Sonya Pervez

08627758

Regional & Urban Economics (EC3511)

[Regional Location Theory Essay]

“(I) USING THE HOTELLING MODEL,

DISCUSS IN DETAIL HOW FIRMS CAN USE

THEIR LOCATION DECISIONS AS PART OF

THEIR STRATEGIC BEHAVIOUR.

(II) IS THE RESULTING NASH

EQUILIBRIUM THE SOCIALLY OPTIMAL

SOLUTION?”

CONTENTS

1. INTRODUCTION

2. KEY THEORIES.............................................(i) Regional Location Theory

...........................................................................(ii) Game Theory & Nash

Equilibrium

...........................................................................(iii) Spatial Competition

3. THE MODEL..................................................(i) Outline and Assumptions

……………………………………….……….….…(ii) Construction of Model

...........................................................................(iii) The Case of Oligopoly

(Fixed Prices)

...........................................................................(iv) Price Competition

4. CONCLUSIONS

5. REFERENCES

INTRODUCTION

Location plays a crucial role in determining the life cycle of any

organisation. No firm or business unit can profit maximise, or survive,

without having initially determined its optimal operating location. A firm will

seek to position itself geographically where it can foresee economic (and

perhaps social) benefits. Ideal locations are ones which are, for example, in

proximity to input markets and resources (offering lower transport costs to

move raw materials); in proximity to output markets (which offer reduced

delivery costs to consumers); in regions with similar industries (allowing for

knowledge spill-overs and other mutual benefits) or in regions that offer

lower prices for input factors (lower wages, rental rate of capital, land). A

firm can also derive locational benefit from being positioned closely to its

direct competitors. Positioned near rivals allows firms to have some degree

of Monopoly Power.

The study of firms and their Location Behaviour as a whole is referred to

as Regional Location Theory, which looks specifically at why certain

economics activities are located (or in some cases, concentrated) in a

specific region or why they are more dispersed. It comprises a set of theories

and models that seek to explain why firms choose to locate and/or relocate

themselves where they do, and how they behave in their respective locations

to ensure survival. One model of this sort is called the Hotelling Model.

KEY THEORIES

(i) THE HOTELLING MODEL AND THE THEORY OF SPATIAL COMPETITION

The Hotelling Model is fundamentally a “Location Decision Model” that is

based on a paper written by an American Economist, Harold Hotelling, in

1929. The paper came about as a result of Hotelling’s observations on

competition, and its associated stability. The model was an attempt to try

and explain why certain businesses (shops, restaurants) are located so close

together rather than being dispersed equally along space and is based on

the basic premise that firms are acting (competing) to profit Maximise.

Spatial Competition, then, refers to competition over a designated space,

in this case, to obtain the maximum possible Market Share.

Hotelling states that firms that produce undifferentiated goods and services

benefit from being in close proximity to their competitors. Firms with

identical products have identical prices (when the good or service is

consumed at the location), so spatial competition is the only way firms can

maximise profits. The model shows that competing in such a way also gives

each firm some Monopoly Power over their respective locations (market

areas), which would not be possible if the firms were to dispersed further

away.

(ii) GAME THEORY AND NASH EQULIBRIUM

Game Theory is a core component of the Hotelling model and is related to

the competition involvevd. In essence, firms in the model are behaving as

opposing players in a “non co-operative game” situation. Each is competing

for a greater share and monopoly power over the market area which is

referred to as a Location Game. Each firms’ respective location behaviour is

characterised by a set of sequential movements where location itself is

the strategy.

In this framework, the final location of either firm, or firms, will be

determined by Nash Equilibrium, which in a game setting, is a stable

state (equilibrium) for players in a game. In this state, neither player (firm)

can gain any more from changing its strategy (e.g. relocating), unless the

other player changes its strategies first. Not engaging in this competitive

game can lead to one firm sacrificing its monopolistic power and subsequent

market share.

BUILDING THE MODEL

(i)ASSUMPTIONS AND OUTLINE

For simplicity, assume that there are two firms in the market: Firm A and Firm B

- Both firms are located along a Linear Market Area, OL (E.g. Beach)

- Both produce identical goods (so there is no product differentiation)

- Each firm’s final good has to be transported to the consumer’s

location, so the final prices reflect the cost of delivery

- The final price is then the price of the good plus the delivered price

- The further away the consumer, the higher the delivered price

- The lower the delivered price the more of the market that can be

captured

- Transport Costs overall, therefore, are crucial to the model

- A firm can only capture ALL of the market if its Transport Cost are

equal to zero

- Consumers are Evenly Distributed across the O-L Space and will

buy from the firm that supplies the good to their location at the lowest

price

- The lower the firm’s production costs and transport costs, the

larger the firm’s market share

- In terms of Game Theory, each firm assumes that it will move, but the

other firm will remain at its location

(ii) CONSTRUCTING THE MODEL

- The Production Costs of producing Good A is equal to the Distance

of a (a)

- The Production Costs of producing Good B is equal to the Distance

of b (b)

- Transport Costs are diagrammatically represented by the Slope(s) of

the “Transport Rate Function” (tA) and (tB)

- The Transport Costs (tA) and (tB) are upward sloping because of the

fact that the further away you are from Firm A or Firm B, the higher the

final price, or costs to transport

- At any location, the Delivered Price of firm A’s good is

PA + tA dA

Where:

PA = The Production Price of Good AtA = The Cost of Transporting Good A t where the Consumer isdA = The Distance to the Consumer

As a starting point, the Linear O-L Space is divided into two equal regions

- The O-X Space and the X-L Space, where each represents Market Share

- Firm A has Monopoly over the O-X Space

- Firm B has Monopoly over the X-L Space

NB: Within the O-X space, the price of Firm A’s good is always lower than

the price of Firm B’s good and within the X-L space, Firm B’s good is always

lower than Firm A’s:

(PA + tA dA) < (PB + tB dB) (PA + tA dA) > (PB + tB dB)

[Consumers pay less for respective goods in their respective locations]

It is important to note here, that in the diagram, Firm A has a lower

production cost of producing one unit of Good A, than Firm B. (PA) < (PB).

The distances vary slightly.

From an economic perspective this implies that Firm A is more efficient than

Firm B (as it is producing the same level of output, but at lower costs).

In theory, this would lead to Firm A having a relatively larger market share.

However, as mentioned earlier, transport costs are crucial to the model and

despite this discrepancy each firm, initially, has equal market share.

This is because location itself gives each firm some Monopoly Power over the

area it has. Consumers will buy the good from the firm closest to him/her, as

the further away he/she is, the more he/she will have to pay for the good. So

Monopoly Power derives fundamentally from the associated Transport Costs.

Furthermore, in reality, it is normally the case that transport functions are

not equal,

tA ≠ tB so the model should reflect different slopes for tA and tB.

(iii) THE CASE OF OLIGOPOLY (NO PRICE COMPETITION)

With the basic model constructed, we can look at how the two firms use their

Location Decisions as part of their Strategic Behaviour. As a starting point in

the analysis, we have:

- Two firms, A and B, Producing undifferentiated products

- PA = PB (Both firms have equal production costs)

- tA = tB (Both firms have equal transport costs)

- Each has an equal market share

- A has more monopoly over O-X

- B has more Monopoly over X-L

- Each firm makes a competitive decision based on the assumption

that the other firm will NOT change its behaviour

So here the firms are not engaging in price competition and (prices are fixed)

and the only way a Firm can gain Market Share is by changing its Location.

For the purposes of simplicity, we also assume here that there are zero

relocation costs

Figure 2. reflects the initial position of both firms.

Wanting to maximise profits, A makes move to increase its market share:

- At a Time Period, T1, Firm A makes the decision to relocate

- Firm A moves to point C (Figure 3.)

- In doing so, Firm A increases its market share from O-X to O-C

- Subsequently, Firm B’s market share decreases from X-L to C-L

This first movement triggers the beginning of the Location Game:

- At time period T2, Firm B assumes that A will now remain at location C- Firm B tries to regain its share of the market- Firm B moves to point D (Figure 4.)- Firm B now has a larger market share, A’s share has reduced

significantly

Both firms know that is they don’t participate in this spatial game each time, they will sacrifice market share, and with Firm A’s market share reduced, the location game continues until both have no more to lose:

- At time period T3, A responds to this move by moving left of firm B- A moves to E- D moves to F- This movement continues until both firms are located at X (Figure 5.)- Leads to a Nash Equilibrium

At point X, the location game stops. Both firms have effectively returned to having equal market share. Neither of the firms now has any incentive to relocate. In a game situation, Point X is the Optimal Solution in the location problem.

Hotelling’s model, therefore, demonstrates that location decisions are used by firms as a strategy to determine optimal operating locations. This type of market analysis also serves as a basis for explaining the reasons why certain restaurants, supermarkets and multi-national firms locate in proximity of each other. Spatial arrangements of this sort allow firms to monopolise the area that they position themselves in, which would be difficult if the firms were spread out further apart.

(iv) PRICE COMPETITION AND THE HOTELLING MODEL

The result above is not the same when firms are competing in terms of price.

In firms operating under price competition:

- Firms decrease their prices at each time period to stimulate demand- Both firms are now engaging in a price game.- This will continue until both firms end up selling at zero profits, where

both are still located at X- However, neither firm wants to sell at zero profits- Rather than move closer together, firms need to disperse further apart

to avoid losses- In the interest of protecting itself, neither firm’s is willing to make a

move first, as the first firm that moves first will lose its market share- In this situation, both firms make a deal to move at the same time,

which eventually leads to “Prisoners Dilemma” - In Prisoners Dilemma, neither firm will do what it says it will do, so

neither will move- Therefore, under price competition, you do not get an optimal solution

- Firms can only locate in close proximity of each other if there is NO price competition

THE SOCIAL OPTIMUM AND CONSUMER WELFARE

So Hotelling’s model successfully explains how firms use location to determine an Optimal Operating Position for themselves, a stable Nash Equilibrium at point X. However, the move to point X leads to a fall in Consumer Welfare. So this optimal solution to the location game is optimal only to the firms.

The reasons behind this are:

- That a relatively larger proportion of consumers now pay higher prices than those that pay lower prices

- At X, consumers are worse off than they were when they were located at their the original positions

- Consumers located at the periphery (Nearer to O, or Nearer to L in the O-L Space) are much further away from both firms

- Welfare Loss at the periphery is greater than anywhere else- They will have to pay higher delivered prices due to the increased

distance they now face- Consumers situated at the centre of the O-L Space are relatively well-

off as they are situated in proximity to both firms so they gain from lower delivered prices

- Though there are gains, the total welfare loss is greater than the gains

The Hotelling model offers an optimal solution that unfortunately is not a socially optimal one. It is important to note, however, that this only holds when there is price competition, so in oligopoly, the location game has losses

Figure 6. Shows this diagrammatically:

- Consumer 1 (C1) is closer to firm A, so originally he/she paid, Price

P1C1

- At point X, C1 now pays a higher price of P2C1

- The consumer is worse off at the edge of the model

- Consumer 2 (C2)

- Consumer 2 is closer to both firms now so will face lower delivered

prices

- At point X, C2 pays much less now than if it were buying from Firm A at

its original location

So while optimal point X has a few gains, it also has relatively many more losses, especially for consumers at the ends of the O-L Space.

CRITICISMS OF THE HOTELLING MODEL

On the whole, the model is a successful one in explaining firms’ location

behavior and is a general tool for market analysis. Drawbacks, however, do

exist:

1. The Model has been criticized for being oversimplified and too general

2. The set of assumptions as a whole are too relaxed. For example:

- It is assumed that there are no relocation costs, which is

unrealistic

- Consumers are assumed to be evenly distributed

- The market space is linear

- Each firm assumes that the other is not going to participate in

the location game initially which is misrepresentative, firms are

not naïve about the behavior of its competitors as the model

suggests1.

3. The Game Theory involved tells us what players are likely to do, but it

is clear that the results are not socially optimal.

4. There are many other factors associated with location decisions such

as personal preferences, environmental constraints, personal

circumstances2. The model could benefit from incorporating these

factors.

5. Vickrey (1964), d’Aspremont, Gabszewicz and Thisse (1979) show

further that hotelling’s argument is flawed stating that “no pure

strategy price equilibrium exists for such locations”3

6. The model only incorporates two firms only whereas in actuality many

firms tend to compete on a spatial level. It has also been shown

empirically that when larger numbers of firms are incorporated into the

model, the firms that lie on the periphery in fact have a greater

advantage in terms of market power than those located in the central

region4. This contradicts the results above.

7. Increases is networking possibilities (more and more is becoming

computerized), telecommunications and globalization have changed

the nature of location theory itself5. Transport possibilities have

increased, and firms can transport goods over larger distances and at

lower costs to a wide variety of developments5.

8. Lastly, empirical research has also shown that the model exhibits a

circular structure as opposed to a 1 dimensional flat 1, where demand

and market areas are represented by a “Demand Cone”, which is a

more realistic approach1.

CONCLUSIONS

Overall the model is useful tool for market analysis and successful in

demonstrating how markets behave in regions they choose to locate in.

The key elements of the model have served as a foundation for further

locational analysis. The model has also been adapted and applied into

Politics, by Anthony Downs in 1957, from his book “An Economic Theory

of Democracy”. The spatial theory in the model has been used to make

predictions on how Politicians will interact during elections, and has so far

also been successfully applied.

However, the optimal solution for firm locations are clearly not always

solutions that offer social benefits for society as a whole, and the basic

assumptions of the model have limited its applicability to real life

economic situations. With changes in socio-economic trends, classical

location theories are in need of updating to incorporate the shift in

location decision making, which if done, could prove as a very powerful

economic tool of analysis.

BIBLIOGRAPHY

1. Hoover, E. M, and Giarratani, F., (1984), An Introduction To Economics, [3rd Ed.], Harvard Press.http://www.rri.wvu.edu/regscweb.htm

2. Williams, H. C. W. L. and Senior, M. L.(1976), “A Retail Location Model with Overlapping Market Areas: Hotelling’s Problem Revisited.”, Working Paper 137, Dept. of Geography, University of Leeds January 1976

3. Osborne, M. J. and Pitchik, C., (1987), “Equilibrium in Hotelling Model of Spatial Competition”, Econometrica, Vol. 55, No. 4, pp. 911-922.

4. Brenner, S. (2005), “Hotelling Games with Three, Four and More Players”, Journal of Regional Science , Vol. 45, No. 4, pp. 851 – 864:

5. Assink, M and Groenendijk, N, (2009) “ Spatial quality, Location Theory & Spatial Planning”, Paper presented at Regional Studies Association Annual Conference 2009; “Understanding and Shaping Regions: Spatial, Social and Economic Futures” Leuven, Belgium, April 6 – 8, 2009

6. Fujita, M, Schweizer, U, Gabszewicz, J. J, Thisse, J. F. (1986) “Spatial Competition and the Location of firms”; Harwood Academic Publishers GmbH

7. McCann, P. (2001) Urban and Regional Economics; Oxford University Press, pp. 27 – 35

8. http://www.horton.ednet.ns.ca/staff/jptrites/APHumanGeog12Folder/ notes_presentations/notes_econ_location_models.pdf

Advancecd Global Geography12, Advanced Placement Human Geography, Location theories and models. [Accessed 2/3/11]

9. Dickie, H; (2011). Regional Location Theory [Topic 2 – Lectures 5 & 6]. University of Aberdeen