Transcript p(y)

25
© 2010 W. W. Norton & Company, Inc.
Monopoly Behavior
How Should a Monopoly Price?
 So
far a monopoly has been thought
of as a firm which has to sell its
product at the same price to every
customer. This is uniform pricing.
 Can price-discrimination earn a
monopoly higher profits?
© 2010 W. W. Norton & Company, Inc.
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Types of Price Discrimination
 1st-degree:
Each output unit is sold
at a different price. Prices may differ
across buyers.
 2nd-degree: The price paid by a
buyer can vary with the quantity
demanded by the buyer. But all
customers face the same price
schedule. E.g., bulk-buying
discounts.
© 2010 W. W. Norton & Company, Inc.
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Types of Price Discrimination
 3rd-degree:
Price paid by buyers in a
given group is the same for all units
purchased. But price may differ
across buyer groups.
E.g., senior citizen and student
discounts vs. no discounts for
middle-aged persons.
© 2010 W. W. Norton & Company, Inc.
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First-degree Price Discrimination
 Each
output unit is sold at a different
price. Price may differ across buyers.
 It requires that the monopolist can
discover the buyer with the highest
valuation of its product, the buyer with
the next highest valuation, and so on.
© 2010 W. W. Norton & Company, Inc.
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First-degree Price Discrimination
$/output unit
Sell the y th unit for $p( y ).
p( y )
MC(y)
p(y)
y
© 2010 W. W. Norton & Company, Inc.
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First-degree Price Discrimination
$/output unit
p( y )
Sell the y th unit for $p( y ). Later on
sell the y th unit for $ p( y ).
p( y )
MC(y)
p(y)
y
y
© 2010 W. W. Norton & Company, Inc.
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First-degree Price Discrimination
$/output unit
p( y )
p( y )
Sell the y th unit for $p( y ). Later on
sell the y th unit for $ p( y ). Finally
sell the y th unit for marginal
cost, $ p( y ).
MC(y)
p( y )
p(y)
y
y
© 2010 W. W. Norton & Company, Inc.
y
y
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First-degree Price Discrimination
The gains to the monopolist
on these trades are:
p( y )  MC( y ), p( y )  MC( y )
and zero.
$/output unit
p( y )
p( y )
MC(y)
p( y )
p(y)
y
y
y
y
The consumers’ gains are zero.
© 2010 W. W. Norton & Company, Inc.
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First-degree Price Discrimination
$/output unit
So the sum of the gains to
the monopolist on all
trades is the maximum
possible total gains-to-trade.
PS
MC(y)
p(y)
y
© 2010 W. W. Norton & Company, Inc.
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First-degree Price Discrimination
$/output unit
The monopolist gets
the maximum possible
gains from trade.
PS
MC(y)
p(y)
y
y
First-degree price discrimination
is Pareto-efficient.
© 2010 W. W. Norton & Company, Inc.
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First-degree Price Discrimination
 First-degree
price discrimination
gives a monopolist all of the possible
gains-to-trade, leaves the buyers
with zero surplus, and supplies the
efficient amount of output.
© 2010 W. W. Norton & Company, Inc.
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Third-degree Price
Discrimination
 Price
paid by buyers in a given group
is the same for all units purchased.
But price may differ across buyer
groups.
© 2010 W. W. Norton & Company, Inc.
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Third-degree Price
Discrimination
A
monopolist manipulates market
price by altering the quantity of
product supplied to that market.
 So the question “What discriminatory
prices will the monopolist set, one for
each group?” is really the question
“How many units of product will the
monopolist supply to each group?”
© 2010 W. W. Norton & Company, Inc.
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Third-degree Price
Discrimination
 Two
markets, 1 and 2.
 y1 is the quantity supplied to market 1.
Market 1’s inverse demand function is
p1(y1).
 y2 is the quantity supplied to market 2.
Market 2’s inverse demand function is
p2(y2).
© 2010 W. W. Norton & Company, Inc.
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Third-degree Price
Discrimination
 For
given supply levels y1 and y2 the
firm’s profit is
( y1 , y2 )  p1 ( y1 )y1  p2 ( y2 )y2  c( y1  y2 ).
 What
values of y1 and y2 maximize
profit?
© 2010 W. W. Norton & Company, Inc.
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( y1 , y2
Third-degree Price
Discrimination
)  p ( y ) y  p ( y ) y  c( y
1
1
1
2
2
2
1  y2 ).
The profit-maximization conditions are


 c( y1  y2 )  ( y1  y2 )


p1 ( y1 )y1  
 y1  y1
 ( y1  y2 )
 y1
0
© 2010 W. W. Norton & Company, Inc.
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( y1 , y2
Third-degree Price
Discrimination
)  p ( y ) y  p ( y ) y  c( y
1
1
1
2
2
2
1  y2 ).
The profit-maximization conditions are


 c( y1  y2 )  ( y1  y2 )


p1 ( y1 )y1  
 y1  y1
 ( y1  y2 )
 y1
0


 c( y1  y2 )  ( y1  y2 )


p 2 ( y2 )y2  
 y2  y2
 ( y1  y2 )
 y2
0
© 2010 W. W. Norton & Company, Inc.
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Third-degree Price
Discrimination
y )
 (y  y )
 ( y1
2 1
and
 y1
1
 y2
2 1
so
the profit-maximization conditions are

 c( y1  y2 )
p1 ( y1 )y1  
 y1
 ( y1  y2 )

 c( y1  y2 )
and
.
p 2 ( y2 )y2  
 y2
 ( y1  y2 )
© 2010 W. W. Norton & Company, Inc.
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Third-degree Price
Discrimination

 c( y1  y2 )

p1 ( y1 )y1  
p 2 ( y2 ) y2  
 y1
 y2
 ( y1  y2 )
© 2010 W. W. Norton & Company, Inc.
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Third-degree Price
Discrimination

 c( y1  y2 )

p1 ( y1 )y1  
p 2 ( y2 ) y2  
 y1
 y2
 ( y1  y2 )



MR1(y1) = MR2(y2) says that the allocation
y1, y2 maximizes the revenue from selling
y1 + y2 output units.
E.g., if MR1(y1) > MR2(y2) then an output unit
should be moved from market 2 to market 1
to increase total revenue.
© 2010 W. W. Norton & Company, Inc.
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Third-degree Price
Discrimination

 c( y1  y2 )

p1 ( y1 )y1  
p 2 ( y2 ) y2  
 y1
 y2
 ( y1  y2 )



The marginal revenue common to both
markets equals the marginal production
cost if profit is to be maximized.
© 2010 W. W. Norton & Company, Inc.
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Third-degree Price
Discrimination
Market 1
Market 2
p1(y1)
p1(y1*)
p2(y2)
p2(y2*)
MC
y1
y1*
MR1(y1)
MC
y2*
y2
MR2(y2)
MR1(y1*) = MR2(y2*) = MC
© 2010 W. W. Norton & Company, Inc.
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Third-degree Price
Discrimination
Market 1
Market 2
p1(y1)
p1(y1*)
p2(y2)
p2(y2*)
MC
y1
y1*
MR1(y1)
MC
y2*
y2
MR2(y2)
MR1(y1*) = MR2(y2*) = MC and p1(y1*)  p2(y2*).
© 2010 W. W. Norton & Company, Inc.
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Third-degree Price
Discrimination
 In
which market will the monopolist
cause the higher price?
© 2010 W. W. Norton & Company, Inc.
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Third-degree Price
Discrimination
 In
which market will the monopolist
cause the higher price?
 Recall that

1
and
MR1 ( y1 )  p1 ( y1 ) 1  
1 


1
MR 2 ( y2 )  p2 ( y2 ) 1   .
2

© 2010 W. W. Norton & Company, Inc.
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Third-degree Price
Discrimination
 In
which market will the monopolist
cause the higher price?
 Recall that

1
and
 But,
MR1 ( y1 )  p1 ( y1 ) 1  
1 


1
MR 2 ( y2 )  p2 ( y2 ) 1   .
2

MR1 ( y*1 )  MR 2 ( y*2 )  MC( y*1  y*2 )
© 2010 W. W. Norton & Company, Inc.
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So
p1
Third-degree Price

1
1
* Discrimination
* 
(y ) 1 
 p (y ) 1 
.
1 
© 2010 W. W. Norton & Company, Inc.

 1 
2
2 

 2 
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So
p1
Third-degree Price

1
1
* Discrimination
* 
(y ) 1 
 p (y ) 1 
.
1 

 1 
2
2 

 2 
Therefore, p1 ( y*1 )  p2 ( y*2 ) if and only if
1
1
1
 1
1
2
© 2010 W. W. Norton & Company, Inc.
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So
p1
Third-degree Price

1
1
* Discrimination
* 
(y ) 1 
 p (y ) 1 
.
1 

 1 
2
2 

 2 
Therefore, p1 ( y*1 )  p2 ( y*2 ) if and only if
1
1
1
 1
1
2
© 2010 W. W. Norton & Company, Inc.

1   2 .
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So
p1
Third-degree Price

1
1
* Discrimination
* 
(y ) 1 
 p (y ) 1 
.
1 

 1 
2
2 

 2 
Therefore, p1 ( y*1 )  p2 ( y*2 ) if and only if
1
1
1
 1
1
2

1   2 .
The monopolist sets the higher price in
the market where demand is least
own-price elastic.
© 2010 W. W. Norton & Company, Inc.
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Two-Part Tariffs
A
two-part tariff is a lump-sum fee,
p1, plus a price p2 for each unit of
product purchased.
 Thus the cost of buying x units of
product is
p1 + p2x.
© 2010 W. W. Norton & Company, Inc.
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Two-Part Tariffs
 Should
a monopolist prefer a twopart tariff to uniform pricing, or to
any of the price-discrimination
schemes discussed so far?
 If so, how should the monopolist
design its two-part tariff?
© 2010 W. W. Norton & Company, Inc.
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Two-Part Tariffs
p1 + p2x
 Q: What is the largest that p1 can be?

© 2010 W. W. Norton & Company, Inc.
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Two-Part Tariffs
p1 + p2x
 Q: What is the largest that p1 can be?
 A: p1 is the “market entrance fee” so
the largest it can be is the surplus
the buyer gains from entering the
market.
 Set p1 = CS and now ask what
should be p2?

© 2010 W. W. Norton & Company, Inc.
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Two-Part Tariffs
$/output unit
Should the monopolist
set p2 above MC?
p(y)
p2  p( y)
MC(y)
y
© 2010 W. W. Norton & Company, Inc.
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Two-Part Tariffs
$/output unit
Should the monopolist
set p2 above MC?
p1 = CS.
p(y)
CS
p2  p( y)
MC(y)
y
© 2010 W. W. Norton & Company, Inc.
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Two-Part Tariffs
$/output unit
Should the monopolist
set p2 above MC?
p1 = CS.
PS is profit from sales.
p(y)
CS
p2  p( y)
PS
MC(y)
y
© 2010 W. W. Norton & Company, Inc.
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Two-Part Tariffs
$/output unit
Should the monopolist
set p2 above MC?
p1 = CS.
PS is profit from sales.
p(y)
CS
p2  p( y)
PS
MC(y)
Total profit
y
© 2010 W. W. Norton & Company, Inc.
y
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Two-Part Tariffs
$/output unit
p(y)
Should the monopolist
set p2 = MC?
MC(y)
p2  p( y)
y
© 2010 W. W. Norton & Company, Inc.
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Two-Part Tariffs
$/output unit
p(y)
p2  p( y)
Should the monopolist
set p2 = MC?
p1 = CS.
CS
MC(y)
y
© 2010 W. W. Norton & Company, Inc.
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Two-Part Tariffs
$/output unit
p(y)
CS
Should the monopolist
set p2 = MC?
p1 = CS.
PS is profit from sales.
MC(y)
p2  p( y) PS
y
© 2010 W. W. Norton & Company, Inc.
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Two-Part Tariffs
$/output unit
p(y)
CS
Should the monopolist
set p2 = MC?
p1 = CS.
PS is profit from sales.
MC(y)
p2  p( y) PS
Total profit
y
© 2010 W. W. Norton & Company, Inc.
y
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Two-Part Tariffs
$/output unit
p(y)
CS
Should the monopolist
set p2 = MC?
p1 = CS.
PS is profit from sales.
MC(y)
p2  p( y) PS
y
© 2010 W. W. Norton & Company, Inc.
y
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Two-Part Tariffs
$/output unit
p(y)
CS
Should the monopolist
set p2 = MC?
p1 = CS.
PS is profit from sales.
MC(y)
p2  p( y) PS
y
y
Additional profit from setting p2 = MC.
© 2010 W. W. Norton & Company, Inc.
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Two-Part Tariffs
 The
monopolist maximizes its profit
when using a two-part tariff by
setting its per unit price p2 at
marginal cost and setting its lumpsum fee p1 equal to Consumers’
Surplus.
© 2010 W. W. Norton & Company, Inc.
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Two-Part Tariffs
A
profit-maximizing two-part tariff
gives an efficient market outcome in
which the monopolist obtains as
profit the total of all gains-to-trade.
© 2010 W. W. Norton & Company, Inc.
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Differentiating Products
 In
many markets the commodities
traded are very close, but not perfect,
substitutes.
 E.g., the markets for T-shirts,
watches, cars, and cookies.
 Each individual supplier thus has
some slight “monopoly power.”
 What does an equilibrium look like
for such a market?
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Differentiating Products
entry  zero profits for each
seller.
 Free
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Differentiating Products
entry  zero profits for each
seller.
 Profit-maximization  MR = MC for
each seller.
 Free
© 2010 W. W. Norton & Company, Inc.
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Differentiating Products
entry  zero profits for each
seller.
 Profit-maximization  MR = MC for
each seller.
 Less than perfect substitution
between commodities  slight
downward slope for the demand
curve for each commodity.
 Free
© 2010 W. W. Norton & Company, Inc.
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Price
Differentiating Products
Slight downward slope
Demand
Quantity
Supplied
© 2010 W. W. Norton & Company, Inc.
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Price
Differentiating Products
Demand
Quantity
Supplied
Marginal
Revenue
© 2010 W. W. Norton & Company, Inc.
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Price
Differentiating Products
Marginal
Cost
Demand
Quantity
Supplied
Marginal
Revenue
© 2010 W. W. Norton & Company, Inc.
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Price
Differentiating Products
Profit-maximization
MR = MC
Marginal
Cost
p(y*)
Demand
y*
© 2010 W. W. Norton & Company, Inc.
Quantity
Supplied
Marginal
Revenue
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Price
Differentiating Products
Zero profit
Price = Av. Cost
Profit-maximization
MR = MC
Average
Cost
Demand
p(y*)
y*
© 2010 W. W. Norton & Company, Inc.
Marginal
Cost
Quantity
Supplied
Marginal
Revenue
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Differentiating Products
 Such
markets are monopolistically
competitive.
 Are these markets efficient?
 No, because for each commodity the
equilibrium price p(y*) > MC(y*).
© 2010 W. W. Norton & Company, Inc.
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Price
Differentiating Products
Zero profit
Price = Av. Cost
Profit-maximization
MR = MC
Marginal
Cost
Average
Cost
Demand
p(y*)
MC(y*)
y*
© 2010 W. W. Norton & Company, Inc.
Quantity
Supplied
Marginal
Revenue
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Price
Differentiating Products
Zero profit
Price = Av. Cost
Profit-maximization
MR = MC
Marginal
Cost
Average
Cost
Demand
p(y*)
MC(y*)
y*
© 2010 W. W. Norton & Company, Inc.
ye
Quantity
Supplied
Marginal
Revenue
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Differentiating Products
 Each
seller supplies less than the
efficient quantity of its product.
 Also, each seller supplies less than
the quantity that minimizes its
average cost and so, in this sense,
each supplier has “excess capacity.”
© 2010 W. W. Norton & Company, Inc.
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Price
Differentiating Products
Zero profit
Price = Av. Cost
Profit-maximization
MR = MC
Average
Cost
Demand
p(y*)
MC(y*)
Excess
capacity
y*
© 2010 W. W. Norton & Company, Inc.
Marginal
Cost
ye
Quantity
Supplied
Marginal
Revenue
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Differentiating Products by
Location
 Think
a region in which consumers
are uniformly located along a line.
 Each consumer prefers to travel a
shorter distance to a seller.
 There are n ≥ 1 sellers.
 Where would we expect these sellers
to choose their locations?
© 2010 W. W. Norton & Company, Inc.
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Differentiating Products by
Location
0
1
x
 If
n = 1 (monopoly) then the seller
maximizes its profit at x = ??
© 2010 W. W. Norton & Company, Inc.
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Differentiating Products by
Location
½
0
1
x
 If
n = 1 (monopoly) then the seller
maximizes its profit at x = ½ and
minimizes the consumers’ travel
cost.
© 2010 W. W. Norton & Company, Inc.
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Differentiating Products by
Location
½
0
1
x
 If
n = 2 (duopoly) then the equilibrium
locations of the sellers, A and B, are
xA = ?? and xB = ??
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A
Differentiating Products by
Location
½
0
B
1
x
 If
n = 2 (duopoly) then the equilibrium
locations of the sellers, A and B, are
xA = ?? and xB = ??
 How about xA = 0 and xB = 1; i.e. the
sellers separate themselves as much
as is possible?
© 2010 W. W. Norton & Company, Inc.
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A
Differentiating Products by
Location
½
0
B
1
x
 If
xA = 0 and xB = 1 then A sells to all
consumers in [0,½) and B sells to all
consumers in (½,1].
 Given B’s location at xB = 1, can A
increase its profit?
© 2010 W. W. Norton & Company, Inc.
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Differentiating Products by
Location
½
A
0
x’
B
1
x
 If
xA = 0 and xB = 1 then A sells to all
consumers in [0,½) and B sells to all
consumers in (½,1].
 Given B’s location at xB = 1, can A
increase its profit? What if A moves
to x’?
© 2010 W. W. Norton & Company, Inc.
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Differentiating Products by
Location
½
A
0
x’
x
B
x’/2
1
 If
xA = 0 and xB = 1 then A sells to all
consumers in [0,½) and B sells to all
consumers in (½,1].
 Given B’s location at xB = 1, can A
increase its profit? What if A moves
to x’? Then A sells to all customers
in [0,½+½ x’) and increases its profit.
© 2010 W. W. Norton & Company, Inc.
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Differentiating Products by
Location
½
A
0
x’
B
1
x
xA = x’, can B improve its profit
by moving from xB = 1?
 Given
© 2010 W. W. Norton & Company, Inc.
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Differentiating Products by
Location
½
A
0
x’
x
x’’
B
1
xA = x’, can B improve its profit
by moving from xB = 1? What if B
moves to xB = x’’?
 Given
© 2010 W. W. Norton & Company, Inc.
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Differentiating Products by
Location
½
A
0
x’
x
(1-x’’)/2
x’’
B
1
xA = x’, can B improve its profit
by moving from xB = 1? What if B
moves to xB = x’’? Then B sells to all
customers in ((x’+x’’)/2,1] and
increases its profit.
 So what is the NE?
 Given
© 2010 W. W. Norton & Company, Inc.
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Differentiating Products by
Location
½
0
x
A&B
1
xA = x’, can B improve its profit
by moving from xB = 1? What if B
moves to xB = x’’? Then B sells to all
customers in ((x’+x’’)/2,1] and
increases its profit.
 So what is the NE? xA = xB = ½.
 Given
© 2010 W. W. Norton & Company, Inc.
73
Differentiating Products by
Location
½
0
x
A&B
1
 The
only NE is xA = xB = ½.
 Is the NE efficient?
© 2010 W. W. Norton & Company, Inc.
74
Differentiating Products by
Location
½
0
x
A&B
1
 The
only NE is xA = xB = ½.
 Is the NE efficient? No.
 What is the efficient location of A and
B?
© 2010 W. W. Norton & Company, Inc.
75
Differentiating Products by
Location
½
¼
¾
0
A
x
B
1
 The
only NE is xA = xB = ½.
 Is the NE efficient? No.
 What is the efficient location of A and
B? xA = ¼ and xB = ¾ since this
minimizes the consumers’ travel
costs.
© 2010 W. W. Norton & Company, Inc.
76
Differentiating Products by
Location
½
0
1
x
 What
if n = 3; sellers A, B and C?
© 2010 W. W. Norton & Company, Inc.
77
Differentiating Products by
Location
½
0
1
x
 What
if n = 3; sellers A, B and C?
 Then there is no NE at all! Why?
© 2010 W. W. Norton & Company, Inc.
78
Differentiating Products by
Location
½
0
1
x
What if n = 3; sellers A, B and C?
 Then there is no NE at all! Why?
 The possibilities are:
– (i) All 3 sellers locate at the same point.
– (ii) 2 sellers locate at the same point.
– (iii) Every seller locates at a different
point.

© 2010 W. W. Norton & Company, Inc.
79
Differentiating Products by
Location
½
0
1
x
 (iii)
Every seller locates at a different
point.
 Cannot be a NE since, as for n = 2,
the two outside sellers get higher
profits by moving closer to the
middle seller.
© 2010 W. W. Norton & Company, Inc.
80
Differentiating Products by
½
Location
A
0
C
x
B
1
C gets 1/3 of the market
 (i)
All 3 sellers locate at the same
point.
 Cannot be an NE since it pays one of
the sellers to move just a little bit left
or right of the other two to get all of
the market on that side, instead of
having to share those customers.
© 2010 W. W. Norton & Company, Inc.
81
Differentiating Products by
½
Location
A B
0
x
C
1
C gets almost 1/2 of the market
 (i)
All 3 sellers locate at the same
point.
 Cannot be an NE since it pays one of
the sellers to move just a little bit left
or right of the other two to get all of
the market on that side, instead of
having to share those customers.
© 2010 W. W. Norton & Company, Inc.
82
Differentiating Products by
½
Location
A B
0
x
C
1
A gets about 1/4 of the market
2
sellers locate at the same point.
 Cannot be an NE since it pays one of
the two sellers to move just a little
away from the other.
© 2010 W. W. Norton & Company, Inc.
83
Differentiating Products by
½
Location
A
0
x
B
C
1
A gets almost 1/2 of the market
2
sellers locate at the same point.
 Cannot be an NE since it pays one of
the two sellers to move just a little
away from the other.
© 2010 W. W. Norton & Company, Inc.
84
Differentiating Products by
½
Location
A
0
x
B
C
1
A gets almost 1/2 of the market
2
sellers locate at the same point.
 Cannot be an NE since it pays one of
the two sellers to move just a little
away from the other.
© 2010 W. W. Norton & Company, Inc.
85
 If
Differentiating Products by
Location
n = 3 the possibilities are:
– (i) All 3 sellers locate at the same
point.
– (ii) 2 sellers locate at the same
point.
– (iii) Every seller locates at a
different point.
 There is no NE for n = 3.
© 2010 W. W. Norton & Company, Inc.
86
 If
Differentiating Products by
Location
n = 3 the possibilities are:
– (i) All 3 sellers locate at the same
point.
– (ii) 2 sellers locate at the same
point.
– (iii) Every seller locates at a
different point.
 There is no NE for n = 3.
 However, this is a NE for every n ≥ 4.
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87