A profit-maximizing firm produces a final good y by using an input r. The techno

Question

A profit-maximizing firm produces a final good y by using an input r. The technologically feasible production plans are described by the following function: y = f(x) x The output price is denoted by p and the input price by w. 1. What returns to scale does the technology display? 2, Determine the marginal product ?? 3. Write down the firm's profit maximization problem 4. Derive the firm's demand function for input *(p, x) and the supply function y"(p, ) 5. Derive the firm's optimal profit at (p, w) and y'(p,w)

Explanation / Answer

y = x1/3

(1) Let us double both inputs so that new production function becomes

y1 = (2x)1/3 = 21/3x1/3 = 21/3y

y1/y = 21/3 < 2

Since doubling the input less than doubles output, there is decreasing returns to scale.

(2) MP = dy/dx = (1/3) / x2/3 = 1 / (3x2/3)

(3) Total revenue (R) = p.y and Total cost (C) = w.x

Profit (z) = R - C

Firm's profit maximization problem is

Maximize z = p.y - w.x

Maximize z = p.x1/3 - w.x

(4) Input demand is optimal when profit is maximized, i.e. dZ/dx = 0

dZ/dx = [p / (3x2/3)] - w = 0

p / 3x2/3 = w

x2/3 = = (p/3w)

x* = (p/3w)3/2

Substituting in production function,

y* = [(p/3w)3/2]1/3 = [(p/3w)1/2] [Supply function]

NOTE: As per Chegg Answering Policy, first 4 parts are answered.

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