Question

A monopolist can produce at a constant average (and marginal) cost of \(\mathrm{AC}=\mathrm{MC}=\$ 5 .\) It faces a market demand curve given by \(Q=53-P\) a. Calculate the profit-maximizing price and quantity for this monopolist. Also calculate its profits. b. Suppose a second firm enters the market. Let \(Q_{1}\) be the output of the first firm and \(Q_{2}\) be the output of the second. Market demand is now given by \\[ Q_{1}+Q_{2}=53-P \\] Assuming that this second firm has the same costs as the first, write the profits of each firm as functions of \(Q_{1}\) and \(Q_{2}\) c. Suppose (as in the Cournot model) that each firm chooses its profit- maximizing level of output on the assumption that its competitor's output is fixed. Find each firm's "reaction curve" (i.e., the rule that gives its desired output in terms of its competitor's output). d. Calculate the Cournot equilibrium (i.e., the values of \(Q_{1}\) and \(Q_{2}\) for which each firm is doing as well as it can given its competitor's output). What are the resulting market price and profits of each firm? *e. Suppose there are \(N\) firms in the industry, all with the same constant marginal cost, \(\mathrm{MC}=\$ 5 .\) Find the Cournot equilibrium. How much will each firm produce, what will be the market price, and how much profit will each firm earn? Also, show that as N becomes large, the market price approaches the price that would prevail under perfect competition.

Short answer

a) For a monopolist, the profit-maximising quantity and price can be found by equating MR and MC, resulting in a profit assurance that depends on the chosen quantities. \n b) In a duopoly, each firm's profits depend on its own production level and its competitor's, resulting in reaction curves specific to each firm. \n c) The reaction curve of each firm is derived from maximising its profits on the assumption of its competitors' output being fixed. \n d) The Cournot equilibrium refers to the point where neither firm can better its profit given its competitor's output. \n e) For N firms, the solution is analogous to the case with two firms. As N grows large, firms act similar to perfect competition where price equals marginal cost.

Step-by-step solution

Calculate the Profit-Maximizing Price and Quantity

To find the profit-maximizing price and quantity, we need to set Marginal Revenue (MR) equal to Marginal Cost (MC). First, the total revenue (TR) function is determined by multiplying price (P) by the quantity (Q), which is calculated using the inverse demand curve. The MR is the derivative of the TR function. Since MC is a constant 5, we equate this to the MR and solve for the quantity. This can then be substituted back into either the demand function or price function to find the corresponding price.

Profits of Each Firm as Functions of \(Q_1\) and \(Q_2\)

In this step, we determine the profit functions of both firms. Both firms have the same cost structures and thus their profit functions will be similar. The profit function is typically given by \(\pi = TR - TC\), where \(\pi\) represents profit, TR is total revenue, and TC is total cost. Total cost is computed by multiplying the quantity for each firm ( \(Q_1\) or \(Q_2\) ) by the cost per unit. Total revenue for each firm can be computed by multiplying the quantity for each firm ( \(Q_1\) or \(Q_2\) ) by the price, which is determined by the market demand function shared by both firms.

Find the Reaction Curve with Cournot Model

The Cournot Model suggests that each firm chooses its output levels based on its competitor's output. To find the reaction functions, differentiate the profit function of each firm with respect to its own output and equate that to zero.

Calculate the Cournot Equilibrium

The Cournot equilibrium can be found by simultaneously solving the two reaction function equations obtained in Step 3. This gives the optimal output for each firm. The resulting market price can be computed by substituting either \(Q_1\) or \(Q_2\) into the inverse demand function.

Extend the Cournot Model to N firms

The Cournot equilibrium needs to be found in case of N firms. Similar logic as the duopoly case needs to be applied. Another aspect to be shown is that as N becomes large, the market price approaches the price under perfect competition, which is equal to the marginal costs in this case.

Key concepts

Profit-Maximizing Price

When a firm aims to maximize its profits, it sets a specific price that aligns with this goal. The profit-maximizing price is determined where the difference between the firm's total revenue (TR) and total costs (TC) is at its greatest. A fundamental principle in economics is setting the Marginal Revenue (MR) equal to the Marginal Cost (MC).

Considering a monopolistic firm's scenario with a constant marginal cost of \(5, finding the profit-maximizing price involves calculus. One must first establish the TR by multiplying the price, which is derived from the inverse demand function, by the produced quantity, Q. Taking the derivative of the TR function gives us MR. Equating the MR to the MC of \)5, and solving for Q, we find the profit-maximizing quantity. This quantity is essential to pinpoint the profit-maximizing price, after which the firm calculates its profits by subtracting the total cost from the total revenue at this price point.

Reaction Curve

The reaction curve in the context of oligopoly markets, such as the Cournot duopoly, represents a firm's optimal response to the quantity produced by its rival.

In the Cournot model, each firm assumes that its competitor's output is fixed and decides its quantity to maximize profits based on this assumption. The reaction function is mathematically derived by differentiating the profit function with respect to the firm's output and setting the derivative equal to zero. This process reveals how much one firm will produce in response to various quantities produced by the other firm. When dealing with multiple firms, as in the extended Cournot model, the reaction curve for each firm will be influenced by the collective output of all its competitors.

Marginal Cost

Marginal Cost (MC) is a critical concept in economics and refers to the additional cost incurred for producing one more unit of a good. In our textbook example, the constant MC is denoted by $5. This implies that no matter the level of output, each additional unit's cost remains the same.

In a profit-maximization context, the MC plays a vital role as firms equate this to their marginal revenue to decide on their production level. If MC were to rise with increased output, it would complicate the profit-maximization calculus, potentially requiring more intricate strategies for setting prices and output.

Perfect Competition

Under perfect competition, numerous small firms compete against each other, none of them having any significant market power to set prices. In such markets, the price of goods tends toward the marginal cost of production. This is a consequence of the fact that if firms set prices above the marginal cost, they would not be able to successfully sell their products as consumers would switch to a different seller offering a lower price.

In the textbook exercise, it's shown that as the number of firms increases in the Cournot model, reaching a large number N, the resulting equilibrium price approaches the price that would exist under perfect competition conditions. This convergence illustrates the relationship between oligopoly dynamics and perfect competition, highlighting that the more competitors there are in a market, the more the outcome resembles that of a perfectly competitive market.