Question

Assume for simplicity that a monopolist has no costs of production and faces a demand curve given by \\[ Q=150-P \\] a. Calculate the profit-maximizing price-quantity combination for this monopolist. Also calculate the monopolist's profits. b. Suppose a second firm enters the market. Let \(q_{1}\) be the output of the first firm and \(q_{2}\) the output of the second. Market demand is now given by \\[ q_{1}+q_{2}=150-P \\] Assuming this second firm also has no costs of production, use the Cournot model of duopoly to determine the profit-maximizing level of production for each firm as well as the market price. Also calculate each firm's profits. c. How do the results from parts (a) and (b) compare to the price and quantity that would prevail in a perfectly competitive market? Graph the demand and marginal revenue curves and indicate the three different price-quantity combinations on the demand curve.

Short answer

Answer: The key differences between the three market structures are as follows: 1. Monopolist: The profit-maximizing price-quantity combination for a monopolist is P = 75 and Q = 75. This is where the marginal revenue equals marginal cost. 2. Cournot duopoly: In the Cournot model of duopoly, the market price is the same as in a perfectly competitive market (P = 0) but the quantity produced is less than what would be produced in a perfectly competitive market (Q = 75). 3. Perfectly competitive market: In a perfectly competitive market, price equals marginal cost (P = 0) and the quantity produced is equal to the total demand (Q = 150). The graph of the demand and marginal revenue curves shows these differences, with separate points at which each market structure's price-quantity combination lies on the demand curve.

Step-by-step solution

Calculate the profit-maximizing price-quantity combination for a monopolist

First, we need to calculate the inverse demand function, which will allow us to determine the marginal revenue: Inverse demand: P = 150 - Q Now, let's find the marginal revenue (MR) by taking the derivative of the total revenue (TR) function with respect to Q. Total revenue (TR) = P * Q = (150 - Q) * Q Marginal revenue (MR) = d(TR)/d(Q) = 150 - 2Q To find the profit-maximizing level of output for the monopolist, we need to set the MR equal to the marginal cost (MC). Since the monopolist has no costs of production, MC = 0. Therefore, 150 - 2Q = 0 Solving for Q gives us: Q = 75 Now, we can find the optimal price using the inverse demand function: P = 150 - Q = 150 - 75 = 75 So the profit-maximizing price-quantity combination for the monopolist is P = 75 and Q = 75.

Calculate the profit of the monopolist

We can now calculate the profit of the monopolist by using the following profit formula: Profit = (Price - Cost) * Quantity Since the monopolist has no costs, the profit is: Profit = (75 - 0) * 75 = 75 * 75 = 5625

Use the Cournot model of duopoly to determine the profit-maximizing level of production for each firm and the market price

In the Cournot model, each firm's output is a function of the other firm's output: q1 = (150 - P)/2 q2 = (150 - P)/2 Market demand is given by: q1 + q2 = 150 - P Adding the optimal output functions of both firms, we get: (150 - P)/2 + (150 - P)/2 = 150 - P Simplifying this equation, we obtain: 150 - P = 150 - P Therefore, the profit-maximizing level of production for each firm is the same as the monopolist's production: q1 = q2 = 75 Now, we can find the market price using the inverse demand function: P = 150 - (q1 + q2) = 150 - (75 + 75) = 150 - 150 = 0

Calculate each firm's profits

As both firms have no costs of production, their profits are: Profit_1 = (0 - 0) * 75 = 0 * 75 = 0 Profit_2 = (0 - 0) * 75 = 0 * 75 = 0

Compare the results with a perfectly competitive market

In a perfectly competitive market, the price equals the marginal cost, which in this case is zero, and the quantity produced is equal to the total demand. In this exercise, the Cournot model of duopoly has the same price as a perfectly competitive market (P = 0), but the quantity produced is less than what would be produced in a perfectly competitive market (Q = 75).

Graph the demand and marginal revenue curves indicating the price-quantity combinations

To graph the demand and marginal revenue curves, follow these steps: 1. Plot the inverse demand function of P = 150 - Q 2. Plot the marginal revenue function of MR = 150 - 2Q 3. Mark the point for the monopolist's profit-maximizing price-quantity combination (P = 75, Q = 75), and label it as 'M' (monopolist) 4. Mark the point for the Cournot model of duopoly (P = 0, Q = 75), and label it as 'C' (Cournot) 5. Mark the point for the perfectly competitive market (P = 0, Q = 150), and label it as 'PC' (perfectly competitive) The graph should show the demand curve decreasing as the quantity increases and the marginal revenue curve being a steeper downward slope than the demand curve. The different price-quantity combinations (M, C, and PC) should be indicated on the demand curve, highlighting the differences between the three market structures.

Key concepts

Understanding Monopoly Pricing

Monopoly pricing occurs when a single firm supplies the entire market, allowing it to control prices and output levels. In our example, the monopolist can maximize profits by setting a price of 75, as calculated using the inverse demand and marginal revenue functions. The inverse demand function, \(P = 150 - Q\), helps determine the price at which a specific quantity can be sold. Marginal revenue, found by differentiating the total revenue function \((150-Q)Q\), gives us \(MR = 150 - 2Q\).

To find the profit-maximizing level of output, set \(MR = MC\), where \(MC\) is the marginal cost of production. Here, since costs are zero, it's straightforward. The calculation \(150 - 2Q = 0\) results in \(Q = 75\), giving a price of \(P = 75\) through the inverse demand equation. The concept behind monopoly pricing is that the firm can decide the price and quantity independently to maximize profits, leading to a profit depicted by \(75*75 = 5625\) in the example. Thus, monopolies can control markets without competition driving prices down to marginal costs.

Exploring Duopoly with the Cournot Model

The Cournot model illustrates a duopoly where two firms influence market prices based on their production output. Each firm's decision impacts the other's profits, unlike in monopoly markets. In our problem, each firm assumes the other’s output as given and sets its output to maximize its profit.

The formula \(q_1 = (150-P)/2\) and \(q_2 = (150-P)/2\) reflects how each firm perceives the market demand \(q_1 + q_2 = 150 - P\). When you plug in optimal output levels into the demand equation, both firms choose a production level of \(Q = 75\), where they collectively saturate the market. However, this results in a market price of zero, unlike monopolistic settings, as each firm's reaction causes prices to drop to marginal costs.

Competing firms in duopoly markets often lead to less profit per firm due to share division. Here, each firm earns zero profit \((0 - 0)*75\), showing how competition can force prices to be lower compared to monopolistic markets. Cournot's model helps understand interdependency among firms, capturing the essence of strategic decision-making in duopolistic frameworks.

The Dynamics of Perfect Competition

In perfectly competitive markets, numerous small firms exist without any single entity controlling prices. Products are homogenous and firms act as price takers, meaning they have to sell at market-determined prices.

Here, the market price equals the marginal cost, which is zero in our scenario. This implies prices continue to fall until they match marginal costs, diverging from monopoly settings where prices exceeded marginal costs. As calculated, the market produces at \(Q = 150\) where supply meets demand. This fulfils total market demand at the lowest possible cost.

The perfectly competitive outcome achieves allocative efficiency, reflecting consumer preferences perfectly at the price where \(P = MC\). Unlike monopolies and duopolies, no single firm possesses market power to set prices. Thus, all economic profits are eliminated in the long run as any profits attract new entrants. The result is a more consumer-favorable environment with prices equating to production costs, maximizing societal welfare and guaranteeing no excess profit for firms.