Maximization of Production Output Subject to a Cost

Question
Exercise 1:
7.15 Maximization of Production Output Subject to a Cost Constraint, Advanced
Material
The output (Q) of a production process is a function of two inputs (L and K) and is
given by the following relationship:
Q = 0.50 LK ? 0.10 L? ? 0.05K?
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 ? ? (CLL + CKK ? C)
b. Take the partial derivatives of LQ with respect to L, K and ?, and set them equal
to zero.
c. Solve the set of simultaneous equations in Part (b) for the optimal values of L, K
and ?.
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 ? value determined in Part (c).
f. Check to see whether the optimality condition (Equation 7A.9) is satisfied for the
solution you obtained.

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