Question 14.1.2: A single-product firm intends to produce 30 units of output ......
A single-product firm intends to produce 30 units of output as cheaply as possible. By using K units of capital and L units of labour, it can produce \sqrt{K }+ L units. Suppose the prices of capital and labour are, respectively, $1 and $20. The firm’s problem is, then:
minK + 20L s.t. \sqrt{K} + L = 30
(a) Find the optimal choices of K and L.
(b) What is the additional cost of producing 31 rather than 30 units?
(a) The Lagrangian is
\mathcal{L}=K+20L-\lambda(\sqrt{K}+L-30)so the first-order conditions are:
\mathcal{L}_{K}^{\prime}=1-\lambda/2\sqrt{K}=0,\;\mathcal{L}_{L}^{\prime}=20-\lambda=0,\;\;\mathrm{and}\;\;\sqrt{K}+L=30The second equation gives λ = 20, which inserted into the first equation yields 1 = 20/2\sqrt{K}. It follows that \sqrt{K}= 10, and hence K = 100. Inserted into the constraint this gives \sqrt{100} + L = 30, and hence L = 20. The 30 units are therefore produced in the cheapest way when the firm uses 100 units of capital and 20 units of labour. The associated cost is K + 20L=500.^{3}
(b) Solving the problem with the constraint \sqrt{K} + L = 31, we see that still λ = 20 and K = 100, while L = 31 − 10 = 21. The associated minimum cost is 100 + 20 · 21 =520, so the additional cost is 520 − 500 = 20. This is precisely equal to the Lagrange multiplier! Thus, in this case the Lagrange multiplier tells us by how much costs increase if the production requirement is increased by one unit from 30 to 31.^{4}
^{3} Theorem 14.5.1 will tell us that this is the constrained minimum, because \mathcal{L} is convex in (K, L).
^{4} Section 14.2 will tell us why this is not entirely coincidental.