1

Product Variety under Monopoly

ECO 171 2

Introduction

Most firms sell more than one product Products are differentiated in different ways

horizontally goods of similar quality targeted at consumers of

different types how is variety determined? is there too much variety

ECO 171 3

Horizontal product differentiation

Suppose that consumers differ in their tastes firm has to decide how best to serve different types of consumer offer products with different characteristics but similar qualities

This is horizontal product differentiation firm designs products that appeal to different types of consumer products are of (roughly) similar quality

Questions: how many products? of what type? how do we model this problem?

ECO 171 4

A spatial approach to product variety The spatial model (Hotelling) is useful to consider

pricing design variety

Has a much richer application as a model of product differentiation “location” can be thought of in

space (geography) time (departure times of planes, buses, trains) product characteristics (design and variety)

consumers prefer products that are “close” to their preferred types in space, or time or characteristics

ECO 171 5

An example of product variety

McDonald’s Burger King Wendy’s

ECO 171 6

A Spatial approach to product variety 2 Assume N consumers living equally spaced along

Montana Street – 1 mile long. Monopolist must decide how best to supply these

consumers Consumers buy exactly one unit provided that price plus

transport costs is less than V. Consumers incur there-and-back transport costs of t per

mile The monopolist operates one shop

reasonable to expect that this is located at the center of Montana

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The spatial model

z = 0 z = 1

Shop 1

t

x1

Price Price

All consumers withindistance x1 to the leftand right of the shopwill by the product

All consumers withindistance x1 to the leftand right of the shopwill by the product

1/2

V V

p1

t

x1

p1 + t.x p1 + t.x

p1 + t.x1 = V, so x1 = (V – p1)/t

What determinesx1?

What determinesx1?

Suppose that the monopolist sets a price of p1

Suppose that the monopolist sets a price of p1

ECO 171 8

The spatial model 2

z = 0 z = 1

Shop 1

x1

Price Price

1/2

V V

p1

x1

p1 + t.x p1 + t.x

Suppose the firmreduces the price

to p2?

Suppose the firmreduces the price

to p2?

p2

x2 x2

Then all consumerswithin distance x2

of the shop will buyfrom the firm

Then all consumerswithin distance x2

of the shop will buyfrom the firm

ECO 171 9

The spatial model 3

Suppose that all consumers are to be served at price p. The highest price is that charged to the consumers at the ends of

the market Their transport costs are t/2 : since they travel ½ mile to the shop So they pay p + t/2 which must be no greater than V. So p = V – t/2.

Suppose that marginal costs are c per unit. Suppose also that a shop has set-up costs of F. Then profit is (N, 1) = N(V – t/2 – c) – F.

ECO 171 10

Monopoly pricing in the spatial model What if there are two shops? The monopolist will coordinate prices at the two shops With identical costs and symmetric locations, these prices

will be equal: p1 = p2 = p Where should they be located? What is the optimal price p*?

ECO 171 11

Location with two shops

Suppose that the entire market is to be servedSuppose that the entire market is to be servedPrice Price

z = 0 z = 1

If there are two shopsthey will be located

symmetrically a distance d from theend-points of the

market

If there are two shopsthey will be located

symmetrically a distance d from theend-points of the

market

Suppose thatd < 1/4

Suppose thatd < 1/4

d

V V

1 - dShop 1 Shop 2

1/2

The maximum pricethe firm can chargeis determined by the

consumers at thecenter of the market

The maximum pricethe firm can chargeis determined by the

consumers at thecenter of the market

Delivered price toconsumers at the

market center equalstheir reservation price

Delivered price toconsumers at the

market center equalstheir reservation price

p(d) p(d)

Start with a low priceat each shop

Start with a low priceat each shop

Now raise the priceat each shop

Now raise the priceat each shop

What happens ifmove to the right?

What happens ifmove to the right?

The shops should bemoved inwards

The shops should bemoved inwards

Price can be increased without losing market

Price can be increased without losing market

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Location with two shops 2

Price Price

z = 0 z = 1

Now suppose thatd > 1/4

Now suppose thatd > 1/4

d

V V

1 - dShop 1 Shop 2

1/2

p(d) p(d)

Start with a low priceat each shop

Start with a low priceat each shop

Now raise the priceat each shop

Now raise the priceat each shop

The maximum pricethe firm can charge is now determined by the consumers at the end-points

of the market

The maximum pricethe firm can charge is now determined by the consumers at the end-points

of the market

Delivered price toconsumers at theend-points equals

their reservation price

Delivered price toconsumers at theend-points equals

their reservation price

Now can gainshifting left and

raising price

Now can gainshifting left and

raising price

The shops should bemoved outwards

The shops should bemoved outwards

ECO 171 13

Location with two shops 3

Price Price

z = 0 z = 11/4

V V

3/4Shop 1 Shop 2

1/2

It follows thatshop 1 shouldbe located at

1/4 and shop 2at 3/4

It follows thatshop 1 shouldbe located at

1/4 and shop 2at 3/4

Price at eachshop is thenp* = V - t/4

Price at eachshop is thenp* = V - t/4

V - t/4 V - t/4

Profit at each shopis given by the

shaded area

Profit at each shopis given by the

shaded area

Profit is now (N, 2) = N(V - t/4 - c) – 2FProfit is now (N, 2) = N(V - t/4 - c) – 2F

c c

ECO 171 14

Gain from introducing extra shop Price Price

z = 0 z = 11/4

V V

3/4Shop 1 Shop 2

1/2

P2 = V - t/4 V - t/4

c c

P1 = V - t/2

• Total surplus increases as transportation costs are now lower

• Producer surplus increases but consumer surplus decreases

•So producer surplus increases more than total surplus

Profits with one firm

Profits with two firms

ECO 171 15

Three shops

Price Price

z = 0 z = 1

V V

1/2

What if there are three shops?

What if there are three shops?

By the same argumentthey should be located

at 1/6, 1/2 and 5/6

By the same argumentthey should be located

at 1/6, 1/2 and 5/6

1/6 5/6Shop 1 Shop 2 Shop 3

Price at eachshop is now

V - t/6

Price at eachshop is now

V - t/6

V - t/6 V - t/6

Profit is now (N, 3) = N(V - t/6 - c) – 3FProfit is now (N, 3) = N(V - t/6 - c) – 3F

ECO 171 16

Optimal number of shops A consistent pattern is emerging.

• Assume that there are n shops.

• We have already considered n = 2 and n = 3.

• When n = 2 we have p(N, 2) = V - t/4

• When n = 3 we have p(N, 3) = V - t/6

• They will be symmetrically located distance 1/n apart.

• It follows that p(N, n) = V - t/2n

• Aggregate profit is then (N, n) = N(V - t/2n - c) – n.F

How manyshops should

there be?

How manyshops should

there be?

ECO 171 17

Optimal number of shops 2

Profit from n shops is (N, n) = (V - t/2n - c)N - n.Fand the profit from having n + 1 shops is:

*(N, n+1) = (V - t/2(n + 1)-c)N - (n + 1)F

Adding the (n +1)th shop is profitable if (N,n+1) - (N,n) > 0

This requires tN/2n - tN/2(n + 1) > F

which requires that n(n + 1) < tN/2F.

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An example

Suppose that F = $50,000 , N = 5 million and t = $1

Then t.N/2F = 50

For an additional shop to be profitable we need n(n + 1) < 50.

This is true for n < 6

There should be no more than seven shops in this case: if n = 6 then adding one more shop is profitable.

But if n = 7 then adding another shop is unprofitable.

ECO 171 19

Some intuition

What does the condition on n tell us? Simply, we should expect to find greater product variety

when: there are many consumers. set-up costs of increasing product variety are low. consumers have strong preferences over product characteristics

and differ in these consumers are unwilling to buy a product if it is not “very close” to

their most preferred product

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How much of the market to supply Should the whole market be served?

Suppose not. Then each shop has a local monopoly Each shop sells to consumers within distance r How is r determined?

it must be that p + tr = V so r = (V – p)/t so total demand is 2N(V – p)/t profit to each shop is then = 2N(p – c)(V – p)/t – F differentiate with respect to p and set to zero: d/dp = 2N(V – 2p + c)/t = 0 So the optimal price at each shop is p* = (V + c)/2 If all consumers are served price is p(N,n) = V – t/2n

Only part of the market should be served if p(N,n) < p* This implies that V < c + t/n.

ECO 171 21

Partial market supply

If c + t/n > V supply only part of the market and set price p* = (V + c)/2

If c + t/n < V supply the whole market and set price p(N,n) = V – t/2n

Supply only part of the market: if the consumer reservation price is low relative to marginal

production costs and transport costs if there are very few outlets

ECO 171 22

Market density and prices

What should this model imply of market prices and density?

Higher density is equivalent in this model to reducing transportation cost: If consumers are concentrated in half the interval, can

reach the same fraction of the consumers at half the cost.

Same effect as if transportation cost were cut in two Implications for prices?

ECO 171 23

Market density and prices

From previous slides, the number of firms n is approximately determined by: n(n + 1) = tN/2F.

As t decreases n goes down From the equation above, t/n is proportional to (n

+ 1). So as n decreases, t/n must decrease. Optimal price is given by p(N,n) = V – t/2n. This decreases as t/n increases.

ECO 171 24

Social optimum Are there toomany shops or

too few?

Are there toomany shops or

too few?What number of shops maximizes total surplus?What number of shops maximizes total surplus?

Total surplus is therefore N.V - Total CostTotal surplus is therefore N.V - Total Cost

Total surplus is then total willingness to pay minus total costsTotal surplus is then total willingness to pay minus total costs

Total surplus is consumer surplus plus profit

Consumer surplus is total willingness to pay minus total revenue

Profit is total revenue minus total cost

Total willingness to pay by consumers is N.V

So what is Total Cost?So what is Total Cost?

ECO 171 25

Social optimum 2

Price Price

z = 0 z = 1

V V

Assume thatthere

are n shops

Assume thatthere

are n shops

Consider shopi

Consider shopi

1/2n 1/2n

Shop i

t/2nt/2nTotal cost istotal transport

cost plus set-upcosts

Total cost istotal transport

cost plus set-upcosts

Transport cost foreach shop is the areaof these two triangles

multiplied byconsumer density

Transport cost foreach shop is the areaof these two triangles

multiplied byconsumer density

This area is t/4n2 This area is t/4n2

ECO 171 26

Social optimum 3

Total cost with n shops is, therefore: C(N,n) = n(t/4n2)N + n.F

= tN/4n + n.F

Total cost with n + 1 shops is: C(N,n+1) = tN/4(n+1)+ (n+1).F

Adding another shop is socially efficient if C(N,n + 1) < C(N,n)

This requires that tN/4n - tN/4(n+1) > F

which implies that n(n + 1) < tN/4F

The monopolist operates too many shops and, more generally, provides too much product variety

The monopolist operates too many shops and, more generally, provides too much product variety

If t = $1, F = $50,000,N = 5 million then this

condition tells usthat n(n+1) < 25

If t = $1, F = $50,000,N = 5 million then this

condition tells usthat n(n+1) < 25

There should be five shops: with n = 4 adding another shop is efficient

There should be five shops: with n = 4 adding another shop is efficient

ECO 171 27

Optimal variety and monopoly In previous example get excess variety Social value of increase in varieties < increase in

monopoly profits What is the key ingredient? Variety makes markets more homogeneous Monopoly is able to extract higher surplus Is this always true?

ECO 171 28

Example 1: Excess variety

3 consumers: {b,g,s} 3 possible goods: bitter (B),

generic(G), sweet(S) Preferences: Mc=0 Offer generic only:

p=2, =6 Offer (B,S)

pB = pG = 3

B G S

b 3 2 1

g 3 3 3

s 1 2 3• Profits increase by 3

• Total surplus increases by 2

• CS decreases by 1 (for g)

• Suppose additional investment to offer both brands: 2<c<3

- It is socially efficient to offer g only

- But monopolist will offer B,S

- Excessive product variety

• Variety partitions consumers into more homogeneous groups.

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Example 2: Too little variety

4 consumers: {b,gb,gs,s} 3 possible goods: bitter (B),

generic(G), sweet(S) Preferences: Mc=0 Offer generic only:

p=2, =8 Offer (B,S)

pB = pG = 2

B G S

b 3 2 1

gb 2 2 1

gs 1 2 2

s 1 2 3• Profits do not increase

• CS increases by 2

• Suppose additional investment to offer both brands: 0<c<2

- It is socially efficient to offer both brands

- But monopolist will offer only G

- Too little product variety

• Variety reduces homogeneity in groups

ECO 171 30

Product variety and price discrimination Suppose that the monopolist delivers the product.

then it is possible to price discriminate What pricing policy to adopt?

charge every consumer his reservation price V the firm pays the transport costs this is uniform delivered pricing it is discriminatory because price does not reflect costs

Should every consumer be supplied? suppose that there are n shops evenly spaced on Main Street cost to the most distant consumer is c + t/2n supply this consumer so long as V (revenue) > c + t/2n This is a weaker condition than without price discrimination. Price discrimination allows more consumers to be served.

ECO 171 31

Product variety and price discrimination 2 How many shops should the monopolist operate now?

Suppose that the monopolist has n shops and is supplying the entire market.

Total revenue minus production costs is N.V – N.c

Total transport costs plus set-up costs is C(N, n)=tN/4n + n.F

So profit is (N,n) = N.V – N.c – C(N,n)

But then maximizing profit means minimizing C(N, n)

The discriminating monopolist operates the socially optimal number of shops.