I borrowed some of the figures and equations in this lecture from Yohanes E. Riyanto, an associate professor at Nanyang Technological University in Singagpore

Addresses the nonexistence of HotellingequilibriumMajor changes to the model

a. Market space is a circle, not a line (Salop says an infinite line yields the same results). Circle is of unit mass. One imagery is a tropical island with a mountain in the middle

b. Firms choose to enter, or not, at a specific location. These are the differentiated goods

c. Salop has a second, undifferentiated, competitively supplied good that a consumer can buy instead of the differentiated good

Allows us to look at optimal and equilibrium product diversity

Firms

Firms are located around the circle market, equidistance from each other. Subscript i indicates a specific firm at location i. There are Nfirms.

There is a fixed cost and constant marginal cost.

Firms choose output and price to maximize profit given by

i i i iq p c q f

( )c q cq f

Consumers

Consumers are located uniformly around the circle.

A consumer’s location represents her most preferred brand.

Each consumer purchases one or no units of the differentiated good (excess income is spent on the homogenous good).

Transportation costs per unit of distance is t. There are no transportation costs for the homogenous good

The consumer derive utility from consuming her one unit of the good

Salop has L consumers. I normalize L1

V p tx

5

x

Salop’s Circle Model (example: N=6)

Firm 1

Firm 2

Firm 3

Firm 4

Firm 5

Firm 6

1/6 1/6

1/6

1/61/6

1/6

Firm 1

Firm 2Firm 6

p1

p px

1/6

Yohanes E. Riyanto

Entry and PricingTwo stage game

1. Firms choose whether to enter and where to locate. Assume the principle of maximum differentiation – if firms enter they try to be as far as possible from the closest competitors, to increase market power. If N firms enter, they locate equidistance around the circle, so there is 1/N distances between them

2. Firms set price to maximize profit, given the number of other firms (ie, given the distance to their closest competitors)

Assume a symmetric equilibrium, so p1=p2=…=pN=p

Firm has only two competitors, the firms located immediately to its right or left. Why?

Consumer choice and firm demand

Given price p charged by the adjacent firms (left and right) and price p1charge by firm 1 the indifferent consumer is at distance so

and this give firm 1’s market share (demand) as

0,1/x N

11

1 ( / )

2

p p t NV p tx V p t x x

N t

1 11( / )

, 22

p p t ND p p x

t

Firm’s profit

Profit of firm 1 is

Remember p is the price of the adjacent firm. All firms are the same, hence p=p1 and so

Like Hotelling, price and profit margin increase with transportation cost. Moreover, they decrease with the number of firms in the market

1

11 1

1

1

/

2

0 2 2

t p c

p

p p t Np c f

t

p N

- t t

p c p cN N

Local monopoliesSo far assumed firms are close enough that the compete for the same consumers. However, if firms are far apart, some consumers may choose to buy none of the differentiated good. This occurs when

So that price plus transportation cost exceeds the value of the good

10V p tx

Yohanes E. Riyanto 10

Firm 1

Firm 2

p1

p

1/N

xm1 xm1

xm2xm

2

these

consumers do

not buy since

price is higher

than the value

obtained (p>V)

10 V p tx

1

V p

xc

1 1 12

2 D p x V pc

Indifferent consumer gets 0 surplus value from the differentiated good, so

The demand for firms 1 and 2 is given by

because they have monopoly over consumers on either side

1V pxt

1

1 1

2( ) 2 ( )D p x V p

t

Yohanes E. Riyanto 11

Price

Quantity

mp

monopoly region

competitive region

Gives a kinked demand curve

mp is the price given that

consumers located at the locations on the previous slide are indifferent between buying the good or not. As any lower price the firm is competing with adjacent firms for those consumers and all consumers .

xm

Where is the kink?

Demand can never exceed monopoly demand, so the kink must be on the monopoly demand curve. As N increases, the kink moves to the left off the monopoly demand curve

Symmetric Zero Profit Equilibrium (SZPE)SZPE is a price, p, and number of firms (brands), N, such that every equally spaced Nash price setters’ maximum profit price earns zero profit.

In equilibrium, a firm’s demand curve and average cost curve will be tangent (assures zero profit is maximum profit). Three equilibria are possible

Monopoly equilibrium, some consumers do not buy the differentiated good. Markets of neighboring firms do not overlap, each firm acts like a monopolist.

At a kinked equilibrium markets just touch (there is a clear marginal consumer). The monopoly demand lies above the AC curve so what would be the monopoly price (shown by pm) is below the kinked-equilibrium price (pk)

Competitive equilibrium all markets overlap, not really concerned with it here

SZPE SZPE satisfies two conditions:1. MR MC

2. P=AC

From symmetry if the equilibrium has no gaps q=1/N (remember we normalized L=1)

Point G is dominated by F so the equilibrium is at the kink unless AC is exceedingly flat – giving the monopoly equilibrium, or steep, giving the competitive equilibrium (2 slides above)

and /dp

p q c p c f qdq

Number of firms: Free entry

Entry takes place until profit is fully dissipated. Since

Solve for N gives

Price exceeds MC, but profit=0

2

10

i

tp c f f

N N

-t

p cN

and c ct

N p c tff

Equilibrium with sufficient firms for competition

An increase in fixed cost causes a decrease in the equilibrium number of firms (Nc) and an increase in each firms’ profit margin (p-c).

When FC 0, Nc

An increase in transportation cost (t) causes an increase in the equilibrium number of firms (Nc) and an increase in each firms’ profit margin (p-c).

Social optimumChoose N to minimize total production and transportation costs

So

Profits are positive and equal 3F

Competitive market has too much entry (too much differentiation) compared to social optimum. Total transportation costs = which decrease in N. Since Ns

What is important in the modelWhich of the model innovations were most important?

The second good and the circle (or infinite line) were tools to remove the discontinuities in the profit function that plagued the Hotellingmodel. They were innovations, but not the focus of the paper

• Notice how little Salop talks about the outside good. It just enabled demand for the differentiated good to be elastic (buy 1 or 0, depending on price)

• Equivalent, in a way, to the congestion costs of Ahlin and Ahlin

• Focus is on product differentiation and competition. That is the research issue of this paper

Most important conclusions/contributions1. Competition results in too much differentiation

2. Explains where a kinked-demand curve comes from

3. In equilibrium, P>MC but profit still equal 0

4. Monopoly demand is more elastic than competitive demand

How important are these conclusions/contributions? Evaluation is not just stating something, it is telling us whether it is good or bad, and why. For example, how valuable is it that we understand where kinked-demand curves come from? Why? Evaluation means assessing what it adds to our knowledge, whether that is valuable, and maybe, as a weakness, where it misleads us