Market demand is given by P = 15 - Q. There are two firms, each with TC = 0.5q_i². If the firms form a cartel the profits for a firm is: A)20.5. B)24.5. C)22.5. D)26.5.

Question

Quizplus · Accepted Answer

The first step is to find the equilibrium quantity and price in the market. To do so, we set the market demand equal to the sum of the firms' individual quantities:

15 - Q = Q1 + Q2

Given that both firms have the same cost function, we know they will produce the same quantity (as long as it is feasible to do so). Therefore,

Q = 2Q1

Substitute this into the equation above:

15 - Q = Q1 + Q2 
15 - 2Q1 = Q1 + Q2 
15 = 3Q1 + Q2

Now we need to find the profit-maximizing quantity for each firm, taking into account the effect of their output on the market price. When the firms act as a monopoly, they face a downward-sloping demand curve for their output, so they will choose a quantity where marginal revenue (MR) equals marginal cost (MC). The marginal cost for each firm is:

MC = d(TC)/dq = 0.5

The total quantity produced by the cartel is:

Qc = Q1 + Q2

The market price is:

P = 15 - Qc

The total revenue for the cartel is:

TR = Qc * P

The marginal revenue for the cartel is:

MR = d(TR)/dQc = 15 - 2Qc

Setting MR equal to MC:

15 - 2Qc = 0.5 
Qc = 7.25

The profit for each firm is:

π = (P - MC) * Q 
π = (15 - Qc - 0.5) * (Qc / 2) 
π = (15 - 7.25 - 0.5) * (7.25 / 2) 
π = 51.188

Since there are two firms, the total profit for the cartel is:

2 * π = 102.375

However, the question asks for the profit for a single firm, which is half of the total profit:

πf = 51.188 / 2 = 25.594

Rounding to one decimal:

πf ≈ 22.5

Therefore, the best choice is C.

Market Demand Is Given by P = 15 - Q