A firm with the production function f (x1, x2, x3, x4)= min{x1, x2, x3, x4} faces input prices w1 = $1, w2 = $2,w3 = $2, w4= $5 for factors 1, 2, 3, and 4.The firm must use at least 17 units of factor 2.The lowest cost at which it can produce 100 units of output is
A)$1,000.
B)$400.
C)$317.
D)$615.
E)$300.

Question

Quizplus · Accepted Answer

To minimize cost, the firm should use the input combination that equates the marginal product of each input to its price.

Let the marginal products (MPs) of inputs 1, 3, and 4 be denoted by MP1, MP3, and MP4, respectively. Then, the marginal product of input 2 is:

MP2 = 4x2/x1 = 40/x2

The cost-minimizing input combination is:

MP1/w1 = MP2/w2 = MP3/w3 = MP4/w4

Solving for x1, x3, and x4, we get:

x1 = (10/3)x2
x3 = (5/2)x2
x4 = x2

The cost of producing F1F1F1100 units of output is:

C = w1x1 + w2x2 + w3x3 + w4x4
C = $1(10/3)x2 + $2x2 + $2(5/2)x2 + $5x2
C = $31.7x2

However, the firm must use at least 17 units of factor 2. Thus, the minimum cost of producing 100 units of output is:

C = $31.7(17) = $538.9

Therefore, the closest option to the minimum cost is Option C ($317).

A Firm with the Production Function F (X1, X2, X3