Question

Suppose that firms' marginal and average costs are constant and equal to \(c\) and that inverse market demand is given by \(P=a-b Q\) where \(a, b > 0\) a. Calculate the profit-maximizing price-quantity combination for a monopolist. Also calculate the monopolist's profit. b. Calculate the Nash equilibrium quantities for Cournot duopolists, which choose quantities for their identical products simultaneously. Also compute market output, market price, and firm and industry profits. c. Calculate the Nash equilibrium prices for Bertrand duopolists, which choose prices for their identical products simultaneously. Also compute firm and market output as well as firm and industry profits. depose now that there are \(n\) identical firms in a Cournot model. Compute the Nash equilibrium quantities as functions of \(n\). Also compute market output, market price, and firm and industry profits. e. Show that the monopoly outcome from part (a) can be reproduced in part (d) by setting \(n=1\), that the Cournot duopoly outcome from part (b) can be reproduced in part (d) by setting \(n=2\) in part (d), and that letting \(n\) approach infinity yields the same market price, output, and industry profit as in part (c).

Short answer

In equilibrium, a monopolist (single seller) sets a profit-maximizing price and quantity, leading to a relatively high price and relatively low output compared to more competitive market structures. The monopolist's profit is greater than zero. In a Cournot duopoly (two firms compete in quantities), the Nash equilibrium results in lower prices, higher output, and lower profits for each firm compared to a monopolist. The industry as a whole has a larger output and lower market price compared to the case of monopolist. In a Bertrand duopoly (two firms compete in prices), the Nash equilibrium prices equal the constant marginal costs, resulting in an output allocation between the firms and zero profits for each firm. This outcome converges to perfect competition as the number of firms in the market increases (as shown in part (d) when n approaches infinity).

Step-by-step solution

Write the Profit Function

The profit function is given by the difference between total revenue and total cost: \[\pi(Q)=R(Q)-C(Q)=(P \cdot Q)-(c \cdot Q)\] Since \(P=a-bQ\), we have \[\pi(Q)=(a-bQ)Q-cQ\]

Find the Profit-Maximizing Quantity

Differentiate the profit function with respect to Q and set the derivative equal to zero: \[\frac{d\pi(Q)}{dQ}=0\] \[\frac{d}{dQ}((a-bQ)Q-cQ)=0\] Solving for Q: \[Q^*=\frac{a-c}{2b}\]

Find the Profit-Maximizing Price

Plug the profit-maximizing quantity back into the inverse demand function: \[P^*=a-b\left(\frac{a-c}{2b}\right)\]

Calculate the Monopolist's Profit

Plug the profit-maximizing price and quantity back into the profit function: \[\pi^*=(P^*\cdot Q^*)-cQ^*\] b.

Write the Reaction Functions

The Cournot reaction functions for Firm 1 and Firm 2 are: \[Q_1^*=\frac{a-c+bQ_2}{2b}\] \[Q_2^*=\frac{a-c+bQ_1}{2b}\]

Solve the Reaction Functions Simultaneously

Solve for the equilibrium quantities \(Q_1^*\) and \(Q_2^*\): \[Q_1^*=\frac{a-c}{3b}\] \[Q_2^*=\frac{a-c}{3b}\]

Compute the Market Output, Market Price, and Firm and Industry Profits

Compute Q, P, and profits: \[Q=Q_1^*+Q_2^*=\frac{2(a-c)}{3b}\] \[P=a-bQ\] \[\pi_1=\pi_2=(P \cdot Q_i)-cQ_i\] \[\pi_{industry}=\pi_1+\pi_2\] c.

Write the Profit Functions for Bertrand Duopolists

The profit functions are given by: \[\pi_1=(P_1-c)\cdot Q_1(P_1,P_2)\] \[\pi_2=(P_2-c)\cdot Q_2(P_1,P_2)\]

Find Nash Equilibrium Prices

The Nash equilibrium prices for Bertrand Duopolists are both equal to the constant marginal cost: \[P_1^*=P_2^*=c\]

Compute Firm and Market Output, and Firm and Industry Profits

Compute outputs and profits given the equilibrium prices: \[Q_1(Q_1^*,Q_2^*)=Q_2(Q_1^*,Q_2^*)\] \[Q_1+Q_2=Q\] \[\pi_1=\pi_2=(P^*\cdot Q_i)-cQ_i\] \[\pi_{industry}=\pi_1+\pi_2\] d.

Write the Reaction Function for the nth Firm

The Cournot reaction function for the nth firm is: \[Q_n^*=\frac{a-c+\sum_{i=1}^{n-1}bQ_i}{(n+1)b}\]

Solve the Reaction Function for the Equilibrium Quantity as a Function of n

The equilibrium quantity for each firm is: \[Q_i^*=\frac{a-c}{(n+1)b}\]

Compute Market Output, Market Price, and Firm and Industry Profits

Compute Q, P, and profits: \[Q=\sum_{i=1}^n Q_i^*\] \[P=a-bQ\] \[\pi_i=(P \cdot Q_i^*)-cQ_i^*\] \[\pi_{industry}=\sum_{i=1}^n \pi_i\] e.

Show that Part (a) can be Reproduced with n=1

Replace n with 1 in the formulas from part (d) and show that the results are the same as in part (a).

Show that Part (b) can be Reproduced with n=2

Replace n with 2 in the formulas from part (d) and show that the results are the same as in part (b).

Show that As n Approaches Infinity, the Result Is the Same as Part (c)

As n becomes very large, the equilibrium prices and quantities converge to those of part (c). To show this, take the limit of the formulas from part (d) as n approaches infinity.

Key concepts

Profit Maximization

Profit maximization is a fundamental objective for businesses aiming to determine a price and output combination that yields the highest possible profit. This is achieved by analyzing the total revenue and total cost functions. The profit, represented as \[\pi(Q) = R(Q) - C(Q),\]is the difference between total revenue (R) and total cost (C). For a monopolist facing an inverse market demand curve given by \(P = a - bQ\), where \(a\) and \(b\) are constants and \(Q\) is quantity, profit maximization involves finding the level of \(Q\) that maximizes this profit function.
To find the profit-maximizing output, differentiate and equate the derivative to zero:\[\frac{d\pi(Q)}{dQ} = 0.\]Solving this, a monopolist obtains an optimal quantity \(Q^*\) and sets a corresponding price \(P^*\) based on this output level. With constant marginal and average costs \(c\), substitute \(Q^*\) back into the price function to find \(P^*\). The monopolist’s profit is then calculated using these values in the profit function.
This fundamental analysis underscores the importance of marginal cost and demand elasticity in dictating a firm's pricing strategy for profit maximization.

Cournot Duopoly

The Cournot duopoly model explores a market structure where two firms compete on quantities. Each firm chooses its output level independently and simultaneously, assuming the opponent's decision remains constant. This creates a strategic interdependence characteristic of oligopolistic markets.
To determine Nash equilibrium in a Cournot duopoly, derive and solve the reaction functions. For firms 1 and 2, their respective reaction functions are:\[Q_1^* = \frac{a-c+bQ_2}{2b}, \quad Q_2^* = \frac{a-c+bQ_1}{2b}.\]By solving these equations simultaneously, each firm's optimal output \(Q_1^*\) and \(Q_2^*\) can be found.
In equilibrium, each firm’s output decision aligns with the other’s choice, leading the market to have a total output \(Q = Q_1^* + Q_2^*\). The market price \(P\) aligns with the demand function, and firm profits are derived from their quantities and costs.
The Cournot model provides insights into how firms in an oligopoly can affect market outcomes through strategic quantity decisions, contrasting the monopolistic approach where one firm dominates and sets the market output and price.

Bertrand Competition

In Bertrand competition, two firms vie to set prices rather than quantities for identical products. Unlike Cournot's quantity strategy, Bertrand focuses on pricing, assuming constant marginal cost for both competitors. Firms anticipate their rivals' prices remain fixed, leading to fascinating competitive dynamics.
The Nash equilibrium in Bertrand competition is reached when both firms set their prices equal to marginal cost, specifically:\[P_1^* = P_2^* = c.\]No firm can gain by changing prices, resulting in minimal profits comparable to those found in perfectly competitive markets. This drastically contrasts with Cournot duopolies, where non-zero profits emerge due to quantity restrictions.
Despite its simplicity, the Bertrand model vividly demonstrates how price competition in oligopoly markets can drive down prices, benefiting consumers but leading to reduced profitability for firms. The model clearly differs from Cournot's outcome, illustrating diverse impacts of strategic choices in oligopolistic industries.

Nash Equilibrium

Nash Equilibrium is a central concept in game theory, indicating a scenario where no player can benefit by altering their strategy while others keep theirs unchanged. This balance of strategy applies both in the Cournot and Bertrand competition frameworks.
In a Cournot context, Nash equilibrium is obtained when each firm's chosen quantity reflects the optimal response considering the rival's output, yielding stable market output and prices. Contrarily, with Bertrand competition, the equilibrium arises when each firm's pricing strategy, set at marginal cost, leaves none incentivized to deviate.
Understanding Nash equilibrium is pivotal for analyzing strategic interactions in economic models, allowing predictions of firm behavior in competitive environments. Whether determining output quantities or sales prices, Nash equilibrium aids in grasping how firms reach a stable competitive balance, despite their strategic interdependence. This underscores its relevance across various economic theories and market contexts.