18. Assume that a firm produces its product in a system described in the followi
ID: 1249997 • Letter: 1
Question
18. 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
where 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. Is it 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? Using this, what is the maximum amount of output that can be produced with a budget of $1000?
c. How much will output increase if the budget rises to $1500?
d. Suppose that the firm is interested in producing 14777 units of output. How much X and Y will be used and what is the cost?
Explanation / Answer
a) R = PQ = 3PX + 5PY +PXY => R = 9 + 30 + 6X = 9 + 30 + 3Y => 6X = 3Y => 2X = Y (Irrespective of level of output) (ANSWER) b) MR(X) = dR/dX = 6 = MC(X) MR(Y) = dR/dY = 3 = MC(Y) C(X) = 6X + a, when X=0, C(X) =0 so a =0 C(Y) = 3Y + b, when Y=0, C(Y) =0 so b =0 => C = 6X +3Y (cost can be expressed by this) => C = 1000 = 6X +6X = 12X => X = 1000/12 = 250/3 = 83.33 => Y = 2X = 500/3 = 166.67 => Q = 3X + 5Y +XY = 250+2500/3 + 250*500/9 = 14972.2 = say 14972 (ANSWER) c) C = 1500 = 12 X, so X = 1500/12 = 375/3 and Y = 750/3 Q = 375 + 3750/3 +375*750/9 = 32875 (ANSWER) d) Q = 14777 = 3X + 10X + 2X^2 = 13X + 2X^2 so, 2X^2 + 13X - 14777 = 0 => X = 82.8 and Y = 165.6 C = 12X = 12*82.8 = 993.6 ($) (ANSWER)
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