Suppose the market for dog parachutes, commonly known as Air Buds, has a single
producer and the demand curve is given as P = 100 –Q. Furthermore, suppose the MC
curve for the firm can be written as MC = 40 + 4q. The firm’s TC can be expressed as TC
= 40q + 2q2 + 300. Use this information to answer this set of questions.
a. What is the profit maximizing price and quantity for this monopolist given the
above information? (3pts)
b. Calculate the monopolist’s profit. (2 pts)
c. Calculate the monopolist’s consumer surplus (CS), producer surplus (PS), and
deadweight loss (DWL). (6 pts)
a. To find the profit maximizing price and quantity, we first need to find the monopolist's marginal revenue (MR) and set it equal to their marginal cost (MC).
Given that P = 100 - Q, we can find the monopolist's total revenue (TR) as TR = P * Q = (100 - Q) * Q = 100Q - Q^2.
Therefore, the monopolist's marginal revenue (MR) is the derivative of total revenue with respect to quantity, which is MR = dTR/dQ = 100 - 2Q.
Setting MR equal to MC, we have:
100 - 2Q = 40 + 4Q
60 = 6Q
Q = 10
So, the profit maximizing quantity is Q = 10. Plugging this quantity back into the demand curve, we can find the profit maximizing price:
P = 100 - Q
P = 100 - 10
P = 90
Therefore, the profit maximizing price is $90 and the profit maximizing quantity is 10 units.
b. To calculate the monopolist's profit, we need to find the total revenue, total cost, and then subtract total cost from total revenue.
Total Revenue = P * Q = 90 * 10 = $900
Total Cost = TC = 40 * 10 + 2 * 10^2 + 300 = $540
Profit = Total Revenue - Total Cost = $900 - $540 = $360
Therefore, the monopolist's profit is $360.
c. To calculate the monopolist's consumer surplus, producer surplus, and deadweight loss, we first need to find the equilibrium price and quantity in a perfectly competitive market.
In a perfectly competitive market, price is equal to marginal cost:
100 - Q = 40 + 4Q
60 = 5Q
Q = 12
P = 100 - 12 = $88
Consumer Surplus (CS) in a perfectly competitive market is the area under the demand curve but above the price line, up to the quantity of 12. This can be calculated as 0.5 * 12 * (100-88) = $72.
Producer Surplus (PS) in a perfectly competitive market is the area under the price line but above the marginal cost curve. This can be calculated as 0.5 * 12 * (88-40) = $288.
Deadweight Loss (DWL) is the loss in total surplus due to the monopolist's pricing power. It can be calculated as the difference in total surplus between the monopoly pricing and perfectly competitive pricing.
Total Surplus (TS) = CS + PS = $72 + $288 = $360
DWL = TS (monopoly) - TS (perfectly competitive) = ($900 - $540) - ($72 + $288) = $0
Therefore, the monopolist's consumer surplus is $0, producer surplus is $360, and there is no deadweight loss in this case.