It is worth noting that the problem studied here is related to the one studied in [1]. In [1], they investigate the existence of an equilibrium in a two-dimensional economy, with a large population of agents endowed with heterogeneous endowments, trading two goods. Their result depends on a property of their model that the marginal utilities of consumption depend only on consumption relative to the average. Our framework differs in two aspects. First, we consider a dynamic environment in which agents use a history of past allocations in order to coordinate their decisions. Second, our model features a continuum of agents, rather than a finite number of agents.
In our model, we show that any Markov perfect equilibrium must involve a constant price and a constant consumption allocation. Thus, we prove the existence of an equilibrium where there are stationary strategies. This does not rule out the existence of other, nonstationary, equilibria. If the initial wealth distribution is Pareto efficient, then the competitive equilibrium is stationary. If the initial allocation is not Pareto optimal, there may be incentives for agents to trade over time. Thus, our work is complementary to theirs, and we are able to derive the unique steady state equilibrium.
${ }^{2}$ See e.g. [3]. the seller, although a higher price generates more revenue for each unit sold. With an increase in the quantity produced, the demand curve determines the price, which influences the producer’s revenue. Thus, the producer is trading off higher revenue from additional output against the increase in costs associated with that higher level of production.
Let us provide a numerical example based on the previous discussion. Assume that there are $n$ producers in the market. Denote by $q{i}$ the quantity produced by firm $i$ and $Q=sum{i=1}^{n} q{i}$ is the total quantity produced. Also, we denote the demand function and cost function of firm $i$ as $P{i}(Q)$ and $C{i}left(q{i}right)$, respectively. We can then define the profit for the $i$-th firm to be given by
[
pi{i}left(q{i}, Q{-i}right)=P(Q) cdot q{i}-C{i}left(q{i}right),
]
where $Q_{-i}$ represents the sum of quantities produced by all firms other than firm $i$.
To simplify calculations, we will consider a Cournot duopoly in which the inverse demand function is linear, such that the market price $p$ is determined by
[
P(Q)=a-b Q, quad a, b>0,
]
where $Q=q{1}+q{2}$ is the aggregate quantity produced by the two firms. Assume the firms face the same constant marginal cost $c$. Firm $i$ ‘s profit function is then
[
pi{i}left(q{i}, q{j}right)=q{i} Pleft(q{i}+q{j}right)-c q{i}=q{i}left(a-bleft(q{i}+q{j}right)right)-c q_{i}
]
Differentiating with respect to $q_{i}$, we obtain firm $i$ ‘s best-response function:
[
a-2 b q{i}-b q{j}-c=0,
]
or
[
q{i}=frac{a-c-b q{j}}{2 b}
]
In a Nash equilibrium, each firm’s choice is a best response to the choice of the other firm, so we have:
[
q{1}^{*}=frac{a-c-b q{2}^{*}}{2 b}
]
${ }^{2}$ I have omitted the proof that $a-c geq 0$ which is necessary for the quantities calculated below to be positive.
[
q{2}^{*}=frac{a-c-b q{1}^{*}}{2 b}
]
Solving the system of equations above, we obtain the optimal quantities:
[
q{1}^{*}=q{2}^{*}=frac{a-c}{3 b}
]
and the market price will be
[
p=a-bleft(q{1}^{*}+q{2}^{*}right)=a-bleft(frac{a-c}{3 b} times 2right)=frac{a+2 c}{3}
]
section{The Bertrand Model}
In Bertrand competition, firms compete on price rather than quantity. The model makes the following assumptions:
begin{enumerate}
item There are at least two firms producing homogeneous (undifferentiated) products.
item Firms do not cooperate.
item Firms compete by setting prices simultaneously.
item Consumers buy everything from a firm with a lower price. If all firms charge the same price, consumers randomly select among them.
end{enumerate}
Let $p{1}$ and $p{2}$ denote the prices charged by Firm 1 and Firm 2 , respectively, and let $c$ be the constant marginal cost of production for both firms. The demand curve for each firm, denoted as $q_{i}$, is as follows:
begin{itemize}
item If $p{i}<p{j}$, then firm $i$ captures the entire market, so $q{i}=a-b p{i}$ and $q_{j}=0$ (a firm will not sell anything if its price is higher).
item If $p{i}=p{j}=p$, then $q_{i}=frac{a-p}{2 b}$ (each firm gets half of the demand).
item Firms produce exactly what is demanded.
item The firms’ marginal and average costs are constant at $c$.
end{itemize}
The profit function of firm $i$ is given by:
[
pi{i}left(p{i}, p{j}right)=left(p{i}-cright) q{i}left(p{i}, p{j}right)= begin{cases}left(p{i}-cright)left(a-b p{i}right) & text { if } p{i}<p{j} left(p{i}-cright)left(frac{a-p{i}}{2 b}right) & text { if } p{i}=p{j} 0 & text { if } p{i}>p_{j}end{cases}
]
In a Nash equilibrium of this game, each firm correctly anticipates its rivals’ price decisions and chooses its own price to maximize its profit given its rivals’ choices.
The price-setting game has a unique Nash equilibrium: $p{1}^{*}=p{2}^{*}=c$. That is, the equilibrium outcome is the same as perfect competition. To see this, note that no firm will ever set a price below marginal cost $c$ because it would make a loss. Suppose firm 1 sets a price $p{1}>c$. Then firm 2 can slightly undercut $p{1}$ (by $epsilon$), take the entire market, and earn a profit. Firm 1 will then want to undercut firm 2 by setting a slightly lower price, and so on. This undercutting stops only when both firms set the price equal to marginal cost $c$. If only one firm sets $p{i}=c$, the other firm can set a slightly lower price, capture the entire market, and earn a small positive profit. If both firms set $p{i}=p_{j}=p>c$, then each firm has the incentive to deviate by slightly undercutting its rival’s price.
begin{itemize}
item If $p{i}>c$, a slight reduction in price to $p{i}-epsilon$ will lead to a discontinuous jump in profits since the firm will capture the whole market.
item If $p_{i}=c$, the firm is making zero profits.
end{itemize}
This phenomenon is known as Bertrand’s paradox. It is a paradox because two competing firms are enough to push prices down to marginal cost level, thereby eliminating all profits. This result holds more generally when the goods are homogeneous, firms have constant marginal costs, and there is no capacity constraint. With capacity constraints or heterogeneous products, the result no longer holds.
subsection{Cournot Competition}
In the Cournot model, firms compete on the basis of quantity rather than price. The quantity produced by each firm is chosen to maximize its own profit, given the quantity chosen by its competitors. Each firm treats the output of its competitors as fixed.
begin{itemize}
item There are $n$ firms producing a homogeneous product. Let $q{i}$ be the quantity of the good produced by firm $i$, and let $Q=q{1}+ldots+q_{n}$ be the aggregate quantity.
item The market clearing price is given by an inverse demand function $P(Q)$.
item All firms have the same cost function, $Cleft(q{i}right)=c q{i}$ for $i=1, ldots, n$, where $c>0$ is the constant marginal cost.
item The strategy for firm $i$ is to choose a quantity $q_{i}$.
item Profits for firm $i$ are given by $pi{i}left(q{1}, q{2}, ldots, q{n}right)=q{i} P(Q)-c q{i}$.
end{itemize}
In the Cournot equilibrium, each firm maximizes its profits given the output of other firms. Thus, each firm solves
[
max {q{i}} q{i} Pleft(q{i}+sum{j neq i} q{j}right)-c q_{i} .
]
The first-order condition is
[
frac{partial pi{i}}{partial q{i}}=P(Q)+q_{i} P^{prime}(Q)-c=0
]
In a symmetric equilibrium, $q{1}^{*}=q{2}^{}=ldots=q_{n}^{}=q^{}$, so $Q^{}=n q^{*}$. Substituting into the first-order condition, we have:
[
Pleft(n q^{}right)+q^{} P^{prime}left(n q^{*}right)-c=0 .
]
If we assume a linear inverse demand function: $P(Q)=a-b Q$, where $a>0$ and $b>0$, then
[
a-b n q^{}-b q^{}-c=0
]
or
[
q^{*}=frac{a-c}{(n+1) b}
]
Thus, total output is
[
Q^{}=n q^{}=frac{n}{n+1} frac{(a-c)}{b}
]
and the market price is
[
p^{}=Pleft(Q^{}right)=a-bleft(frac{n(a-c)}{(n+1) b}right)=frac{a}{n+1}+frac{n}{n+1} c
]
The equilibrium profits of firm $i$ are
[
pi_{i}^{}=left(p^{}-cright) q^{*}=left(frac{a}{n+1}+frac{n}{n+1} c-cright) frac{a-c}{(n+1) b}=frac{(a-c)^{2}}{(n+1)^{2} b}
]
section{Numerical examples of the Prisoner’s Dilemma}
Let the following be the payoff matrix of a game in normal form:
begin{center}
begin{tabular}{c|c|c|}
& multicolumn{1}{c}{Left} & multicolumn{1}{c}{Right}
cline { 2 – 3 }
Top & 4,4 & 1,5
cline { 2 – 3 }
Bottom & 5,1 & 2,2
cline { 2 – 3 }
& &
cline { 2 – 3 }
end{tabular}
end{center}
In this game, (Bottom, Right) is the only pure strategy Nash Equilibrium, as can be seen from the fact that neither player would want to unilaterally deviate.
If the game is repeated infinitely many times, the following strategies form a Nash equilibrium:
begin{itemize}
item Cooperate and play (Top, Left) in the first round.
item In subsequent rounds, if the outcome of the previous round was (Top, Left), then play (Top, Left). If either player deviated from (Top, Left) in the previous round, play (Bottom, Right) forever after.
end{itemize}
subsection{Public goods game}
The public goods game is a classic example in game theory of the free-rider problem. It is an $n$-player game in which each player can choose to contribute to the public good or to free-ride on the contributions of others. In our example, we will consider a very simple version of this game where each player can choose to contribute or not contribute a fixed amount $C$ to a public good. The public good is produced only if at least one player contributes, and provides a benefit $B$ to every player in the group, where $B>C$. If no player contributes, then the public good is not provided and no one receives any benefit.
A simple example with two players is presented below.
begin{center}
begin{tabular}{|c|c|c|}
cline { 2 – 3 }
multicolumn{1}{c|}{} & Player 2 contributes & Player 2 does not contribute
hline
Player 1 contributes & $B-C, B-C$ & $-C, B$
hline
Player 1 does not contribute & $B,-C$ & 0,0
hline
end{tabular}
end{center}
In this game, (Don’t Contribute, Don’t Contribute) is the unique Nash equilibrium, even though (Contribute, Contribute) is better for both players. This game illustrates the free-rider problem: each individual player is better off not contributing, even though everyone would be better off if everyone contributed.
section{Sequential Games}
In sequential games, players make decisions one after another, and later players can observe the actions of earlier players. The extensive form representation, often using a game tree, is a useful tool for analyzing such games. We use the concept of subgame perfect Nash equilibrium (SPNE) to solve these games, which requires that each player’s strategy be a Nash equilibrium in every subgame of the original game. This essentially means that players’ strategies are credible – they will follow through with their planned actions even if they are not on the equilibrium path.
subsection{Example: Entry Game}
Consider a simple entry game where there is an incumbent firm (player 1) in a market, and a potential entrant (player 2) is deciding whether to enter the market or not. If player 2 enters, player 1 can either fight or accommodate. The payoffs are as follows:
begin{itemize}
item If player 2 stays out, player 1 gets a payoff of 2, and player 2 gets 0 .
item If player 2 enters and player 1 fights, both players get a payoff of -1.
item If player 2 enters and player 1 accommodates, both players get a payoff of 1.
end{itemize}
This game can be represented by the following game tree:
begin{center}
includegraphics[max width=textwidth]{2023_10_18_aa41930d13799f44e65bg-082_1038_1483_730_1389}
end{center}
To find the SPNE, we use backward induction. If player 2 enters, player 1 will choose to accommodate since $1>-1$. Knowing this, player 2 will choose to enter since $1>0$. Therefore, the SPNE of this game is (Enter, Accommodate).
subsection{Stackelberg Duopoly}
The Stackelberg duopoly model is a sequential game where one firm (the leader) chooses its quantity first, and then the other firm (the follower) chooses its quantity after observing the leader’s choice. The market price is then determined by the total quantity produced.
The inverse demand function is given by $P(Q)=a-b Q$, where $Q=q{1}+q{2}$ is the total output of the two firms. Both firms have the same constant marginal cost $c$.
Firm 2 (the follower) observes $q{1}$ and then chooses $q{2}$ to maximize its profit:
[
max {q{2}}left(a-bleft(q{1}+q{2}right)right) q{2}-c q{2} .
]
The first-order condition is
[
a-b q{1}-2 b q{2}-c=0,
]
so the best response function for firm 2 is
[
q{2}left(q{1}right)=frac{a-c-b q_{1}}{2 b} .
]
Firm 1 (the leader) anticipates this reaction and chooses $q_{1}$ to maximize its profit:
[
max {q{1}} q{1}left(a-bleft(q{1}+frac{a-c-b q_{1}}{2 b}right)-cright) .
]
The first-order condition for firm 1 is
[
a-2 b q{1}-bleft(frac{a-c-b q{1}}{2 b}right)-c=0
]
Solving for $q_{1}^{*}$, we get
[
q_{1}^{*}=frac{a-c}{2 b} .
]
Plugging this back into the best response function of firm 2, we get
[
q_{2}^{*}=frac{a-c-bleft(frac{a-c}{2 b}right)}{2 b}=frac{a-c}{4 b} .
]
The total quantity is
[
Q^{}=q_{1}^{}+q_{2}^{*}=frac{a-c}{2 b}+frac{a-c}{4 b}=frac{3(a-c)}{4 b} .
]
The market price is
[
p^{}=a-b Q^{}=a-frac{3(a-c)}{4}=frac{a+3 c}{4} .
]
The profits are
[
begin{aligned}
& pi{1}=left(p^{*}-cright) q{1}^{}=left(frac{a+3 c}{4}-cright)left(frac{a-c}{2 b}right)=frac{(a-c)^{2}}{8 b}
& pi_{2}=left(p^{}-cright) q_{2}^{*}=left(frac{a+3 c}{4}-cright)left(frac{a-c}{4 b}right)=frac{(a-c)^{2}}{16 b}
end{aligned}
]
Total industry profits are $frac{3(a-c)^{2}}{16 b}$. Notice that in the Stackelberg model, the leader produces more output and has higher profits than the follower. Also note that total output is higher in Stackelberg than Cournot, but less than Bertrand.
subsection{Infinitely Repeated Games}
In infinitely repeated games, players consider the impact of their current actions on the future actions of other players. This can lead to outcomes that are different from the Nash equilibrium of a one-shot game.
For example, consider the Prisoner’s Dilemma game described above. If the game is played only once, the dominant strategy for both players is to confess, resulting in a Nash equilibrium of (Confess, Confess). However, if the game is repeated indefinitely, cooperation (both players remaining silent) can be sustained as a Nash equilibrium if players are sufficiently patient. One such strategy is the “grim trigger” strategy:
begin{itemize}
item Cooperate (remain silent) in the first round.
Continue cooperating as long as the other player cooperates.
item If the other player defects (confesses), then defect (confess) in all future rounds.
end{itemize}
Under certain conditions, both players choosing to cooperate in every round can be a subgame perfect Nash equilibrium.
section{Finite extensive-form game}
The following game is an extensive form game in which player 1 moves first, choosing either $mathrm{U}$ or $mathrm{D}$. Player 2 observes player 1’s choice and then chooses $mathrm{L}$ or R. The payoffs are given in the terminal nodes of the game tree. In this game, subgame perfect Nash equilibrium can be found by backward induction.
begin{center}
includegraphics[max width=textwidth]{2023_10_18_aa41930d13799f44e65bg-083_1579_2053_840_1285}
end{center}
begin{itemize}
item If Player 1 chooses U, player 2’s best response is L, yielding player 1 a payoff of 1.
item If player 1 chooses D, player 2’s best response is L, yielding player 1 a payoff of 3 .
end{itemize}
Therefore, player 1’s best strategy is to choose D. The subgame perfect Nash equilibrium is (D, L). The outcome (U,R) with payoffs $(2,2)$ is not a Nash equilibrium because player 1 can improve its payoff by deviating to D. It is also not a subgame perfect equilibrium because it is not a Nash equilibrium in the subgame starting at the node after player 1 chooses U.
section{Example: Entry deterrence/accommodation}
Two firms compete in a market by setting quantities. Firm 1, the incumbent firm, is already in the market. Firm 2, the potential entrant, must decide whether to enter the market or stay out. If Firm 2 enters, then both firms simultaneously and independently decide on the quantity to produce. The market price is then determined by the inverse demand curve
[
P(Q)=a-Q
]
where $Q$ is the total output. Each firm has a marginal cost of $c$. There is an entry cost of $K$ for Firm 2 if it enters the market.
subsection{Solution}
We analyze this game using backward induction. First, we consider the subgame that arises if Firm 2 enters the market. In this case, the firms compete in a Cournot duopoly. We know that the equilibrium quantities are
[
q{1}^{*}=q{2}^{*}=frac{a-c}{3 b}=frac{a-c}{3}
]
and profits are
[
pi{1}^{*}=pi{2}^{*}=frac{(a-c)^{2}}{9 b}=frac{(a-c)^{2}}{9}
]
If firm 2 decides to enter the market, it earns $frac{(a-c)^{2}}{9}-K$, where $K$ is the entry cost. If it does not enter, it earns 0 . Therefore, Firm 2 will enter if and only if
[
frac{(a-c)^{2}}{9} geq K
]
or
[
a geq c+3 sqrt{K} .
]
This condition tells us that if the demand is high enough or the fixed cost of entry is low enough, then firm 2 will enter the market.
Now, consider the decision of the incumbent firm, Firm 1. It can deter entry by choosing a quantity $q{1}$ such that Firm 2’s best response would yield a profit less than or equal to 0 . In other words, Firm 1 wants to choose $q{1}$ such that
[
pi{2}left(q{1}, q{2}^{*}left(q{1}right)right) leq 0
]
From the previous analysis of Cournot duopoly, we know that for a given $q{1}$, Firm 2 ‘s best response is $q{2}=frac{a-c-b q_{1}}{2 b}$. Plugging this into Firm 2’s profit function, we get
[
begin{aligned}
pi{2}left(q{1}, q{2}^{*}left(q{1}right)right) & =left(frac{a-c-b q{1}}{2 b}right)left(a-bleft(q{1}+frac{a-c-b q{1}}{2 b}right)-cright)-K
& =left(frac{a-c-b q{1}}{2}right)^{2} frac{1}{b}-K
end{aligned}
]
Setting $pi_{2} leq 0$ yields the deterring quantity
[
q_{1} geq frac{a-c-2 sqrt{b K}}{b}
]
If Firm 1 chooses this quantity, Firm 2 will not enter. In this case, the market price will be $P=a-b q{1}$ and Firm 1’s profit is $left(a-b q{1}-cright) q{1}$. To maximize this profit, Firm 1 will choose the lowest value of $q{1}$ that deters entry, i.e., $q_{1}=frac{a-c}{b}-2 sqrt{frac{K}{b}}$. The profit of Firm 1 is then
[
pi{1}=left(a-b q{1}-cright) q_{1}=left(a-bleft(frac{a-c}{b}-2 sqrt{frac{K}{b}}right)-cright)left(frac{a-c}{b}-2 sqrt{frac{K}{b}}right)=4 K
]
If Firm 1 decides to accommodate entry, it produces the Cournot output $q{1}^{*}=frac{a-c}{3 b}$, and its profit is $pi{1}^{*}=frac{(a-c)^{2}}{9 b}$. Thus, Firm 1 will deter entry if
[
begin{aligned}
4 K & geq frac{(a-c)^{2}}{9 b}
36 b K & geq(a-c)^{2}
6 sqrt{b K} & geq a-c
a & leq c+6 sqrt{b K}
end{aligned}
]
Therefore, if $a leq c+6 sqrt{b K}$, firm 1 can deter entry by setting $q_{1}=frac{a-c}{b}-2 sqrt{frac{K}{b}}$. If $a>c+6 sqrt{b K}$, then firm 1 cannot deter entry.
section{Problem 2}
Final Answer: The final answer is $boxed{1}$
