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

For a monopolist the profit-maximizing price and quantity are approximately $24 and 29 units respectively, with profits around $551. In a duopoly, the Cournot equilibrium quantities for firm 1 and firm 2 are approximately 16 units each, and the price is around $21. Each firm gets a profit of $256. If N firms are equally producing in a competitive market, each will produce amount 48/N units, the market price will be $5 + 48/N and each firm's profit will be (48/N)^2. As N becomes infinitely large, the market price tends to $5, equivalent to the perfect competition scenario.

Step-by-step solution

Monopoly Profit Maximization

Start by setting the marginal cost (MC) and marginal revenue (MR) equal,since a monopolist maximizes its profits where MC = MR. Given that AC = MC = $5 and the market demand is \(Q=53-P\), we can calculate MR by first expressing the demand equation in terms of P = 53 - Q. The total revenue, TR, is then P * Q = Q * (53 - Q). Differentiating this with respect to Q gives the MR, which is 53 - 2Q. Setting this equal to MC results in 53 - 2Q = 5. Solving this for Q will give us the profit-maximizing quantity, and substituting this into the P = 53 - Q equation will yield the corresponding price.

Profit Function of Each Firm in Duopoly

To express the profits for each firm, use the total revenue minus total cost formulation. As given, the total market demand is Q1+Q2=53-P. The total cost for each firm is the quantity produced multiplies by the constant marginal cost, i.e. 5Q1 and 5Q2. The total revenue for each firm is their quantity produced times the market price - which is determined by the total quantity supplied.

Reaction Functions of Firms

A firm's reaction function signifies its best response, that is, its optimal output given the output of the rival firm. We take the derivative of the profit function of each firm with respect to its own quantity, set it equal to zero and solve for the firm's quantity. This will give us firm 1's reaction function as a function of Q2, and vice versa for firm 2.

Cournot Equilibrium

To find the Cournot equilibrium, set the reaction functions equal to each other and solve for Q1 and Q2. Substitute these values back into the demand curve to find the market price P. The profits for each firm can then be determined by substituting Q1 and Q2 into the respective profit functions.

Market Outcome with N Firms

If the market increases to N firms with constant marginal cost of 5, derive the reaction curve for an individual firm as a function of aggregate output from firms other than itself and solve for its output in terms of N. Substituting the expression into the market demand equation gives the Cournot equilibrium price and each firm's profit can be calculated accordingly. As N tends to infinity (large number of firms), observe the nature of the market price and compare it to that of perfect competition.

Key concepts

Profit Maximization in Monopolistic Competition

In a monopolistic competitive market, profit maximization is a crucial concept that enables firms to determine the most advantageous level of production and pricing. Understanding profit maximization involves dissecting how a firm can achieve the highest profit possible with its product or service offering.

As an example, consider a monopolist producing at a constant average and marginal cost of $5. With the market demand curve set as \(Q = 53 - P\), the firm's goal is to find the price and quantity that maximizes its profits. The profit-maximizing quantity is where marginal revenue (MR) equals marginal cost (MC), and for monopoly markets, the MR curve is derived from the demand curve. This approach aligns with the economic principle that the incremental gain from selling one more unit (MR) must match the incremental cost (MC) of producing that unit.

By expressing the demand equation in terms of price and then calculating total revenue (TR = P * Q), we subsequently obtain the MR by differentiating TR with respect to Q. Setting MR equal to MC gives us the optimal output level. The corresponding market price is then deduced from the demand curve, ensuring that the firm maximizes profit at this quantity and price point.

Cournot Equilibrium and Its Significance

The Cournot equilibrium is another fundamental aspect of oligopoly markets, where several firms compete with each other by deciding the amount of output to produce independently. It represents a stable state where no firm has an incentive to change its output, given the output levels of its competitors.

Following the exercise with two firms entering a market, the Cournot equilibrium is achieved by determining the optimal output for both firms under the assumption that each firm's output decision is based on the other's output being fixed. This is illustrated by finding each firm's reaction function, which maps its best response in output to the quantity produced by its competitor.

Once the reaction functions for both firms are understood, we can calculate the Cournot equilibrium outputs by setting these functions equal to one another and solving for the quantities \(Q_1\) and \(Q_2\). After determining these output levels, we can then derive the market price and calculate each firm's profits based on this equilibrium state. This Cournot framework is vital for analyzing strategic decision-making in oligopolistic competitions.

The Role of the Reaction Function in Oligopolistic Markets

A reaction function in oligopoly theory encapsulates the relationship between a firm's optimal output and its competitor's output. It is essentially the firm's strategic response to the quantity produced by the rival firm, hence the term 'reaction' function.

In the provided exercise, each firm's reaction function was determined by taking the derivative of the profit function with respect to its own output and then solving it while holding the competitor’s output constant. For example, if firm 1 increases its output, firm 2's reaction function would guide firm 2 in adjusting its own output to maintain profitability.

Understanding Firm 1's Reaction Function

Considering firm 1, its reaction function would prescribe the level of \(Q_1\) in response to different levels of \(Q_2\) from firm 2. This strategic move is like a game of chess where each player must anticipate their opponent's move and plan their strategy accordingly. The concept of the reaction function is integral to forming a Cournot equilibrium, as it lays the groundwork for firms to adjust their outputs strategically until neither can improve their situation by changing their production level unilaterally.