Q2) Given a firm’s demand function, P = 24 - 0.5Q and the average cost function, AC = Q2 – 8Q + 36 + 3/Q, calculate the level of output Q which:

a) maximizes total revenue
b) maximizes profits

A) Total revenue is the product of the quantity sold and the price at which they were sold.

(total revenue) = Q*P
= Q*(24 - .5Q)
= -.5Q^2 + 24Q
This quadratic function will have a maximum at Q = -24/(2*(-.5)) = 24
Total revenue will be maximized when Q = 24.

B) The profit on each unit is the difference between the selling price and the cost. The total profit will be the product of the unit profit and the number of units sold.
(total profit) = Q*(P - AC)
= Q*((24 - .5Q) - (Q^2 - 8Q + 36 + 3/Q))
= 24Q - .5Q^2 - Q^3 + 8Q^2 - 36Q - 3
= -Q^3 + 7.5Q^2 - 12Q - 3
This will be maximized when the derivative is zero.
-3Q^2 + 15Q - 12 = 0
Q^2 - 5Q + 4 = 0 (divide by -3)
(Q - 1)(Q - 4) = 0
This suggests that quantities of 1 or 4 will maximize profit. Using our equation for total profit, we find that profit for Q=1 is negative. Profit will be maximized when Q = 4.
Here is a plot of profit versus quantity sold.

To find the level of output that maximizes total revenue and maximizes profits, we will need to analyze the demand function, the average cost function, and understand their relationship.

a) To maximize total revenue:
Total revenue is calculated by multiplying the price per unit (P) by the quantity sold (Q). So, Total Revenue (TR) = P * Q.

Given the demand function P = 24 - 0.5Q, we can substitute this value of P into the Total Revenue equation: TR = (24 - 0.5Q) * Q.

To maximize total revenue, we need to take the derivative of the Total Revenue equation with respect to Q and set it equal to zero to find the critical points. Then, we can solve for Q to find the level of output that maximizes total revenue.

b) To maximize profits:
Profits are calculated by subtracting the total cost (TC) from the total revenue (TR). So, Profit (π) = TR - TC.

Given the average cost function AC = Q^2 – 8Q + 36 + 3/Q, we can find the total cost by multiplying the average cost (AC) by the quantity sold (Q): TC = AC * Q.

To maximize profits, we need to find the level of output that maximizes the difference between total revenue and total cost. This can be done by taking the derivative of the profit equation with respect to Q and setting it equal to zero to find the critical points. Then, we can solve for Q to find the level of output that maximizes profits.

Note: To calculate the total cost function, multiply the average cost (AC) by the quantity (Q), not the average cost squared as shown in the original equation.

By solving these equations, we can determine the level of output that maximizes total revenue and maximizes profits.