Intro to Econ: Seventh Lecture – Competition

Economists tend to think that competition between firms is a good thing. In fact most countries (all?) have some anti-trust regulation in some form or another. Anti-trust means against “trusts”, where trusts are here meant to be cartels (groups of firms) that collude especially by determining prices together and thus avoid competitive pricing. But how would competition improve matters in the first place?

Let us go back to the fundraising efforts of the student organization from the previous lecture. The setup is the same: the student organization wants to show a movie in a lecture hall that can seat up to 500 people and the demand function is such that at a ticket price of P (in euros) the number of people interested in a ticket is given by 700-100P (that is 600 interested people at a price of 1 €, 500 at a price of 2 €, and so on).

But now suppose that instead of there being one student organization there are actually two with very independent interests. Each student organization is given 250 tickets which they can sell at any price they want. Suppose that they share the cost for cleaning and the movie equally before the whole thing begins. And suppose that you are in charge of exactly one of these student organizations. How should you determine the price of your tickets now? Should you charge the 3,50 that were profit-optimal when there was only one student organization or should you charge another amount?

In some ways “ideally”, for you and the other student organization, you would both charge a price of €3,50 and so roughly split the €1225 revenue that you would make together. You would get a revenue of €612,50 and from this you would have to pay €350 for the cleaning and the movie.

But what if the other student organization charges €3,50 and you charge slightly less, say €3. We first need to think about what we think the consumers would do in this case. Well, I guess if they see the two prices they would first try to buy your tickets, after all they are cheaper and a ticket is a ticket regardless of who they buy it from (and we believe that people prefer, ceteris paribus, having more money over less). So suppose this is more or less the case. Recall that at €3 the total demand for tickets is 400. You can only sell 250 as this is how many you have. So your revenue would now be 3 times 250 which is equal to €750. So this is €137,50 more than in the previous case! Undercutting the price would pay off for you!

How much would your opponent “make” in this case? This is actually a tricky question and we will need to make another assumption. We need to think about who the people are that bought tickets from you and who the people are that did not in order to determine what kind of people are still left with a desire to buy a ticket. In order to make sense of what I just wrote, we need to go a bit deeper into our demand function (in fact, very much as in the market experiment). There are two ways to think about a demand function. The one that I typically favor provides the number of tickets that people would buy as a function of the price. But we can also turn this upside down if you like. We can also think of the demand function as providing the price that is necessary so that a certain number of tickets can be sold as a function of the number of tickets. This is often called the inverse demand function, although many people also just call it the demand function. If you think of it this way, with number of tickets demanded on the x-axis and price on the y-axis, then how do we interpret the intercept, the value at which the (inverse) demand function intersects the y-axis? This is the price that is just so high that no one would want to buy a ticket. If we decrease the price we would get one person interested in buying a ticket. What characterizes this person? It is the person with the highest maximal willingness to pay for a ticket! Here this highest willingness to pay is €7. If we go down a Euro-cent we hit the maximal willingness to pay for a ticket from the next most interested consumer, and so on.

So at a price of €3 we get 400 people interested but only 250 can buy (remember my post on rationing). So who buys? It must be some selection of all the 400 people with a maximal willingness to pay of €3 or higher. You can play with different scenarios if you like, but I will here assume that the people with the highest willingness to pay (that is all with a willingness to pay of €4,50 or higher) get a ticket. This scenario makes the profits to the other student organization as low as possible. But please do play with other scenarios: the other extreme would be that the people with a willingness to pay between €3 and €5,50 get the 250 tickets. In my scenario, what is then the demand “left over” for the other student organization at their price of €3,50? It is all the 100 people with a willingness to pay between €3,50 and €4,50. The other student organization then gets a revenue of 3,50 times 100 equal to €350 (just enough to cover their costs). If you played with the other extreme case, I believe you would have found that the other student organization would be getting 3,50 times 150 equal €525. This is the best they can hope for and is still considerably less than the €612,50 that they would have expected if both of you had charged a price of €3,50.

So if both student organizations really do choose prices independently and only care about the money that they themselves raise, then this starts a race to the bottom. I worked out the whole table of possible revenues for the two organizations for each possible price combination (assuming that only multiples of 50 euro-cents are considered). Here you are:

€ 1,00
€ 1,50
€ 2,00
€ 2,50
€ 3,00
€ 3,50
€ 4,00
€ 4,50
€ 5,00
€ 1,00
€ 250,00
€ 250,00
€ 250,00
€ 250,00
€ 250,00
€ 250,00
€ 250,00
€ 250,00
€ 250,00
€ 1,50
€ 450,00
€ 375,00
€ 375,00
€ 375,00
€ 375,00
€ 375,00
€ 375,00
€ 375,00
€ 375,00
€ 2,00
€ 500,00
€ 500,00
€ 500,00
€ 500,00
€ 500,00
€ 500,00
€ 500,00
€ 500,00
€ 500,00
€ 2,50
€ 500,00
€ 500,00
€ 500,00
€ 562,50
€ 625,00
€ 625,00
€ 625,00
€ 625,00
€ 625,00
€ 3,00
€ 450,00
€ 450,00
€ 450,00
€ 450,00
€ 600,00
€ 750,00
€ 750,00
€ 750,00
€ 750,00
€ 3,50
€ 350,00
€ 350,00
€ 350,00
€ 350,00
€ 350,00
€ 612,50
€ 875,00
€ 875,00
€ 875,00
€ 4,00
€ 200,00
€ 200,00
€ 200,00
€ 200,00
€ 200,00
€ 200,00
€ 600,00
€ 1.000,00
€ 1.000,00
€ 4,50
€ 0,00
€ 0,00
€ 0,00
€ 0,00
€ 0,00
€ 0,00
€ 0,00
€ 562,50
€ 1.125,00
€ 5,00
€ 0,00
€ 0,00
€ 0,00
€ 0,00
€ 0,00
€ 0,00
€ 0,00
€ 0,00
€ 500,00

The way to read this is as follows. The blue left-most column has the possible prices that you could choose for your student organization. The green up-most row has the possible prices that the other student organization could choose. The euro amount in each cell is then your revenue if you choose your corresponding price in blue and the other student organization their price in green. So for instance, you can find your revenue of €750 when you choose a price of €3 while your “opponent” chooses a price of €3,50. I have marked your best “reactions” to possible opponent prices in red in the table. If you stare at the whole thing for a few minutes you will see that this setting gives rise to a certain repeated undercutting logic. If your opponent chooses a price of €5 you would optimally choose one of €4,50. To your opponent’s choice of €4,50 you would choose €4. To your opponent’s €4 you would choose €3,50. To your opponent’s €3,50 you would choose €3. To your opponent’s €3 you would choose €2,50. And, interestingly finally, to your opponent’s €2,50 you would still choose €2,50. [In fact, if we considered any multiple of 10 euro-cents you would now optimally react with €2,40, and then all the way down to €2,10 both.] So, while a price of €2,50 each is not as good as a price of €3,50 each for both of you, the undercutting logic can easily lead you to €2,50 and this is much better for your consumers! In fact this is what economists tend to think and hope that competition does: it leads to lower prices and to an allocation that is thus much closer to the Pareto-efficient allocation (with 450 sold tickets at a price of €2,50 we are not far from the Pareto-efficient 500 sold tickets at a price of €2 – see previous post).

Will this really happen if we have two such firms in a market? Well, it all depends. In my class I managed to let my students play this game against each other in a slightly confusing way and a relative majority of them ended up choosing a price of €2,50. But I did not actually pay them any money, so I do not know how indicative this is of what would go on in the real world. In the real world firms sometimes manage to avoid this form of competition. In Austria fairly regularly big construction companies are caught colluding, that is they are caught meeting and discussing setting high prices. This is illegal in Austria and therefore punished in the form of heavy(?) fines. By the way, in the language of game theory, the theory of strategic interaction, what I have just described is called a “Bertrand game” (after some person Bertrand who wrote about this in 1883) with capacity constraints (you may want to look at a paper by Kreps and Scheinkman, Bell Journal of Economics, 1983) and the €2,50 would be the (not weakly dominated – whatever this means) “Nash equilibrium” strategy for both “players” in the game. A “Nash equilibrium” (after a person called John Nash who wrote about this in 1950) can here be defined as a price pair (one price for each student organization) such that no student organization would want to deviate to another price if they believe that the other student organization chooses according to the original price pair. Note that a price of €2,50 for both satisfies this definition. Actually, so does a price of €2 for both (but it is weakly dominated – as I said this is not so interesting at this point). Note that both choosing a price of €3,50 is not a Nash equilibrium exactly because if you assume your opponent really does choose a price of €3,50 then you do not want to do so, you want to choose €3 in this case.

To study collusion researchers have also considered what could happen in the “repeated game”. After all in most cases firms compete for the same customers not just once, but every week, day, month, hour, year anew (the frequency may depend, among other things, on how “durable” the good under consideration is or how often production decisions can be made). And it turns out that in the repeated game – meaning that the competition is a long-lasting one, which after all often seems more plausible, it may be quite easy or at least possible for firms to collude without even speaking to each other. But I should really leave this for another lecture, probably in another course on game theory. But just briefly, how could this work in our setting? Suppose the two student organizations hold a similar event once every month. They could follow the following repeated game “strategy”: choose a price of €3,50 always unless at some point your opponent chose differently, in which case choose a price of €2,50 forever after. If both student organizations follow this so-called “grim-trigger” strategy, then both have no incentive to charge any price other than €3,50. In reality such “tacit collusion” is not always that easy because the demand function is probably not always the same over time and the two (or more) firms do not necessarily observe what the other firm’s prices are.

So you have seen the reason why economists tend to like competition. It tends to bring prices down and this is good for the consumers and (at least closer to) Pareto-efficient. But whether or not the presence of two or three firms is enough to have proper competition is not always clear and should probably be investigated on a case per case basis.

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