Texts: Consider a cost-minimizing firm with a production function Q = f(K,L) = [min {L, K}]^(1/3), where L is labor and K is capital. The firm is a price taker in the input markets and pays w for each unit of labor hired and r for each unit of capital, and faces no other costs.
(1) Sketch a representative isoquant.
(2) Calculate the Marginal product for each input and indicate whether they are diminishing, constant, or increasing.
(3) Indicate whether the production function exhibits constant, increasing, or diminishing returns to scale.
(4) Derive expressions for the cost-minimizing conditional input demands L+ (r,w, Q) and K* (r,w, Q). Confirm that the conditional input demand functions are "homogeneous of degree zero" in w and r; that is L+ (tr, tw, Q) = L* (r,w, Q) and K* (tr, tw, Q) = K+ (r,w, Q).
(5) What happens to the conditional demand for labor if there is an increase in the wage rate, assuming that r and Q remain the same.
(6) Derive the Cost function C (r,w, Q) from your answer in 4. Is this function homogeneous of degree 1, i.e. C (tr, tw, Q) = t * C(r,w, Q), for all t > 0?