Directed reading in Industrial Organization

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Chapter 7: Product Variety and Quality under Monopoly

Product Variety and Quality under Monopoly

Chapter 7: Product Variety and Quality under Monopoly

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
vertically
consumers agree on quality
differ on willingness to pay for quality
how is quality of goods being offered determined?

Chapter 7: Product Variety and Quality under Monopoly

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?

Chapter 7: Product Variety and Quality under Monopoly

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

Chapter 7: Product Variety and Quality under Monopoly

An geographic example of product variety

McDonald’s

Burger King

Wendy’s

Chapter 7: Product Variety and Quality under Monopoly

A Spatial approach to product variety 2

Assume N consumers living equally spaced along Main 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 Main Street

Chapter 7: Product Variety and Quality under Monopoly

The spatial model

z = 0

z = 1

Shop 1

Price

All consumers within

distance x1 to the left

and right of the shop

will by the product

1/2

p1 + tx

p1 + t.x

p1 + tx1 = V, so x1 = (V – p1)/t

What determines

x1?

Suppose that the monopolist

sets a price of p1

Chapter 7: Product Variety and Quality under Monopoly

The spatial model 2

z = 0

z = 1

Shop 1

Price

1/2

p1 + t.x

Suppose the firm

reduces the price

to p2?

Then all consumers

within distance x2

of the shop will buy

from the firm

Chapter 7: Product Variety and Quality under Monopoly

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 p(N, 1) = N(V – t/2 – c) – F.

Chapter 7: Product Variety and Quality under Monopoly

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*?

Chapter 7: Product Variety and Quality under Monopoly

Location with two shops

Suppose that the entire market is to be served

Price

z = 0

z = 1

If there are two shops

they will be located

symmetrically a

distance d from the

end-points of the

market

Suppose that

d < 1/4

1 - d

Shop 1

Shop 2

1/2

The maximum price

the firm can charge

is determined by the

consumers at the

center of the market

Delivered price to

consumers at the

market center equals

their reservation price

p(d)

Start with a low price

at each shop

Now raise the price

at each shop

What determines

p(d)?

The shops should be

moved inwards

Chapter 7: Product Variety and Quality under Monopoly

Location with two shops 2

Price

z = 0

z = 1

Now suppose that

d > 1/4

1 - d

Shop 1

Shop 2

1/2

p(d)

Start with a low price

at each shop

Now raise the price

at each shop

The maximum price

the firm can charge

is now determined

by the consumers

at the end-points

of the market

Delivered price to

consumers at the

end-points equals

their reservation price

Now what

determines p(d)?

The shops should be

moved outwards

Chapter 7: Product Variety and Quality under Monopoly

Location with two shops 3

Price

z = 0

z = 1

1/4

3/4

Shop 1

Shop 2

1/2

It follows that

shop 1 should

be located at

1/4 and shop 2

at 3/4

Price at each

shop is then

p* = V - t/4

V - t/4

Profit at each shop

is given by the

shaded area

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

Chapter 7: Product Variety and Quality under Monopoly

Three shops

Price

z = 0

z = 1

1/2

What if there

are three shops?

By the same argument

they should be located

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

1/6

5/6

Shop 1

Shop 2

Shop 3

Price at each

shop is now

V - t/6

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

Chapter 7: Product Variety and Quality under Monopoly

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 p(N, n) = N(V - t/2n - c) – nF

How many

shops should

there be?

Chapter 7: Product Variety and Quality under Monopoly

Optimal number of shops 2

Profit from n shops is p(N, n) = (V - t/2n - c)N - nF

and the profit from having n + 1 shops is:

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

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

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

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

Chapter 7: Product Variety and Quality under Monopoly

An example

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

Then tN/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.

Chapter 7: Product Variety and Quality under Monopoly

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

Chapter 7: Product Variety and Quality under Monopoly

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 p = 2N(p – c)(V – p)/t – F
differentiate with respect to p and set to zero:
dp/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.

Chapter 7: Product Variety and Quality under Monopoly

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

Chapter 7: Product Variety and Quality under Monopoly

Social optimum

Are there too

many shops or

too few?

What number of shops maximizes total surplus?

Total surplus is therefore NV - Total Cost

Total 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?

Chapter 7: Product Variety and Quality under Monopoly

Social optimum 2

Price

z = 0

z = 1

Assume that

there

are n shops

Consider shop

1/2n

Shop i

t/2n

Total cost is

total transport

cost plus set-up

costs

Transport cost for

each shop is the area

of these two triangles

multiplied by

consumer density

This area is t/4n2

Chapter 7: Product Variety and Quality under Monopoly

Social optimum 3

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

= tN/4n + nF

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

If t = $1, F = $50,000,

N = 5 million then this

condition tells us

that n(n+1) < 25

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

Chapter 7: Product Variety and Quality under Monopoly

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

Chapter 7: Product Variety and Quality under Monopoly

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

Chapter 7: Product Variety and Quality under Monopoly

Product variety and price discrimination 2

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.

Chapter 7: Product Variety and Quality under Monopoly

Product variety & price discrimination 3

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 NV – Nc

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

So profit is p(N,n) = NV – Nc – C(N,n)

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

The discriminating monopolist operates the socially optimal number of shops.

Chapter 7: Product Variety and Quality under Monopoly

Monopoly and product quality

Firms can, and do, produce goods of different qualities
Quality then is an important strategic variable
The choice of product quality determined by its ability to generate profit; attitude of consumers to q uality
Consider a monopolist producing a single good
what quality should it have?
determined by consumer attitudes to quality
prefer high to low quality
willing to pay more for high quality
but this requires that the consumer recognizes quality
also some are willing to pay more than others for quality

Chapter 7: Product Variety and Quality under Monopoly

Demand and quality

We might think of individual demand as being of the form
Qi = 1 if Pi < Ri(Z) and = 0 otherwise for each consumer i
Each consumer buys exactly one unit so long as price is less than her reservation price
the reservation price is affected by product quality Z
Assume that consumers vary in their reservation prices
Then aggregate demand is of the form P = P(Q, Z)
An increase in product quality increases demand

Chapter 7: Product Variety and Quality under Monopoly

Demand and quality 2

Begin with a particular demand curve

for a good of quality Z1

Price

Quantity

P(Q, Z1)

If the price is P1 and the product quality

is Z1 then all consumers with reservation

prices greater than P1 will buy the good

R1(Z1)

These are the

inframarginal

consumers

This is the

marginal

consumer

Suppose that an increase in

quality increases the

willingness to pay of

inframarginal consumers more

than that of the marginal

consumer

Then an increase in product

quality from Z1 to Z2 rotates

the demand curve around

the quantity axis as follows

R1(Z2)

Quantity Q1 can now be

sold for the higher

price P2

P(Q, Z2)

Chapter 7: Product Variety and Quality under Monopoly

Demand and quality 3

Price

Quantity

P(Q, Z1)

R1(Z1)

Suppose instead that an

increase in

quality increases the

willingness to pay of marginal

consumers more

than that of the inframarginal

consumers

Then an increase in product

quality from Z1 to Z2 rotates

the demand curve around

the price axis as follows

P(Q, Z2)

Once again quantity Q1

can now be sold for a

higher price P2

Chapter 7: Product Variety and Quality under Monopoly

Demand and quality 4

The monopolist must choose both
price (or quantity)
quality
Two profit-maximizing rules
marginal revenue equals marginal cost on the last unit sold for a given quality
marginal revenue from increased quality equals marginal cost of increased quality for a given quantity
This can be illustrated with a simple example:

P = Z( - Q) where Z is an index of quality

Chapter 7: Product Variety and Quality under Monopoly

Demand and quality 5

P = Z(q - Q)

Assume that marginal cost of output is zero: MC(Q) = 0

Cost of quality is C(Z) = aZ2

This means that quality is

costly and becomes

increasingly costly

Marginal cost of quality = dC(Z)/d(Z)

= 2aZ

The firm’s profit is:

p(Q, Z) =PQ - C(Z)

= Z(q - Q)Q - aZ2

Chapter 7: Product Variety and Quality under Monopoly

Demand and quality 6

Again, profit is:

p(Q, Z) =PQ - C(Z)

= Z(q - Q)Q - aZ2

The firm chooses Q and Z to maximize profit.

Take the choice of quantity first: this is easiest.

Marginal revenue = MR =

Zq - 2ZQ

MR = MC 

Zq - 2ZQ = 0 

Q* = q/2

 P* = Zq/2

Chapter 7: Product Variety and Quality under Monopoly

Demand and quality 7

Total revenue = P*Q* =

(Zq/2)x(q/2) =

Zq2/4

So marginal revenue from increased quality is

MR(Z) = q2/4

Marginal cost of quality is

MC(Z) = 2aZ

Equating MR(Z) = MC(Z) then gives

Z* = q2/8a

Does the monopolist produce too high or too low quality?

Chapter 7: Product Variety and Quality under Monopoly

Demand and quality: multiple products

What if the firm chooses to offer more than one product?
what qualities should be offered?
how should they be priced?
Determined by costs and consumer demand

Chapter 7: Product Variety and Quality under Monopoly

Demand and quality: multiple products 2

An example:
two types of consumer
each buys exactly one unit provided that consumer surplus is nonnegative
if there is a choice, buy the product offering the larger consumer surplus
types of consumer distinguished by willingness to pay for quality
This is vertical product differentiation

Chapter 7: Product Variety and Quality under Monopoly

Vertical differentiation

Indirect utility to a consumer of type i from consuming a product of quality z at price p is Vi = qi(z – zi) – p
where qi measures willingness to pay for quality;
zi is the lower bound on quality below which consumer type i will not buy
assume q1 > q2: type 1 consumers value quality more than type 2
assume z1 > z2 = 0: type 1 consumers only buy if quality is greater than z1:
never fly in coach
never shop in Wal-Mart
only eat in “good” restaurants
type 2 consumers will buy any quality so long as consumer surplus is nonnegative

Chapter 7: Product Variety and Quality under Monopoly

Vertical differentiation 2

Firm cannot distinguish consumer types
Must implement a strategy that causes consumers to self-select
persuade type 1 consumers to buy a high quality product z1 at a high price
and type 2 consumers to buy a low quality product z2 at a lower price, which equals their maximum willingness to pay

Firm can produce any product in the range

MC = 0 for either quality type

z, z

Chapter 7: Product Variety and Quality under Monopoly

Vertical differentiation 3

For type 2 consumers charge maximum willingness to pay for the low quality product: p2 = q2z2

Suppose that the firm offers two products with qualities z1 > z2

Now consider type 1 consumers: firm faces an incentive compatibility constraint

q1(z1 – z1) – p1 > q1(z2 – z1) – p2

Type 1 consumers prefer the high quality to the low quality good

q1(z1 – z1) – p1 > 0

Type 1 consumers have nonnegative consumer surplus from the high quality good

These imply that p1 < q1z1 – (q1 - q2)z2

There is an upper limit on the price that can be charged for the high quality good

Chapter 7: Product Variety and Quality under Monopoly

Vertical differentiation 4

Take the equation p1 = q1z1 – (q1 – q2)z2
this is increasing in quality valuations
increasing in the difference between z1 and z2
quality can be prices highly when it is valued highly
firm has an incentive to differentiate the two products’ qualities to soften competition between them
monopolist is competing with itself
What about quality choice?
prices p1 = q1z1 – (q1 – q2)z2; p2 = q2z2
check the incentive compatibility constraints
suppose that there are N1 type 1 and N2 type 2 consumers

Chapter 7: Product Variety and Quality under Monopoly

Vertical differentiation 5

Profit is

P = N1p1 + N2p2 =

N1q1z1 – (N1q1 – (N1 + N2)q2)z2

This is increasing in z1 so set z1 as high as possible: z1 =

For z2 the decision is more complex

(N1q1 – (N1 + N2)q2) may be positive or negative

Chapter 7: Product Variety and Quality under Monopoly

Vertical differentiation 6

Case 1: Suppose that (N1q1 – (N1 + N2)q2) is positive

Then z2 should be set “low” but this is subject to a constraint

Recall that p1 = q1z1 – (q1 - q2)z2

So reducing z2 increases p1

But we also require that q1(z1 – z1) – p1 > 0

Putting these together gives:

The equilibrium prices are then:

Chapter 7: Product Variety and Quality under Monopoly

Vertical differentiation 7

Offer type 1 consumers the highest possible quality and charge their full willingness to pay
Offer type 2 consumers as low a quality as is consistent with incentive compatibility constraints
Charge type 2 consumers their maximum willingness to pay for this quality
maximum differentiation subject to incentive compatibility constraints

Chapter 7: Product Variety and Quality under Monopoly

Vertical differentiation 8

Case 1: Now suppose that (N1q1 – (N1 + N2)q2) is negative

Then z2 should be set as high as possible

The firm should supply only one product, of the highest possible quality

What does this require?

From the inequality offer only one product if:

Offer only one product:

if there are not “many” type 1 consumers

if the difference in willingness to pay for quality is “small”

Should the firm price to sell to both types in this case? YES!

Chapter 7: Product Variety and Quality under Monopoly

Empirical Application: Price Discrimination and Imperfect Competition

Although we have presented price discrimination and product design (versioning) issues in the context of a monopoly, these same tactics also play a role in more competitive settings of imperfect competition

Imagine a two-store setting again

Assume N customers distributed evenly between the two stores, each with maximum willingness to pay of V .

No transport cost—Half of the consumers always buys at nearest store. Other half always buys at cheapest store.

Chapter 7: Product Variety and Quality under Monopoly

Price Discrimination and Imperfect Competition 2

If both stores operated by a monopolist, set price = V.

Cannot set it higher of there will be no customers.

If Store 1 cuts its price  below V.

It loses N/2 from all current customers

Setting it lower though gains nothing.

What if stores operated by separate firms?

Imagine P1 = P2 = V. Store 1 serves N/4 price-sensitive customers and N/4 price-insensitive ones. The same is true for Store 2.

It gains N(V - )/4 by stealing all price-sensitive customers from Store 2

Chapter 7: Product Variety and Quality under Monopoly

Price Discrimination and Imperfect Competition 3

MORAL 1: Both firms have a real incentive to cut price.

This ultimately proves self-defeating

Cutting their price does not increase their likelihood
of shopping at a particular place. It just loses revenue.

MORAL 2: Unlike the monopolist who sets the same price to everyone, these firms have an incentive to discriminate and so continue to charge a high price to loyal consumers while pricing low to others.

In equilibrium, both still serve N/2 customers but now do so at a price closer to cost.

This is especially frustrating in light of the “brand-loyal” or price-insensitive customers

Chapter 7: Product Variety and Quality under Monopoly

Price Discrimination and Imperfect Competition 4

The intuition then is that price discrimination may be associated with imperfect competition and become more prominent as markets get more competitive (but still less than perfectly competitive).

This idea is tested by Stavins (2001) with airline prices.

Restrictions such as a required Saturday night stay-over or an advanced purchase serve as screening mechanism for price-sensitive customers. Hence, restrictions lead to lower ticket price.

Stavins (2001) idea is that price reduction associated with flight restrictions will be small in markets that are not very competitive.

Chapter 7: Product Variety and Quality under Monopoly

Price Discrimination and Imperfect Competition 6

Stavins (2001) looks at nearly 6,000 tickets covering 12 different city-pair routes in September, 1995.

She finds strong support for the dual hypothesis that:

In highly competitive (low HHI) markets, a Saturday night restriction leads to a $253 price reduction but only a $165 reduction in less competitive ones.

a) passengers flying on a ticket with restrictions pay less;

b) price reduction shrinks as concentration rises

In highly competitive (low HHI) markets, an Advance Purchase restriction leads to a $111 price reduction but only a $41 reduction in less competitive ones.

Chapter 7: Product Variety and Quality under Monopoly

Price Discrimination and Imperfect Competition 5

Variable Coefficient t-Statistic Coefficient t-Statistic

Saturday

Night Stay – 0.408 – 4.05 ----- -----

Required

Saturday

Night Stay 0.792 3.39 ----- -----

RequiredxHHI

Advance Purchase ----- ----- – 0.023 –5.53 Required

Advance Purchase ----- ----- 0.098 8.38
RequiredxHHI

NOTE: HHI is the Herfindahl Index. A Saturday Night Stay or an Advance Purchase lowers the price significantly. But the HHI terms show that this effect weakens as market concentration increases.

Chapter 7: Product Variety and Quality under Monopoly

Demand and quality A1

Price

Quantity

Z1q

P(Q,Z1)

How does increased quality

affect demand?

Z2q

P(Q, Z2)

MR(Z1)

MR(Z2)

q/2

P1 = Z1q/2

P2 = Z2q/2

When quality is Z1

price is

Z1q/2

When quality is Z2

price is

Z2q/2

Chapter 7: Product Variety and Quality under Monopoly

Demand and quality A2

Price

Quantity

Z1q

Z2q

q/2

P1 = Z1q/2

P2 = Z2q/2

An increase in quality from

Z1 to Z2 increases

revenue by this area

So an increase is quality from

Z1 to Z2 increases surplus

by this area minus the

increase in quality costs

The increase in total

surplus is greater than

the increase in profit.

The monopolist produces

too little quality

Social surplus at quality Z1

is this area minus quality

costs

Social surplus at quality Z2

is this area minus quality

costs

Chapter 7: Product Variety and Quality under Monopoly

Demand and quality

Derivation of aggregate demand

Order consumers by their reservation prices

Aggregate individual demand horizontally

Price

Quantity

Chapter 7: Product Variety and Quality under Monopoly

Location choice 1

d < 1/4

We know that p(d) satisfies the following constraint:

p(d) + t(1/2 - d) = V

This gives:

p(d) = V - t/2 + td

 p(d) = V - t/2 + td

Aggregate profit is then: p(d) = (p(d) - c)N

= (V - t/2 + td - c)N

This is increasing in d so if d < 1/4 then d should be increased.

Chapter 7: Product Variety and Quality under Monopoly

Location choice 2

d > 1/4

We now know that p(d) satisfies the following constraint:

p(d) + td = V

This gives:

p(d) = V - td

Aggregate profit is then: p(d) = (p(d) - c)N

= (V - td - c)N

This is decreasing in d so if d > 1/4 then d should be decreased.

Chapter 7: Product Variety and Quality under Monopoly

Directed reading in Industrial Organization

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