A firm has the production function
q = KL + L
where q is output and K and L are quantities of two inputs.
(a) Derive the firmâ??s expansion path.
(b) Explain what the expansion path tells us. For example, what happens to make the firm move along its expansion path?
(c) Indicate how you would proceed to derive the conditional factor demands for K and L. (You do not need to actually derive them.)
(d) Explain whether or not this firmâ??s production technology is homothetic. Give your reasoning(s).
a) The firm's production function can be written into:
Q = (K+1)L
This is a Cobb-Douglas production function.
Marginal product of capital (MPK) = dQ/dK = L
Marginal product of labour (MPL) = dQ/dL = K+1
Assume the unit prices of capital (K) and labour (L) are r and w respectively. The question is then to minimize the total cost given the production level Q.
Min C = rK + wL
s.t. Q = (K+1)L
The total cost is minimized if the marginal rate ...
This solution thoroughly exemplifies the firm's expansion path.