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, quantities, and profits for each scenario are calculated by maximizing the profit functions using the demand functions provided. The preferred strategy depends on which scenario results in the highest profit for each firm.

Step-by-step solution

Calculating the Nash Equilibrium for Simultaneous Pricing

Firstly both firms maximize their profits, supposing the other firm's action is given. For zero marginal costs, the profit of a firm is just the firm’s price times the quantity it sells, namely \( \Pi_{i}=P_{i}Q_{i} \). Firm 1 maximizes: \( \Pi_{1}=P_{1}(20-P_{1}+P_{2}) \) and Firm 2 maximizes: \( \Pi_{2}=P_{2}(20+P_{1}-P_{2}) \). To find the best responses, take the first derivatives with respect to own price and setting equal to zero. This allows us to find the prices \( P_{1} \) and \( P_{2} \) that maximize the profit for each firm.

Substituting and Calculating the Equilibrium Prices and Profits

Solving the equations from step 1, allows us to find the Nash equilibrium which is the price combination ( \( P_{1} \), \( P_{2} \)) where neither firm would want to deviate, given the price of the other. The equilibrium quantities and profits can also be calculated using these equilibrium prices.

Calculating for Sequential Pricing

Here, we suppose Firm 1 sets its price first (it becomes the leader) and then Firm 2 sets its price. Find the best response of Firm 2 as in step 1 but by treating \( P_{1} \) as a given. Substitute this into the profit function of firm 1 and maximize it with respect to \( P_{1} \) to obtain \( P_{1} \). The resulting equilibrium prices, quantities and profits can also be calculated for this scenario.

Selecting Optimal Strategy Given three Options

Analyzing the results of the previous steps, decide which scenario yields the highest profit for a firm. Consider the three available options: (i) Both firms set price at the same time; (ii) Firm sets price first; or (iii) competitor sets price first.

Key concepts

Cournot Competition

In a Cournot competition, companies determine their output quantity in order to maximize their profits, considering the quantity choices of their competitors.
This model is built on the assumption that companies choose how much to produce, and not the price itself.
The result is a Nash equilibrium, where each firm's output decision is optimal given the output decision of other firms.

• Firms simultaneously decide their output levels.
• The market price is determined by the total output of all firms.
• Each firm decides the best production level by assuming competitors' production will remain constant.

In our original exercise, the concept of Cournot competition can be indirectly applied by analyzing each firm's output or pricing decision with respect to the other.
Essentially, though it appears like pricing competition, each firm carefully chooses their pricing strategy by estimating how the other firm will react.
This strategic interaction leads to a stable state or Nash equilibrium where neither firm wishes to change their decision.

Sequential Pricing

Sequential pricing occurs when firms take turns setting prices, moving away from simultaneous decision-making.
This results in a dynamic strategic interaction, typically modeled using the Stackelberg leadership model.

• The first firm (the leader) sets its price first.
• The second firm (the follower) observes the leader's decision and optimizes its price accordingly.
• Sequential decisions often give the leader an advantage, as they can anticipate and incorporate the follower's reactions.

In the exercise, Firm 1 sets its price first, making it the leader, and Firm 2 sets its price after observing Firm 1's choice.
This strategic positioning allows Firm 1 to potentially enjoy higher profits by factoring in Firm 2's likely response into its price setting.
The result is a new equilibrium different from the one found in simultaneous pricing, often resulting in a more advantageous position for the leader.

Profit Maximization

Profit maximization entails choosing the price or quantity that leads to the highest possible profit.
Companies in competitive markets constantly analyze and adjust their pricing strategies to balance supply and demand, all while considering costs.

• Profits are maximized by setting the derivative of the profit function with respect to price or quantity to zero.
• The demand function of a firm, reliant on its pricing and that of its competitor, plays a crucial role.
• Firms aim to set prices where total revenue is highest above the total cost, leading to maximum profit.

In the exercise, both firms optimize their profits based on zero marginal costs, thus profit is simply the firm's price times the quantity sold.
By solving equations to find optimal pricing strategies, each firm determines a strategic price that maximizes its own profit given the competitor's probable actions.
The process showcases how businesses arrive at pricing decisions by leveraging calculus to find where profits plateau, which is the goal in profit maximization.