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Chapter 10: Problem 7

Consider an asymmetric Cournot duopoly game, where the two firms have different costs of production. Firm 1 selects quantity \(q_{1}\) at a production cost of \(2 q_{1}\). Firm 2 selects quantity \(q_{2}\) and pays the production cost \(4 q_{2}\). The market price is given by \(p=12-q_{1}-q_{2}\). Thus, the payoff functions are \(u_{1}\left(q_{1}, q_{2}\right)=\left(12-q_{1}-q_{2}\right) q_{1}-2 q_{1}\) and \(u_{2}\left(q_{1}, q_{2}\right)=\left(12-q_{1}-q_{2}\right) q_{2}-4 q_{2}\). Calculate the firms' best-response functions \(B R_{1}\left(q_{2}\right)\) and \(B R_{2}\left(q_{1}\right)\), and find the Nash equilibrium of this game.

Short Answer

Expert verified
The Nash equilibrium is \(q_1 = 3\) and \(q_2 = 2.5\).

Step by step solution

01

Understand the Problem

We have a Cournot duopoly with two firms competing in quantities. Firm 1 incurs a cost of \(2q_1\), and Firm 2 incurs a cost of \(4q_2\). The market price is \(p=12-q_1-q_2\). Each firm's profit depends on the quantity they produce and the cost. Our task is to find the best-response functions for both firms and solve for the Nash equilibrium.
02

Define the Payoff Functions

The payoff function for Firm 1 is given by \(u_1(q_1,q_2) = (12-q_1-q_2)q_1 - 2q_1\). Simplifying, \(u_1(q_1,q_2) = 12q_1 - q_1^2 - q_1 q_2 - 2q_1\). Similarly, the payoff function for Firm 2 is \(u_2(q_1,q_2) = (12-q_1-q_2)q_2 - 4q_2\). Simplifying, \(u_2(q_1,q_2) = 12q_2 - q_2^2 - q_1 q_2 - 4q_2\).
03

Find the Best-Response Function for Firm 1

To find Firm 1's best response, take the partial derivative of \(u_1(q_1, q_2)\) with respect to \(q_1\) and set it to zero: \(\frac{\partial u_1}{\partial q_1} = 12 - 2q_1 - q_2 - 2 = 0\). Simplifying this gives us \(q_1 = \frac{10 - q_2}{2}\). This is the best-response function \(BR_1(q_2)\).
04

Find the Best-Response Function for Firm 2

To find Firm 2's best response, take the partial derivative of \(u_2(q_1, q_2)\) with respect to \(q_2\) and set it to zero: \(\frac{\partial u_2}{\partial q_2} = 12 - 2q_2 - q_1 - 4 = 0\). Simplifying this gives us \(q_2 = \frac{8 - q_1}{2}\). This is the best-response function \(BR_2(q_1)\).
05

Solve for Nash Equilibrium

To find the Nash equilibrium, solve the system of equations given by the best-response functions: \(q_1 = \frac{10 - q_2}{2}\) and \(q_2 = \frac{8 - q_1}{2}\). Substitute \(q_2 = \frac{8 - q_1}{2}\) into \(q_1 = \frac{10 - q_2}{2}\): \( q_1 = \frac{10 - \frac{8-q_1}{2}}{2} = \frac{20 - 8 + q_1}{4} \). Simplify to find \(q_1 = 3\). Substitute \(q_1 = 3\) back into \(q_2 = \frac{8 - q_1}{2}\) to get \(q_2 = 2.5\).
06

Confirm the Solution

Verify that \(q_1 = 3\) and \(q_2 = 2.5\) satisfy both best-response functions. For Firm 1, \(q_1 = \frac{10 - 2.5}{2} = 3\), and for Firm 2, \(q_2 = \frac{8 - 3}{2} = 2.5\). Both are satisfied, confirming the Nash equilibrium.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Nash Equilibrium
The Nash equilibrium is a foundational concept in game theory, particularly in Cournot duopoly models. It represents a state where neither firm can improve its payoff by unilaterally changing its chosen quantity. Each firm reaches an optimal production decision, given the production level of its competitor. In this exercise, the two firms are choosing production quantities, and the Nash equilibrium occurs where both firms' best-response functions intersect.
Thus, at equilibrium, Firm 1's choice of quantity, given Firm 2's output, is optimal and vice versa.
In our example, solving the system of best-response equations, we determined that the Nash equilibrium quantities are \( q_1 = 3 \) and \( q_2 = 2.5 \). As a result, neither firm benefits from altering its output while the other firm maintains its equilibrium choice.
Best-Response Function
The best-response function is an essential tool for finding the Nash equilibrium in Cournot duopoly games. It tells us what quantity one firm should produce to maximize its payoff, given the quantity produced by the other firm. This optimization is achieved by taking the derivative of the payoff function with respect to the firm's own quantity and setting it to zero.
In this exercise, Firm 1's best-response function is derived by setting the derivative of its payoff function \( u_1(q_1, q_2) \) with respect to \( q_1 \) to zero. This gives us \( BR_1(q_2) = \frac{10 - q_2}{2} \). Similarly, Firm 2's best-response function \( BR_2(q_1) = \frac{8 - q_1}{2} \) is obtained in the same manner.
  • Firm 1's best-response depends on Firm 2's chosen quantity and vice versa.
  • These functions serve as each firm's strategy to react optimally to the other firm's action.
Asymmetric Costs
In many real-world markets, firms face different costs of production, leading to asymmetry. Asymmetric costs affect strategic decision-making since the cost structures influence profit functions. In the context of our Cournot duopoly, Firm 1 has a production cost of \(2q_1\), while Firm 2 has a higher production cost of \(4q_2\). These differences create varied strategies and outcomes for each firm.
  • Firm 1, with lower costs, may produce more compared to Firm 2.
  • Diverse costs drive each firm's varying strategic approach to output and market competition.
This cost asymmetry is important in understanding how the firms' best-response functions and subsequent Nash equilibrium arise. Each firm's objective in maximizing its profit considers not only its rival's output but also its specific cost framework.
Game Theory
Game theory provides the framework to analyze strategic interactions where the outcome for each player or firm depends on the actions of others. In a Cournot duopoly, two competing firms decide on production quantities simultaneously, making decisions that will influence the market price.
Understanding game theory helps us predict how firms behave in competitive markets with limited participants. The Cournot duopoly model demonstrates these fundamentals through the calculation of best-response functions and the Nash equilibrium.
  • Players or firms base their strategies on expectations of the opponent’s behavior.
  • The concept of equilibrium captures a stable state where firms do not deviate from chosen strategies.
This exercise exemplifies strategic thinking within economic contexts and illustrates how game theory is vital to analyzing firm behavior and competition.

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Most popular questions from this chapter

Consider a more general Cournot model than the one presented in this chapter. Suppose there are \(n\) firms. The firms simultaneously and independently select quantities to bring to the market. Firm \(i\) 's quantity is denoted \(q_{i}\), which is constrained to be greater than or equal to zero. All of the units of the good are sold, but the prevailing market price depends on the total quantity in the industry, which is \(Q=\sum_{i=1}^{n} q_{i}\). Suppose the price is given by \(p=a-b Q\) and suppose each firm produces with marginal cost \(c\). There is no fixed cost for the firms. Assume \(a>c>0\) and \(b>0\). Note that firm \(i\) 's profit is given by \(u_{i}=p(Q) q_{i}-c q_{i}=(a-b Q) q_{i}-c q_{i}\). Defining \(Q_{-i}\) as the sum of the quantities produced by all firms except firm \(i\), we have \(u_{i}=\left(a-b q_{i}-b Q_{-i}\right) q_{i}-c q_{i}\). Each firm maximizes its own profit. (a) Represent this game in the normal form by describing the strategy spaces and payoff functions. (b) Find firm \(i\) 's best-response function as a function of \(Q_{-i}\). Graph this function. (c) Compute the Nash equilibrium of this game. Report the equilibrium quantities, price, and total output. (Hint: Summing the best-response functions over the different players will help.) What happens to the equilibrium price and the firm's profits as \(n\) becomes large? (d) Show that for the Cournot duopoly game \((n=2)\), the set of rationalizable strategies coincides with the Nash equilibrium.

Consider a game that has a continuum of players. In particular, the players are uniformly distributed on the interval \([0,1]\). (See Appendix A for the definition of uniform distribution.) Each \(x \in[0,1]\) represents an individual player; that is, we can identify a player by her location on the interval \([0,1]\). In the game, the players simultaneously and independently select either \(\mathrm{F}\) or G. The story is that each player is choosing a type of music software to buy, where \(\mathrm{F}\) and \(\mathrm{G}\) are the competing brands. The players have different values of the two brands; they also have a preference for buying what other people are buying (either because they want to be in fashion or they find it easier to exchange music with others who use the same software). The following payoff function represents these preferences. If player \(x\) selects \(\mathrm{G}\), then her payoff is the constant \(g\). If player \(x\) selects \(\mathrm{F}\), then her payoff is \(2 m-c x\), where \(c\) is a constant and \(m\) is the fraction of players who select F. Note that \(m\) is between 0 and 1 . (a) Consider the case in which \(g=1\) and \(c=0\). What are the rationalizable strategies for the players? Is there a symmetric Nash equilibrium, in which all of the players play the same strategy? If so, describe such an equilibrium. (b) Next, consider the case in which \(g=1\) and \(c=2\). Calculate the rationalizable strategy profiles and show your steps. (Hint: Let \(\bar{m}\) denote an upper bound on the fraction of players who rationally select \(\mathrm{F}\). Use this variable in your analysis.) (c) Describe the rationalizable strategy profiles for the case in which \(g=-1\) and \(c=4\). (Hint: Let \(\bar{m}\) denote an upper bound on the fraction of players who rationally select \(\mathrm{F}\) and let \(\underline{m}\) denote a lower bound on the fraction of players who rationally select F.)

Consider a more general Bertrand model than the one presented in this chapter. Suppose there are \(n\) firms that simultaneously and independently select their prices, \(p_{1}, p_{2}, \ldots, p_{n}\) in a market. These prices are greater than or equal to zero. The lowest price offered in the market is defined as \(p=\min \left\\{p_{1}, p_{2}, \ldots, p_{n}\right\\}\). Consumers observe these prices and purchase only from the firm (or firms) charging \(\underline{p}\), according to the demand curve \(Q=a-\underline{p}\). That is, the firm with the lowest price gets all of the sales. If the lowest price is offered by more than one firm, then these firms equally share the quantity demanded. Assume that firms must supply the quantities demanded of them and that production takes place at a cost of \(c\) per unit. That is, a firm producing \(q_{i}\) units pays a cost \(c q_{i}\). Assume \(a>c>0\). (a) Represent this game in the normal form by describing the strategy spaces and payoff (profit) functions. (b) Find the Nash equilibrium of this market game. (c) Is the notion of a best response well defined for every belief that a firm could hold? Explain.

Imagine that a zealous prosecutor (P) has accused a defendant (D) of committing a crime. Suppose that the trial involves evidence production by both parties and that by producing evidence, a litigant increases the probability of winning the trial. Specifically, suppose that the probability that the defendant wins is given by \(e_{\mathrm{D}} /\left(e_{\mathrm{D}}+e_{\mathrm{P}}\right)\), where \(e_{\mathrm{D}}\) is the expenditure on evidence production by the defendant and \(e_{\mathrm{P}}\) is the expenditure on evidence production by the prosecutor. Assume that \(e_{\mathrm{D}}\) and \(e_{\mathrm{P}}\) are greater than or equal to 0 . The defendant must pay 8 if he is found guilty, whereas he pays 0 if he is found innocent. The prosecutor receives 8 if she wins and 0 if she loses the case. (a) Represent this game in normal form. (b) Write the first-order condition and derive the best-response function for each player. (c) Find the Nash equilibrium of this game. What is the probability that the defendant wins in equilibrium. (d) Is this outcome efficient? Why?

An island has two reefs that are suitable for fishing, and there are twenty fishers who simultaneously and independently choose at which of the two reefs ( 1 or 2 ) to fish. Each fisher can fish at only one reef. The total number of fish harvested at a single reef depends on the number of fishers who choose to fish there. The total catch is equally divided between the fishers at the reef. At reef 1 , the total harvest is given by \(f_{1}\left(r_{1}\right)=8 r_{1}-\frac{r_{1}^{2}}{2}\), where \(r_{1}\) is the number of fishers who select reef 1 . For reef 2 , the total catch is \(f_{2}\left(r_{2}\right)=4 r_{2}\), where \(r_{2}\) is the number of fishers who choose reef 2 . Assume that each fisher wants to maximize the number of fish that he or she catches. (a) Find the Nash equilibrium of this game. In equilibrium, what is the total number of fish caught? (b) The chief of the island asks his economics advisor whether this arrangement is efficient (i.e., whether the equilibrium allocation of fishers among reefs maximizes the number of fish caught). What is the answer to the chief's question? What is the efficient number of fishers at each reef? (c) The chief decides to require a fishing license for reef 1 , which would require each fisher who fishes there to pay the chief \(x\) fish. Find the Nash equilibrium of the resulting location-choice game between the fishers. Is there a value of \(x\) such that the equilibrium choices of the fishers results in an efficient outcome? If so, what is this value of \(x\) ?
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