The demand and cost function for a company are estimated to be as follows: P = 100 - 8Q TC = 50+80Q-10Q^2 +0.6Q^3 a. what price should the company
The demand and cost function for a company are estimated to be as follows:
P = 100 - 8Q
TC = 50+80Q-10Q^2 +0.6Q^3
a. what price should the company charge if it wants to maximize its profit in the short run?
b. what price should it charge if it wants to maximize its revenue in the short run?
c. suppose the company lacks confidence in the accuracy of cost estimates expressed in a cubic equation and simply wants to use a linear approximation. Suggest a linear representation of this cubic equation. What difference would it make on the recommended profit-maximizing and revenue-maximizing prices?
a)
Profit is maximized when MR = MC
Since P = 100-8Q
TR = P x Q
TR = (100-8Q)Q
= 100Q – 8Q2…….Equation 1
MR = 
………Equation 2
= 100-16Q
TC = 
….. Equation 3
MC = 
……..Equation 4
= 80-20Q+1.8Q2
At MR = MC
100-16Q = 80 – 20Q + 1.8Q2
80-20Q+1.8Q2-100+16Q = 0
1.8Q2 – 4Q – 20 = 0
Using the Quadratic formula:
Q = 
Q= 
= 
= 4.62
Since
P = 100 – 8Q
= 100-(8 × 4.62)
= 100 – 36.97
= 63.02
To maximize profit in the short-run, the company should charge $63.02
b)
In the short run, a revenue-maximizing firm should set the price where:
MR = 0
Since:
MR = 100 – 16Q …….(Equation 2 above)
Therefore:
100 – 16Q = 0
16Q = 100
Q = 6.25
Therefore:
P = 100 – (8 × 6.25)
P = 100 – 50
P = 50
To maximize revenue in the short-run, the firm should charge $50
c)
A Linear Function is in the form a + bx
Hence, to simplify a cubic function, we leave out the squared and cubic terms and the linear function becomes:
50 + 80Q
There will be no significant difference, the revenue maximizing price would remain the same but the profit maximizing price increases to $90
