Question

#7. (30%) Eagle Manufacturing, Inc., produces telecommunication equipments. The pro- duction function of Eagle has been estimated as Q=K0.3 0.8 Eagle currently pays $10 per unit of the labor and $20 per unit of the capital input. (a) (5%) Find the algebraic expressions for the marginal product of labor (MPL) and for the marginal product of capital (MPK). (b) (5%) What can you say about returns to scale of the production function? (c) (10%) If Eagle wishes to produce 400 units of output, how much of each input should be used to minimize the cost of producing 400 units? What would that cost be? (d) (10%) If the labor union demands wage increases and other benefits that push the cost per unit of labor upto $20, how would this influence the cost minimization problem? Rework part (c) to show how the added labor costs would influence the employment of labor and cost of production. Explain also the change in the employment of labor graphically.

Answer 1

a. The production function is:
$\dpi{80} \fn_phv \small Q=K^{0.3}L^{0.8}$
Marginal products of labor and capital are:
$\dpi{80} \fn_phv \small MPL=0.8K^{0.3}L^{-0.2}, MPK=0.3K^{-0.7}L^{0.8}$

b. To check for returns to scale, we increase the inputs by $\dpi{80} \fn_phv \small \alpha$ times and observe the increase in output:
$\dpi{80} \fn_phv \small f(\alpha K,\alpha L)=(\alpha K)^{0.3}(\alpha L)^{0.8}=\alpha^{1.1}f(K,L)$
Since increasing inputs by $\dpi{80} \fn_phv \small \alpha$ times increased output by more than $\dpi{80} \fn_phv \small \alpha$ times, therefore, this production function exhibits increasing returns to scale.

c. At equilibrium, marginal rate of technical substitution is equal to the factor price ratio:
$\dpi{80} \fn_phv \small MRTS=\frac{MPL}{MPK}=\frac{0.8K^{0.3}L^{-0.2}}{0.3K^{-0.7}L^{0.8}}=\frac{8K}{3L}=\frac{10}{20}\rightarrow L=\frac{16K}{3}$
Substituting this into the production function:
$\dpi{80} \fn_phv \small Q=K^{0.3}(\frac{16K}{3})^{0.8}\rightarrow Q=3.815*K^{1.1}\rightarrow K=(\frac{Q}{3.815})^{0.91},L=5.33*(\frac{Q}{3.815})^{0.91}$
For 400 units of output:
$\dpi{80} \fn_phv \small K=68.9, L=367.65$
Cost of this input combination is:
$\dpi{80} \fn_phv \small 10*367.65+20*68.9 =5054.9$