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Chapter 10: Problem 2

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.

Short Answer

Expert verified
The Nash equilibrium is when all firms set \(p_i = c\). The best response is well-defined.

Step by step solution

01

Define Strategy Spaces

In this Bertrand competition model, each firm has a strategy space that consists of selecting a price \(p_i\), where \(0 \leq p_i < a\). This is because the demand \(Q = a - \underline{p}\) goes to zero when prices reach \(a\). Since the cost of production of each unit is \(c\), the price must also be \(p_i > c\) for profitability, but \(p_i = c\) is a feasible choice even if it results in zero profit.
02

Determine Payoff Functions

The payoff for each firm is determined by its profit, which is revenue minus cost. If a firm sets the lowest price \(\underline{p}\), its profit is \((\underline{p} - c)(a - \underline{p})\). If \(\underline{p}\) is shared by multiple firms, the profit for each firm is equally divided: \((\underline{p} - c)(a - \underline{p})/k\), where \(k\) is the number of firms sharing the lowest price. If a firm does not set the lowest price, its payoff is zero since it makes no sales.
03

Identify Nash Equilibrium

In Nash equilibrium, each firm chooses a strategy that maximizes its payoff given the strategies of others. The equilibrium is reached when each firm sets a price where \(\underline{p} = c\). If a firm charges more than \(c\), competing firms can slightly undercut this price and capture all residual demand. All firms setting \(p_i = c\) means each firm makes zero profit, discouraging any deviation since lowering below \(c\) would incur losses.
04

Assess Best Response Definition

For every belief a firm might hold about the prices of the other firms, the best response is to slightly undercut \(\underline{p}\) unless the price is already \(c\). If firms believe others will set prices above \(c\), undercutting to \(c\) captures all the demand. If all firms choose \(c\), any lower price leads to losses, so \(p_i = c\) always aligns with the best response strategy, confirming that the notion of best response is well-defined.

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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 in the context of the Bertrand competition model is a key concept. It occurs when each firm selects a pricing strategy where they cannot improve their profit by unilaterally changing their price.
When firms reach a Nash Equilibrium, they are making the best decision possible taking into account the pricing strategies of their competitors.
In the Bertrand model, this state is found when all firms set their prices equal to the cost of production, denoted as \( p_i = c \).
This is because if any firm charges more than this cost, other firms can slightly undercut that price and capture the entire market share by attracting all consumers.
Hence, all firms pricing at \( c \) means each has zero profit, but no incentive to deviate since going below \( c \) would result in a loss.
The competitive push to minimize prices leads to all firms converging at this equilibrium point.
Strategy Spaces
In a Bertrand competition scenario, strategy spaces refer to the range of pricing options each firm can choose.
Firms decide on a price \( p_i \) such that \( 0 \leq p_i < a \).
This range is determined because consumer demand, expressed as \( Q = a - \underline{p} \), becomes zero when prices reach the value of \( a \).
Additionally, from a profitability perspective, a price greater than the production cost \( c \) is necessary, though technically \( p_i = c \) is feasible even if it yields zero profit.
Thus, firms in the Bertrand model must carefully choose within these bounds to remain competitive and maximize market share.
Payoff Functions
Payoff functions in this model represent the profit firms make based on their chosen pricing strategies and competitive interactions.
For a firm setting the lowest price \( \underline{p} \), the profit is calculated as \((\underline{p} - c)(a - \underline{p})\).
If several firms offer the same lowest price, they split the demand equally: the profit becomes \((\underline{p} - c)(a - \underline{p})/k\), with \( k \) being the number of firms sharing this lowest price.
Conversely, if a firm does not offer the lowest price, its sales, and thus profits, drop to zero.
These payoff functions highlight the high-stakes nature of pricing in Bertrand competition, where even the smallest price changes can impact market outcomes significantly.
Normal Form Representation
The normal form representation is a structured way to visualize this strategic interaction among firms.
This representation organizes the strategies and payoffs in a tabular format, showcasing each firm's potential pricing decisions and corresponding profits.
It highlights how each firm's strategy is not an isolated choice but directly influenced by the choices of their competitors.
In this context, by clearly laying out all possible moves and outcomes, the normal form helps identify the strategic interdependencies that characterize the Nash Equilibrium.
It brings to light the decision-making challenges where firms must anticipate competitors' responses to achieve optimal outcomes.

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Most popular questions from this chapter

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 game in which, simultaneously, player 1 selects a number \(x \in[0,6]\) and player 2 selects a number \(y \in[0,6]\). The payoffs are given by: $$ \begin{aligned} &u_{1}(x, y)=\frac{16 x}{y+2}-x^{2} \\ &u_{2}(x, y)=\frac{16 y}{x+2}-y^{2} \end{aligned} $$ (a) Calculate each player's best-response function as a function of the opposing player's pure strategy. (b) Find and report the Nash equilibrium of the game. (c) Suppose that there is no social institution to coordinate the players on an equilibrium. Suppose that each player knows that the other player is rational, but this is not common knowledge. What is the largest set of strategies for player 1 that is consistent with this assumption?

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.

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?

Suppose that the speed limit is 70 miles per hour on the freeway and that \(n\) drivers simultaneously and independently choose speeds from 70 to 100 . Everyone prefers to go as fast as possible, other things equal, but the police ticket any driver whose speed is strictly faster than the speeds of a fraction \(x\) of the other drivers, where \(x\) is a parameter such that \(0 \leq x \leq 1\). More precisely, for a given driver, let \(m\) denote the number of drivers that choose strictly lower speeds; then, such a driver is ticketed if and only if \(m /(n-1)>x\). Note that by driving 70, a driver can be sure that he will not be ticketed. Suppose the cost of being ticketed outweighs any benefit of going faster. (a) Model this situation as a noncooperative game by describing the strategy space and payoff function of an individual player. (b) Identify the Nash equilibria as best you can. Are there any equilibria in which the drivers choose the same speed? Are there any equilibria in which the drivers choose different speeds? How does the set of Nash equilibria depend on \(x\) ? (c) What are the Nash equilibria under the assumption that the police do not ticket anyone? (d) What are the Nash equilibria under the assumption that the police ticket everyone who travels more than 70 ? (e) If the same drivers play this game repeatedly, observing the outcome after each play, and there is some noisiness in their choices of speed, how would you expect their speeds to change over time as they learn to predict each other's speeds when \(x\) is near 100 and when \(x\) is near 0 ? Explain your intuition.
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