Question

Two firms compete by choosing price. Their demand functions are $$Q_{1}=20-P_{1}+P_{2}$$ and $$Q_{2}=20+P_{1}-P_{2}$$ where \(P_{1}\) and \(P_{2}\) are the prices charged by each firm, respectively, and \(Q_{1}\) and \(Q_{2}\) are the resulting demands. Note that the demand for each good depends only on the difference in prices; if the two firms colluded and set the same price, they could make that price as high as they wanted, and earn infinite profits. Marginal costs are zero. a. Suppose the two firms set their prices at the same time. Find the resulting Nash equilibrium. What price will each firm charge, how much will it sell, and what will its profit be? (Hint: Maximize the profit of each firm with respect to its price. b. Suppose Firm 1 sets its price first and then Firm 2 sets its price. What price will each firm charge, how much will it sell, and what will its profit be? c. Suppose you are one of these firms and that there are three ways you could play the game: (i) Both firms set price at the same time; (ii) You set price first; or (iii) Your competitor sets price first. If you could choose among these options, which would you prefer? Explain why.

Short answer

The Nash equilibrium prices, sales and profits will be determined by solving equations obtained from differentiating firms' profit functions. The strategic preference will depend on which scenario yields the highest profit for the firm.

Step-by-step solution

Solve for the Nash Equilibrium in simultaneous price-setting game

Where firms set their prices simultaneously, we need to calculate the Nash equilibrium. Firm 1’s profit will be \(π_1 = P_1Q_1\) and similarly, Firm 2’s profit will be \(π_2 = P_2Q_2\). Substituting the demand functions into the profit functions for both firms yields: \(π_1 = P_1(20-P_1+P_2)\) and \(π_2 = P_2(20+P_1-P_2)\). Differentiating these profit functions with respect to their own prices and solving for optimum values of prices yields the Nash equilibrium.

Solve for the Nash Equilibrium in sequential price-setting game

In this scenario, Firm 1 sets its price first, therefore it ought to consider how Firm 2 will react to any given price. Solve for the reaction function of Firm 2 by differentiating its profit function with respect to P2, holding P1 fixed. Then, substitute this reaction function into the profit function of Firm 1 and differentiate with respect to P1 to attain Firm 1’s optimal price. Use this value of P1 and substitute it back into Firm 2's reaction function to get P2.

Analyse strategic preference

We compare profits of each firm under the three scenarios: simultaneous price-setting, firm 1 setting price first, and firm 2 setting price first. The preference of one of the firms will depend on which scenario yields the highest profit for it.

Key concepts

Price Competition

The concept of price competition is central to understanding the dynamics of markets, especially in microeconomics. In a situation where multiple firms offer similar products or services, they must determine the price at which to sell their offerings.

Typically, a lower price can attract more customers, but it may also reduce profit margins; conversely, a higher price might mean higher margins but fewer sales. The challenge for firms is finding the sweet spot that maximizes profits while remaining competitive.

In the given scenario with two firms, the competition revolves around strategically setting prices to capture a greater share of the market without compromising profitability. The complexity increases as each firm’s demand is affected not only by its own price but also by the price set by its competitor, leading to a highly interdependent and strategic environment.

Demand Functions

Demand functions are mathematical representations that show how the quantity demanded of a good or service varies with price. In microeconomic models, they are crucial for predicting consumer behavior and for firms to plan their pricing strategies.

In the provided exercise, each firm’s demand function includes terms for both the firm’s own price and the price set by its competitor, signifying that the demand for each firm's product is sensitive to the relative pricing between them. For instance, if one firm sets a higher price than the other, the demand shifts in favor of the more competitively priced product.

These functions disclose that demand can be swayed by not just the absolute price, but also the comparative pricing, a dynamic which is particularly significant in markets with few competitors or in oligopolies.

Marginal Costs

Marginal cost is the cost of producing one additional unit of a good or service. In traditional economic analysis, setting prices based on marginal costs is fundamental as it informs businesses on how to adjust production in response to changes in demand and price.

In the exercise, it is given that marginal costs are zero, which implies that the cost of producing an additional unit is nil. This could be a portrayal of digital goods or industries where the initial investment is substantial, but additional units can be produced at no extra cost. A zero marginal cost scenario simplifies the profit maximization calculation as firms do not need to subtract any costs from their revenue to calculate profits.

Profit Maximization

Profit maximization is the process through which a firm determines the price and output level that returns the highest profit. Firms achieve profit maximization when they are able to find the price where the difference between total revenue and total costs is at its maximum.

In the case of the two competing firms with zero marginal costs, profit maximization involves determining the optimal price point that balances demand attraction with revenue generation. This entails deriving and solving the profit functions with respect to their prices, as evidenced in the exercise, to determine where profits are maximized given the demand functions and competition dynamics.

The concept is critical in strategic decision-making, where firms must anticipate the likely actions of rivals and choose their own actions accordingly to achieve the best financial outcome.

Sequential Price-Setting Game

A sequential price-setting game is a strategic scenario in which firms take turns setting prices, with one firm making its pricing decision only after observing the decision of another. This can give a 'first-mover' advantage, as the latter firm can optimize its price in response to the already established price of the competitor.

The exercise poses a variation of this game where Firm 1 has the opportunity to set its price before Firm 2. In this setting, Firm 1 must anticipate how Firm 2 will react to its price setting. For Firm 1, understanding Firm 2's reaction function—how Firm 2 will set its price in relation to different prices set by Firm 1—is critical. By doing so, it can indirectly influence the market dynamics, potentially capturing a more significant market share and securing a higher profit.