Question

Suppose a productiin function is given by Q=KL2, and that the price of labor is $15 and the price of capital $10.
A. What combination of labor an d capital minimizes the cost of producing any given output?
B. Formulate the dual problem and solve it. Use substitution, equal slopes, and lagrange to solve the problem.

Answer 1

Answer on Question #42157, Economics, Economics of Enterprise

A. If $L = 1$ and $K = 1$, the cost of producing any given output is minimized and $Q = 1 * 1^2 = 1$. Since the lower bound is valid for every $y \geq 0$, we can search for the best one, that is, the largest lower bound:

$$p^* \geq d^* := \max_{y \geq 0} g(y).$$

The problem of finding the best lower bound:

$$d^* := \max_{y \geq 0} g(y)$$

is called the dual problem associated with the Lagrangian defined above. It is optimal value $d^*$ if the dual optimal value is not used. As noted above, $\mathcal{G}$ is concave. This means that the dual problem, which involves the maximization of $\mathcal{G}$ with sign constraints on the variables, is a convex optimization problem.

**Example:** For the problem of minimum distance to a polyhedron above, the dual problem is

$$d^* = \max_{y \geq 0} g(y) = \max_{y \geq 0} -b^T y - \frac{1}{2} \|A^T y\|_2^2.$$