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Chapter 3: Problem 65

Consider the extent to which the analysis depends upon the demand function \(D\) taking the specific form \(D(p)=\alpha-p .\) Suppose that \(D\) is any function for which \(D(p) \geq 0\) for all \(p\) and there exists \(\bar{p}>c\) such that \(D(p)>0\) for all \(p \leq \bar{p} .\) Is \((c, c)\) still a Nash equilibrium? Is it still the only Nash equilibrium?

Short Answer

Expert verified
Yes, (c, c) remains a Nash equilibrium and is likely the only Nash equilibrium, provided no other (d, p :\barp, counterpart ensures profits translating zero shift with cost-marginal aligned fails otherwise validating only conceptual reference optimal outcome.

Step by step solution

01

Understand the Nash Equilibrium Concept

A Nash equilibrium occurs when each player's strategy is the best response given the other player's strategies. In this context, consider the strategies of setting the price equal to the marginal cost, Given this, explore if c , c) could potentially be a Nash equilibrium.
02

Define the Demand Function

Generally, verify that the given Demand functions satisfy the conditions:1. D(p) is non-negative for all prices p. 2.There exists a price - such that -D(-p) is positive if -p is smaller or equal to p.
03

Consider Profit Function

The profit Pi for a firm 1 can be represented as : -pq)p D(p)= - c) given.c) where - q = firm i (p - c implies zero profit.
04

Validate If p=c is Optimal

other price increases p> c reduces demand ,hence profits become negative .That means firm 1 cannot do better than c ) - Nash equilibrium given that price would shift c as long as decision remains same .If any other firm increases to p<c will derive determination ,this won鈥檛 raise p too. Essentially it satisfies the definition terms.
05

Evaluate Uniqueness

Explores the possibility of any other equilibrium -(demand and prices are positive for all p<-p). If any alternative equilibrium -( -( involves prices higher or lower minimal optimal deviation,is no (c , c ) the point proves only equilibrium.

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Key Concepts

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

Demand Function
The demand function represents the relationship between the price of a good and the quantity demanded by consumers. For this exercise, consider the demand function given as \(D(p)=\alpha - p\). This is a linear function where demand decreases as price increases.
To generalize, we assume \(D(p)\) is non-negative for any price \(p\) and that for some \(\bar{p}>c\), demand remains positive when the price is at or below \(\bar{p}\).
Key points to remember:
  • \(D(p)\) must always be \(\geq 0 \).
  • There exists a threshold price \(\bar{p}\) such that for prices less than or equal to \(\bar{p}\), demand remains positive.
Understanding the demand function is crucial since it affects the firm's pricing strategy and ultimately its profit.
Profit Function
The profit function for a firm quantifies its earnings after accounting for costs. The profit \(\Pi_i\) for firm 1 can be represented as:
\(\Pi_1 = p \times D(p) - c \times D(p)\).
Here, \(p\) is the price set by the firm, \(c\) is the marginal cost, and \(D(p)\) is the demand function.
Key components:
  • Revenue: \(p \times D(p)\), which is the price multiplied by the quantity sold.
  • Cost: \(c \times D(p)\), which is the marginal cost multiplied by the quantity produced.
Net profit is thus:
\(\text{Profit} = (p - c) \times D(p)\).
If a firm sets \(p=c\), the profit is zero because:
\(\text{Profit} = (c - c) \times D(c) = 0\).
Setting a higher price than \(c\) results in reduced demand and, consequently, potential negative profits.
Uniqueness of Equilibrium
In game theory, a Nash equilibrium occurs when each player's strategy is optimal, considering the other players' strategies.
To determine if \((c, c)\) is a unique Nash equilibrium, we examine if any other price strategy can yield a higher payoff.
Key observations:
  • If firm 1 increases \(p > c\), demand decreases, lowering profits possibly to negative.
  • If firm 1 charges \(p < c\), it may attract more demand, but since \(c\) is the marginal cost, selling at \(p < c\) results in losses.
Thus, the optimal strategy for both firms is to set \(p = c\).
Since any deviation from this strategy does not benefit either firm, \((c, c)\) remains the unique Nash equilibrium.
This uniqueness is paramount in ensuring predictability in competitive markets where firms may set prices based on cost structures and demand functions.

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Most popular questions from this chapter

(Electoral competition with three candidates) Consider a variant of Hotelling's model in which there are three candidates and each candidate has the option of staying out of the race, which she regards as better than losing and worse than tying for first place. Use the following arguments to show that the game has no Nash equilibrium. First, show that there is no Nash equilibrium in which a single candidate enters the race. Second, show that in any Nash equilibrium in which more than one candidate enters, all candidates that enter tie for first place. Third, show that there is no Nash equilibrium in which two candidates enter the race. Fourth, show that there is no Nash equilibrium in which all three candidates enter the race and choose the same position. Finally, show that there is no Nash equilibrium in which all three candidates enter the race, and do not all choose the same position.

(Electoral competition with asymmetric voters' preferences) Consider a variant of Hotelling's model in which voters's preferences are asymmetric. Specifically, suppose that each voter cares twice as much about policy differences to the left of her favorite position than about policy differences to the right of her favorite position. How does this affect the Nash equilibrium? In the model considered so far, no candidate has the option of staying out of the race. Suppose that we give each candidate this option; assume that it is better than losing and worse than tying for first place. Then the Nash equilibrium remains as before: both players enter the race and choose the position \(m\). The direct argument differs from the one before only in that in addition we need to check that there is no equilibrium in which one or both of the candidates stays out of the race. If one candidate stays out then, given the other candidate's position, she can enter and either win outright or tie for first place. If both candidates stay out, then either candidate can enter and win outright. The next exercise asks you to consider the Nash equilibria of this variant of the model when there are three candidates.

Consider Cournot's game in the case of an arbitrary number \(n\) of firms; retain the assumptions that the in-verse demand function takes the form (54.2) and the cost function of each firm \(i\) is \(\mathrm{C}_{i}\left(q_{i}\right)=c q_{i}\) for all \(q_{i}\), with \(c<\alpha\). Find the best response function of each firm and set up the conditions for \(\left(q_{1}^{*}, \ldots, q_{n}^{*}\right)\) to be a Nash equilibrium (see \(\left.(34.3)\right)\), assuming that there is a Nash equilibrium in which all firms' outputs are positive. Solve these equations to find the Nash equilibrium. (For \(n=2\) your answer should be \(\left(\frac{1}{3}(\alpha-c), \frac{1}{3}(\alpha-c)\right)\), the equilibrium found in the previous section. First show that in an equilibrium all firms produce the same output, then solve for that output. If you cannot show that all firms produce the same output, simply assume that they do.) Find the price at which output is sold in a Nash equilibrium and show that this price decreases as \(n\) increases, approaching \(c\) as the number of firms increases without bound. The main idea behind this result does not depend on the assumptions on the inverse demand function and the firms' cost functions. Suppose, more generally, that the inverse demand function is any decreasing function, that each firm's cost function is the same, denoted by \(C\), and that there is a single output, say \(q\), at which the average cost of production \(C(q) / q\) is minimal. In this case, any given total output is produced most efficiently by each firm's producing \(q\), and the lowest price compatible with the firms' not making losses is the minimal value of the average cost. The next exercise asks you to show that in a Nash equilibrium of Cournot's game in which the firms' total output is large relative to \(q_{\prime}\) this is the price at which the output is sold.

(Electoral competition between candidates who care only about the winning position) Consider the variant of Hotelling's model in which the candidates (like the citizens) care about the winner's position, and not at all about winning per se. There are two candidates. Each candidate has a favorite position; her dislike for other positions increases with their distance from her favorite position. Assume that the favorite position of one candidate is less than \(m\) and the favorite position of the other candidate is greater than \(m .\) Assume also that if the candidates tie when they take the positions \(x_{1}\) and \(x_{2}\) then the outcome is the compromise policy \(\frac{1}{2}\left(x_{1}+x_{2}\right) .\) Find the set of Nash equilibria of the strategic game that models this situation. (First consider pairs \(\left(x_{1}, x_{2}\right)\) of positions for which either \(x_{1}m\) and \(x_{2}>m .\) Next consider pairs \(\left(x_{1}, x_{2}\right)\) for which either \(x_{1}

(A fight) Each of two people has one unit of a resource. Each person chooses how much of the resource to use in fighting the other individual and how much to use productively. If each person \(i\) devotes \(y_{i}\) to fighting then the total output is \(f\left(y_{1}, y_{2}\right) \geq 0\) and person \(i\) obtains the fraction \(p_{i}\left(y_{1}, y_{2}\right)\) of the output, where $$ p_{i}\left(y_{1}, y_{2}\right)= \begin{cases}1 & \text { if } y_{i}>y_{j} \\\ \frac{1}{2} & \text { if } y_{i}=y_{j} \\ 0 & \text { if } y_{i}
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