Assume that a firm produces its product in a system described in the following production function and price data:

Q = 3X + 5Y + XY

PX = $3

PY = $6

Here, X and Y are two variable input factors employed in the production of Q.

A. What are the optimal input proportions for X and Y in this production system? Is this combination rate constant regardless of the output level?

B. It is possible to express the cost function associated with the use of X and Y in the production of Q as Cost = PXX + PYY or Cost = $3X + $6Y. Use the Lagrangian technique to determine the maximum output that the firm can produce operating under a $1,000 budget constraint for X and Y. Show that the inputs used to produce that level of output meet the optimality conditions derived in Part A.

C. What is the additional output that could be obtained from a marginal increase in the budget?

D. Assume that the firm is interested in minimizing the cost of producing 14,777 units of output. Use the Lagrangian method to determine what optimal quantities of X and Y to employ. What will be the cost of producing that output level? How would you interpret λ, the Lagrangian multiplier, in this problem?