# Economic Classroom Experiments/Bertrand Competition

Bertrand (1883) [1] modelled firms competing on price (as oppossed to quantity). The experiment demonstrates the Bertrand model and tests the equilibrium prediction.

## Theory

Examine the case with two firms where both firms choose prices simultaneously and have constant marginal cost ${\displaystyle c}$. Firm one chooses ${\displaystyle p_{1}}$. Firm two chooses ${\displaystyle p_{2}}$. Consumers buy from the lowest price firm. (If ${\displaystyle p_{1}=p_{2}}$, each firm gets half the consumers.) An equilibrium is a choice of prices ${\displaystyle p_{1}}$ and ${\displaystyle p_{2}}$ such that firm 1 wouldn’t want to change his price given ${\displaystyle p_{2}}$ and firm 2 wouldn’t want to change her price given ${\displaystyle p_{1}}$.

Take firm 1’s decision if ${\displaystyle p_{2}}$ is strictly bigger than c: If he sets ${\displaystyle p_{1}>p_{2}}$, then he earns 0. If he sets ${\displaystyle p_{1}=p_{2}}$, then he earns ${\displaystyle 1/2*D(p_{2})*(p_{2}-c).}$ If he sets ${\displaystyle p_{1}}$ such that ${\displaystyle c he earns ${\displaystyle D(p_{1})*(p_{1}-c).}$ For a large enough ${\displaystyle p_{1}}$ that is still less than ${\displaystyle p_{2}}$, we have: ${\displaystyle D(p_{1})*(p_{1}-c)>1/2*D(p_{2})*(p_{2}-c)}$. From this we see that each has incentive to slightly undercut the other. Thus, an equilibrium is that both firms charge ${\displaystyle p_{1}=p_{2}=c}$.[2] Note with three or more firms, in a (pure strategy) equilibrium, we need only two of the firms to set price equal to ${\displaystyle c}$, the others can charge a higher price.

## Hand Run

Andreas Ortmann (2003) has published a hand-run version.[3] On varying the number of firms in experiments on Bertrand competition see Martin Dufwenberg and Uri Gneezy (2000) [4]

There are some elements to Bertrand competion in the Twenty-pound auction.

## Computerized Version

There is a computerized version of this experiment available on both Veconlab and Exeter games site. The main advantage of the Exeter games version is the graphing feature which allows graphing of the average sale price in addition to average price. Looking at the graph of the data, this makes the results much clearer (since for three or more firms average price is not a reflection of proximity to equilibrium).

You can quickly log in as a subject to try out this group participation experiment, by pretending to be one of the original participants in a real session. You may also find the sample instructions helpful.

## Results

This experiment has fairly consistent results. At Exeter, it has been run on sixth form (high school) students and Korean executives (from Korean Gas Corporation). When there are two firms in the market, prices are above marginal cost. This persists even when matching is random. Once we increase the number to 4 or 5 firms and random matching, the sales price drops to marginal cost. Note that the average price does not always do so. Three firms and random matching usually also goes to marginal cost.

File:Bertrand2.jpg
Results from an experiment at Exeter. The first 10 rounds were groups of 2 and fixed pairing. The second 10 rounds were groups of 2 and random pairing. The final 10 rounds were groups of 4 with random matching.

### Student Quotes

These quotes are from Exeter microeconomics students.

In reference to the two firm, Bertrand competition experiment:

“I learnt that collusion can take place in a competitive market even without any actual meeting taking place between the two parties.”

In reference to the four firm, random matching, Bertrand competition experiment:

“Some people are undercutting bastards!!! Seriously though, it was interesting to see how the theory is shown in practise.”

## Discussion

We see cooperation (collusion) between firms in setting prices in practice as well. This can occur without direct communication.

### Cooperation in Bertrand Competition

Case: The New York Post vs. the New York Daily News

 time Post Price Daily News Price January 1994 40¢ 40¢ February 1994 50¢ 40¢ March 1994 25¢ (in Staten Island) 40¢ July 1994 50¢ 50¢

Until Feb 1994 both papers were sold at 40¢. Then the Post raised its price to 50¢ but the News held to 40¢ (since it was used to being the first mover). So in March the Post dropped its Staten Island price to 25¢ but kept its price elsewhere at 50¢, until News raised its price to 50¢ in July, having lost market share in Staten Island to the Post. No longer leader. So both were now priced at 50¢ everywhere in NYC.

### Theoretical Cooperation

If firms get together to set prices or limit quantities, what would they choose?

As in your experiment, demand for the goods is ${\displaystyle D(p)=15-p}$ and each firm has the cost function of ${\displaystyle c(q)=3q}$ (marginal cost equal to 3).

#### Price Cooperation

Let us say the firms try to coordinate on a price that maximizes their profits. Here the firms' joint problem is ${\displaystyle \max _{p}(p-3)\cdot (15-p)}$. What is the choice of p? It is ${\displaystyle p=9}$. (The expression ${\displaystyle (p-3)(15-p)}$ is an upside-down parabola that starts at 3 and ends at 15 reaching its peak midway.) This is the monopoly price and D(p) is the monopoly quantity!

#### Quantity Cooperation

The joint firms' problem is ${\displaystyle \max _{q_{1},q_{2}}(15-q_{1}-q_{2})\cdot (q_{1}+q_{2})-3(q_{1}+q_{2})}$. All that matters in the solution is that ${\displaystyle q_{1}+q_{2}=6}$. (We can call ${\displaystyle q:=q_{1}+q_{2}}$ and then the problem becomes ${\displaystyle \max _{q_{1},q_{2}}(12-q)\cdot (q)}$.) This is the monopoly quantity and ${\displaystyle D^{-1}(6)}$ is the monopoly price.

[Slides] for teaching Bertrand Competition.

## Footnotes

1. Bertrand, J. (1883) Book review of theorie mathematique de la richesse sociale and of recherches sur les principles mathematiques de la theorie des richesses. Journal de Savants 67: 499–508.
2. [Kaplan & Wettstein (2000)] show that there may be other equilibria with positive profits if there aren’t restrictions on D(p).
3. Ortmann, A., “Bertrand Price-undercutting: A Brief Classroom Demonstration,” Journal of Economic Education 34.1., 2003, 21-26.
4. Dufwenberg, M and U Gneezy, “Price Competition and Market Concentration: An Experimental Study.,” International Journal of Industrial Organization 18 (2000), 7-22.

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This page was first created by Toddkaplan 07:46, 26 April 2007 (UTC).