Question

4. The research department of Cardinal Novelties has estimated the demand function facing the firm for price increases and price declines from the prevailing price are, respectively: Q = 250 -12.5P and Q’ = 208-9P The marginal and average total cost functions of the firm were also estimated to be, respectively: MC =1.75 + (1/50)Q and ATC = 1.75 + (1/25)Q Determine the best level of output of the firm, the price at which the firm sells its output, as well as the total profit.

Answer 1

The research department of Cardinal Novelties face 2 distinct demand curves for deviations from the prevailing price.
For price increases:
Q = 250 - 12.5P
or, 12.5P = 250 - Q
or, P = 20 - 0.08Q
TR : PQ = 20Q - 0.08Q^2
MR =  20 - .16Q  
The profit maximizing output is obtained by equating MR with MC.
So, 20-0.16Q = 1.75 -0.02Q
or, 18.25 = 0.14Q
or, Q = 130.35
So, Total revenue = 130.35 x 9.572 = 1247.71
total cost = Total average cost x Q = 1.75Q + 0.04Q^2 = 907.75
Total profit in this case = 1247.71 - 907.75 = 339.96

For the price increase
Q' = 208-9P
or, P = 23.11 - .11Q'
TR = PQ' = 23.11Q' - .11Q'^2
MR: 23.11 -.22Q'
The profit maximizing output is obtained by equating MR with MC.
So, 23.11 -.22Q' = 1.75 -0.02Q'
or, 21.36 = 0.2Q'
or, Q' = 106.8
So, Total revenue = 106.8 x 11.36 = 1213.46
Total Cost = 1.75Q' + 0.04Q'^2= 643.14
So, Total profit = 1213.46 - 643.14 = 570.32

So total profit is higher when the price increases from the prevailing price.

SO the best level of output is at 106.8, sold at the price of 11.36 to get a total profit of 570.32