Markups And Entry In A Circular Hotelling Model

>> John B. Taylor: Okay, let's get started for all of you for coming and being here. He's the Paul Warburg professor. Are you still there?

>> John B. Taylor: That's very important.

>> Speaker 2: Come on, he was on the Federal Reserve Board.

>> John B. Taylor: Okay, forgot about that. PhD at Harvard, BS from Caltech. And the paper is Markups and Entry in a Circular Hotelling Model by Complex.

Anyway, we're interested to hearing what you have to say and there'll be lots of commentary, I'm sure. From all over the place, back and forth, so welcome, Robert.

>> Robert J. Barro: Okay, thanks. This work is based on the classic hotelling model from 1929, I think is one of the all-time most cited papers, actually.

Anyway, so hotelling constructed this locational model which he used to think about pricing behavior in particular as well as business locations. And this model was based on a finite straight line. But I'm gonna use a circular version of this model which is usually attributed to Salip, although, there's actually an earlier version of it that exists.

But the circle model is much more tractable because it's completely symmetric, doesn't have endpoints that you have to worry about. And I think it brings out the main points. And I'm gonna think about this as a conceptual framework especially to analyze price markups, which I think the model is very well suited for.

And of course there's been a lot of interest in markups, but this is basically a conceptual paper of where markups come from. And the results are gonna relate particularly to the well known learner formula for markups based on the elasticity of demand and in a sense the hotelling approach.

You can think of it as either supplementing or substituting for the learner kind of formula, which relates particularly to the elasticity of demand. So I'm gonna think about this kind of circle framework which has a circumference which represents the overall size of the economy but that's not particularly important. The stores are gonna be arranged around the circle. You can think about the buyers as being uniformly distributed around the circumference. And that's gonna be taken as given and the store. There's gonna be a finite number of stores which service these consumers. So I'm gonna focus on some store called store 1, which is at the bottom.

And there'll be capital N stores in total. I'm gonna assume at the beginning that they're evenly spaced around the circle, but that's gonna be actually an equilibrium outcome in this model that they'll be located like that. So there'll be some spacing between the stores, which I call 2H.

I call it too little H, because half of that is gonna end up being the distance or market that the store is gonna service on either side of its location. So you can think about store 1 as competing directly with store 2 to the right and store capital N to the left.

And that's gonna generalize to the whole market, but I'll focus on this individual store 1, anyway. Okay, so there's some cost of production here, C Hotelling had that being zero. So there's some positive marginal cost of production, which is gonna be the same for all of the stores.

And there's gonna be in addition to that, some costs that people pay to transport goods from the store where they buy the stuff to where they're living, which is. And they're uniformly distributed around the circle. So little Z is gonna represent the distance of a customer from the store.

And there's gonna be some transport cost parameter analogous to hoteling little T, which is gonna tell you how much you have to pay to do this. Transporting little T times the distance Z is gonna be the cost of transport. It's of some importance actually, this is a cost per unit transported.

So the total transportation cost is gonna be proportional to the quantity and the quantity is gonna be determined in the model I have here. It's not gonna be like everybody buys one unit, which is sometimes the way this model is applied. So TZ is gonna effectively add to the cost per unit of buying stuff.

So stores are gonna price at some amount and store J is gonna have some price that it sets P sub J. It'll be P sub 1 for store 1. And effectively the customers, they're all gonna pay the same price, but they're gonna have a different transport cost depending on where they're located. So each customer is gonna pay the explicit price plus the TZ associated with the transport cost.

>> Speaker 4: That's a assumption or a conclusion that they will pay the same price?

>> Robert J. Barro: That'll be a conclusion of the model. But what I meant by paying the same price here is a given store, say store 1, is not gonna discriminate in terms of where the customers are coming from.

>> Speaker 4: Yeah, but that's an assumption, or.

>> Robert J. Barro: So it's an assumption that stores that they don't even know where the customers are coming from. You're posting a price, and anybody who shows up is paying that explicit price. But it'll be an equilibrium outcome that the price charge is gonna be the same for all the stores. So I'm not imposing that at the beginning, I didn't know which of those two things are.

>> Speaker 4: I mean, you were opposing the fact that they can't tell where the customer's coming.

>> Robert J. Barro: Right, right, right. Yes.

>> Speaker 4: They can't-

>> Robert J. Barro: I agree.

>> Speaker 4: Set up a contract with the customer signals or something like that.

>> Robert J. Barro: Yeah, they might wanna have some kind of discrimination. That's not what's in this way I'm doing it. Now I think it's important in applications, as hotelling stresses in his original paper, that this transportation cost thing is basically a metaphor. And it's supposed to represent things aside from the price that people care about when they're buying goods from stores.

And there might be lots of distinguishing characteristics of goods or stores that people care about. So you can have potentially people paying different prices compensated by this other characteristic. So here it's all about transportation costs, and then it can work out explicitly the whole equilibrium within that model.

>> Speaker 5: So this is literally a model of horizontal differentiation. Do you think about it this way and why do you think it's a compelling framework? Obviously, there are models of vertical differentiation too. So, you go back to an old timer just, I was curious about the diary. What you love this particular model crisis person.

>> Robert J. Barro: So, I'm not sure I'm getting this. If I think about some other framework like from Lancaster, where you think about some kind of other differentiating characteristics, is that what you would think of this?

>> Speaker 5: Yeah, exactly.

>> Robert J. Barro: So, I thought that it was analogous to this, and I didn't think it was different.

And that I thought meant that this model had more application than it might seem. That's how I was thinking about it.

>> Kenneth Judd: How do you compare it to Dixit Stiglitz? The way I always thought about.

>> Robert J. Barro: So I'm thinking about Atkinson and Burstein, which I think is related to that. And then you can think about, they think about substitutions within a sector and across sectors. And that I think bears some relation to the hotelling type model. I think you can make that parallel. Anyway, so capital N is the number of stores and capital H is the whole circumference.

So this distance that I care about between two adjacent stores is 2H is given by this simple formula here. And ultimately, as I said, half of that is what is gonna end up being relevant half of 2H, which is what I'm going to call H. So for the moment, let me think about store one and let me suppose that it's gonna service a market out some distance going towards store 2.

And I'll call that H sub 1, 2, meaning that it's the distance going toward store 2 from store 1. And ultimately that's gonna equal H. But I'm not gonna impose that condition on the problem. So I'll just call it some market distance, H12, which is gonna be covered to the right.

And I'm gonna think about the analysis applied in that range from store 1 to store 2. But it's gonna be completely parallel, going to the left towards store, capital N. And I don't have to do that kind of separately. So here I'm gonna think about the market between store 1 and store 2.

>> Speaker 5: Let me ask you, maybe now I can ask you my same question again. So, this is a model in which the differentiation sits on the firm side and from the consumer side is perceived as a cost to accessing the store because there's a geographical distance among them.

But you can think about a story for price dispersion coming from differentiation on the consumer side, like consumers different in the marginal willingness to pay for the good. And so, I was using the Trita taxonomy between horizontal and vertical differentiation, and I was curious.

>> Robert J. Barro: So maybe when we finish this, we can think about how that may or may not apply to that broader, maybe deeper context.

>> Speaker 5: Thank you.

>> Robert J. Barro: Okay, so initially, I'm thinking of the number of stores in the market capital N is given, but ultimately that will be determined from a free entry condition. So, I'll get back to that a bit later.

>> Kenneth Judd: Now, you say that things are symmetric between 1 and 2 in 1 and n. However, n may be at a different distance than 2.

>> Robert J. Barro: They won't be in equilibrium but at the moment they could be.

>> Kenneth Judd: They may be the result of equilibrium. But when you're looking, analyzing it before equilibrium, you have to take that into account possibility.

>> Robert J. Barro: Right.

>> Kenneth Judd: That the distances are different but you're not doing that.

>> Robert J. Barro: Well, at the moment, I'm assuming that the stores are evenly spaced around the surface.

>> Kenneth Judd: Is that an assumption or is that a claim for an equilibrium?

>> Robert J. Barro: It's an assumption at the moment, and then I'm gonna claim later on that that's an equilibrium in the model at the bottom.

>> Kenneth Judd: But then to make that claim, you've got to analyze the case where there's not conjecture.

>> Speaker 5: This is a metric equilibrium.

>> Kenneth Judd: Okay, but the verification requires looking at the case where the distances are not the same.

>> Robert J. Barro: Nobody wants to show.

>> Kenneth Judd: Well, he's gonna talk about an entry process, he says.

>> Robert J. Barro: Okay, so let me think about this market between store 1 and store 2 for the moment, okay? And then the effective price that consumers at distance Z pay for buying from store one is what's given by equation two. It's the explicit price that store one charges for the good P1 plus the TZ.

And of course, that's going to vary across distances, z going from zero out to this thing H sub 12, which is what I'm denoting it for the market distance. Okay, this is just some statement that this is analogous to the Lancaster and Baumol type of models that were around going back to the 1960s.

So I'm not doing anything with that at the moment.

>> Speaker 7: Can store 3 participate in store 1?

>> Kenneth Judd: Yeah.

>> Robert J. Barro: Actually, yes but it'll turn out not to be part of the equilibrium that that's gonna happen. The question will be, can store 1 price in such a way as to sort of go beyond store 2 in terms of the customers that it's attracting?

>> Kenneth Judd: Yeah.

>> Robert J. Barro: So, I consider that later. So at the moment, initially I'm going to assume that this H12 is in the interior between 0 and 2h. So, the distance out to store 2 is 2h at the moment. And I'm going to assume the market distance is interior at the moment.

And then I'm gonna see whether that in fact is the equilibrium. I'm going to allow it to be different from that and show that that produces a contradiction that suggests that it's the equilibrium. Okay, so for the moment, let me think about the market distance as interior between 0 and 2h.

That is gonna be a certain quantity that people are buying. So they're not just buying either 0 or 1 unit of the good, but rather there's some constant elasticity demand function dependent on the effective price, which is this P1 star of Z. So that the quantity bought by consumers at distance z is going to be given by equation three.

So there's two parameters in there. One is the scale of demand by individuals, which is the capital A. So that that's gonna be some parameter you can move around. And the other more important thing is the elasticity of demand eta, which is taken to be constant. So the standard learner type analysis of this problem is going to require that the elasticity be bigger than one.

And that's not gonna be required in this model. In this hotelling model that's not gonna be required. So, I think that's important because it doesn't seem like typically it would be the case that this elasticity of demand would be bigger than one in magnitude. Anyway at the moment, I'm just assuming it's some positive ETA.

>> Kenneth Judd: That demand function assumes that you purchase something from all stores at all distances.

>> Robert J. Barro: This is the demand within this market area from buying stuff from.

>> Kenneth Judd: What is the demand function over this for store one at people at location two, location three, etc. What's the demand function for those people?

>> Robert J. Barro: The goods provided are gonna be viewed as perfect substitutes, except for the fact they're different distances. So people who are beyond are gonna be buy zero of store one's goods. People who are beyond this market distance, which I call this H12 going to the right, they're gonna buy nothing from store one.

>> John B. Taylor: Unless store one undercuts to such.

>> Robert J. Barro: Yeah, so then I consider whether that might be desirable for store one to do that kind of price undercutting. They don't.

>> John B. Taylor: So Ken, what you're asking for is analysis with a set of HIJs, more general analysis and then showing that a bunch of things can be eliminated, likely being symmetric.

>> Kenneth Judd: Because symmetry is not a natural outcome of this kind of stuff.

>> Robert J. Barro: The equilibrium that I propose and focus on is completely symmetrical. And I find that compelling myself, but at the moment, let me see if I can construct that.

>> Speaker 9: Can you give us a hint on its elasticity? Because as I'm remembering, the problem with the elasticity lesson one is that you charge an infinite price, right?

>> Robert J. Barro: Exactly.

>> Speaker 9: So how are you gonna get around that?

>> Robert J. Barro: It's because the market border is gonna react in a way to make the effective elasticity bigger than one.

>> Speaker 9: Yeah, not just, so that's not the demand curve.

>> Robert J. Barro: Exactly.

>> Speaker 9: That's not the demand people in a specific epsilon, right.

>> Robert J. Barro: This is the quantity that would be demanded by some individual given that they're buying from this store one.

>> Speaker 9: Epsilon buying from somebody.

>> Robert J. Barro: Yes, and ETA is actually gonna be the elasticity of demand in the whole market. It's actually going to be that. But that's a different point, let me put that off. Okay, so in hotelling, assumed ETA was equal to zero. A lot of people have versions of the model where that's true, where ETA is equal to zero, which has important implications going forward with the results.

Okay, so I'm gonna think about, we have these purchases by people all along this circle at these distances. I'm gonna think about the quantity by each individual is contributing essentially an infinitesimal amount to the total amount sold by each store in particular store one. And then I'm gonna think about the total quantity that store 1 sells, which is over an interval z from 0 to h12 going to the right from its location.

It's gonna correspond to that integral there, which is the integral of these infinitesimal amounts sold to each of the bars who are designated by their location Z. Okay, so given the demand curve that I wrote down, this is all simple enough that you can integrate it out, and you get equation 4.

That's how the total quantity sold by store 1 going to the right up to the market border H12. That's what Q1 is, capital Q1. And it's gonna depend on the price that's set P1, which is the price at the store's location. And it's gonna depend on. So TH12 is gonna be the transport cost per unit at the border at the market border H12. So the effective price at the border is gonna be P1 plus TH12. And that appears also in this expression.

>> Speaker 10: This is assuming that P1 is equal to P2.

>> Robert J. Barro: No, I didn't, I'm not assuming that, but the question is what is H12? And P2 is gonna matter for that, right?

>> Speaker 10: So P2, shouldn't P2 be in there somewhere?

>> Robert J. Barro: Well, not once I know H12, but in order to figure out H1. Okay, so at the moment I'm gonna assume 8 is bigger than 1. But that's just so the algebra isn't messed up. You really don't need that condition here. You only need it to be non-negative. But at the moment I'm gonna assume eight is bigger than one so that I don't have problems thinking about this integral. But I don't need that condition which I think is important. Okay, so then you can figure out what's the store is going to price.

It's gonna wanna know how does the total quantity it sells react to a change in the price. And that's gonna be given by equation 3. And that's gonna depend on what this H12 is. But as long as that's given, that's all that the store has to care about.

So there's a critical expression as part of this equation 5 which is the effect of a change in the price on the market border H12. That term is part of this expression purely for expository purposes. I don't know, people may not like this. I'm gonna pretend for the moment that H12 is given when you change the price, it doesn't change.

But that's really just because I wanna see how the ultimate result relates to the learner formula. So the learner formula for the monopoly price is gonna result when you take the market border as fixed. And that's gonna be completely counterfactual. But I'm gonna pretend that that holds for the moment, just so I have this result that I can compare with the full hotelling type result.

So if I pretend that H12 is given, then this is a very simple monopoly pricing problem to figure out what the price is going to be. And that's gonna be what you need to maximize the profit in equation 6, which includes a fixed cost. And then there's gonna be some expression for a first order condition for choosing the price P1 from equation 7, and that's going to ultimately result in this equation 8.

Equation 8 is gonna be the first order condition for the price if the market border is fixed. That's this H12 thing. Now, it's probably not obvious from that expression because you have all these demands coming from people located at different points. If it's true that the main part of the price is the explicit price P1, rather than the transport costs, if the price is much bigger than the transport cost in particular at the market border, which is TH12.

Then you can get a very nice linear approximation to this first order condition, and that's equation 9, which is the standard learner formula. So if the transport cost is not that big a deal relative to the explicit price, then you're gonna get the standard Lerner formula. Which is only well defined if 8 is bigger than 1, as has already been remarked.

Okay, so I'm only putting that up here for background to compare with the actual solution. But it makes sense if the market border is fixed and it's gonna be the standard learner problem, where you have a monopoly pricing thing, and you're gonna get the usual learner formula. Okay, now the key thing in this model is the behavior at the border and how the border position changes when you change the price.

So that's what I'm gonna work out next. Okay, so the critical thing is that at the border between stores one and two in particular, the effective price has to be the same for both stores. Because people are gonna be just indifferent between buying from one store or the other at the border.

And that's what equation 11 says. So the term on the left is the effective price for buying from store 1, given that the distance from the store is H12. So TH12 is the transport cost per unit, and the expression on the right is the effective price for buying from store 2, which brings in P2, and it also brings in the distance from store 2.

So if h12 is in the interior, then 2h minus h12 is the distance, and that gives the transport cost for buying from store 2. So these two things have to be equal at the border in order for it actually to be the border. Okay, so that in particular tells me, if I'm doing a Bertrand kind of analysis here, where store one is now gonna pick a price.

Picking is given the price of store 2 and also the location of store 2. So those things are being held fixed from the perspective of store one. And that tells me how the market border changes if store one changes its price. That's equation 12. So there are effects coming on both sides from that change.

If you're gonna move the border, you're gonna get closer. If you move it to the left, you're gonna get closer to store one and further away from store two, which means that there are two effects coming in which is gonna bring in a two when you work out the answer.

And the answer to that is equation 12, this tells me how the border changes if you raise the price. And of course if you raise the price, the border is gonna contract because you're only gonna be able to sell to people who are closer to you when you're charging a higher price.

And this tells you exactly how much compensation you need to get a higher price in terms of the location has to be closer to the store. So that was the term I needed in the first order condition before which I just assumed that that was zero, which doesn't really make any sense.

So this is what it actually is in the hotelling context.

>> Speaker 11: So Robert, question. I would have thought the DH12DP1, that would involve entry of new stores rather than just competition between an existing set of stores and that somehow that would be relevant for the derivative. It just-

>> Robert J. Barro: I mean for the moment I was holding fixed the number of stores and then thinking about how those stores interact in particular what is the border position between adjacent stores. And that's the thing I'm looking at here. Then ultimately I wanna think about the number of stores, capital N.

And that's gonna be determined by a free entry condition. But at the moment I'm holding that fixed.

>> Speaker 11: That's gonna be Nash equal, right. So you're not gonna think about the Nash equilibrium with the price of the other store and the decision of the other store B.

>> Kenneth Judd: And the locations.

>> Robert J. Barro: Okay, so now I can plug that result into what I did before in terms of store one choosing its price to maximize profits. And when you do that you get a revised first order condition which is equation 14 which has this funny minus a half thing on the right hand side which is involving this effect on the market border position.

So, if I again assume that the price dominates over the transport cost, which is about the P1 being much bigger than TH12, I can again approximate what this equation is saying in terms of a linearization which gives me equation 15. And I'm gonna look later quantitatively about the nature of this approximation as to how good it is and how that depends on the underlying parameters of the model.

>> Speaker 12: 14 is also competing to the N4 on the other side. The change of price, the change in order on both sides through the same equation. Can I just look at one side of the month?

>> Robert J. Barro: Yeah, that's exactly how I was thinking of it. Yes, yeah, yeah, it's gonna be the same trade off.

Everything's gonna be there twice. It's not gonna affect the relative amounts you're gonna end up with the same answer, but we could work that out in annoying detail, I suppose.

>> Speaker 5: But that's where the twos come from.

>> Robert J. Barro: No, no, no, this two is different from that. This two here is about this competition with one adjacent store.

But when you move the border, you're getting closer to this one and further away from that one. That's what that two is. That has nothing to do with store N on the other side.

>> Speaker 12: But I was thinking that was good at the same time.

>> Robert J. Barro: Well, you don't have to move both market borders at the same time, but.

>> Kenneth Judd: When you change the price, you do.

>> John B. Taylor: Yeah.

>> Kenneth Judd: Aren't you just changing the price that you're offering to anyone coming from either side?

>> Robert J. Barro: Right, yes, you're right. That's true.

>> Speaker 5: But unless I thought you were going on a rotary fashion. So when you go and cartridge and not that. The firm that I'm next on the left side is the last firm you may be going over in constructing these best responses. So you've gotten that at the end.

>> Robert J. Barro: So maybe I should work the whole thing out with the two sides at once. It's gonna be the same thing, but maybe it's better to do it all.

>> Speaker 13: Aren't you just saying that the first order conditions are the same. If you calculate gross profit-

>> Robert J. Barro: It's gonna look the same.

>> Speaker 13: But you're not, you're just trying to figure out what the first order condition is.

>> Robert J. Barro: Yes, it's gonna look the same.

>> Speaker 14: So kinda going back to the way I usually think of this hotelling model where everybody just has elasticity of 0, everybody's buying, I don't know, one unit or something.

Then we get the price is proportional to the transferred costs.

>> Robert J. Barro: Right, the price is proportional to the transport cost.

>> Speaker 14: I mean, the simplest is if we have two people, there's one here and one there. Two supermarkets, one's here and one's there, there's transport costs, and both supermarkets charge the same price.

And they're each going to get half of the market. Basically, the margin that both supermarkets are gonna get is proportional to the transport cost, right. And here, what you've got in addition is people don't have an elasticity. It's not everybody's buying the same amount, regardless of the price.

>> Robert J. Barro: But they could be, I mean, eta could be 0.

>> Speaker 14: Right, in which case everything solves out to just right the price, is proportional to the transport cost, right?

>> Robert J. Barro: You mean P minus C is proportional to the transport.

>> Speaker 14: Yeah, you pay C plus.

>> Robert J. Barro: Okay, so equation 15 is what the linear approximation looks like in this model so that the excess from marginal cost is just gonna be involving this transport thing.

>> Speaker 14: I'm trying to understand-

>> Robert J. Barro: Along with the distance, the H12, the market distance.

>> Speaker 14: Well, unlike these other guys, I have no problem with the way you've developed your equilibrium.

>> Robert J. Barro: But I don't know whether that's comforting or not.

>> Speaker 14: I'm an old guy, I have it all wrong.

But, you know, I would see the price being. If we didn't have the constant elasticity demand overlay just, right, we just have the price everybody charged.

>> Robert J. Barro: I mean, that's important for some of the results.

>> Speaker 14: No, no, no, I got that, but one step at a time.

>> Robert J. Barro: Okay.

>> Speaker 14: The price would be C plus T, basically.

>> Robert J. Barro: Yeah.

>> Speaker 14: Right, so I actually had a question, so I'm trying to understand here. So I said if I start there with the C plus T, okay? Can you just tell me basically what happens to the margins when we put in the constant elasticity piece?

>> Robert J. Barro: So you like equation 15, but you're saying your intuition for that depends on 8 equals 0. That what you're saying?

>> Speaker 14: I've dealt with two kinds of cases in my life. One is like essentially the 8 equals 0 case, right?

>> Robert J. Barro: Right, Right.

>> Speaker 14: And the other is where the way we have a differentiation is we have a whole bunch of firms. And the way you decide how much you like each product, it's drawn from some kind of a demand curve which could be linear or constant elasticity.

>> Robert J. Barro: Okay.

>> Speaker 14: And then you just choose among the N firms which one gives you the product that you like. And so I kind of have a sense of how to solve those two things and you've got something a little different.

And I'm trying to understand, okay, I'm trying to understand how, how to think about it.

>> Robert J. Barro: I think I should go a little further cuz I need to think about this market border thing being determined. This H12 thing, which I don't know how you would treating that I'm fine with.

>> Speaker 13: So why is eta missing from equation 15?

>> Robert J. Barro: Well, it's an approximation, which I'm gonna try to quantify the nature of that approximation.

>> Speaker 13: This is a small eta approximation or it's a full t approximation?

>> Robert J. Barro: Well, both things come in. If eta is really big, this approximation is no good. If eta is small, it doesn't have to be less than 1, but in some range the approximation is gonna be pretty good. I can quantify that in a little while.

>> Speaker 11: I'd have thought of t equals 0-

>> Robert J. Barro: But it also involves how big th is. That's gonna also matter.

>> Speaker 15: If you quantify it relative to the true solution, right, just use the true solution.

>> Robert J. Barro: Because I get a lot of results out of this that I don't know how to get out of that true solution unless I do the whole thing numerically.

>> Speaker 15: And what would be wrong with that? All these questions in that case.

>> Robert J. Barro: In the end, you wanna see what I get out of this and then you can tell me why it's crap. But I think it's premature at the moment. Go a little further here. The second order condition in this model, which has a hotelling border effect in it, only requires that eta be greater than or equal to 0.

It doesn't require eta to be bigger than 1. And I can explain that more in a little while exactly why that's true, but I think that's important. But let me think first about the border position. Now I wanna think about how that's determined, right? I mean, you know in your heart that it should be halfway along from 0 to 2h.

So that's what I'm trying to derive here. Okay, so we have equation 15, that's for pricing of store 1. There's an analogous condition if you bring in pricing by store 2, which involves another distance. 2h minus h 12 is the distance going the other way. So you have equations 15 and 16, and you also have equation 11, which is that at the border people are indifferent from which store that they're buying from.

If you put those three equations together, you get the result h 12 equals h, which is sort of intuitive, which is completely symmetric. And corresponding to that, you get that the prices for the two stores, here I'm looking at P1 and P2, are the same. And they're given now by C+2th because there's just a 1H.

Now there's not the H12. So all the stores are gonna price in the same way and they're gonna involve this excess over marginal cost depending on this transport cost thing. So little H is given now as long as the number of stores in the whole market is given, which I have to go back to the intuition of this.

>> Speaker 15: So H is the distance between stores, right, and the borders?

>> Robert J. Barro: 2h, so h is the market distance.

>> Speaker 15: Okay, so we're trying to give you perfect competition. They're selling at the marginal cost, including the transport cost of the guys at the edge and then making money from the guys.

>> Robert J. Barro: Well 2th is the transport cost. If you wanted to move goods completely from store one to store two, not to the market border but all the way to store two, you'd have to pay 2th. That's what comes into play when you think about price undercutting because then you'd have to move all the way to the next store.

And the transport cost associated with that is 2th.

>> Speaker 16: John, the way you described it would be right if there's like an auction for each customer. And so you made money off of the customers who were near you and you've made nothing off of border customer. But since you don't know who's exactly where, everybody's paying the same price and you're making the same money from all the customers.

But the ones who are on the border get less. Less consumer surplus because they have big transport costs.

>> Robert J. Barro: They're not getting C plus their own transport costs. They're paying even more than that.

>> Speaker 16: Yeah, that's to take it all the way to the next door.

>> Robert J. Barro: Why is that? You don't have to go h. I don't have to go 2h.

>> Speaker 14: That way that the next store can't find a profitable to get the guy that's closest to you. But the other store was charging C. That customer would still be in different between paying C+2h at your store or C at the other store.

I think there might be the intuition about thinking about the customer that's closest to you having that customer go to the other place and charge.

>> Robert J. Barro: Yeah, I need to have more into why it's 2th rather than th. That's the question.

>> Speaker 15: Because if you're sitting in between the indifferent customer sitting between the two stores, if somebody undercuts by epsilon, he just goes the other way. And yet you've got the whole H. You can undercut him by, right?

>> Robert J. Barro: I didn't follow that. In order to undercut the next door, you're gonna have to cut by 2th and that's not going to be profitable.

>> Speaker 15: That's the result. But the part that's hard to understand is take that guy who's indifferent normally you just undercut him by epsilon, right? Then you would get a whole h. You wouldn't get 0 normally undercut pipes you get 0. Here you're getting the whole h. So that seems odd because it's too.

>> Speaker 16: Your market is too major from the end to 1 and 1 to 2. So you got 2h.

>> Robert J. Barro: H on each side, yeah.

>> Speaker 17: Yeah. Your price point, two pennies.

>> Speaker 16: You actually only get another issue. So you're assuming prices are 51.

>> Robert J. Barro: The effective price is linear in the distance.

>> Speaker 16: No, but there's potential here to discriminate across customers by offering them price quantity schedule. Because they will be if I understand correctly.

>> Robert J. Barro: Then you can figure out implicitly where they're coming from.

>> Speaker 16: Yes, at least you can move in that direction.

>> Robert J. Barro: I guess so.

>> Speaker 16: Okay.

>> Robert J. Barro: You can assume a way.

>> Speaker 16: You can't assume that away, that's an explicit. That's an unstated assumption and everything to this point stated assumption.

>> Robert J. Barro: Yeah, I'm assuming that the story can just charge one price at its location.

>> Kenneth Judd: Yeah and he said that ahead of time. He said it right away.

>> Speaker 16: No, no, what he said is he can't tell directly who's coming from where. But you don't need to do that to engage in price quantity discrimination. You can just operate.

>> Kenneth Judd: This whole literature is price competition.

>> Speaker 16: I understand. Well, you can compete on price quantity schedules too.

>> John B. Taylor: But that's off that table.

>> Kenneth Judd: It's an interesting addition.

>> John B. Taylor: Continue, Robert.

>> Robert J. Barro: Okay, there was this question about undercutting. So in order to undercut you'd have to cut your price.

If you think about store one and for example, trying to get customers all the way who are at the location of Store 2, in order to do that you'd have to reduce the price by this 2th. And that means you had to be pricing at the below marginal cost at or below marginal cost.

And that's not gonna be profitable. So that's the argument for why you're not gonna be doing that. Okay, so I looked at the approximations that I made in pricing formula. You can express the results fully in terms of two parameters that matter in this model. One is the elasticity of demand eta and the other one is this object that looks like 2th is going to be this, the markup.

And that relative to the marginal cost is gonna be the second parameter that is gonna describe the solution. So 2th over C and eta. You can express the results completely in terms of those two parameters. And then what I show by working out the results exactly for what the first order condition says without the linear approximation.

That if the elasticity of demand isn't bigger than about three or so and this other object isn't bigger than about 0.2 and that the approximate solution is within 3% of the full solution. In some sense that the approximate solution is pretty accurate. That's what's worked out numerically there.

>> Speaker 18: You can solve the model, assert that you've got a solution and describe that doing side calculations that embody the approximations.

>> Robert J. Barro: So I guess a question, how you can characterize the solutions. If you wanna do the full non linear solution, you need to develop a set of assumptions like linear.

>> Speaker 2: But you need that at least to get this paper published.

>> Kenneth Judd: No, no, no, Bob, you're way too optimistic, way too complimentary to the journal system.

>> Robert J. Barro: Well, what I work out in this table that has these exact results, first I compare the full solution with this approximate hoteling type formula. And then you can also compare the solution with the the Lerner formula, which is only defined if 8 is bigger than 1. And what's true there is that unless the elasticity is really large or this transport cost is really large relative to the cost of production, the approximation in the hotelling formula is very accurate.

And the Lerner formula is terrible is what that shows in those tables. That's what's claimed to be true. So one reason I found this of interest is there's a lot of empirical literature looking at markups and how they're determined. And many of these papers sort of write down the Lerner formula based on the elasticity of demand.

And then make no use of that formula whatsoever in the analysis. This seems to be a common research strategy here. So that fits well with this analysis because this analysis says the Lerner formula is gonna be useless and then it's a good thing that it's not used in this empirical analogy.

>> Robert J. Barro: That's the way I was looking at it.

>> Speaker 2: Maybe they had used it and got rejected. They might have written this paper a bit earlier.

>> Robert J. Barro: I don't know about that. Another way to look at this result, which is fairly straightforward, if you look at this pricing formula, equation 17, which is this hotelling kind of markup result.

Is that the effect of the marginal cost on the price is one-to-one, which Sanghani refers to in his paper as 100% pass through in levels. Whereas if you think about the Lerner formula, which is equation nine that says that the price is proportional to the marginal cost. Which is 100% pass through in logs or in a proportionate sense.

And Sanghani argues in his empirical analysis that that doesn't fit very well for the cases that he looks at. So that's a contrast in terms of these predictions.

>> Speaker 19: In the appearance of this literature, I don't know what kinds of goods are people looking at. So if I think of milk, milk would seem to be good, that's probably not in your good approximation set.

Cause it spoils quickly, the cost of going to the store and getting the milk seems high relative to the milk itself. So you potentially got a way to sort goods into categories that do or don't fit into your approximation range. And then you have, if I understand correctly, you have different implications about how cost passed through to price at the test of.

I don't know, whether the literature breaks down that way in terms of what we currently know about the pass-through from cost to prices.

>> Robert J. Barro: Well, he looks at cases where it's straightforward what the sort of marginal cost is like in terms of retail gasoline involving wholesale gasoline or oil prices. Or there's a lot of literature on excise taxes as a cost.

>> Speaker 19: So for people who live in rural areas, time to get to the gas station is probably high. It's a source of cost to the cost of the gas itself. Whereas in urban areas, gas stations.

>> Robert J. Barro: Yeah, I mean the transfer would have to be pretty high for that anyway.

>> Speaker 19: Well, I'm just trying to take this to the data in a way.

>> Robert J. Barro: The big thing is, this is a model of that undifferentiated products sold by different stores. So, something like gasoline, where it's really the same product and it's sold by a bunch of different stores and the margins are basically gonna be determined by location.

And the alternative model would be something like women's dresses where, there are a whole bunch of different stores. And it's entirely possible that as you have more stores, more different dresses, differentiated, that you're actually going to end up with higher prices.

>> Speaker 20: I took your transport cost. It's just a different thing.

>> Robert J. Barro: Yeah, but maybe it's not gonna be linear.

>> Speaker 20: Products versus the same product.

>> Robert J. Barro: The transport cost is just comparable.

>> Speaker 19: Yeah, it's intended that way.

>> John Cochrane: Very nice to spare us the motivation, but I think I now I'm seeing it, which is in standard models. There's this dicks at Stiglitz thing with the elasticity substitution and to get a huge markup, you go, I can't believe that Pete's Coffee and Starbucks Coffee has that huge elasticity of substitution.

Now I understand why you got ETA out of the formula. Just saying it is completely irrelevant has nothing to do with the elasticity substitution of the two goods. It's entirely this question of how well do I match with the particular location that sells an identical good.

>> Robert J. Barro: So let me talk about that in the context of this full elasticity of demand, because there's a sense in which this model can recover the learner formula.

But you have to reinterpret what you mean by the elasticity of demand to include in the hotelling model the market border effect that I talked about before. So when the approximation that I described is satisfactory, the quantity soldier by a store like store one on let's say one side of a market is given by equation 18.

Which involves this market border size little H, and otherwise it involves the price to the minus eta, which is the elasticity, the usual elasticity of demand.

>> Speaker 5: What's the approximation here?

>> Robert J. Barro: The approximation is when that pricing formula that I worked out before is satisfactory, which relates to the underlying parameters in the way I described before.

But the Lerner formula will work if you think about the full elasticity of demand, which includes the market border effect. So that elasticity is given by equation 19. So this is the elasticity demand for an individual store holding fixed what all the other stores are given, including their prices.

So that involves the usual minus ETA term, which is the standard elasticity. But it also involves this effect where little H is gonna decrease when you raise the price. And that term is the last term on the right hand side here and you can see that the magnitude of the last term has to be bigger than one.

And it's the fact that the overall, the full elasticity of demand is bigger than one in magnitude that gets around the problem of having a store charge an infinite price. And that's why store won't be motivated to charge an infinite price. And moreover, if it's true that the marginal cost is much more important than the transport cost, which looks like 2th, then that magnitude is gonna be quite large.

So that's gonna be like the substitution between products that are pretty close substitutes, essentially.

>> Speaker 22: So is this the intuition? In the standard model, every browser is one unit. When you move the price, you're giving up one unit for everybody, right? Whereas now the people close to you, you're giving up much more than.

Because they're buying more.

>> Robert J. Barro: Well, you've got the quantity thing, the minus eta, but at the border you lose all the business by having an infinitesimal increase in the price. It's an infinitesimal quantity sold at the border, but you're losing all of it.

>> Speaker 22: But that's true in the hotelling model, but that's true in the hotelling model too. If you-

>> Robert J. Barro: Yeah, in the hoteling mom.

>> Speaker 22: So you're getting that extra bit here because when you lose the customers close to you, you're losing. You're not losing just a single unit you're using because they're buying more. Isn't that what's-

>> Robert J. Barro: That's the eta part. So that's why, I mean, that gonna be a small part of the relevant overall elasticity.

So in that sense the eta is not gonna be critical for the results in terms of the pricing formula. And as an approximation, the eta is not gonna matter for the markup. And that's not true if eta becomes very large, but it's true over some range.

>> Speaker 23: You asserted that what small meant.

>> Robert J. Barro: Well, I worked out some numerical results to find out how close the approximate solution was to the full one in some range.

>> Speaker 11: Yeah, what's the range?

>> Robert J. Barro: So I was thinking if the elasticity demand isn't bigger than for example, three, and if this transport cost is not more than like 20% of the cost of production, something like that, that kind of.

>> John Cochrane: Okay, so you need to convince the audience that that's an interesting-

>> Robert J. Barro: Okay, fair enough.

>> Speaker 24: Okay, is it a Taylor approximation with respect to transport costs that holds exactly for a range of units or is it a Taylor approximation with respect to eta and transport cost evaluation?

>> John Cochrane: The solution holds exactly if eta is equal to zero, and then if eta is a little bit bigger than zero, it's very close.

>> Robert J. Barro: Taylor approximation with respect to eta and t, both of them at zero. Well, I'm looking at p plus t H basically, so it's about tH relative to P in terms of the approximation.

>> Kenneth Judd: If you had solved this numerically with a well defined set of algebraically expressed equations, then you could directly do what John just suggested, a Taylor series approximation in the parameters that you care about. And there wouldn't be all these discussions about this or that. And this is just basic standard uncertainty quantification methods that are hated by economists, by the way.

But no, if you did a numerical solution to this with fully expressed equations, then all of the formulas that you really care about are easily attainable. Without this business of first order approximations here, first order approximations there, and then we're confused about what results are.

>> Robert J. Barro: Okay, all right.

>> Kenneth Judd: No, you, if you do it numerically and then, and then examine the basically the gradient of the system of equations at the solutions, you will get out expressions like that. And it won't depend on the, this small assumption. That small assumption you will see exactly. It will depend on the base at which you do the Taylor series approximation but you'll be able to answer these questions from Bob and John and others.

And as opposed to all of this hand waving, I mean, I'd like to remind you that this is not how we got men in the moon. It was by numerically solving equations that we got men in the moon in 1960.

>> Robert J. Barro: I'd like to go a little bit further, cuz I wanted to describe the free entry solution.

Let me just note that if you look at the overall market, the total quantity sold by all the stores in the market, that's given by this expression A times capital H P to the minus eta. So capital H is the circumference of the whole thing which is given.

So the elasticity associated with that is in fact just eta. So it's not just the sort of micro, it is the aggregate elasticity that is here. You can show that you wanna do equal spacing of the stores. So you can contemplate one store moving away from its location, say away from store N and towards store two.

And then you can show that the store is gonna lose on net by doing that. But that result depends on the downward sloping demand curve and on the fact that the price per unit includes the distance. Then you can show that what you're losing is more than what you're gaining by moving away from one store and toward the other one.

Vickery, in 1964 in his book, had basically this result.

>> Kenneth Judd: If you have uniformly spread out the stores, then yes, then I agree with your argument about this being an equity.

>> Robert J. Barro: I'm trying to deviate from the uniformity and see whether that.

>> Kenneth Judd: Yeah, but that's just a marginal deviation.

And you're saying you don't wanna do a marginal deviation. No, it's gonna be any deviation from that is going to be sub optimal for an individual store given what all the other ones are doing. It's not just a small deviation.

>> Speaker 5: Okay, so that's for my own increase.

>> Kenneth Judd: But my point is that there are other distributions of the stores which also satisfy your tests.

>> Robert J. Barro: Have to show that.

>> Kenneth Judd: No, you have to show that those aren't possible. What you've shown is that the uniforms distribution can be an equilibrium. You have not shown that a non-uniform-

>> Robert J. Barro: An individual store won't wanna deviate given what given what-

>> Kenneth Judd: Given a uniform distribution, yes. But you have not shown that if you have some hypothetical non-uniform distribution that somebody's gonna wanna deviate. You haven't shown that. And also the thing is, if you look at papers like Prescott and Vischer on the entry process and markets, and I think Ulrich Doraszelski has a paper on entry processes in differentiated markets, you don't get this kind of result coming out.

And particularly if you're talking about the entry process, it's who moves first is going to affect things. And also then there's the Spalinzy analysis in the serials case in the late 70s where he talks about the importance of what if somebody builds two stores or three stores and how they're grouped.

So the assumption that there's only one store per economic actor is very limiting and unreasonable in this case. So, and the entry process is gonna often give you something very much more complicated than this.

>> Robert J. Barro: Let me describe what the free-entry condition is that I have.

>> Kenneth Judd: Yeah, what's entry if it's not a sequential process?

>> Robert J. Barro: No, I understand.

>> Kenneth Judd: Yeah, it's a dynamic nash.

>> Speaker 25: I thought that Prescott and Fisher got symmetry, because they're taking account, you're entering first and then you know how store two is gonna act.

>> Kenneth Judd: Well, if you do a full dynamic version, which I think you don't get symmetry.

>> Speaker 5: Can I ask a question?

>> Kenneth Judd: Okay.

>> Robert J. Barro: I wanted to describe the entry result within the model that I had. So with the hotelling markup solution for price that I had, the total quantity sold by each store is gonna be given by 18, which involves sales on both sides.

And the profit for that store is gonna be given by equation 19 and it depends on the market border distance, little h. So I'm just gonna be looking for a free entry solution in terms of what does the number of stores and correspondingly this market border distance little h have to be in order to generate 0 profits for each store.

That's what I'm working out here and neglecting the fact that there would be an integer constraint on the number of stores which is gonna be more satisfactory if their number is larger. If you neglect that, you get the square root formula that's given by equation 20. So that's gonna be the value of the distance between stores, which is 2H, which is gonna correspond to a certain number of stores, capital N.

And that's gonna be what generates zero profit for each of the stores.

>> Kenneth Judd: But the 0 profit assumption is not necessarily picking out an equilibrium. You could have you take your N firm to have zero profits, but then now suppose you have N minus 1 firms distributed uniformly.

My guess is that you'll find that nobody wants to enter because then to enter they'd have to stick themselves in between two firms and lose equilibrium. Yeah, what's the dynamics that lead to zero profits?

>> Robert J. Barro: This only works if there's a rearrangement when the new store comes in so that they're evenly distributed.

>> Kenneth Judd: But that's a very different game that there's a rearrangement.

>> Robert J. Barro: Well-

>> Kenneth Judd: So what's the assumptions here about the dynamics of entry? Re-arrangements are not multiple storage or single front storage.

>> Robert J. Barro: The assumption is that the dynamics doesn't involve any costs, that the shifting of the location of the stores doesn't involve any costs. Then I think this is gonna be the right answer.

>> Kenneth Judd: By the way, the other thing is if there are fixed costs of entry, you're still gonna have a problem with multiple numbers of firms. But you have 0 cost of entry, which is unusual assumption in this literature, which you're also assuming.

>> Robert J. Barro: Well, each store is paying a fixed cost to be in business. That's part of the solution. That's a little f.

>> Kenneth Judd: Okay, the thing is that my point about n minus one firms being an equilibrium also if there's a fixed cost.

>> Robert J. Barro: Okay, so what matters for the market distance is how big the fixed cost is relative to the scale of quantity demanded by each customer in the market, which is this ac to the minus A to.

So bigger fixed cost relative to that is gonna give you a bigger spacing and the transport cost, little t is also gonna enter into that. Given that free entry solution, you get the price solution corresponding to equation 22. So that the excess of price over marginal cost is gonna involve the term on the right, which is gonna have to do with this entry margin, which is determining this market distance h.

So just one comment on these. This is empirical study by Chevalier et al. Which particularly looks at how markups relate to whether there's peak demand or not. I think their main finding is that in periods of peak demand like holidays or lent, they find that markups are lower than usual, which is sort of counterintuitive, I suppose.

So they argue in their paper that this can't be explained based on what the elasticity of demand is in different periods like holidays versus non holidays. So you can get this result in the model that I worked out, if you have something that looks like entry during peak demand.

Peak demand means, in the model that I had, that the scale of demand, which is the capital A is unusually high, which is gonna motivate more entry in the way I looked at it. And I think you might wanna think about entry in this context as having to do with store hours rather than numbers of stores in terms of what would vary seasonally, for example.

So I think then that matches up with the empirical findings that are in this Chavalier paper. So I also compare the free entry with the socially optimal entry, which relates to some other literature here, including the book by Tyrol. So defining, as Tyrol worked it out, which basically has an elasticity of demand of 0, which is eta of equal to 0, he has that there's too much entry in terms of the free entry solution.

So there are basically two margins of distortion that occur in this model. One is that people don't buy the right quantity of goods because there's essentially monopoly pricing. And that depends on this elasticity of demand eta as to how important this is. And when ADA is equal to 0, that's gonna end up not mattering, which is true in Tyrol's model.

The other distortion comes from what's usually called a business stealing effect, which gives people too much incentive to enter because they're partly transferring profits from the incumbents. So that's the only force that operates in the model that Tyrol has. And that's why he gets that there's excessive entry in the equilibrium.

The model that I worked out, when I have the approximation that I've already described being true. I get this distortion from the quantity being not the socially optimal quantity associated with this elasticity of demand. But in the range that I assumed applied, that works when the elasticity is not too big.

And that condition ensures that the business dealing effect still dominates over the other inefficient quantity effect. So you still get excessive entry in this model. But that needn't be true if you look in ranges where the approximation I made is not satisfactory. And then you can get the opposite kind of result in this model.

Okay, so that's the main thing I wanna talk about. I wanna think about this as a framework particularly for thinking about markups, but it also has some implications with respect to these entry decisions.

>> Speaker 5: Okay, I asked the question I was trying. So what you showed us is that if you restrict attention to symmetric equilibrium and you derive the equilibrium price equation under this bivariate for sort of Taylor expansion on eta and t, because eta is small and t small.

And then you wonder at the solution, what happens if you start start increasing eta? T still dominates. So the effect in the in determine.

>> Robert J. Barro: What do you mean T still dominate?

>> Speaker 5: That if I look at the pricing equation for the segment, eta doesn't drop, right?

>> Robert J. Barro: Well, that was an approximation.

>> Speaker 5: I thought your statement on the 3%, if you start increasing eta a little bit, but you are close to zero, but you start making it positive. The th term still what's the intuition is that in that segment, since consumer the products are undifferentiated, consumers are. And differentiate, except for distance somehow is that the age size of the market is off.

>> Robert J. Barro: It's because there's this margin with very close substitution which in this model applies at the market border. And that gives you a very large effect.

>> Speaker 5: That's why it's dominant, okay.

>> Robert J. Barro: So I mean in Atkinson and Burstein, they look within sectors and across sectors and they argue that the within sector elasticity tends to be much bigger.

So it's somewhat analogous to that.

>> Speaker 5: Although they haven't differentiated with setup. Okay, sorry.

>> Robert J. Barro: Okay.

>> John B. Taylor: You have any big conclusions?

>> Robert J. Barro: I'm trying to think about this framework as a useful way to do markups. And I think ultimately in terms of thinking about markups, you wanna know how markups relate to fundamentals. And I think the papers, at least that I'm familiar with, many of them follow Bob Hall's suggestion about how do you measure markups empirically. But I don't think those papers really look at how markups relate to fundamentals. So fundamentals in this model is stuff like elasticity of demand and things affecting transport costs or more generally the substitutions between products, so I would have wanted to apply it. Think about how markups relate to those things and whether there's empirical mileage and looking at it that way.