Answer to Question #224251 in Macroeconomics for Natty Des

Question

Suppose a perfectly competitive firm with total cost function given as:
TC= 400+20Q-2Q2+2
3Q
3.
A. Find profit maximizing level of output and the maximum profit if the average revenue
equals $180.
B. Calculate the shutdown level of output and price.

Answer 1

Solution:

A.). A perfectly competitive firm will find its profit-maximizing level of output where MR = MC.

For a perfectly competitive firm: AR = MR = P

AR = 180

Therefore, MR = 180

Derive MC from Total Cost (TC):

TC = 400+20Q-2Q²+23Q³

MC = $\frac{\partial TC} {\partial Q} =$ 20 – 4Q + 69Q² = 69Q² – 4Q + 20

MC = 69Q² – 4Q + 20

MC = MR

69Q² – 4Q + 20 = 180

69Q² – 4Q + 20 – 180 = 180 – 180

69Q² – 4Q – 160 = 0

Solve through quadratic function:

Q = 1.55

The profit maximizing level of output = 1.55

 

Profit = TR – TC

TR = P $\times$ Q = 180 $\times$1.55 = 279

TC = 400+20Q-2Q²+23Q³

Substitute Quantity in the TC function to derive TC.

TC = 400 + 20(1.55) - 2(1.55²) + 23(1.55³)

TC = 400 + 31 – 4.81 + 85.65 = 400 + 31 + 85.65 – 4.81 = 511.84

Profit/Loss = TR – TC

Profit/Loss = 279 – 511.84 = (232.84)

Profit/Loss = (232.84)

B.). The shutdown price occurs at the minimum of the average variable (AVC) curve, a point where MC = AVC.

First, derive MC:

MC = $\frac{\partial TC} {\partial Q} =$ 20 – 4Q + 69Q² = 69Q² – 4Q + 20

MC = 69Q² – 4Q + 20

Then, derive the AVC:

AVC = $\frac{VC} {Q} =\frac{20Q - 2Q^{2} + 23Q^{3} } { Q} = 20 - 2Q + 23Q^{2}$

Set MC = AVC

69Q² – 4Q + 20 = 20 – 2Q + 23Q²

69Q² – 4Q + 20 – 23Q² + 2Q = 20

46Q² – 2Q = 0

Solve through quadratic equation:

Q = 0.04

The shutdown level of output = 0.04

Derive shutdown price:

P = MC

P = 69Q² – 4Q + 20

P = 69(0.04)² – 4(0.04) + 20

P = 11.04 – 0.16 + 20

P = 19.95

The shutdown price = 19.95