Question 14.7.3: A firm uses K units of capital and L units of labour to prod......
A firm uses K units of capital and L units of labour to produce F(K, L) units of a commodity. The prices of capital and labour are r and w, respectively. Consider the cost minimization problem
min C(K, L) = rK + wL s.t. F(K, L) = Q
where we want to find the values of K and L that minimize the cost of producing Q units. Let C^{∗} = C^{∗}(r,w,Q) be the value function for the problem. Find ∂C^{∗}/∂r, ∂C^{∗}/∂w, and ∂C^{∗}/∂Q.
Including the output requirement Q and the price parameters r and w, the Lagrangian is
\mathcal{L}(K,L,r,w,Q)=r K+w L-\lambda(F(K,L)-Q)whose partial derivatives are ∂L/∂r = K, ∂L/∂w = L, and ∂L/∂Q = λ. According to Theorem 14.7.1,
{\frac{\partial C^{*}}{\partial r}}=K^{*},\quad{\frac{\partial C^{*}}{\partial w}}=L^{*},\quad\mathrm{and}\quad{\frac{\partial C^{*}}{\partial Q}}=\lambda (∗)
The first two equalities are instances of Shephard’s lemma. The last equation shows that λ must equal marginal cost, the rate at which minimum cost increases w.r.t. changes in output.