91Ó°ÊÓ

Suppose that there are infinitely many firms, all of which have the same cost function \(C\). Assume that \(C(0)=0\), and for \(q>0\) the function \(C(q) / q\) has a unique minimizer \(q ;\) denote the minimum of \(C(q) / q\) by \(\underline{p}\). Assume that the inverse demand function \(\bar{P}\) is decreasing. Show that in any Nash equilibrium the firms' total output \(Q^{*}\) satisfies $$ P\left(Q^{*}+\underline{q}\right) \leq \underline{p} \leq P\left(Q^{*}\right) $$ (That is, the price is at least the minimal value \(\underline{p}\) of the average cost, but is close enough to this minimum that increasing the total output of the firms by \(q\) would reduce the price to at most \(\underline{p}\).) To establish these inequalities, show that if \(P\left(Q^{*}\right)<\underline{p}\) or \(P\left(Q^{*}+q\right)>\underline{p}\) then \(Q^{*}\) is not the total output of the firms in a Nash equilibrium, because in each case at least one firm can deviate and increase its profit. 3.1.5 A generalization of Cournot's game: using common property In Cournot's game, the payoff function of each firm \(i\) is \(q_{i} P\left(q_{1}+\cdots+q_{n}\right)-C_{i}\left(q_{i}\right)\). In particular, each firm's payoff depends only on its output and the sum of all the firm's outputs, not on the distribution of the total output among the firms, and decreases when this sum increases (given that \(P\) is decreasing). That is, the payoff of each firm \(i\) may be written as \(f_{i}\left(q_{i}, q_{1}+\cdots+q_{n}\right)\), where the function \(f_{i}\) is decreasing in its second argument (given the value of its first argument, \(q_{i}\) ).

Step by step solution

01

- Understand the Cost Function

The cost function is given as \( C \) with \( C(0) = 0 \). For any \( q > 0 \), the average cost \( \frac{C(q)}{q} \) has a unique minimizer at \( q \). Identify the minimum value of \( \frac{C(q)}{q} \) as \( \underline{p} \).

02

- Consider the Inverse Demand Function

The inverse demand function \( \bar{P} \) is decreasing, meaning that as the total output increases, the price decreases.

03

- Establish Nash Equilibrium Conditions

In any Nash equilibrium, no firm can increase its profit by unilaterally changing its output. Denote the total output of all firms in Nash equilibrium by \( Q^* \). We need to show that \( P(Q^*+\underline{q}) \leq \underline{p} \leq P(Q^*) \).

04

- Analyze the Case \( P(Q^*) < \underline{p} \)

If \( P(Q^*) < \underline{p} \), then any firm can produce the minimizer quantity \( q \) at a cost of \( \underline{p} q \) and sell it for \( P(Q^*) \), thus making a profit. Hence, \( Q^* \) cannot be in Nash equilibrium because firms would have an incentive to increase production.

05

- Analyze the Case \( P(Q^*+q) > \underline{p} \)

If \( P(Q^*+q) > \underline{p} \), the total output can be increased by \( q \) without pushing the price below \( \underline{p} \). Thus, a firm could profit by producing a bit more, implying that \( Q^* \) could not be an equilibrium output because firms would want to increase their output.

06

- Conclude the Inequality

Since both \( P(Q^*) < \underline{p} \) and \( P(Q^*+q) > \underline{p} \) lead to contradictions, it must be that \( P(Q^*+\underline{q}) \leq \underline{p} \leq P(Q^*) \). Thus, the Nash equilibrium condition is satisfied.

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Key Concepts

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

Cost function in game theory

The cost function is a fundamental concept in game theory, particularly when analyzing the strategies of firms in a competitive market. A cost function, denoted as \(C(q)\), represents the total cost incurred by a firm to produce a quantity \(q\) of goods or services. In this context, it's essential to understand the relationship between the cost and output.
The given exercise specifies that the cost function satisfies \(C(0)=0\) and that for any \(q > 0\), the average cost \(C(q)/q\) has a unique minimizer at \(q\). This means there is a specific output level where the average cost of production is the lowest. This point is crucial because it influences the firm's production decisions to minimize costs and maximize profit.
The minimum value of the average cost is denoted by \(\bar{p} \). In simple terms, \(\bar{p} \) is the lowest possible cost per unit of output that the firm can achieve. This minimization point is vital in determining the firm's strategy in a competitive market, especially under Cournot competition where several firms decide on their output simultaneously.

Inverse demand function

The inverse demand function, denoted as \(\bar{P}(Q)\), plays a crucial role in determining the market price of goods based on total output. Unlike the standard demand function, which maps price to quantity demanded, the inverse demand function shows how price changes as output changes. This function is decreasing, meaning that as total output \(Q\) increases, the market price \(P\) decreases.
This relationship is intuitive because higher supply in the market leads to lower prices due to competition among firms. Understanding the inverse demand function is crucial for analyzing how firms' output decisions impact market prices. The exercise requires showing that in a Nash equilibrium, the total output \(Q^*\) produces a price \(P(Q^*)\) that satisfies specific inequalities involving \(\bar{p} \).
Given that all firms face the same cost structure and the market price depends on total output, the inverse demand function is central to understanding price dynamics in a competitive market. It provides a link between the firms' production decisions and the resulting market price, which is essential for achieving equilibrium.

Cournot competition

Cournot competition is a foundational model in game theory that describes how firms compete by choosing quantities. Each firm decides its output level to maximize its profit, given the output levels of its competitors. The firms make their decisions simultaneously, and the resulting output levels determine the market price through the inverse demand function.
In Cournot competition, the payoff of each firm depends on its output and the total output of all firms. The payoff function of firm \(i\) can be written as \(q_{i} P\bigg(\bigg) q_{1}+\theta+q_{n}\bigg)-C_{i}\bigg(q_{i}\bigg ) \). This means firm \(i\)'s profit is the revenue from selling its output at market price minus its cost of production. The market price itself is determined by the total output of all firms.
The Nash equilibrium in this context occurs when no firm can increase its profit by unilaterally changing its output. The given exercise's solution shows that in equilibrium, the total output \(Q^*\) must meet specific conditions ensuring all firms' profits are optimized. If the price \( P(Q^*)\) is less than \(\bar{p} \) or greater than \(\bar{p} \) after increasing output slightly, firms would have an incentive to change their production, violating the equilibrium condition.
Cournot competition, thus, captures the strategic interactions among firms in an oligopoly market and helps predict the production levels and prices in equilibrium.