(d) What is the maximum profit the firm can achieve? (e) Calculate the point price elasticity of demand at the profit-maximizing price and quantity.
need d and e rest part are solved
answer is here
Solution:-
Given
Monthly demand function: Q=300-4P .... (1)
Cost function: TC=15Q+1000 ..... (2)
Part a
Revenue =
Q=300-4P⇒4P=300-Q⇒P=75-0.25Q
So
R=PQR=75-0.25QQR=75Q-0.25Q2 ... (3)
To maximize revenue, differentiate the equation 3 with respect to Q
dRdQ=75-0.5Q Put dRdQ=075-0.5Q=0⇒0.5Q=75⇒Q=150
Second-order condition for revenue maximization:
d2RdQ2=-0.5<0
Revenue is maximization quantity is 150
From equation 3
R=75×150-0.25×1502R=5625
The maximum revenue is $5625
part b
Differentiate equation 1 with respect to Q
dQdP=-4
Price elasticity is calculated as
e=dQdP×PQe=-4×75-0.25×150150e=-4×0.25e=-1
Price elasticity at revenue-maximizing point is -1
part c
Total cost of producing 150 units:
TC=15×150+1000TC=3250
Profit at revenue-maximizing level of output:
Profit = revenue - costProfit =5625-3250Profit =2375
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