Question #59921

A competitive firm has the following production function:
: ;(<) 400< 60< 6 6<?
Where y = output, x = input. The firm faces an output price of P = 10 and an input
price of W = 5440.
i) Write a profit function of this firm in terms of output and input prices and the
input level.
ii) What is the profit maximizing level of input for this firm? Verify that the input
level you choose is the profit maximizing point.
iii) Find the marginal product (MPx) of the variable input.
iv) Verify that P(MPx) = W at the profit maximizing input level.
b) What economics interpretation would be you attribute to the method of finding the
‘time-path’ using a difference equation? Explain your answer with the help of the
Cobweb model.

Expert's answer

y = 400x + 60x^2 - 6x^3, P = 10, W = 5440.

i) A profit function of this firm in terms of output and input prices and the input level will be:

TR = TR - TC = P*y - W*x = 4000x + 600x^2 - 60x^3 - 5440x = -60x^3 + 600x^2 - 1440x.

ii) The profit maximizing level of input for this firm will be in the point, for which TP' = 0, so:

(-60x^3 + 600x^2 - 1440x)' = 0,

-180x^2 + 1200x - 1440 = 0,

3x^2 - 20x + 24 = 0

x1 = (20 + 10.58)/6 = 5.1 units.

x2 = (20 - 10.58)/6 = 1.57 unit.

iii) The marginal product (MPx) of the variable input is MPx = y' = 400 + 120x - 18x^2.

iv) At the profit maximizing input level P(MPx) = W, so to prove it let's set the the value of x1 into the MPx equation:

MPx = 400 + 120*5.1 - 18*5.1^2 = 5440.

i) A profit function of this firm in terms of output and input prices and the input level will be:

TR = TR - TC = P*y - W*x = 4000x + 600x^2 - 60x^3 - 5440x = -60x^3 + 600x^2 - 1440x.

ii) The profit maximizing level of input for this firm will be in the point, for which TP' = 0, so:

(-60x^3 + 600x^2 - 1440x)' = 0,

-180x^2 + 1200x - 1440 = 0,

3x^2 - 20x + 24 = 0

x1 = (20 + 10.58)/6 = 5.1 units.

x2 = (20 - 10.58)/6 = 1.57 unit.

iii) The marginal product (MPx) of the variable input is MPx = y' = 400 + 120x - 18x^2.

iv) At the profit maximizing input level P(MPx) = W, so to prove it let's set the the value of x1 into the MPx equation:

MPx = 400 + 120*5.1 - 18*5.1^2 = 5440.

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