A business firm has the following total revenue, R = f(Q) = 15Q – 6Q² and total cost, C = f(Q) = 2Q³ – 3Q² + 2 functions

(a) Find the output (Q) that maximizes the profit of the business firm.

To find the equilibrium set market demand equal to market supply: 1000 – 2Q =
100 + Q. Solving for Q, you get Q = 300. Plugging 300 back into either the
market demand curve or the market supply curve you get P = 400.From part (a) you know the equilibrium market price is $400. You also know that
the firm profit maximizes by producing that level of output where MR = MC.
Since the equilibrium market price is the firm’s marginal revenue you know that
MR = $400. Setting MR = MC gives you 400 = 2q + 1, or q = 199.5. Thus, the
profit maximizing level of output for the firm is 199.5 units when the price is
$400 per unit. Using this information it is easy to find total revenue as the price
times the quantity: TR = ($400 per unit)(199.5 units) = $79,800. Total cost is
found by substituting q = 199.5 into the TC equation: TC = $40,099.75. Profit is
the difference between TR and TC: Profit = TR – TC = 79,800 – 40,099.75 =
$39,700.25. Since profit is not equal to zero this cannot be