Competition Between Firms with Multiple
Locations in a Hotelling Game
Author: Eric Metelka
Advisor: Mark Witte
Mathematical Methods in the Social Sciences
Spring 2007
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Abstract
The Hotelling model has been a standard in analyzing linear firm competition for over a
decade. It has spawned numerous papers on the extrapolation of its concepts. Yet
none of these have ever considered the effect of multiple agents controlling multiple
locations.
In this paper we explore the classic Hotelling model and some of its
implications. Then we reflect on the differences between the classic model and the new
model we are building, and present three different cases of how firms can align
themselves along our new linear space. A careful analysis will then show how these
cases create a general equilibrium of how firms place their locations to control the
extremities of the linear space. Finally, we will then explore the implications this new
equilibrium has on political theory, competing chains of stores, and transportation
infrastructure.
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Introduction
Competition between firms is a cornerstone of economic analysis. It is the basis
on how consumers are able to choose their goods from a variety of producers and
likewise how producers determine what and how to produce to meet consumer demand.
Any discussion of supply must include a breakdown of the suppliers and how they make
their economic decisions to maximize their profit.
A number of models have been
formalized to this effect. For example, Bertrand and Cournot competition are models on
how firms compete against each other in prices and quantities respectively. Similarly
Harold Hotelling also produced a model of how firms compete but his model included
dimensional space for the first time in the competitive analysis. Two firms compete and
their prices and locations are determined endogenously over a uniform space. Just like
the Betrand and Cournot models, the simplest way to analyze competition is looking at
the effect of two firms competing. From there it is not hard to extrapolate these models
to include N firms. However, all these models analyze firms competing individually
while we know in reality that owners of firms usually control more than one store. We
see chains and franchises in reality: owners of firms controlling more than one location
of their store to sell their product. The crux of this paper will be to explain what happens
when we apply the Hotelling Model to this competition – that is when there are two
players who each control two locations in the simplest case. We will attempt to find the
answer to the question of how firms make pricing and location decisions when they are
chains. From there the effects of the new model we construct will be extrapolated into
different arenas from firm positioning to the model’s effects on political analysis.
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Hotelling conceived his model as a reaction to the instability in the Bertrand and
Cournot models. He saw that in the Betrand there is an equilibrium, but if one player
undercut his price by a minimal amount he would capture all the profit and thus create
instability. What Hotelling sought was a stable equilibrium point and thus he added
another dimension to the analysis of completion: distance. By creating a linear space
over which firms compete, he brought completion from the first to the second
dimension. Furthermore, it would not be much harder to extend the theory to the third
dimension as well if we used a planar space instead of a line. It is not hard then to
construct an equation that solves for the profits of each firm given their location and
price. What Hotelling did next was find the stable equilibrium he was looking for that
solved the instability problem. The key to the whole problem was finding the point of
indifference.
“The point of division between the regions served by the two
entrepreneurs is determined by the condition that at this place it is a matter of
indifference whether one buy from A or from B.” (Hotelling, 46) We can visualize the
problem as thus:
Where a and b are the distances from each end point and x and y are the differences
from points A and B to the indifference point. To find the point of indifference, first he
sets the demand equations for each product equal to one another under the constraint
that our individual segments a, b, x, and y must equal the length of the whole segment l.
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The derivation of Hotelling’s Model can be found in Appendix A. The final profit for both
firms are:
Hotelling found that profits are directly related to the cost of transportation and where
each firm positions itself. Additionally, the greater the value of a for Player 1 and the
greater the value of b for Player 2 results in greater profits. (Hotelling, 46, 47, 50)
The framework that Hotelling created served as a platform for applications of the
model to different areas of study along with the new found implications of the loosening
of certain assumptions.
Hotelling himself grasped some of the immediate
consequences of his model. First, let us note again that profits increase as the cost of
travel increases.
Thus, firms will not want to encourage easy travel and invest in
infrastructure for consumers.
Rather they will put up barriers to trade.
Hotelling’s
model is actually a good argument for protectionist policies. (Hotelling, 50) In another
example, Hotelling noted that if instead of determining prices endogenously they were
fixed, such as in a socialist regime, firms now locate themselves at the center of the
space. Each firm tries to control a greater portion of the space since they now only
compete in the amount of the space they control. “As more and more sellers of the
same commodity arise, the tendency is not to become distributed in the socially
optimum manner but to cluster unduly.” (Hotelling, 53) This clustering effect at the
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center is rather striking since it has many real world analogues. Say that instead of the
space representing distance, it represented an axis of possible differentiations in the
given product. We see that the firms cluster at the center, explaining why when we see
two similar goods there may be slight differentiation such as an improvement to attract
more buyers, but they are for the most part the same. (Hotelling, 54) Take another
analogue – this time in the political landscape. Our space is now a political ideology.
Both candidates in a given election will want to focus on the center and most of their
platforms will be relatively similar with a few exceptions that still pander to their core
base. (Hotelling, 54-55) This is actually a rather dismal conclusion as it leaves the
consumer with very little choice. It purports that consumers want the middle option and
competing firms will offer them alternatives that are barely different. Obviously while we
do see a plethora of undifferentiated products being sold in reality, we so also see
products that vary widely.
So the model cannot truly account for every scenario.
Loosening of further assumptions must be in order to get an even clearer picture of
reality.
Since we will be exploring a case of two players with two locations, it is only
prudent to look at other examples of Hotelling theory with more than two locations.
Eaton & Lipsey explored some of these cases. For example, they take the case of
three firms and find that there is no pure equilibrium in this game. (Eaton, 47)
Additionally they look at the case of what happens in general to equilibrium when a firm
is added and define what they call the “principle of local clustering.” This is defined as
“When a new firm enters a market, or when an existing firm relocates, there is a strong
tendency for that firm to locate as close as possible to another firm.” (Eaton, 46) This is
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done because firms can capture more market share by locating close to each other
rather than from a distance where it leaves their opponent free to capture a whole
region to themselves. It is better to split a local region in half by locating next to one
another rather than leave it all to another player.
Thus while there is no stable
equilibrium in the three firm case, the principle of local clustering helps us define how
firms locate when n > 3 and in fact helps explain what we find in the four firm case. The
four firm case does indeed have a stable equilibrium and two firms locate at ¼ and ¾
each according to the principle of local clustering. Additionally, all firms receive an
equal payoff of ¼ of the profit. (Huck, 232-233) Once we consider more than four firms,
no more pure strategy equilibriums can be found. “With five the unique equilibrium
configuration is asymmetric and implies unequal payoffs; and with six and more agents
the equilibrium configurations cease to be unique.” (Huck, 231-232) Of note in these
results is whether our new model will follow the same principles. While it may be
doubtful that we receive the same result as the four firm case since all firms are
independent of one another, it is likely that the principle of local clustering may still
apply. This research that has come before lays a solid foundation of which we can build
and understand if our model makes intuitive sense.
Model Assumptions
Unlike the classic Hotelling model or the four firm case, situating two firms with
two different locations presents a more complicated scenario. Like the Hotelling model
we have only two players. However, since they each have two locations, the outcome
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will be decisively different except for one case which will be explored later.
While
similar to the four player model, the fact that the two firms have knowledge of where
their other location is along the finite space always them access to more information
than four individual players would have. The key is how this access to more information
is reflected in the model we build. The solution is that the firms compete on profits. This
is a fact that the classic Hotelling model and any of its extrapolations can easily agree
on. Firms compete on profits and the player that wins is the one that makes the most
money. Knowing this fact, we can now fold in the knowledge of their other location into
each firm’s profit motive.
In our model, instead of competing according to profit
maximization at one location, the firms will strive to maximize the joint profit they obtain
at both locations and they do so simultaneously by calculating their opponents reaction
functions.
Now that there is a clear idea about how the firms will compete against each
other, it is prudent to explore the underlying assumptions that are inherent to this model.
•
Firms compete over a closed linear space. This space is one unit in length and
its endpoints are 0 and 1.
•
The demand functions are strictly concave and decreasing. This invalidates the
case of any plateaus in demand and also insures that the demand functions of
the different players must intersect at some point in out linear space.
•
All demand functions are quadratic. This is done since they are a more realistic
analogue of consumer demand than linear functions. (Tirole, 279)
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•
No more than one store can operate at any given point. The implication of this is
that no more than one location can occupy a given space and there will be no
clustering in this model. This is done to make the model more realistic since no
more than one store can exist at one physical location. (Eaton, 48)
•
Locations exist at the points a, a+c, 1-b, and 1-b-d. Additionally, all variables are
positive. This is done to impose linear order on the space. Since c must be
positive, a+c must be to the right of point a. Likewise, 1-b-d must be to the left of
point b.
•
a+c < 1-b-d to preserve order between a+c and 1-b-d much like in our last
assumption.
•
. This insures that at the minimum a=b=0 and at the maximum
a+b=1.
•
. This states that the maximum value of a+c is 1.
•
. This states that the maximum value of 1-b-d is 0.
•
Consumers choose firms based upon prices and transportation costs only
(Eaton, 48)
•
Cost is denoted as co to differentiate it from the point c. This cost refers to both
the fixed and variable costs of setting up a store at this location. We assume this
costs are similar all over the linear space and thus keep them constant.
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With these assumptions in place we now have a realistic model, a defined space
which that model exists on, and we know how our demand functions act on that
space.
We can now start analyzing the model that had been built. The first thing that
becomes apparent is how the two players arrange their locations over this space. In
fact, there are three distinct combinations of how this can happen.
By working
through each case it can be determined which one produces the greatest joint profit
for both players and that will be the case that would be emulated in real competition.
In Case 1, one player controls the extremes of the space while the other controls
a central “power core” over which his two locations are placed. This creates an
interesting problem since we can explore which is more powerful to control: the
center or the extremes.
0
a
a+c
1-b-d
a 1-b
1
Case 2 presents a scenario similar to the classic Hotelling model in which each
player’s two locations exist on the same side.
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0
a
a+c
1-b-d
a 1-b
1
Finally, in Case 3, the players mix their locations by alternating with each other over
the
space.
0
a
a+c
1-b-d
a 1-b
1
Case 1
Case 1 starts out much like the traditional Hotelling model. We want to find the
indifference point between the two demand curves to find the demand at that point.
Once we know demand, we can then maximize profit based on price and costs. In this
model, prices are the same at each location.
While prices can be determined
endogenously, the decision to leave them constant for now is for the purpose of
simplification.
The interesting aspects of this model is how the players place
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themselves in this unique game.
The prices charged to maximize profits at these
locations can be determined later.
First, let us determine the demand at point a. To do this we find the indifference
point between D1 and D2 by setting them equal. In our equations, s is consumer surplus
(a constant), p is price, t is the cost of transportation for the consumer, and x is the point
of indifference.
We have found that the point of indifference is half way between a and a+c. We
can only use this result to find the demand for D1 since it is bounded on the other end by
the endpoint 0. To find D2 we would also need to find the indifference point between D2
and D3. However, since we are looking to maximize joint profit and D2 and D3 are
controlled by the same player, it is easier to look at them both as the joint demand D1 +
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D3. Just as we found it easy to solve for D1 since it is bounded on one end by an
endpoint, it is just as easy to find D4 since it is bounded on one side by the endpoint 1.
We do this by equating D3 and D4.
Again, we obtain the logical answer that the indifference point is halfway between
the two points 1-b and 1-b-d. Now we have found the solutions for D1 and D4 which are
controlled by the same player. Maximizing the joint profit we find:
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The joint profit D2+D3 must be the leftover demand on over the linear unit space or 1(D1+D4).
Since for our purposes, p and co are fixed, the term (p - co) can be treated as a
constant and we only need to focuses on the latter half of the profit equation. This still
leaves us with a indiscriminant answer since we do not yet know the values of our
variables a, b, c, and d. We do know though, that since player i chooses locations a
and 1-b, he will choose these to maximize his profits. As such, he chooses the highest
value of a and the lowest possible value of b. He would choose to locate a point 1 and
would also choose to locate 1-b at point 1 with a value of 0 for b. Due to the ordering of
our locations, both a+c and 1-b-d would also have to be at 1 with a value of 0 for both c
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and d. We can see this more clearly on a graph that shows the relationship between
location of the variables and profit.
We then run the same analysis on how player j will choose his locations. Player j
chooses the values of c and d. He chooses the lowest possible value for c since it is
inversely proportional to profit. However, he wants to maximize the value of d. Due to
the fact of what values were chose for a, b, and c, whatever value he chooses for d is
negligible since all the locations are located at the endpoint 1 by definition.
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Inputting these points we obtain:
Player i clearly is in the better position since player j is not even making positive
profits. It seems as if controlling the outsides of the space – or adhering to the extremes
– is the more powerful situation to hold since player i is essentially putting player j out of
business. However, as stated in the assumptions and as logic would dictate, no two
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stores can occupy the same location, let alone four stores occupying the same space.
Clustering is not allowed in this model by definition. Nevertheless, if the stores cannot
occupy the same location, then they will choose to operate the minimum distance ε
apart from one another. Since ε is a distance as small as physically possible to locate
one store next to another, it is analogous to the case we just explored where the four
stores are located at the same space. Therefore, our result is the same: it is better to
control the extremes than try to maintain a hold on the center.
Case 2
Because of how the players align their locations, there is only one point of indifference
that occurs. The locations at the extremes of the space are negligible since the other
location by the same player would capture that demand if the extreme location did not
exist anyway. I posit that the result is the same as the classic Hotelling model with just
two locations competing. Instead of using a+c and 1-b-d to find the different point, since
they contain extra variables that are not needed in this case, we will use a and 1-b
instead. It greatly simplifies the calculation.
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This is a purely logical result. Player i controls the left side and thus as a
increases, so does his profit since he controls more of the space. Likewise, Player j
controls the right side and his profit increases when 1-b increases. The only aspect that
is different from the classic Hotelling model is that we have fixed prices for both players.
However, that fact should not be an impediment since when we solve that model the
prices come out equal for both players. As such, the result here should be the same and
players should position themselves both at ½
The other location each player controls
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will be between each endpoint and ½. When we use these location to determine profit,
we discover that they split the profit equally between them.
Case 3
Case 3 is similar to Case 1. We start with calculating the demand for D1 and D4
the same way. Because both these demand curves are bounded on one side by an
endpoint this is easy to calculate, and the result is the same except for in this instance
one player controls D1 and the other controls D4. The challenge now is determine the
derived demand for D2 and D3. We first consider using the same approach to find the
demand as we did to find D1 and D4. The answer for this approach contains several
unknowns – many of them quadratic. Instead, it would be easier to shrink the interval
our demand curves exist on. Since it is known that that D1 covers from 0 to
covers
and D4
to 1, we know that the demand for D2 and D3 must exist over the rest of
the interval. In effect, we can look at a small piece of the interval whose endpoints are
and
.
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a+c
1-b-d
Working over this smaller interval now makes it easier to solve for the demand.
Just like in the previous cases, we know that between two locations with constant prices
the indifference point will be the exact midpoint between the locations. Since we have a
discrete interval over which only these demand curves exist, it is quite easy to find this
midpoint. The length of the interval will be denoted as l.
Now that we have the point of indifference, it is a simple matter to find the
demand over the interval by just taking the distance between the endpoints and the
indifference point.
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a+c
1-b-d
D2 and D3 are equal which is to be expected since all we did is find the distance
between the endpoints to the midpoint. Now solving for the joint profit:
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Player i controls the position of points a and d. Player j controls the position of
points b and c. With this information, players then choose their points to maximize their
profits. Player i chooses to maximize a and to minimize d while Player j choose to
minimize both b and c. We can once again better visualize this in a graph.
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This means that like in Case 1, players choose to cluster around the right
endpoint with a value of 1 for point a and a value of 0 for the rest of the points. This
outputs:
In Case 3 both players make equal profits. Once again, this is not the ideal
situation since the players have once again clustered at a single location. But the
negligible distance of ε is so small that it does not effect the final profits. The insight that
is gleamed from Case 3 is that alternating over the linear space will only serve to divide
profits and gives neither player an advantage. It is also more profitable to mix like this
than it is to split the continuum as in Case 2. However, it is still not as profitable as
controlling the outside of the space in Case 1.
General Equilibrium
The results that we have found in the previous sections have allowed us to learn
about which scenario is the most profitable way to locate firms along a linear space. For
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example, Case 2 is strictly dominated by Case 3 since in Case 3 each firm can make
twice as much profit as in Case 2. However, Case 1 is still the most profitable if the
player maintains control of the extremes of the space. Case 1 gives one player a
distinct advantage over the other by allowing him control over the extremes of the linear
space. In this section, we will conduct a comparative analysis of which case is the most
profitable as we move each location (a, b, c, and d) individually along the continuum
while holding all else equal. We will also explore what happens when each player tries
to compete to be the most profitable as in Case 1 and how this settles in equilibrium
Up until now we have predicted how players will act under the profit maximizing
rule. This has led us to find that our solutions have placed us at endpoints of our linear
space. Now, let us step back a step back and analyze what happens when players do
not act completely rational and do not end up at one of these endpoints. As a player
decides where to place his firm, where is each individual location maximized? To see
this we plot profit as a function of location of each variable along the unit space. For
location a, the leftmost location, profit is maximized over the interval from [0, .35] for
Case 3 Player j and from [.35, 1] for Case 1 player i.
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It would be irrational to play any other scenario such as being Player j in Case 1 since
Player j can make more profit by playing Case 3. We can then work through the graphs
for the other locations to come to similar conclusions.
In this case, players mix over Case 3 Player j form [0, .65] and Case 1 Player j from
[.65, 1].
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Case 3 Player j [0, .65] and Case 1 Player i [.65, 0] are the mixing strategies for c.
Playing Case 3 player j dominates any other placement of d.
Given these facts, it is possible to not only choose a location from these graphs
and know which player and case it is best to be, this information can also be used to
undercut your opponent. These graphs give you the best options for reaction functions.
Say you know that the other player is placing his first location .30. So his a location (as
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long as you place both your firms right of his and you induce him to also place his other
location to the right) will make the minimum amount of profit if you make him play Case
3. Since his location is the leftmost, he must be Player i. So to make him worst off you
play Case 3 which makes significantly less profit at that location than Case 1. Knowing
that you are now Player j in Case 3, you can choose c to be 0, or rather ε remembering
our constraints.
Now, whatever value Player i chooses for b (likely also ε), you will
make maximum profit for any value of d since you are Player j playing Case 3. Just like
the scenario that we just walked through, using these plots it is possible to choose
strategies based on information and signals from your opponent to choose how to react.
The graphs we’ve just examined give good strategies on how to choose
locations – especially in the case where a player knows some information about his
opponent.
But how about the case where we know no information?
walkthrough how players will act until they settle to an equilibrium.
We can
As we know
extensively by this point, the most desirable scenario is to be Player i and play Case 1.
Therefore, we must assume that both players will try to position their locations on the
extremes of the interval in order to be this player. However, by placing both locations at
the most extreme points allows the other player to capture the rest of the interval by
being an ε closer to the center of both locations. But if the player moves too far in, the
other player will take the outside position and grab the desirable outside spot. We must
now find the optimal point between being too close to the endpoints of the interval and
being too close to the center.
In this case, we are dealing with the placement of two points: a and 1-b. The
other player will play the same locations but with an ε increment either towards the
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outside or the inside of the interval depending on which is more profitable. The player
who captures the outside of the interval will control a total distance of a+b. The player
that captures the inside will receive 1-a-b.
We want to find the point where
because this is where the player will be Player i in Case 1 and receive
maximum profit by capturing as much of the space as possible. It is trivial to discover
that
While
otherwise it is more desirable for the other player to play the outside.
, this does not specify over what interval on our unit space this must
be true. Variables a and b can take on any values as long as they sum to slightly less
than ½. With this constraint in place, we can now choose the individual values of a and
b. Just like in our solution for Case 1, profit is maximized when a is maximized and b is
minimized. This means that Player i chooses a to be ½ and b to be 0. Player j chooses
to locate an ε to the right of a and an ε to the left of 1-b meaning his choice of c and d
are essentially 0. Knowing the values of all our variables we can now solve for profit:
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Finally, we have a result for the general equilibrium. Previously we looked at
specific cases or ways to maximize based on information. Bringing all this information
together allowed us to come to the conclusion of the best case to play. Knowing that
both players will try to play the best case allows us to analyze how the game settles in
equilibrium and the result is certainly surprising. One player locates himself at the
midpoint of the interval and at the far endpoint while his opponent captures the middle
of that space. By doing so, the outside player makes positive profits and forces his
opponent to take a loss and in the long run, to even possibly shut down. How does one
player capture the outside in this case since they are both trying to locate at the same
space? This is determined by variables outside our sphere of influence. Perhaps it is
more costly for one firm to open at the outside location over the other firm. Perhaps both
firms bid on the real estate that is on the outside of the interval and the firm that
presents the better deal wins. Or perhaps one firm just gets there first. Either way, one
player will come out on top at the expense of the other due to some exogenous
variables outside of this model. The consequences of this result will be the focus of the
rest of this paper.
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Model Implications
The political implications of the Hotelling model delve into campaign
management and elections. Recall that the regular Hotelling model shows us where
two candidates will both attempt to locate at the center of the political spectrum. The
analogous concept in our revised model has similar results. In this case the two players
are the two political parties: one liberal and one conservative. The two locations each
party controls refers to any race in which there is a primary candidate and they also run
with a deputy under them. The most relevant example is focusing on a presidential
election with both a presidential and vice presidential candidate on the same ticket. Our
new model can tell us where we should place both of them along the political spectrum
in order to win the election. Before we begin this analysis, it must be noted that in most
elections the ideology of the vice presidential candidate does not carry as much weight
as the ideology of the president. Nevertheless, in our analysis we will act as if voters
care equally about the political views of both the president and vice president on the
ticket.
We start with the general result we found for our two player two firm case. This
says that both candidates choose to locate at ½ and 1 respectively. However, politics is
decidedly different from our original model in that there are two ends of the spectrum
and people are generally not willing to cross to the other side. A liberal is not likely to
vote for a candidate who runs on a conservative ticket. Furthermore, it is just unrealistic
and we do not see this occur in reality. Therefore, if both candidates ran at ½ and 1,
there would be an outcry from voters. Instead, a liberal candidate would be better off
running at 0 and ½ to capture the voters of his ideology. Note that he still maintains
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control over ½ the length of the space which was crucial in finding our equilibrium.
However, we are no longer playing a Case 1 game and instead have moved to a Case
3. It is a Case 3 because Case 3 dominates Case 2 so that’s what both parties will
play. Therefore, the liberals will locate at 0 and ½+ε and the conservatives will locate at
1 and ½-ε. The political spectrum will look like this:
liberal
conservative
The conventional wisdom behind what we observe is that liberals are going to want a
liberal candidate – one who ascribes to their views. Likewise, conservatives want to
vote for a conservative. That is why there is one candidate located at each extreme for
each party. However, the reason candidates also locate around the center, as in the
classic Hotelling model, is to capture moderate voters of the opposing political view.
The extremist of either party are going to vote for their own candidate regardless of
where they are along the political spectrum as long as it still reflects their ideology.
Moderates, on the other hand, are much more fickle and often undecided. By locating
at the ½ point, it is possible to garner the votes of these voters by not appearing to be
too extreme in either direction.
The interesting question that we are left with is where do the president and vice
president locate? Is the president the one at the extreme or the one at the center? And
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how does this play into primary party voting? Let us tackle the last question first. In
primary voting, only the party’s registered voters vote for who they want to be
candidate. Therefore, candidates have to appeal to the parties ideology and be less
moderate and more extreme. This would seem to favor the extremist to be the ones
who are president. However, an election is successfully won by not churning out the
party base on voting day, but rather capturing those undecided voters at the center from
your opponent. New models of politics which take into account this kind of behavioral
view move away from the old models of capturing median voters. Specifically, the
Erikson-Romero model finds that “…both candidates are motivated to appeal to
independent voters, as opposed to campaigning for the support of committed partisans
– this because strong partisans can be to some extent ‘taken for granted’…” (Adams,
298) The president, the figure most in the public eye, has to be moderate in order to do
this. This also reflects the fact that after primaries, candidates try to make themselves
look more moderate.
However, to still carry their base, the presidential candidate
choose a vice presidential running mate who complements them and capture voters
they themselves would not carry. The vice president therefore captures the base and
attempts to keep the voters the president might alienate by appearing more moderate.
Consequently, according to this model of political theory, it is best for a president to be
moderate with an extremist running mate in order to win an election. It is interesting to
also note that while the position of the vice president is important, if we think about the
fact that the position of the president is more important than the vice president, we still
essentially see the same theory that results from the classic Hotelling model.
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Presidential candidates locate at the center of the spectrum to capture the undecided
moderates in order to win the election.
Let us now turn our attention to another type of competition we see with the
model we built: competition between chains. This is how firms such as McDonald’s and
Burger King compete with each other. We once again start by looking at the general
case we solved and see that the firms position themselves at ½ and 1. But now we are
going to relax one of the initial assumptions. In this scenario instead of a finite space,
these two firms compete over an extended infinite linear space with an infinite number
of locations. Let us take a step back to see how we arrive at this scenario. The firms
are located at ½ and 1 but what if they want to create a new location as they grow due
to profits? They create a new location, also a half length away to the right. Essentially,
we are just extending the space to the right half a length and now viewing a new space
from ½ to 1½. We continue this ad infinitum. The same principles still apply. The firms
still battle each other to control the extremes of the space. This means that whenever
one firm places a location, the other will try to capture the outside of it and we will see
the two firms both locate at the same point. The principle of local clustering holds in our
model. The question now is how far apart these battleground locations of the firms
competing are going to be from one another. Due to the lengthening of the linear
space, we assumed they were a half length apart from one another. Since we also
know that both firms try to control half the space or otherwise they lose the extreme
position and market share, this assumption holds. What is a half length though in the
real world? It could be half a mile or half a kilometer. Furthermore, different types of
firms will have different standards for what a half length is. Department stores are likely
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to have longer definitions of a half length than coffee shops. The point is that once a
standard is established of how far apart two locations are, this standard distance
between any additional locations holds. Now our space looks like this:
0
½
1
1½
2
½
This explains why we see gas stations across the street from one another or a
Coke machine next to a Pepsi machine. Firms are best off when they place their
product next to their competitor’s product for the consumer to choose the best one
based on the marginal differences in products. By placing their locations anywhere
else, they leave more space for their competitor to capture market share. It is also true
that if one firm creates a new location to capture what they believe to be some market
that was previously not captured, that their competitor can see that and also create a
new location to compete for that market. This holds true for entrance into new markets
such as new cities and new countries. Furthermore, this once again holds true to the
classic Hotelling model where we saw that it is best to locate next to your competitor.
The final implication of our model has to do with traveling costs. Our general
result reinforces the Hotelling model in that it creates a disincentive to improve
transportation infrastructure. As we have seen in this model, profits increase as the
cost of travel increases. Firms also have no incentive to see transportation improved so
that customers could potentially have access to their competitors goods easier. As
access and cost to transportation becomes easier, positioning along the space
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becomes less influential. Firms want to control the extremes of the space for a reason –
it allows them greater market share. In fact, “buyers’ locations are such that nonminimal (including maximal) transportation costs are incurred in order for competition
between sellers to be kept as fierce as possible” (Camacho-Cuena, 91) Additionally,
this means that those consumers who reside on or close to the firms profit greatly since
they do not incur these great transportation costs.
Some of this may be due to
consumers smartly positioning themselves next to desirable locations and some may be
luckily free riding. (Camacho Cuena, 96) However, this accounts for high rent prices in
desirable locations in cities. Due to all this, businesses will not play an active role in
governmental policy to increase transportation infrastructure. Corporate lobbies have a
major influence on public policy so the disincentive to increase the ease of
transportation is somewhat unsettling as it decreases overall societal welfare.
Conclusion
Using Harold Hotelling’s classic model, we were able to build on his theories and
create a better approximation of how firms compete in reality. For after all, while we do
see single firms compete against one another, in today’s world of multinational
corporations firms must compete in a different form. They must consider that they
themselves control multiple locations of their corporation as do their competitors. This
led us to consider three cases of how firms could align themselves on a linear unit
space which were dubbed Case 1, Case 2, and Case 3. We then found that Case 1,
where one player controls the extremes of the space was the most profitable to that
Metelka 35
outside player. Knowing that both players could rationalize that Case 1 was the best
outcome, we analyzed what happens when both players try to play Case 1 and discover
that the firms would end up positioning themselves at the points ½ and 1. We also
looked at how we could use the reaction functions of the cases we analyzed to find the
best positioning of locations under certain restrictions as well.
Using the new general
equilibrium we could explore the effects of our model on other institutions.
As an
analysis of political theory, we find that candidates in a presidential election with a vice
presidential running mate should locate at the center of the space while their vice
president should cater to the extremes of their political party. For competing chains, the
principle of local clustering holds and all firms should endeavor to locate next to their
competitor in order to capture the most profit.
With regards to transportation
infrastructure, we find that there is no incentive for firms to put pressure on governments
to improve ease of transportation as profits rise in direct correlation with transportation
costs. From all of this we can conclude that while it might not be the most logical thing
to locate next to a competitor, it is what we see in reality and the most profitable
stratagem. Furthermore, it is what is reflected in reality. In the future, it would be
prudent to look at how this model, since it assumes the simultaneous actions of firms,
compares to that where firms take terms placing locations against each other over time.
It would also be useful to look at cases of more than two firms and more than two
locations. This model could also be applied to different spaces besides the linear one
which we used which are better approximations of how cities arrange themselves in
reality. All in all, this is a great primer to use in our growing world of hyper-chains and
mega corporations that compete against one another and perhaps even serves as a
Metelka 36
glimpse into the future of how we can expect firms to locate themselves in our ever
increasing global world where corporations can spread their arms to any corner of the
world.
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Appendix A
Setting the demand equations for each firm equal we obtain:
The length of the total segment is l:
By algebra:
Firms now maximize profit by finding the optimal price to charge:
Metelka 38
(Hotelling, 46, 47, 50)
Metelka 39
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(2000): 297-325.
Camacho-Cuena, Eva, et al. "Buyer-Seller Interaction in Experimental Spatial Markets."
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Eaton, B. Curtis and Richard G. Lipsey. "The Principle of Minimum Differentiation: Some
New Developments in the Theory of Spatial Competition." The Review of Economic
Studies, 42 (1975): 27-49.
Hotelling, Harold. "Stability in Competition." The Economic Journal, 39 (1929): 41-57.
Huck, Steffen, Wieland Muller and Nicolaas J Vriend. "The East End, the West End, and
King's Cross: On Clustering in the Four-Player Hotelling Game." Economic Inquiry
(2002): 231-240.
Tirole, Jean. The Theory of Industrial Organization. Cambridge: MIT Press, 1988.