Suppose a competitive firm faces a short-run profit maximization problem. Specif

Question

Suppose a competitive firm faces a short-run profit maximization problem. Specifically it has the production function f(x_1,x_2) = x^1/2_1 x^1/2_2 and input 2 is fixed at x_2 = 16. Let p, w_1, and w_2 be the prices of the output, x_1, and x_2 respectively. (a) Write out the consumer's short-run optimization problem. (b) Solve for the firm's optimal value of x_1. (c) Using your solution for x_1, derive a solution for the optimal output y* by plugging x_1 into the production function and simplifying. (d) Using your solutions for x_1* and y*, derive a solution for the value of profit at the optimal solution. Using this expression, derive the condition on prices needed for the firm to have a nonnegative profit (pi > 0).

Explanation / Answer

Production function : F = x11/2x21/2 and x2 is fixed at 16. Thus F = x11/2(16)1/2 = 4x11/2

b) Optimal value of x1 can be find out by the following:

The firm should have MPofx1/MPofx2 = Price of x1/Price of x2

MP of x1 / MP of x2 = dF/dx1 / dF/dx2 = 2/x11/2

and Px1/Px2 = w1/w2.

Thus 2/x11/2 = w1/w2

4w22/w12 = x1.

c) y = 4x11/2 = 4(4w22/w12)1/2 = 8w2/w1.

d) Profit = TR - TC

TR = P*y = P(8w2/w1)

TC = w1x1 + w2x2 = w1(4w22/w12 ) + 16w2

Profit =  P(8w2/w1) -  w1(4w22/w12 ) - 16w2 = P(8w2/w1) - 4w22/w1 -16W2 = 4w2 (2P/w1 - w2/w1 - 4)

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