Question

Consider a monopolist who sells its product in two distinct markets (and therefore can charge different prices in these different markets). By the way, charging observably different group different prices is called 3rd-degree price discrimination. The cost function is

C(Q1 + Q2) = 0.25[Q1 + Q2]²

The inverse demand curves for these markets are p1(Q1) = 200 – Q1 for market 1 and p2(Q2) = 300 – Q2 for market 2, where Q1 is the quantity sold in market 1 and Q2 is the quantity sold in market 2. At some point below, you may wish to use the marginal cost function, which is MC(Q1 + Q2) = 0.5[Q1 + Q2].

a) State this 3rd-degree price discriminating monopolist’s maximization problem, using the particulars of this problem. Think carefully about what this monopolist is choosing and what it is trying to maximize.

Answer 1

The monopolist's profit maximization problem involves calculating the quantity to sell in each market to maximize profits, considering cost and inverse demand functions. Marginal revenue for each market is set equal to marginal cost to find the optimal quantities, which determines the profit-maximizing prices. The monopolist's profit is the difference between total revenue and total cost.

Step-by-step explanation:

The profit maximization problem for a monopolist able to engage in third-degree price discrimination involves determining the quantities to produce for different markets to maximize profit. Taking into account the cost function C(Q1 + Q2) = 0.25[Q1 + Q2]² and the inverse demand functions p1(Q1) = 200 – Q1 for market 1 and p2(Q2) = 300 – Q2 for market 2, the monopolist will calculate the marginal revenues for each market and equate them to the marginal cost function MC(Q1 + Q2) = 0.5[Q1 + Q2].

The decision variables are the quantities Q1 and Q2, and the goal is to find the combination that maximizes total profit, which is total revenue across both markets minus total cost.

To apply this, the monopolist would set marginal revenue equal to marginal cost in each market (MR1 = MC and MR2 = MC) and solve the resulting system of equations for Q1 and Q2. The optimal prices P1 and P2 can then be found by substituting these quantities back into the respective inverse demand equations.

Once the optimal quantities Q1 and Q2 are established, the monopolist will calculate the total revenue in both markets as TR = P1 × Q1 + P2 × Q2 and subtract the total cost TC = C(Q1 + Q2) to find the profit. The profit is then given by Profit = Total Revenue - Total Cost.

The profit-maximizing monopolist generates ongoing profits that are not eroded by competition due to barriers to entry, an indicator of a monopoly power.