Intro to Econ: Seventh Lecture – Pricing in the presence of a flat demand function

What if you, as a producer or at least seller of some good, face a “flat” demand function? With “flat” demand function I mean any demand function that has a non-infinite slope, that is any demand function where you can vary the price a bit and this does not immediately lead to a demand of more than you can provide (at a slightly lower price) or a demand of zero (at a slightly higher price). This means that in such a case you could choose a price, and different prices will have different consequences for you but also for your consumers.

To fix ideas consider the following situation. You are in charge of a student organization and you are trying to do a bit of fundraising. You are thinking of showing a movie in a university lecture hall at reasonable ticket prices to students. You have convinced the university that they let you have a largish lecture hall with 500 seats for free. You only have to pay for the cleaning cost, which say amounts to €200. You also have to pay for the right to show a movie, which say amounts to €500. You have otherwise convinced some of the other members of the student organization to help with ticket sales, advertising, and other matters, for free. The key question for you is now, what to charge the students for the tickets?

To simplify matters let us assume that all seats are more or less the same, or perhaps better that you sell tickets without a particular seat number and people can choose their seat on a first come first serve basis. Do you have any idea on how to choose a ticket price yet?

Well, at this point, with the information that we are given, it is pretty unclear how you would identify a ticket price. You lack a key bit of information: what is your demand function? How would you know the demand function you are facing? Well, this is pretty hard sometimes. I guess in most cases firms learn their demand function from experience. Think of Lufthansa’s flight ticket pricing problem from a previous post, for instance. Suppose you look at the direct flight from Graz to Dusseldorf, which operates I believe three or four times a week. It may be hard for Lufthansa to work out the demand function for the very first such flight, but the longer they operate this flight the better they probably understand the demand function and how it changes on different days of the week and different months of the year and so on. Well, in fact, Lufthansa’s automatic booking system can automatically detect changes in the demand function. If the automatic booking system experiences a sudden increase in ticket requests (perhaps because of a conference staged in Graz – unknown to Lufthansa) then it automatically “infers” that the demand has risen and reacts accordingly. I am pretty sure that the real booking system Lufthansa uses is pretty sophisticated and pretty good at identifying any changes in demand for any flight.

Or consider a bakery and their decision as to how much bread and how many bread rolls and other things to stock on any given day. I believe that experience, together with some knowledge of the calendar, allows bakeries to make these decisions pretty well on average and that these decisions are based on a pretty good understanding of the demand they face. But of course, one can also easily get this wrong sometimes. I remember one Sunday that was a day of voting for some general election in Graz when I found that the local bakery next to the voting booth was almost devoid of all food. When I asked them, they “admitted” that they underestimated the demand on that day. But, on the whole, I find it pretty amazing that I almost always get what I want (or a close substitute) from my bakery.

Sorry for the long excursions. Back to our problem. Where do you get the demand function from? Actually, let me not get back to our problem just yet, but let me add one last bit of excursion. I believe that correctly identifying (at least a good lower bound) for a product’s demand function is a key ingredient to the possible success of any start-up. It is definitely something that an investor in your start-up will want to be pretty sure about (or at least properly understand its risks – and perhaps demand a premium for such risks – see more on this in a future post).

But now let us really get back to our problem. Suppose in your case that the student organization has done similar things in the past and that you have a reasonably good idea of the demand function that you are facing. Let us assume it is roughly as follows. At a price of zero, that is if you give the tickets away for free, you would find that roughly 700 people would be interested in coming. At a price of €1 you would get 600 people interested, at a price of €2 you get 500 people, at a price of €3 this would be 400 people, at €4 there would be 300 people, at €5 we get 200, at €6 one hundred, and at €7 no one would  come. With this information, what price would you, finally, choose?

Have you made your decision? When I asked my students in the class roughly 60% said they would charge €3, another 20% would charge €4, and another 16% would charge €2, with the remaining 4% going for €1 and €6 equally.

In fact both €3 and €4 give the same “revenue” (and thus the same “profits”) of 3 times 400 = 1200 € for revenue (and 1200 – 700 = 500 € for profits), and this is the best profit we can get if we only allow full euro-amounts. If we actually allowed also euro-cent amounts, we could do even better (assuming the demand function extends naturally (that is linearly) between prices of 3 and 4) profit-wise by charging €3,50 per ticket and then get a revenue of 3,50 times 350 = 1225 € (with profits of 525 €). If all this is approximate, as it most likely is, then this is our best guess as to what price would be the best for your fundraising efforts. But perhaps this is not our (only) goal?

How about our “consumers”? If we compare prices €3 and 4 only, then consumers would of course prefer the lower price, after all an extra 100 people would prefer to come to the movie at the lower price and all those 300 that would come anyway have to pay less now and have a little bit more money to spend on other things. That’s probably why most of my students went for the price of €3 instead of €4.

But, even at a price of €3 per ticket, is the allocation of tickets (that is seats) to people, Pareto-optimal? This is a tricky question, but it actually has a very clear answer. At first glance, and given what we have learnt so far, it seems that the allocation might well be Pareto-optimal. After all we said that letting the price adjust so that there is not too much demand is the only way we get Pareto efficient allocations, when there is money that people could spend on other things. In fact, and maybe I did not stress this enough in a previous post (rationing) this is only a pre-condition for a Pareto efficient allocation. For a Pareto-efficient allocation, when there is money which one could spend on other things, we need that the tickets go to those people who have the highest willingness to pay for these tickets. This is still true. And, of course, we have that here. But this is true for any price, as long as the price is such that the demand is less than the number of possible tickets we could have sold.

So why does a price of €3 (or any higher price, in fact any higher price than €2) not lead to a Pareto-efficient allocation after all? Think of it this way: after you sold the 400 tickets for €3 each, there are still 100 empty seats and many people (300 in fact) who would be happy to take a seat (but for a lower ticket price). If you could somehow manage to sell another 100 tickets at a price of say €1 then we would get a Pareto-improvement [I am not saying that this is easily achievable!]. This is why the allocation with 400 tickets at a price of €3 is not Pareto-efficient. Because both you (the student organization) as well as some consumers would be better off without making anyone worse off. But what then would be Pareto-efficient? The only answer is a price of €2, which then leads to 500 tickets sold. [One should maybe add that it is possible that people do not like being in such a crowded lecture hall, in which case we would say that any ticket sold to someone else creates a negative externality on someone who already holds a ticket. Then this argument is not correct and a price of €2 is not Pareto-efficient after all. See more about externalities in a future lecture and about this particular problem in this post here. On the other hand for football stadiums the externality is possibly reversed and it may be in the at least long-run interest of the football club to charge fairly low prices to make sure that the stadium is not half empty, which might lead to even fewer people interested in coming to the next game.]

Does a price of €2 lead to a Pareto-improvement over a price of €3? Almost. A price of €2 is clearly better than a price of €3 for all consumers. Only one person (party) suffers: you, or the student organization, are clearly worse off at this price. So again, Pareto-efficiency is a subtle concept. A price of €3 leads to a Pareto-inefficient allocation of tickets, a price of €2 leads to a Pareto-efficient allocation of tickets, but the latter case is not a Pareto-improvement over the former.

But note that, especially if we care mostly about the consumers, this is one major problem that firms with some monopoly power, such as an airline who is the only one offering flights from A to B, or the producer of Xboxes or PlayStations, or (interestingly and very upsetting to me) also some publishers of scientific journals, or producers of certain fairly unique drugs, or any other firm with a “flat” demand function, induce: that they charge too high a price and sell too little of their product. With “too high” and “too little” I mean that there are potential customers of this good who have a higher willingness to pay for this good than it would cost the firm to produce this good and yet these customers do not get this good.

5 thoughts on “Intro to Econ: Seventh Lecture – Pricing in the presence of a flat demand function

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