Consider a production function of two inputs, labor and capital, given by Q=( L^1/2+ k^1/2 )^2 The marginal products associated with this production function are as follows:
MPL=[L^1/2+K^1/2]*L^(-1/2)
MPK=[L^1/2+ K^1/2]*k^(-1/2)
Let w =2 and r = 1.
a) Suppose the firm is required to produce Q units of output. Show how the cost-minimizing quantity of labor depends on the quantity Q. Show how the cost-minimiz- ing quantity of capital depends on the quantity Q.
b) Find the equation of the firm's long-run total cost curve.
c) Find the equation of the firm's long-run average cost curve.
d) Find the solution to the firm's short-run cost- minimization problem when capital is fixed at a quantity of 9 units (i.e., K 9).
e) Find the short-run total cost curve, and graph it along with the long-run total cost curve.
f ) Find the associated short-run average cost curve.