1. Consider the production function y = f (L, K ) = LK 2 . Denote the factor prices of L and K by w and r respectively and assume that these are given to the firm. Also assume that the firm is unable to influence the market price p of its output.
(a) Prove that the firm experiences increasing returns to scale.
(b) Consider the problem of profit maximization. Assuming for the moment that this is unconstrained, the decision problem of the firm can be denoted by
Max p (L, K ) = pf (L, K ) - wL - rK .
L,K
Prove that there is no finite level of input use for which the firm maximizes profit.1
Interestingly, there is a finite combination (L*, K*) for which a firm experiencing increasing returns to scale minimizes the cost of production subject to an output constraint. To see this, consider the constrained cost minimization problem
Min. C = wL + rK subject to LK 2 = y .
L,K
(c) Write down the Lagrangean for this problem.
(d) Write down the first order necessary conditions (FOC) for a minimum.
(e) Simply state the mathematical condition that needs to hold for you to be able to solve the FOCs explicitly for values of L and K. 2
(f) Assuming that the condition in part (e) holds, derive the values L and K.
(g) Verify that the values of L and K you obtained in part (f) are, in fact, optimal by showing that the second order sufficient condition for constrained minimization (SOC) holds.3
(h) Write down the indirect objective function or value function for the problem.
(i) Prove that the indirect objective function is homogeneous of degree one in both w and r .
(j) Verify that partially differentiating the indirect objective function with respect to w should give you the input demand function for labor L* = L *(w, r, y) you found in part (f).
(k) Simply write down the name of the theorem which is being applied implicitly in part (j).
(l) Write down the expression for the Lagrange multiplier l * for the problem.
2. Consider a firm which operates with the general concave production function y = f (L, K) ,
factor prices w of L and r of K are given to the firm. The objective of the firm is to maximize profit subject to an output constraint f (L, K) = y .
(a) Write down the Lagrangean for the problem.
(b) Write down the first order necessary conditions (FOC) for a maximum.
(c) Verify that the second order sufficient condition (SOC) for maximum is satisfied in the sense that the Bordered Hessian determinant has the appropriate sign.
(d) If the second order condition in part (c) holds, you should be able to solve the first order conditions derived in part (b) for the optimal values of L, K, and l . Our goal is to see how the conditional demand for labor L* responds to changes in w. To this end substitute the optimal values L*, K* and l * back into the FOC, partially differentiate the three equations with respect to w, and write down the resultant equations in matrix form.
(e) Solve for the partial derivative ¶L * ¶w using Cramer's Rule and show that it is negative.
3. Let us return to the cost minimization problem subject to an output constraint. This time assume the general Cobb-Douglas production function y = La K b , where 0 < a, b < 1 and a + b ¹ 1
(a) Derive the levels of L* and K* that solve the decision problem of the firm and write down the expression for the ratio L * K *.
(b) Differentiate L * K * with respect to (w / r) and write down the value of the derivative.
(c) Find the magnitude of the term σ = d ( L * K *) / d (w r) . w / r L * / K *
(d) The term s you found in part (c) is called the elasticity of substitution between L and K. Write down a brief interpretation of the term. That is, what exactly is it measuring?