123-150

Cost function and conditional input demand functions

cost function: C ( w1 ,..., wn , Q)  min w1 X 1  ...  wn X n
X1 ,..., X n

Q  f ( X 1 ,..., X n )

s.t.

Example:

Cobb-Douglas Technology

C (w1 , w2 , Q)  min w1 X1  w2 X 2

s.t. Q  X 1 X 2

L  w1 X1  w2 X 2 +[Q  X1 X 2 ]
FONC:
a 1
b
LX1  w1   aX 1 X 2  0

(1)

a

X1 , X 2

a

LX 2  w2  bX 1 X 2
a

b 1

b

b

0

(2)

w
aX 2
aX 2 w2
(1)
 1 
 X1 
(2)
w2 bX1
bw1
aX w
b
 output constraint  Q  ( 2 2 ) a X 2
bw1
 X 2 ( w1 , w2 , Q)  Q

1
a b

bw
( 1 ) a b
aw2

1

aw2 a b
)
bw1

 X 1 ( w1 , w2 , Q)  Q a b (

a

conditional input demand functions

b

cost function:
C ( w1 , w2 , Q)  w1 X 1 *  w2 X 2 *

 w1 [Q
1
a b

1
a b

b

1

a

aw
bw
( 2 ) a b ]  w2 [Q a b ( 1 ) a b ]
bw1
aw2
b

1

a

a
b
aw
bw
 Q ( 2 ) a b w1 a b  Q a b ( 1 ) a b w2 a b
b
a
1
b
a
a
b
a a b
b a b
a b
a

b
a

b
 Q w1 w2 [( )  ( ) ]
b
a

123

Proposition: If the production function exhibits constant returns to scale, then the cost function may be
written as C (w, Q)  QC (w,1) where w  (w1 ,..., wn ) .
Let X * be the cheapest bundle to produce 1 unit of output at prices w so that

Proof:

n

C ( w,1)   wi X i * .
i 1

When Q units are produced, the cost minimizing bundle is (QX1 *,..., QX n *) .

Claim;

If not. Let X '  ( X1 ',..., X n ') be the cost minimizing bundle, i.e.

w X
i

i

'   wi (QX i *)   wi (

Xi '
)   wi X i * .
Q

Xi
can be used to produce 1 unit of the good (CRS)
Q
 X * is not the cost-minimizing bundle for 1 unit of the good  contradiction.

Since

124

Supply function, profit function and input demand functions
Profit function:

 (w, P)  max Pf ( X1 ,..., X n )  w1 X1  ...  wn X n

Input demand functions:
Supply function:

X i * (w, P)
Q * (w, P)

Example:

X1 ,..., X n

Cobb-Douglas Technology

C ( w1 , w2 , Q)  KQ

1
a b

w1

a
a b

w2

b

a

a
b
where K  [( ) a b  ( ) a b ]
b
a

b
a b

Profit function:
1

a

b

 ( w, P)  max   PQ  C ( w, Q)  PQ  KQ a b w1 a b w2 a b
Q

FONC:
1
a
b
1
d
K
a b
 P
Q
w1 a b w2 a b  0
dQ
ab
1 a b

a
b
K
P
Q a b w1 a b w2 a b
ab
1 a b
P ( a  b)
 Q a b 
a
b
Kw1 a b w2 a b

 Q*  [

a b
1 a b

P ( a  b)
Kw1

a
a b

w2

b
a b

 supply function

]

Conditional Input demand functions: X 1 ( w1 , w2 , Q)  Q

Kw1

P ( a  b)
a

b

a b

]1 a b  K{[

Kw1 a b w2 a b
1

  * ( w, P)  P 1 a b [

a
a b

w2

b
a b

]

b

aw
( 2 ) a b
bw1
}

1
a b

aw2 a bb
P(a  b) 1a1b aw2 a bb
(
) [
]
(
)
a
b
bw1
bw1
a b
a b
Kw1 w2

 *(w, P)  PQ * C(w, Q*)

Since profit function:

  * ( w, P)  P[

a b
1 a b

P ( a  b)

Input demand function: X 1 ( P, w1 , w2 )  {[

1
a b

a

b

a b

1

a

b

]1 a b }a b w1 a b w2 a b

Kw1 a b w2 a b
a b

ab
a

P ( a  b)

b

]1a b  K



a b
1 a b

a

b

1

[ P(a  b) w1 w2 ]1a b

Kw1 a b w2 a b

125

Properties of the cost function:
1.
(non-decreasing in w )
2.
(homogeneous of degree1 in w )
3.
(concave in w )

If w '  w, then C (w ', Q)  C (w, Q)
C ( w, Q)  C (w, Q)
  0
C[ w  (1   )w ', Q]   C (w, Q)  (1   )C(w ', Q)

Proof:
1.
Method A (for differentiable function)
C ( w, Q)  min

X1 ,..., X n

n

w X
i 1

i

s.t. Q  f ( X 1 ,..., X n )

i

n

L   wi X i +[Q  f ( X 1 ,..., X n )]
i 1

By Envelop Theorem,

C
 Xi*  0
wi

Method B (for any function)
Let X and X ' be the cost-minimizing bundles associated with w and w ' respectively.
i)
ii)

w X  w X '
 w X '   w 'X '
i

i

i

i

i

i

i

i

(X is the cost-minimizing bundle at w)
(

w  w ')

(i) and (ii)   wi X i   wi 'X i '  C (w, Q)  C (w ', Q)

2.

Let X be the cost-minimizing bundle at prices w .
Claim: X is also a cost-minimizing bundle at prices  w .
Suppose not.
Let X ' be the cost-minimizing bundle at prices  w so that

  wi X i '   wi X i

Contradiction

126

 ( w ) X
i

i

'   ( wi ) X i

3.

Let (w, X ) and (w ', X ') be cost-minimizing price-factor combinations.
Let w"   w  (1   )w '

for   (0,1)

Note that C (w", Q)   wi " X i "  [ wi  (1   )wi '] X i "    wi X i "(1   ) wi ' X i "
Since X " is not necessary the cheapest way to produce Q at prices w or w '

  wi X i "  C (w, Q)

and

w ' X
i

i

"  C (w ', Q)

 C (w", Q)    wi X i " (1   ) wi ' X i "   C ( w, Q)  (1   )C( w ', Q)

Properties of the profit function:
1.
(non-decreasing in P )
If P '  P   ( P ', w)   ( P, w)
2.
(non-increasing in w )
If w '  w   ( P, w ')   ( P, w)
3.
(homogenous degree 1 in ( P, w) )
 ( P,  w)   ( P, w)
  0
4.
(convex in ( P, w) )
Let ( P ", w")  ( P  (1   ) P ', w  (1   )w ')
then  ( P ", w")   ( P, w)  (1   ) ( P ', w ')
  (0,1)

127

Perfect competitive market
Properties of a perfect competitive market
1.
many sellers (firms)
2.
many buyers (consumers)
3.
homogeneous product
4.
free entry/exit
Profit-maximizing condition of a competitve firm: P  MC
max   PQ  C (Q)
Q

FOC:

d
 P  MC  0  P  MC
dQ

Short run equilibrium of a perfect competitve firm
 0
$
MC
AC
P1
AVC

profit

Q
Q1*

128

$

 0

Why should the firm produce Q2* if it is
losing money?

MC
AC

It is because the firm has already sign a
lease. Even if the firm closes down, it still
has to pay the rent.

AVC

P2

In general, so long as P>AVC, the firm
should continue to produce until the lease
expires.

loss

Q
Q2*

For example, let Q2*=100, P=$8, AC=$10,
AVC=$5.
  ($8  $10)(100)  $200
If the firm closes down, then
  FC  ( AC  AVC )(Q)
 ($10  $5)(100)  $500

However if P  AVC , then the firm should shut down immediately. Note that in this case, the price
received is not enough to pay for the labor cost and the cost of the raw material.
$
MC
AC
AVC

P3

Q
closed down point

129

Supply curve of a profit-maximizing perfect competitve firm
$
AC
AVC

MC

S

P1

P1

P2

P2

Q

Q

Q2 * Q1 *

Q 2 * Q1 *

Market supply curve
Remark:
The market supply curve is the horizontal sum of the individual supply curves.
P

P
S

P1

SM=Si=nSi

P1

Q

Q

Q1

QM=nQ1

130

Long run equilibrium (when the number of firms adjusts) for a competitive firm
Remark:
In the long run,   0
Whenever   0
 free entry  #firms  Supply  S curve shifts to the right  P , QM    UNTIL   0
$

MC

P

SM=nSi

SM*=n*Si

AC
E
P1

P1

P*

P*

E’

D
Q
Qi*

Q

Qi

QM’

QM

Whenever   0
 free exit  #firms  Supply  S curve shifts to the left  P , QM    UNTIL   0
$

MC

P
SM*=n*Si
AC

SM=nSi

E’
P*

P*

P2

P2

E
D

Q
Qi

Qi*

Q
Q M*

131

QM

Application of the perfect competitive model
Example:

Tax revenue

Suppose the equilibrium point in a perfect competitive market is ( Q*, P * ), if an unit tax of $t is
imposed on the good, how much tax revenue will the government collect?
Note that when an unit tax is imposed, if the demand curve is downward sloping and the supply curve is
upward sloping, then the P will not go up by $t .
Let x be the increase in price, hence PC  P *  x and PS  P * (t  x) .
x
Since PC goes up by $ x, hence the % in PC  
;
P*
tx
and PS goes down by $t  x, hence the % in PS  
.
P*
P
S

PC

x

P*

E
E’

tx

PS

D
D’
Q
Q’

% in Qd  Ed (

x
)
P*

Since

% in Qd  % in QS  Ed (

Q*

% in QS  ES (

tx
)
P*

tES
x
tx
)  ES (
)  xEd  (t  x) Es  x( ES  Ed )  tES  x 
P*
P*
ES  Ed

tES
 x
ES  Ed
 (more negative, more elastic)

 tES
E E
x
d
 % in Qd  Ed ( )  Ed  S
P*
 P*


tES

ES  Ed



  Q '  Q *(1  Ed



132

t
x
Ed
1
ES

 tES
E E
d
 S
 P*




)



ES  x 

tES
 x
ES  Ed

Example:
Price ceiling
The government imposes an effective price ceiling on beef. As a result, there is a shortage of beef.
Suppose the competitive suppliers are willing to supply 1 million pounds at the maximum price P * .
Suppose there are 10,000 families and the government rations the beef by distributing 50 coupons to
each family. Each coupon will entitle a family to purchase a pound of beef at P * and also get one
pound of beef free of charge. Families are free to sell coupons to one another for a competitive market
price. Assume there is no income effect (so that the height of the demand curve represents how much
the consumers are willing to pay for different units).
By means of graphical analysis, determine the market price of coupon, the total cost of the government
(net of the revenue received for distributing coupons) and the consumer surplus of the families if
a)
all families are identical with the same demand function, or
b)
there are two types of families with different demand functions.
a)

all families are identical

As a family with a coupon can purchase a pound of beef at P * and also get one pound of beef free of
charge, that is, the family with a coupon can buy 2 pounds of beef at a unit price of P * / 2 .
Effectively each family get 100 coupons (altogether 100,000 coupons) and each coupon allows the
family to buy 1 pound of beef at P * / 2 .
Pbeef
A
Scoupon
Sbeef

B
price of coupon

P* G

H
Dcoupon

P*/2
.
C

F
Dbeef
Qbeef

1m
consumer surplus:
total cost to government:

Area ABFC
Area GHFC

133

b)

two types of families

Sbeef

Pbeef
A

B

Scoupon

C

G

K
H
P*

Dcoupon
price of coupon
P*/2
.

Dbeef

I

J

H

Dbeef2

L
Dbeef1

Qbeef

Q1

Q2 1m

consumer surplus (type 1 family):

Area BCJI

consumer surplus (type 2 family):

Area AGHI

total cost to government:

Area HKLI

134

Monopoly
Properties of a monopoly market
1.
one seller (firm)
2.
many buyers (consumers)
3.
homogeneous product
4.
no entry/exit
Remark:

Because there is only one seller in the market. The seller is a price-setter and is facing a
downward sloping demand curve.

Sources of monopoly power/barriers to entry
1.
Patent: Traditionally, American patent laws allow an investor the exclusive right to use the
invention for a period of 17 years from the date the patents were granted. However, in 1995, the
US agreed to change its patent law as part of an international agreement. Now, US patents last
for 20 years after the investors file for patent protection.

2.

Economies of scale: New firms have less production capacity than do established firms. New
firms usually have higher average costs than established firms, this inhibits their entry.
A natural monopoly occurs when economies of scale are so large that there is room for only one
firm in the industry. Example: local public utilities that deliver telephone, gas, water and electric
service.

3.

Economies of scope: Sometimes it is cheaper to produce two related products in a single firm
rather than in two separate firms.

4.

Exclusive ownership of raw materials: Established companies may be protected from the entry
of new firms by their control of raw materials. Example: The International Nickel Company of
Canada once owned almost all of the world’s nickel reserves.

5.

Public franchises: Example: US Postal Service.

6.

Licensing: Example: lawyers and medical doctors.

Profit maximizing condition of a monopolist:
max   TR(Q)  TC (Q)
Q

d dTR dTC


 MR  MC  0  MR  MC
dQ dQ
dQ
d 2 dMR dMC


 0 (maximum)  slope of MR curve  slope of MC curve
SOC:
dQ
dQ
dQ 2
FOC:

135

Monopoly equilibrium
 0
$

MC
profit
AC

P*

D

MR
Q
Q*

 0
$

MC
loss

AC

P*

D

Q
MR

Example:
D : P  80  Q,
TC : C (Q)  20Q
2
max   (80  Q)Q  20Q  80Q  Q  20Q  60Q  Q 2
Q

FOC:

d
 60  2Q  0  Q*  30
dQ

  *  60(30)  (30)2  1800  900  900

Example:

D : P  80  Q,

TC : C (Q)  Q2

max   (80  Q)Q  Q 2  80Q  Q 2  Q 2  80Q  2Q 2
Q

FOC:

d
 80  4Q  0  Q*  20
dQ

  *  80(20)  2(20) 2  1600  800  800

136

Note that since there is no entry and
exit, hence there is no distinction
between the short-run equilibrium and
long-run equilibrium.

Welfare cost of monopoly
Remark:
Suppose the monopoly is broken down into many small firms. In this case, the MC is
“divided” into many MCs, one for each firm. Since the MC is the supply curve for each
firm, when we add up all these MCs or Si’s, we get a market supply curve which is the
same as the MC curve for the monopolist.
P
CS

MC
DWL

P*
competitive equilibrium

D
PS
Q
MR

Remark:

The welfare loss is due to “under-production”.

There is no supply curve for monopoly
P

D
D
MR

MR

Q*

Q

137

Imposing an unit tax on a monopoly
P

CS
MC’=MC+tax
A’ A
initial DWL

P’
P*

MC
A
competitive equilibrium
E

E

PS
B
D
B
B’
government
revenue

A’

new DWL

E
MR
Q

Q’ Q*

Remark:

B’

Imposing an unit tax on a monopoly will lead to a higher welfare loss!
Intuitively, the unit tax will lower the output level, which makes a more severe “underproduction” problem.

Granting an unit subsidy to a monopolist ( Q '  QC )
P
B
A

A

CS

P’

B

P’

B

cost to
government

MC
MC’=MC–subsidy

P*
P’

I

E
I
competitive
equilibrium (QC)

F

G
PS

G

G
D

H
H

H

MR

F

Q

B

DWL

Q* Q’
E
I

138

Setting a price ceiling on a monopolist
P

Case 1
D

MC

P*
P#
MR’
competitive price PC
MR

D’

Q
Q* Q#

P

MR’

Case 2
D

MC

P*
P#
MR’

competitive price P C

MR

D’

Q
Q* Q#=QC

MR’

139

P

Case 3
D

MC

P*
P#
MR’

competitive price P C

MR
D’

Q
Q* Q#

P

MR

Case 4
D

MC

P*
P#
competitive price PC

D’

Q
Q# Q*

MR

140

A two-market monopolist
Suppose a monopolist sells its product to US and Asia. If the consumers can resell the good, then by the
arbitrate process, at equilibrium PUS  PAsia .
P

P
MC

P*

DUS
DW = DUS + DAsia

DAsia
MRW
QAsia* QUS*

Q

Q W*

P

Q

P
MC

P*

DUS

DW = DUS + DAsia
MRW

DAsia
QAsia*

QUS*

Q

QW*

141

Q

Example:
Asia : P  20  Q  Q  20  P
US: P  30  Q  Q  30  P
 World demand : Q  50  2 P
Q  30  P
Let TC  10Q
Note that Q  50  2 P  P  25 

when P  20
when 20  P  30
Q
2

Q
Q2
Q2
)Q  10Q  25Q 
 10Q  15Q 
Q
2
2
2
d
Q
15
FOC:
 15  Q  0  Q*  15  P  25   25   $17.5
dQ
2
2
When P  $17.5  QA *  2.5 and QUS *  12.5  QW *  QA * QUS *  2.5  12.5  15
max   TR  TC  (25 

Note:
Asia: TR  (20  Q)Q  20Q  Q2  MR  20  2Q
US:
TR  (30  Q)Q  30Q  Q2  MR  30  2Q

@QA *  2.5  MRA  $15
@QUS *  12.5  MRUS  30  2(12.5)  $5

Hence MRA  MRUS
Since MRA  MRUS    if sales is shifted from US to Asia .
When QA *  by 1 unit and QUS *  by 1 unit, then   $15  $5  $10

142

Remark:

When the firm increases its sale in Asia and decrease its sale in US, PA  and PUS  .
Therefore PA  PUS . In this case, however, the arbitrage process will make the prices
equal again. Hence, in order to increase its profit, the firm must have the ability to
separate the market and charge different consumers different prices in the 2
markets. This is called price discrimination.

1
Since MR  P(1  ) , MRA  MRUS  PA  PUS if  A   US



In general,  A   US 

1



A



1



Since MRUS  MRA , hence PUS (1 



US

1



US

1



A



1



 1

US

)  PA (1 

1

A

1



A

 1

1

 US

)  PA  PUS

Note that   0 ,  A   US   A   US
(i.e. the demand in Aisa is more elastic, or more "price-sensitive,
because the income level of the Asians is lower)

Intuitively, the more price elastic the demand, the lower is the price charged.
2
max   (20  QA )QA  (30  QUS )QUS  10(QA  QUS )  20QA  QA2  30QUS  QUS
 10QA  10QUS

QUS ,QA

2
 10QA  QA2  20QUS  QUS

FOC:


 10  2QA  0  QA *  5  PA *  $15
QA

 20  2QUS  0  QUS *  10  PUS *  $20
QUS

dQ P
15
 (1)  3  more elastic, lower price
dP Q
5
dQ P
20
US 
 (1)
 2  less elastic, higher price
dP Q
10
1
1
1
1
Also MRUS  PUS (1  US )  20(1  )  10  15(1  )  PA (1  A )  MRA
2
3



A 

143

In general, a two-market monopolist which can separate the markets will solve the following
problem.
max   TRA (QA )  TRB (QB )  C (QA  QB )
QA ,QB

FOC:

 MRA  MC  0
QA

 MRB  MC  0
QB

 MRA  MRB  MC

Examples of price discrimination:
1.
Residential phone services vs business phone services.
2.
Manufacturers’ coupons.
3.
Discount airline tickets for passengers which have advanced reservations.
4.
US edition books vs International edition books.
5.
Regular tickets and student tickets.
6.
Education discounts.
Remark:

A monopolist can separate the markets by legal means, geographical regions, etc.

Block discrimination
P

In a regular monopolist, the price is P*
and the quantity is Q*. In doing so, the
consumers can enjoy a CS of ABP*

A B
F

P’
P*

G

MC

In order to make more money, the
monopolist can charge $P’/unit for the
first Q’ unit and then charge $P*/unit for
the remaining (Q*–Q’) units. In doing
so, the firm can extract extra CS of the
size equals to  P’FGP*.

D

Q
Q’

Q*

MR

144

Perfect price discrimination
P
total amount
of money charged

MC

The monopolist will charge a
lump-sum equal to the total
shaded area under the demand
curve for a quantity equal to QC.

D
Q
Q*=QC
Q

max    P( x)dx  C (Q)
Q

0

d
 P(Q)  C '(Q)  0  P  MC
dQ

FOC:

Two-part tariff
P
membership fee
MC

Instead of charging a lump sum for a fixed
quantity, a monopolist can sell Q* units at price
P* and then charge a membership fee equal to
the shaded area. By doing this, the monopolist
can make as much money as practicing perfect
price discrimination.

P*

D

Q

Note that in this case the monopolist will sell
the competitive output QC.

Q*=QC

145

Example:
Let D : P  30  Q and TC (Q)  Q2
TC (Q)  Q 2  MC  2Q
Profit-maximizing condition for a perfect competitive market:P  MC
30  Q  2Q  Q*  10  P*  20
P
S
30

E
20

Membership fee = CS=

D

Q
10

Example:
Grocery
Telephone, gas and electricity
Amusement park

Membership fee
Rent for the equipment
Entrance fee

146



($30  20)(10)
 $50
2

Multi-plant monopoly
max   TR(Q1  Q2 )  TC1 (Q1 )  TC2 (Q2 )
Q1 ,Q2

FOC:


 MR  MC1  0
Q1

 MR  MC2  0
Q2

 MR  MC1  MC2

Example:
max   [20  (Q1  Q2 )](Q1  Q2 )  Q12  2Q2 2  20Q1  20Q2  Q12  2Q1Q2  Q22  Q12  2Q2 2
Q1 ,Q2

 20Q1  20Q2  2Q12  3Q2 2  2Q1Q2
FOC:


 20  4Q1  2Q2  0
Q1

(1)


 20  2Q1  6Q2  0
Q2

(2)

(2) x 2  40  4Q1  12Q2  0

(3)

(3)  (1)  20  10Q2  0  Q2 *  2  Q1*  4

147

Product Variety and Quality Under Monopoly
Definition:

When a firm offers a variety of products in response to different consumer tastes, it is
called horizontal product differentiation.

Example:

The consumer goods giant, Procter and Gamble, offers 12 different versions of its Head
and Shoulders shampoo and another 12 varieties of its Crest toothpaste.

Definition:

When a firm responds to differing willingness to pay among consumers by offering
different qualities of the same product, it is called vertical product differentiation.

Example:

Airline companies offer economy class, business class and first class service.

A spatial model of (horizontal) product differentiation
Assume that there is a town spread out along a single road of one mile in length. There are N identical
consumers spaced evenly along this road. A firm that has a monopoly in, for example, fast food must
decide how to serve these consumers at the greatest profit, i.e., the monopolist must choose the number
of retail outlets that it will operate, where these should be located, and what prices it should charge.
[In the product differentiation analogy to drinks of different sweetness, the monopolist has to decide
how many different drinks it should offer, what their precise degrees of sweetness should be, and what
their prices should be.]
Assume consumers travel to a retail outlet in order to buy the product, incurring transport costs. Assume
that the transportation cost per unit of distance (to-and-back) is t . Assume that in each period each
consumer is willing to buy exactly one unit of the product sold by the monopolist provided that the price
paid, including transportation costs (the full price) is less than the reservation price ( V ).
One single outlet (located at the center)
P
1
P1  t (  x )
2

P1  t ( x 

1
)
2

V

P1 : price of the good
t : transportation cost

V  P1
t

z0

x1

z 

1

x2

2

z 1

1
1 V  P1
1
1 V  P1
and P1  t ( x2  )  V  x2  
P1  t (  x1 )  V  x1  
2
2
t
2
2
t

148

Since the road is 1 mile long, hence there are 2  N 

V  P1

consumers, hence the total demand for the

t

2N
(V  P1 ) .
t
# of retail outlet

monopolist’s product given that it operates just 1 retail outlet is Q( P1 ,1) 

Note that the demand function is a decreasing function of P . When P  , consumers who are
t
further away from the shop will buy the product. As a matter of fact, if P  V  , then all consumers
2
t
will buy the product. Define p( N ,1)  V  .
2
1 V P
1  x2  
2
t
c:
the monopolist’s per unit cost.
1 V P
the setup costs for each retail outlet
F:
 
2
t
t
 V  P
Profit function facing the monopolist which is selling to the whole market:
2
t


t
 ( N ,1)  N  [ p( N ,1)  c]  F  N V   c   F
 P V 
2


2
2 outlets
P
V

z0

1

z 

4

t
p( N , 2)  V  ;
4




t

1

3

2

4




 ( N , 2)  N  V   c   2 F
4

1
1 V  P2
P2  t ( x2  )  V  x2  
4
4
t

1 1 V P
1 V P
t
 
 
 P V 
2 4
t
4
t
4

149

z 1

3 outlets
P
V

z0

1

z 

6

t
p( N ,3)  V  ;
6

1

5

2

6




t




t

 c   nF
2n


z 1




 ( N ,3)  N  V   c   3F
6

n outlets
p ( N , n)  V 

t
;
2n

 ( N , n)  N  V 

t


max  ( N , n)  N V 
 c   nF
n
2n



FOC:

Nt
Nt
 F  0  n2 
n
2
2F
2n

Nt
2F

Note that when n  , the price will be closer to the reservation price, and thereby captures a
greater proportion of consumer surplus.
[Note that we can interpret the spatial model as a model of difference in taste. Instead of “address of a
location”, z can be interpreted as the taste. The “retail store” now becomes a product with a particular
taste, and consumers will lose some “utility” by consuming products differs from his taste. In this case,
n becomes the number of different varieties of products offered by the monopolist.
Also note that the monopolist has the incentive to offer many varieties of a good. Doing so allows the
monopolist to exploit the wide variety of consumer tastes, charging each consumer a higher price
because each is being offered a variety that is very close to the most preferred type. It is not surprising
that we can see extensive product proliferation in real-world markets such as those for cars, soft drinks,
toothpaste, cameras, etc.]

150