Consider a firm that uses two factors, factor 1 and factor 2, to produce a good

Question

Consider a firm that uses two factors, factor 1 and factor 2, to produce a good (output). The quantity of the output, say q, that the firm produces when it uses y_1 units of factor 1 and 1/2 units of factor 2 is specified by the production function q = f (y_1, y_2) = 3y_1^1/3 y_2^1/6. The prices of factor 1 is omega_1 = 2, and the prices of factor 2 is omega_2 = 1. Show that the firm's production technology exhibits decreasing returns to scale. Find the combinations of the inputs y_1 and y_2 that minimizes the firm's cost when producing a given amount q of the output. Derive the (total) cost function C(q) and the marginal cost function MC(q). Assume the firm is price-taking. Derive the supply curve x_s(p).

Explanation / Answer

a. q = 3(y1)^1/3(y2)^1/6

INcreasing every input by

   q' = 3( y1)^1/3( y2)^1/6

   q' = 3^1/3(y1)^1/3^1/6(y2)^1/6

  q' = ^1/2 3(y1)^1/3(y2)^1/6

q' = ^1/2*q

  So q increased only by  ^1/2 when all inputs increased by  . hence decreasing returns to scalw.

b.

   MPy1 = dq/dy1 = (y1)^-2/3(y2)^1/6

MPy2 = dq/dy2 = 1/3(y1)^1/3(y2)^-5/6

MPy1/MPy2 = W1/W2

[(y1)^-2/3(y2)^1/6]/[1/3(y1)^1/3(y2)^-5/6] = 2/1

3y2/y1 = 2

y1 = (3/2)y2

c. Total cost function C = W1*y1 + W2*y2

C = 2*y1 + 1*y2

C = 2* (3/2)y2 + 1*y2

C = 4*y2

  q = 3(y1)^1/3(y2)^1/6

q = 3((3/2)y2)^1/3(y2)^1/6

q^2 = 9*1.31*y2

y2 = q^2/12

So

Cost function C = 4*q^2/12 = q^2/3

MC(q) = 2/3*q

d. If the firm is price taking firm then MC is the suply curve

P = 2/3q

q = 3/2P is the supply function

If you don;t understand anything then comment. I'ill revert back on the same. :)

  

  

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