Question

Frontier Inc. is the only producer of video games in Manbari Country. The market demand for video games is given by P=40−2Qd , where P is the price per video game and Qd is the number of video games purchased by consumers every day.

Frontier Inc. has a cost function given by C(Q)=4Q, and the associated average and marginal cost curves are given by AC(Q)=MC(Q)=4.
(a) If Frontier Inc. wants to maximize its profit, what price should it charge? How many video games will be sold at this price? What will its profit be?
(b) Determine the price elasticity of demand at the profit-maximizing output level. Is demand elastic at that point or not?
(c) Suppose now the government imposes a five percent profit tax on Frontier Inc. What price will Frontier Inc. charge now to maximize its profit? Will this price be higher than, equal to, or lower than the price in (a)? Explain and justify your answer carefully.
(d) Determine the net (after tax) profit of Frontier Inc. in equilibrium.

Answer 1

Frontier Inc. should charge a price of $32 to maximize its profit. At this price, 4 video games will be sold, resulting in a profit of $112.

The price elasticity of demand at the profit-maximizing output level is -4, indicating elastic demand.

If a 5% profit tax is imposed, Frontier Inc. should charge a price of $30 to maximize its profit, resulting in a net profit after tax of $106.40.

Step-by-step explanation:

(a) To maximize its profit, Frontier Inc. should produce at the quantity where marginal revenue (MR) equals marginal cost (MC). Since the marginal cost curve is constant at 4, the profit-maximizing quantity is determined by setting MR equal to 4. The demand function P = 40 - 2Qd can be rewritten as Qd = (40 - P) / 2. Setting MR = MC, we have 4 = (40 - P) / 2, which gives P = 32.

Therefore, Frontier Inc. should charge a price of $32. To find the quantity of video games sold at this price, we substitute P = 32 into the demand function: Qd = (40 - 32) / 2 = 4. So, Frontier Inc. will sell 4 video games at a price of $32. To calculate the profit, we subtract the total cost from the total revenue.

Total revenue is given by P x Qd, which is $32 x 4 = $128. Total cost can be calculated using the cost function C(Q) = 4Q, where Q is the quantity of video games produced. Substituting Q = 4 into the cost function gives C(4) = 4 x 4 = $16. Therefore, the profit is $128 - $16 = $112.

(b) The price elasticity of demand (PED) can be calculated using the formula: PED = (%change in quantity demanded) / (%change in price). At the profit-maximizing output level, the price is $32 and the quantity demanded is 4. If the price increases by 1%, the new price will be $32 + $0.32 = $32.32.

Using the demand function, the new quantity demanded can be calculated as Qd = (40 - 32.32) / 2 = 3.84. The percentage change in quantity demanded is (3.84 - 4) / 4 x 100% = -4%. The percentage change in price is (32.32 - 32) / 32 x 100% = 1%.

herefore, the price elasticity of demand at the profit-maximizing output level is (-4% / 1%) = -4. Since the price elasticity of demand is greater than 1 in absolute value, demand is elastic at the profit-maximizing output level.

(c) When a profit tax is imposed, the firm's profit will be reduced.

Frontier Inc. will aim to maximize its profit after taxes. To find the new price to maximize profit, we follow the same steps as in part (a) and set MR = MC. We have 4 = (40 - P - P*0.05) / 2. Solving for P gives P = 30.

Therefore, Frontier Inc. should charge a price of $30 with the profit tax. This price is lower than the price in (a) because the profit tax reduces the firm's ability to charge a higher price.

(d) To calculate the net profit after the tax, we subtract the tax from the profit calculated in part (a). The profit before tax was $112. The profit tax is 5% of the profit, which is $112 x 0.05 = $5.60.

Therefore, the net profit after the tax is $112 - $5.60 = $106.40.