One of the most usual forms of horizontal differentiation is differentiation in location. It makes a difference for consumers where firms are located. We always prefer to do our shopping and buy our products from the stores that they're located close by to us we have enervation in drive far away and buying products from stores that they are further away from where we are. Sometimes when we have to we do it but we prefer to not have to go with way in order to buy our products. So, what we are going to examine now is a model of horizontal differentiation with location. And then, we will try to extrapolate this model and to understand how this model works in general differentiation. The most usual and simple model of horizontal differentiation is Hotelling's linear city, this a model that was first introduced by Hotelling and it's very simple. It has a linear city considers a linear city, straight line that pretends to be a city of length l like for example this line that has a west end and an east end and in this linear city we have consumers that they uniformly distributed. So, every point of this straight line has one consumer. That's why I'm saying that density of consumers here is equal to one. Now, we will consider a duopoly, two firms that they're selling a homogeneous good with cost c. Consumers cannot understand the difference when they buy these goods from one firm or the other firm. However, what we can understand is how close this firm that they buy from is to them and they will always prefer to buy from the firm that is closest to them. So, if could this homogenous, both firms have the same cost that these lower case c, this is the cost per unit for the product. And firms locate in the beginning at two random points on the line, so firms decide randomly to go to two different points like for example here, I have a firm one that is located at that distance a, lowercase a from the west end and then I have firm two that's located at a distance b from the east end and these are two random distances they do not have to be equal or they do not have to be different they just random and I just call from one the firm that went to towards the west and from two the firm that went towards the east. So, firms compete by setting prices simultaneously. So, once they have located and they are fixed in these locations they will compete with respect to prices. So, firm one, will set the price be one it's that high. We can use these linear sitting now as a graph to show how tall the costs are in this market for the consumers and firm two has a lower price than the red price there, p_2. Now, what is interesting here is that consumers incur a quadratic transportation cost. That these even consumer has to go from a distance on this line in order to shop. The firm is not at the same point where the consumer is located. This means that the consumer incurs some cost and this cost is quadratic that is a consumer at distance x, x let's say a way from a firm incurs a cost of transportation costs of t, which is a constant times the distance x squared. That is the further away you are from the firm the higher the transportation is, so the transportation cost is increasing in this case. This means that over each firm we are going to consider except the price above and beyond the price they will be a quadratic cost over the head of the firm like that, they will be a quadratic cost. So, this parabolic care will go over the head of each firm and will show that costs, the total cost that consumers have to incur in order to shop from this exact firm. So, if a firm one will be the blue one, for firm two will be the red one and you see there that the further away you are from the firm the higher the cost, the overall cost of aquiring the product is. Now, what is really interesting and I want you to observe there in this graph is that consumers look at the overall cost of buying the goods, they're not looking only at the price. And even if you think for yourself what you're doing, there are shops inside the city close by where you live that usually they're a little more expensive. And then, there are some other stores in outside of the city in the suburbs, near the airport usually that there are outlet stores and there are the prices are much cheaper. But most of the people in order to go and buy from these outlet stores they have to drive a distance. When you want to shop, you do not always look at the price. You do not always go to the outlet store. Sometimes you say, "I'm not going to drive for an hour to go there and then an hour back I will just go to the shop near me and I will buy this pair of socks that I need because it's not worth the drive." So, in this case, you look at the overall cost for acquiring the good and the same happens in our model. So, not only price but also transportation cost will play a role to that final cost of their product for the consumer. Now, consumers they will shop from the store that is cheapest to them overall, including the transportation cost. So, if you look at my graph here, you will see that there is a very interesting point in this graph and this point is when there is a consumer that we call him marginal consumer, him or her. And for this consumer, this exact consumer this is the only consumer in the graph that isn't different from which firm will shop. And this is the one that the two parabolic curves above the prices, they intersect because it means that, at that point incurs the same cost from each of the two firms. So, it's this person there which is that distance x from firm one and at this point at this exact location, consumers that are right there, this exact consumer that there is indifferent from shopping from one or from two. We call this exact point, the marginal consumer and it will be very important. Now, for the marginal consumer, they calls from acquiring the good from firm one, which is on the left hand side of the equation I'm giving you and the cost of acquiring the product for firm two are equal. And this is very important and will help me to understand which exact point x is, because this is an equation that is true and I can solve it with respect to x. So, the location of the marginal consumer is a plus x from the west end, were x from my previous equation, from equating their overall cost of acquiring their product from firm one or firm two can be computed to be asked the equation that I'm showing to you here. So, this is there way that we derive, where the marginal consumer is located. So, it is located from the west end to a distance a plus x, where x is given in our equation here. Now, these will split our demand, our market into two parts. There is this bluish part that I'm showing you in the graph and these pinkish part because from the marginal consumer towards the west, ever will shop from firm one and from the marginal consumer towards the east ever is going to shop from the firm two. So, this marginal consumer is very important because splits the market into the two parts provides the market shares, and therefore can show us the demand that each of the two firms will have according to their locations and to the prices that they decided to set. Let's see what happens to the equilibrium prices now, the quantity that firm one will sell is q_1 equal to a plus x. This is the bluish area that I showed you before, the quantity that firm two will sell is q_2 equal to L minus a minus x. That is the rest of what the firm one will not serve. So, profit for each firm will be therefore pi minus c, which is their average profit times the amount that these two firms will sell, I'm giving you the demands right above here so you can understand what this profit function will look like for each firm.And then, we will take these function there too for I and J for firm one and firm two and then we'll maximize them and we'll solve the system of first order condition and we will derive these prices that I'm giving you, right here. I know it's a little complicated in doing the algebra, not complicated it's a little too much calculations but it will be okay you'll know how to do it, It will not be that hard and it's really worth it to follow and try to do those calculations at home because they will show you exactly how the model works behind what I'm showing you under the hood. So, firms do have market power in this model you can see it from the price equations which is c plus something positive so you see that the prices are a little above marginal cost and this is what makes very big importance here, that firms do end up having market power just because they are located in different locations. So, if firms are in different locations, they do have market power and this is what we have hypothesized from the beginning. So, profits will be positive even if we have a Bertrand game. So, in this Bertrand game, we avoid the Bertrand paradox because of the differentiation. Now, what you can do and you will have to do it at some point because we will use it in the future, is the plug p_1 and p_2 as we have derive them up there into the profit function and calculate how much the profit will be in this model. And this is something that we will use right away. Stay with us, we're going to continue this model and make it now dynamic.
