Question

What price-quantity combination maximizes total expenditure along the demand curve: ? = 90 − 3?2? What is the maximum expenditure? Show your work.

Answer 1

The inverse demand function here has been given as P=90-3Q^2 where P and Q are the price and quantity of the concerned good or service, in this case, respectively. The total expenditure=P*Q=(90-3Q^2)*Q=90Q-3Q^3 and hence, marginal or incremental expenditure=d(P*Q)/dQ=90-9Q^2. Now, we know that based on the first-order condition(FOC) to maximize total expenditure, the marginal expenditure function can be set equal to 0.

Therefore, based on the FOC to maximize total expenditure, we can state:-

90-9Q^2=0

-9Q^2=-90

Q^2=-90/-9

Q^2=10

Q=+/-3.16 approximately

Hence, the expenditure maximizing quantity of the concerned good or service purchased by the consumer or buyer would be 3.16 units approximately.

Now, plugging the value of expenditure maximizing quantity of the good or service into the inverse demand function, we can obtain:-

P=90-3Q^2

P=90-3*(3.16)^2

P=90-3*(10)

P=90-30

P=60

Thus, the expenditure maximizing per unit price of the concerned good or service, in this case, would be 60.

Therefore, the maximum total expenditure of the consumer or buyer=60*3.16 units=189.6 approximately.