A cost-minimizing firm\'s production function is given by Q = LK. Suppose you kn
ID: 1176466 • Letter: A
Question
A cost-minimizing firm's production function is given by Q = LK. Suppose you know that when w = $4 and r = $2, the firm's total cost is $160.
A) What is the optimal allocation of K & L?
B) What is the output of the firm at this allocation?
C) You are now told that when input prices change such that the wage rate is 8 times the rental rate, the firm adjusts its input combination but leaves total output unchanged. What would the cost-minimizing input combination be after the price changes?
D) If production changed to Q = 10L^1/3 * K^2/3 what is the new MRTS L,K given that w = $4 and r = $2?
E) What would be the new allocation of labor and capital given the firm's total cost is $160
Explanation / Answer
A. Finding the optimal allocation means you need to find the values for K and L where
Min wL + rK = 160 subject to Q=KL
Take the first derivative with respect to L and set it to 0
=> w- (rQ/L^2) = 0
=> w= rQ/L^2 Therefore, L= sqrt(rQ/w) and K= sqrt(wQ/r)
Given that 4L + 2K = 160
Substitution,
4 * sqrt(rQ/w) + 2 * sqrt(wQ/r) =160
( square both sides of equation)
8Q + 2(8Q) + 8Q= 160 ^2
=> 32Q= 25600
Q= 800 is the optimal output
At this output, the optimal L and K are
L= sqrt(rQ/w)= sqrt(2*800/4)= 20 units of labor and K= sqrt(4*800/2)= 40 units of capital
B. 800 as calculated above
C. Change w to 16 but keep r=2 so that the w is 8 times the r. Do the same calculations again
L = sqrt(2*800)/16= 10 units of labor, and K= sqrt( 16* 800)/2= 80 units of capital.
D. Given
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