpi{80} \fn_phv \small \beta$ times. $\dpi{80} \fn_phv \small f(\alpha L, \alpha K)=min((\beta K^\alpha),(\beta L^\alpha))=\beta^{\alpha}min(K^\alpha,L^\alpha)\rightarrow \alpha=1$

b. The producer treats the two inputs as perfect complements and always employs them in the fixed ratio:
$\dpi{80} \fn_phv \small L^{\alpha}=q\rightarrow L=q^{1/\alpha}, K^\alpha=q\rightarrow K=q^{1/\alpha}$
The firm's cost function is:
$\dpi{80} \fn_phv \small Cost=wq^{1/\alpha}&plus;rq^{1/\alpha}=(w&plus;r)q^{1/\alpha}$

c. The new production function is:
$\dpi{80} \fn_phv \small f(L,K)=min(100,L^{1/2})$
$\dpi{80} \fn_phv \small L^{1/2}=q\rightarrow L=q^2$
Since the firm wants to produce less than 100 units of output, it will employ less than 10,000 units of labor.