Product Variety under Monopoly

1
Product Variety under Monopoly
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

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?

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

5
An example of product variety
McDonalds
Burger King
Wendys
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 Main Street

7
The spatial model
Suppose that the monopolist sets a price
of p1
Price
Price
p1 t.x
p1 t.x
V
V
All consumers within distance x1 to the left and
right of the shop will by the product
t
t
What determines x1?
p1
z 0
z 1
x1
x1
1/2
Shop 1
p1 t.x1 V, so x1 (V p1)/t
8
The spatial model 2
Price
Price
p1 t.x
p1 t.x
Suppose the firm reduces the price to p2?
V
V
Then all consumers within distance x2 of the shop
will buy from the firm
p1
p2
z 0
z 1
x1
x1
x2
x2
1/2
Shop 1
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 p(N, 1) N(V t/2 c) F.

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?

11
Location with two shops
Delivered price to consumers at the market center
equals their reservation price
Suppose that the entire market is to be served
Price
Price
Price can be increased without losing market
If there are two shops they will be
located symmetrically a distance d from
the end-points of the market
p(d)
The maximum price the firm can charge is
determined by the consumers at the center of the
market
p(d)
What happens if move to the right?
Now raise the price at each shop
Start with a low price at each shop
d
1 - d
1/2
z 0
z 1
Shop 1
Shop 2
Suppose that d lt 1/4
The shops should be moved inwards
12
Location with two shops 2
Delivered price to consumers at the end-points
equals their reservation price
The maximum price the firm can charge is now
determined by the consumers at the
end-points of the market
Price
Price
p(d)
p(d)
Now can gain shifting left and raising price
Now raise the price at each shop
Start with a low price at each shop
d
1 - d
1/2
z 0
z 1
Shop 1
Shop 2
Now suppose that d gt 1/4
The shops should be moved outwards
13
Location with two shops 3
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
Price
Price
V - t/4
V - t/4
Profit at each shop is given by the shaded area
c
c
1/4
3/4
1/2
z 0
z 1
Shop 2
Shop 1
Profit is now p(N, 2) N(V - t/4 - c) 2F
14
Gain from introducing extra shop
Price
Price
Profits with two firms
V - t/4
P2 V - t/4
Profits with one firm
P1 V - t/2
c
c
1/4
3/4
1/2
z 0
z 1
Shop 2
Shop 1

• Total surplus increases as transportation costs
  are now lower
• Producer surplus increases but consumer surplus
  decreases
• So producer surplus increases more than total
  surplus

15
Three shops
By the same argument they should be located at
1/6, 1/2 and 5/6
What if there are three shops?
Price
Price
Price at each shop is now V - t/6
V - t/6
V - t/6
z 0
z 1
1/2
1/6
5/6
Shop 1
Shop 2
Shop 3
Profit is now p(N, 3) N(V - t/6 - c) 3F
16
Optimal number of shops

• A consistent pattern is emerging.

• Assume that there are n shops.

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

How many shops should there be?

• 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

• It follows that p(N, n) V - t/2n

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

17
Optimal number of shops 2
Profit from n shops is p(N, n) (V - t/2n - c)N
- n.F
and the profit from having n 1 shops
is p(N, n1) (V - t/2(n 1)-c)N - (n 1)F
Adding the (n 1)th shop is profitable if
p(N,n1) - p(N,n) gt 0
This requires tN/2n - tN/2(n 1) gt F
which requires that n(n 1) lt tN/2F.
18
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) lt 50.
This is true for n lt 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.
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

20
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) lt p
• This implies that V lt c t/n.

21
Partial market supply

• If c t/n gt V supply only part of the market and
  set price p (V c)/2
• If c t/n lt 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

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?

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 decreases.

24
Social optimum
Are there too many shops or too few?
What number of shops maximizes total surplus?
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 surplus is then total willingness to pay
minus total costs
Total willingness to pay by consumers is N.V
Total surplus is therefore N.V - Total Cost
So what is Total Cost?
25
Social optimum 2
Assume that there are n shops
Price
Price
Transport cost for each shop is the area of these
two triangles multiplied by consumer density
Consider shop i
Total cost is total transport cost plus
set-up costs
t/2n
t/2n
1/2n
1/2n
z 0
z 1
Shop i
This area is t/4n2
26
Social optimum 3
Total cost with n shops is, therefore C(N,n)
n(t/4n2)N n.F
tN/4n n.F
If t 1, F 50,000, N 5 million then
this condition tells us that n(n1) lt 25
Total cost with n 1 shops is C(N,n1)
tN/4(n1) (n1).F
There should be five shops with n 4 adding
another shop is efficient
Adding another shop is socially efficient if
C(N,n 1) lt C(N,n)
This requires that tN/4n - tN/4(n1) gt F
which implies that n(n 1) lt tN/4F
The monopolist operates too many shops and, more
generally, provides too much product variety
27
Optimal variety and monopoly

• In previous example get excess variety
• Social value of increase in varieties lt 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?

28
Example 1 Excess variety
B G S
b 3 2 1
g 3 3 3
s 1 2 3

• 3 consumers b,g,s
• 3 possible goods bitter (B), generic(G),
  sweet(S)
• Preferences
• Mc0
• Offer generic only
• p2, p6
• Offer (B,G)
• pB pG 3
• p 9

• Profits increase by 3
• Total surplus increases by 2
• CS decreases by 1 (for g)
• Suppose additional investment to offer both
  brands 2ltclt3
• - It is socially efficient to offer g only
• - But monopolist will offer B,S
• - Excessive product variety
• Variety partitions consumers into more
  homogeneous groups.

29
Example 2 Too little variety
B G S
b 3 2 1
gb 2 2 1
gs 1 2 2
s 1 2 3

• 4 consumers b,gb,gs,s
• 3 possible goods bitter (B), generic(G),
  sweet(S)
• Preferences
• Mc0
• Offer generic only
• p2, p8
• Offer (B,G)
• pB pG 2
• p 8

• Profits do not increase
• CS increases by 2
• Suppose additional investment to offer both
  brands 0ltclt2
• - It is socially efficient to offer both
  brands
• - But monopolist will offer only G
• - Too little product variety
• Variety reduces homogeneity in groups

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) gt c
  t/2n
• This is a weaker condition than without price
  discrimination.
• Price discrimination allows more consumers to be
  served.

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 p(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.