<blockquote>
<p>if there are any papers that study the [e]ffect of n on S[P]NE price between the competitive and monopoly price</p>
</blockquote>
<p>In general, equilibria in (repeated) Bertrand competition would not depend on <span class="math-container">$n$</span>. This is due to the demand structure in such games, i.e. firm(s) with the lowest price gets (or share, in the case of ties) the entire market demand at that price. It may be possible to purposely construct an equilibrium in which the equilibrium price is a function of <span class="math-container">$n$</span>, but those are more the exceptions than the rule.</p>
<p>You correctly mentioned that <span class="math-container">$n$</span> matters in supporting a collusive outcome as an equilibrium. For example, <span class="math-container">$n$</span> may determine the discount factor that the firms must have in order to collude at a particular outcome. However, once such a collusive equilibrium is ensured to exist, <span class="math-container">$n$</span> becomes less relevant in determining the equilibrium price. This is especially true in view of the fact that the most focal collusive outcome is the monopoly price, which does not depend on <span class="math-container">$n$</span>.</p>
<p>Give the above, you're unlikely to find any paper on the effect of <span class="math-container">$n$</span> on the equilibrium price of repeated Bertrand competition.</p>
<blockquote>
<p>whether one can design a closed form model for which there is unique equilibrium price for different levels of n?</p>
</blockquote>
<p>The canonical <span class="math-container">$n$</span>-firm Cournot competition features the equilibrium price as a function of <span class="math-container">$n$</span>: <span class="math-container">$$p^* = \frac{a+nc}{n+1}.$$</span> But this is probably not what you're looking for.</p>
<p>In an infinitely repeated Bertrand competition, you won't get uniqueness of SPE, as repeated games in general tend to have a multiplicity of equilibria. For example, there are already at least two in repeated Bertrand: (i) all firms play marginal cost pricing (i.e. stage game NE) in every period regardless of history; and (ii) all firms play grim trigger strategy with the monopoly price (given suitable discount factor). Again, you are likely able to construct an SPE where price is a function of <span class="math-container">$n$</span>, but that will be one among many SPEs of the game.</p>

CC BY-SA 4.0
2024-08-23T22:11:54.090Z
Thanks for the response! I am more interested in, as you said, constructing a set of rules which lead to price as function of n. Are you familiar with these "exceptions" or any papers that discuss rules for deciding equilibrium in bertrand collusion game with multiplicity? Ultimatly, I was pondering a model of asymmetric firms producing a homogenous good with entry deterrence and collusion. I thought might be analytically pleasing as may allow changes in marginal cost (i.e. innovation) but price pinned down by n symmetric leaders rather than entire distribution of costs.
thow96
CC BY-SA 4.0
2024-08-24T02:19:47.467Z
@thow96: [Baye and Morgan (1999)](http://code.uab.es/xmg/Docencia/IO-en/IOReadings/SharingRules/BayeMorgan99.pdf) is a good place to start. They construct a symmetric mixed strategy equilibrium for the one-shot Bertrand game and the strategy is a function of $n$. Note that even though they call their main result a "folk theorem (FT)", it's an equilibrium for the one-shot (stage) game, and thus should not be confused with the FT(s) in the repeated games literature. The authors do discuss how the two FTs are related and how their FT can be extended to finitely and infinitely repeated games.
Herr K.
CC BY-SA 4.0
2024-08-24T02:26:48.823Z
Another way, which is almost trivial, is to simply use a grim trigger strategy with a price that's dependent on $n$, e.g. $p^*=c+\frac1n(p^M-c)$. In an infinitely repeated Bertrand game, this can be supported with a sufficiently high discount factor. But you may have a hard time justifying why firms would collude on such an ad hoc price instead of the monopoly price which does not depend on $n$.
Herr K.
CC BY-SA 4.0
2024-08-24T11:43:46.023Z
Thank you again! I will continue to read Baye and Morgan (1999) and go from there. From my brief reading of the paper it seems that so long as the choke-price is unbounded or bounded but uncertain, a positive-profit equilibrium can be achieved for some k>0 under mixed strategies with distribution F(p,n,k).  The intuition is that firms weight both probability of collusion and probability of being undercut, leading to p>c. However, in this paper's example, expcted profits are independent of n, hence price will be. But may depend on functional form of F or sampling, ill have to go throught it.
thow96
CC BY-SA 4.0
2024-08-24T13:23:01.287Z
@thow96: You can always let $k $ be a function of $n $
Herr K.
CC BY-SA 4.0
2024-08-25T00:17:33.487Z
I will play around with it. I also found an interesting table in Saporiti and Coloma (2010) covering number of papers which demonstrate games/setups that resolve the Bertrand Paradox. Mixed strategies in one shot games seem most compelling for my own interest. Thanks again!
thow96
CC BY-SA 4.0
2024-08-22T19:19:28.537Z
2024-08-22T19:19:28.537Z
Herr K.
<p>Under bertrand competition with n symmetric firms the price can be equal to marginal cost for finite repeated games but a collusive monopoly price can be sustained for infinitely repeated games. Moreover, sustaining the monopoly price becomes more difficult as n increases.</p>
<p><strong>Q:</strong> However, I want to know if there are any papers that study the affect of n on equilibrium price between the competitive and monopoly price. I am interested in whether one can design a model for which there is unique equilibrium price for different levels of n?</p>

<game-theory><oligopoly>
Any papers on bertrand pricing and number of firms?
thow96
# Any papers on bertrand pricing and number of firms?

(Asked by: username_0 on 2024-08-22)

Under bertrand competition with n symmetric firms the price can be equal to marginal cost for finite repeated games but a collusive monopoly price can be sustained for infinitely repeated games. Moreover, sustaining the monopoly price becomes more difficult as n increases.

**Q:** However, I want to know if there are any papers that study the affect of n on equilibrium price between the competitive and monopoly price. I am interested in whether one can design a model for which there is unique equilibrium price for different levels of n?

**Tags:** game-theory, oligopoly

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## 1 answers

### Answer by username_1 on 2024-08-22. Score: 0

> if there are any papers that study the [e]ffect of n on S[P]NE price between the competitive and monopoly price

In general, equilibria in (repeated) Bertrand competition would not depend on $n$. This is due to the demand structure in such games, i.e. firm(s) with the lowest price gets (or share, in the case of ties) the entire market demand at that price. It may be possible to purposely construct an equilibrium in which the equilibrium price is a function of $n$, but those are more the exceptions than the rule.

You correctly mentioned that $n$ matters in supporting a collusive outcome as an equilibrium. For example, $n$ may determine the discount factor that the firms must have in order to collude at a particular outcome. However, once such a collusive equilibrium is ensured to exist, $n$ becomes less relevant in determining the equilibrium price. This is especially true in view of the fact that the most focal collusive outcome is the monopoly price, which does not depend on $n$.

Give the above, you're unlikely to find any paper on the effect of $n$ on the equilibrium price of repeated Bertrand competition.

> whether one can design a closed form model for which there is unique equilibrium price for different levels of n?

The canonical $n$-firm Cournot competition features the equilibrium price as a function of $n$: $$p^\* = \frac{a+nc}{n+1}.$$ But this is probably not what you're looking for.

In an infinitely repeated Bertrand competition, you won't get uniqueness of SPE, as repeated games in general tend to have a multiplicity of equilibria. For example, there are already at least two in repeated Bertrand: (i) all firms play marginal cost pricing (i.e. stage game NE) in every period regardless of history; and (ii) all firms play grim trigger strategy with the monopoly price (given suitable discount factor). Again, you are likely able to construct an SPE where price is a function of $n$, but that will be one among many SPEs of the game.

> **Comments:**
> **username_0**: Thanks for the response! I am more interested in, as you said, constructing a set of rules which lead to price as function of n. Are you familiar with these "exceptions" or any papers that discuss rules for deciding equilibrium in bertrand collusion game with multiplicity? Ultimatly, I was pondering a model of asymmetric firms producing a homogenous good with entry deterrence and collusion. I thought might be analytically pleasing as may allow changes in marginal cost (i.e. innovation) but price pinned down by n symmetric leaders rather than entire distribution of costs.  
> **username_1**: @thow96: [Baye and Morgan (1999)](http://code.uab.es/xmg/Docencia/IO-en/IOReadings/SharingRules/BayeMorgan99.pdf) is a good place to start. They construct a symmetric mixed strategy equilibrium for the one-shot Bertrand game and the strategy is a function of $n$. Note that even though they call their main result a "folk theorem (FT)", it's an equilibrium for the one-shot (stage) game, and thus should not be confused with the FT(s) in the repeated games literature. The authors do discuss how the two FTs are related and how their FT can be extended to finitely and infinitely repeated games.  
> **username_1**: Another way, which is almost trivial, is to simply use a grim trigger strategy with a price that's dependent on $n$, e.g. $p^*=c+\frac1n(p^M-c)$. In an infinitely repeated Bertrand game, this can be supported with a sufficiently high discount factor. But you may have a hard time justifying why firms would collude on such an ad hoc price instead of the monopoly price which does not depend on $n$.  
> **username_0**: Thank you again! I will continue to read Baye and Morgan (1999) and go from there. From my brief reading of the paper it seems that so long as the choke-price is unbounded or bounded but uncertain, a positive-profit equilibrium can be achieved for some k>0 under mixed strategies with distribution F(p,n,k).  The intuition is that firms weight both probability of collusion and probability of being undercut, leading to p>c. However, in this paper's example, expcted profits are independent of n, hence price will be. But may depend on functional form of F or sampling, ill have to go throught it.  
> **username_1**: @thow96: You can always let $k $ be a function of $n $  
> **username_0**: I will play around with it. I also found an interesting table in Saporiti and Coloma (2010) covering number of papers which demonstrate games/setups that resolve the Bertrand Paradox. Mixed strategies in one shot games seem most compelling for my own interest. Thanks again!  

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