The problem facing a firm is to minimize the cost of producing any given output. Production is achieved by combining inputs in the process illustrated by the production function Q=f(K,L) where for simplicity we restrict ourselves to two inputs. The marginal product for each good is the partial derivative of the function f with respect to each argument. That is, the marginal product of L is given by the formula MUL=¶ f(K,L)/¶ L. Similarly, the marginal product of K is MUK=¶ f(K,L)/¶ K. We assume that the second partial derivatives, ¶ ²f(K,L)/¶ L² and ¶ ²f(K,L)/¶ K² are negative to ensure diminishing marginal productivity in each input. Also assume the function f(K,L) is a smooth, continuous function and that an interior solution to the problem exists.

The firm minimizes costs, C=P_kK+P_lL holding output constant. To find the cost minimizing input combination we use the technique of Lagrangian multipliers. This Lagrange equation has the form L=(P_kK+P_lL)+l [Q-f(K,Y)]. L becomes the new function to be minimized, l is called the Lagrangian multiplier, and Q is the specific quantity we wish to produce.

The constrained maximum is found by treating l as a third variable in the problem, taking the partial derivatives of L with respect to each variable, and setting the results equal to zero. That is, derive each of the following equations:

¶ L/¶ L = l P_l - ¶ f/¶ L = 0 (1)

¶ L/¶ K = l P_k - ¶ f/¶ K = 0 (2)

¶ L/¶ l = Q - f(K,L) = 0 (3)

These are called the first-order necessary conditions. Equation (3) is just the constraint holding Q constant. Equation (1) requires that the MPL=l P_l, and likewise equation (2) requires that MPK=l P_k. Rearranging gives that MPK/P_k=l and MPL/P_l=l . Thus, we find the condition, that MPL/P_l=MPK/P_k. The conditions we put on f(X,Y), smoothness, continuity, and that the function exhibit diminishing marginal productivity, ensures that this is a minimum, not a maximum.

As an example, suppose a firm has the production function Q=2L^½Y^½ and wants to minimize the cost of producing 200 units of output when the price of L is 4 and the price of K is 1. The Lagrangian function is L=(4L+K)+l [200-2L^½K^½]. The first-order necessary conditions for maximization are:

¶ L/¶ K = 4 - ½l K^-½L^½ = 0 (4)

¶ L/¶ L = 1 - ½l K^½L^-½ = 0 (5)

¶ L/¶ l = 2L^½K^½-200 = 0 (6).

Reducing (4) and (5) gives K=4L which can be substituted into (6) to find 200-2(4L)^½L^½=0, or 200=4L which means that L=50 and K=200.

We can also find the cost functions, that is cost as a function of output, by using the results of the cost minimization exercise. For example, when we used the production function 2L^½K^½ with the input prices P_l=4 and P_k=1 we found that cost minimization required that K=4L. Substituting this into the production function tells us that Q=4L, or alternatively, L=Q/4. Now total long run cost, LTC, is given by LTC=K+4L, which by substitution is LTC=8L, or LTC=2Q. Long run marginal cost (LMC) is ¶ LTC/¶ Q, which in this case is equal to 2. LMC is constant, which implies that LAC=LTC/Q=2 is constant also, and equal to LMC.

Short run cost curves are determined by holding K fixed, say at 25. then the short run production function becomes Q=2L^½25^½=10L^½. By rearranging we can find that L=Q²/100. Short run total cost (STC) is STC=25+4L=25+Q²/25, and short run average cost (SAC), short run average variable cost (SAVC), short run average fixed cost (SAFC) and short run marginal cost (SMC) are given by:

1. Using the example production function from the appendix, show that both inputs exhibit diminishing marginal products. 2. Try various prices and outputs for K and L in the example from the appendix, and determine a general rule for allocation of costs across inputs with this type of production function. Also try _Q=L^¼_K^¾.

3. Suppose there is a third input, Z with a price of 2. The production function is 2K^½L^¼Z^¼, and the output is 300. Find the cost minimizing input combination. Determine the changes if output falls to 200, or if the price of Z increases to 4.