The output (Q) of a production process is a function of two inputs (L and K) and is given by the following relationship:

Q = 0.50LK − 0.10L2 − 0.05K2

The per-unit prices of inputs L and K are $20 and $25, respectively. The firm is interested in maximizing output subject to a cost constraint of $500.

a. Formulate the Lagrangian function:

LQ = Q − λ(CLL + CKK − C)

b. Take the partial derivatives of LQ with respect to L, K, and λ, and set them equal to zero.

c. Solve the set of simultaneous equations in Part (b) for the optimal values of L, K, and λ.

d. Based on your answers to Part (c), how many units of L and K should be used by the firm? What is the total output of this combination?

e. Give an economic interpretation of the λ value determined in Part (c).

f. Check to see whether the optimality condition (Equation) is satisfied for the solution you obtained.

MPL/CL = MPK/CK