123

Cost function and conditional input demand functions

11 1 1

,...,

1

cost function: ( ,..., , ) min ...

s.t. ( ,..., )

nn n n

X X

n

C w w Q w X w X

Q f X X

Example: Cobb-Douglas Technology

1 21 2 1 1 2 2 1 2

,( , , ) min s.t.

a b

X XC w w Q w X w X Q X X

1 1 2 2 1 2+ [ ]a b

L w X w X Q X X

FONC:

1

2

1

1 1 2

1

2 1 2

0 (1)

0 (2)

a b

X

a b

X

L w aX X

L w bX X

1 2 2 21

2 1 1

(1)

(2)

w aX aX wX

w bX bw

2 22

1

1

12 1 2

2

1

21 1 2

1

output constraint ( )

( , , ) ( )

( , , ) ( )

ba

a

a b a b

b

a b a b

aX wQ X

bw

bwX w w Q Q

aw

awX w w Q Q

bw

cost function:

1 2 1 1 2 2

1 1

2 11 2

1 2

1 1

2 11 2

1

1 2

( , , ) * *

[ ( ) ] [ ( ) ]

( ) ( )

[( ) ( ) ]

b a

a b a b a b a b

b aa ba b a b a b a b

a b a b

b aa ba b a b a ba b a b

C w w Q w X w X

aw bww Q w Q

bw aw

aw bwQ w Q w

b a

a bQ w w

b a

conditional input demand functions

124

Proposition: If the production function exhibits constant returns to scale, then the cost function may be

written as ( , ) ( ,1)C w Q QC w where 1( ,..., )nw w w .

Proof: Let *X be the cheapest bundle to produce 1 unit of output at prices w so that

1

( ,1) *n

i i

i

C w w X

.

Claim; When Q units are produced, the cost minimizing bundle is 1( *,..., *)nQX QX .

If not. Let 1' ( ',..., ')nX X X be the cost minimizing bundle, i.e.

'' ( *) ( ) *i

i i i i i i i

Xw X w QX w w X

Q .

Since iX

Q can be used to produce 1 unit of the good (CRS)

*X is not the cost-minimizing bundle for 1 unit of the goodcontradiction.

125

Supply function, profit function and input demand functions

Profit function: 1

1 1 1,...,

( , ) max ( ,..., ) ...n

n n nX X

w P Pf X X w X w X

Input demand functions: *( , )iX w P

Supply function: *( , )Q w P

Example: Cobb-Douglas Technology 1

1 2 1 2( , , ) where [( ) ( ) ]

b aa b

a b a b a ba b a ba b

C w w Q KQ w w Kb a

Profit function: 1

1 2( , ) max ( , ) a b

a b a b a bQ

w P PQ C w Q PQ KQ w w

FONC: 1

1

1 2 0a b

a b a b a bd K

P Q w wdQ a b

1

1 2

a ba b

a b a b a bK

P Q w wa b

1

1 2

( )a b

a b

a b

a b a b

P a bQ

Kw w

1

1 2

( )* [ ] supply function

a b

a b

a b

a b a b

P a bQ

Kw w

Conditional Input demand functions:

1

21 1 2

1

( , , ) ( )

b

a b a baw

X w w Q Qbw

Input demand function:1 1

2 21 11 1 2

1 11 2 1 2

( ) ( )( , , ) {[ ] } ( ) [ ] ( )

a b b b

a b a b a b a b a ba b a b

a b a b a b a b

aw awP a b P a bX P w w

bw bwKw w Kw w

Since profit function: *( , ) * ( , *)w P PQ C w Q

1

1 11 2

1 2 1 2

1 1

1 1 1 11 2

1 2

( ) ( )* ( , ) [ ] {[ ] }

* ( , ) [ ] [ ( ) ]

a b a b a ba b a b a b a b a b

a b a b

a b a b a b a b

a b a ba ba b a b a b a b

a b

a b a b

P a b P a bw P P K w w

Kw w Kw w

a bw P P K P a b w w

Kw w

126

Properties of the cost function:

1. (non-decreasing in w ) If ' , then ( ', ) ( , )w w C w Q C w Q

2. (homogeneous of degree1 in w ) ( , ) ( , ) 0C w Q C w Q

3. (concave in w ) [ (1 ) ', ] ( , ) (1 ) ( ', )C w w Q C w Q C w Q

Proof:

1. Method A (for differentiable function)

1

1,...,

1

( , ) min s.t. ( ,..., )n

n

i i nX X

i

C w Q w X Q f X X

1

1

+ [ ( ,..., )]n

i i n

i

L w X Q f X X

By Envelop Theorem, * 0i

i

CX

w

Method B (for any function)

Let and ' be the cost-minimizing bundles X X associated with and 'w w respectively.

i) ' ( is the cost-minimizing bundle at )i i i iw X w X X w

ii) ' ' ' ( ')i i i iw X w X w w

(i) and (ii) ' ' ( , ) ( ', )i i i iw X w X 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 ( ) ' ( )i i i iw X w X

'i i i iw X w X Contradiction

127

3. Let ( , ) and ( ', ')w X w X be cost-minimizing price-factor combinations.

Let " (1 ) ' for (0,1)w w w

Note that ( ", ) " " [ (1 ) '] " " (1 ) ' "i i i i i i i i iC w Q w X w w X w X w X

Since "X is not necessary the cheapest way to produce Q at prices or 'w w

" ( , ) and ' " ( ', )i i i iw X C w Q w X C w Q

( ", ) " (1 ) ' " ( , ) (1 ) ( ', )i i i iC w Q w X w X C w Q 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 ) ( , ) ( , ) 0P w P w

4. (convex in ( , )P w ) Let ( ", ") ( (1 ) ', (1 ) ')P w P P w w

then ( ", ") ( , ) (1 ) ( ', ') (0,1)P w P w P w

128

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 ( )

FOC: 0

QPQ C Q

dP MC P MC

dQ

Short run equilibrium of a perfect competitve firm

$ 0

MC

AC

P1

AVC

profit

Q

Q1*

129

$ 0

MC

AC

AVC

P2 loss

Q

Q2*

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

Why should the firm produce Q2* if it is

losing money?

It is because the firm has already sign a

lease. Even if the firm closes down, it still

has to pay the rent.

In general, so long as P>AVC, the firm

should continue to produce until the lease

expires.

For example, let Q2*=100, P=$8, AC=$10,

AVC=$5.

($8 $10)(100) $200

If the firm closes down, then

( )( )

($10 $5)(100) $500

FC AC AVC Q

130

Supply curve of a profit-maximizing perfect competitve firm

$

AC

MC AVC S P1 P1

P2 P2

Q Q

Q2* Q1* Q2* Q1*

Market supply curve

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

S SM=Si=nSi

P1 P1

Q Q

Q1 QM=nQ1

131

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 curve shifts to the right , UNTIL 0MS P Q

$ MC P SM=nSi SM*=n*Si

AC

E

P1 P1

E’

P* P*

D

Q Q

Qi* Qi QM QM’

Whenever 0

free exit #firms Supply curve shifts to the left , UNTIL 0MS P Q

$ MC P

SM*=n*Si

AC SM=nSi

E’

P* P*

P2 P2 E

D

Q Q

Qi Qi* QM* QM

132

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 * and * ( )C SP P x P P t x .

Since goes up by $ , hence the % in *

C C

xP x P

P ;

and goes down by $ , hence the % in *

S S

t xP t x P

P

.

P

S

PC

x

P* E

E’ t x

PS

D

D’

Q

Q’ Q*

% in ( )*

d d

xQ E

P % in ( )

*S S

t xQ E

P

Since

% in % in ( ) ( ) ( ) ( )* *

Sd S d S d s S d S

S d

tEx t xQ Q E E xE t x E x E E tE x

P P E E

(more negative, more elasti c)

S

S d

tEx

E E

1

S S

S

dS d S d

S

tE tEtx E x x

EE E E E

E

% in ( ) ' *(1 )* * *

S S

S d S dd d d d

tE tE

E E E ExQ E E Q Q E

P P P

133

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 * / 2P .

Effectively each family get 100 coupons (altogether 100,000 coupons) and each coupon allows the

family to buy 1 pound of beef at * / 2P .

beefP

A Scoupon Sbeef

B

price of coupon

*P G H

Dcoupon

* / 2P F

. Dbeef

C

beefQ

1m

consumer surplus: Area ABFC

total cost to government: Area GHFC

134

b) two types of families

beefP

Sbeef

A Scoupon

B C

G

K

H

*P Dcoupon

price of coupon

* / 2P

. Dbeef

I J H L Dbeef2

Dbeef1

beefQ

Q1 Q2 1m

consumer surplus (type 1 family): Area BCJI

consumer surplus (type 2 family): Area AGHI

total cost to government: Area HKLI

135

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 ( ) ( )Q

TR Q TC Q

FOC: 0d dTR dTC

MR MC MR MCdQ dQ dQ

SOC: 2

20 (maximum) slope of MR curve slope of MC curve

d dMR dMC

dQ dQdQ

136

Monopoly equilibrium

0

$ MC

profit AC

P*

D

MR

Q

Q*

0

$ MC

loss AC

P*

D

Q

MR

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

FOC: 60 2 0 * 30

QQ Q Q Q Q Q Q Q

dQ Q

dQ

2* 60(30) (30) 1800 900 900

Example: 2: 80 , : ( )D P Q TC C Q Q 2 2 2 2

2

max (80 ) 80 80 2

FOC: 80 4 0 * 20

* 80(20) 2(20) 1600 800 800

QQ Q Q Q Q Q Q Q

dQ Q

dQ

Note that since there is no entry and

exit, hence there is no distinction

between the short-run equilibrium and

long-run equilibrium.

137

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

138

Imposing an unit tax on a monopoly

P CS

MC’=MC+tax

A’ A

initial DWL

P’ MC

P* A

competitive equilibrium

E E

PS

B

D

B A’ new DWL

B’

government E

revenue MR

Q

Q’ Q* B’

Remark: 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 “under-

production” problem.

Granting an unit subsidy to a monopolist ( ' CQ Q )

P cost to

B A CS government

A MC

P* MC’=MC–subsidy I

P’ E P’ B

I

P’ B F

competitive G

equilibrium (QC) PS

F G G

D H

H

H MR Q B DWL

Q* Q’

E

I

139

Setting a price ceiling on a monopolist P Case 1

D MC

P*

P#

MR’

competitive price PC

MR

D’

Q

Q* Q# MR’

P Case 2

D MC

P*

P#

MR’ competitive price PC

MR

D’

Q

Q* Q#=QC MR’

140

P Case 3

D MC

P*

P#

MR’ competitive price PC

MR

D’

Q

Q* Q#

MR

P Case 4

D MC

P*

P#

competitive price PC

D’

Q

Q# Q* MR

141

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 US AsiaP P .

P P

MC

P*

DUS

DW = DUS + DAsia

DAsia

MRW

QAsia* QUS* Q QW* Q

P P

MC

P*

DUS DW = DUS + DAsia

MRW

DAsia

QAsia* QUS* Q QW* Q

142

Example:

Asia : 20 20

US: 30 30

P Q Q P

P Q Q P

World demand : 50 2 when 20

30 when 20 30

Q P P

Q P P

Let 10TC Q

Note that 50 2 252

QQ P P

2 2

max (25 ) 10 25 10 152 2 2

15FOC: 15 0 * 15 25 25 $17.5

2 2

Q

Q Q QTR TC Q Q Q Q Q

d QQ Q P

dQ

When $17.5 * 2.5 and * 12.5 * * * 2.5 12.5 15A US W A USP Q Q Q Q Q

Note:

Asia: 2(20 ) 20 20 2 @ * 2.5 $15A ATR Q Q Q Q MR Q Q MR

US: 2(30 ) 30 30 2 @ * 12.5 30 2(12.5) $5US USTR Q Q Q Q MR Q Q MR

Hence A USMR MR

Since if sales is shifted from US to AsiaA USMR MR .

When * by 1 unit and * by 1 unit, then $15 $5 $10A USQ Q

143

Remark: When the firm increases its sale in Asia and decrease its sale in US, and A USP P .

Therefore A USP P . 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.

Since 1

(1 )MR P

, if A US

A US A USMR MR P P

1 1 1 1 1 1In general, 1 1

1 1Since , hence (1 ) (1 )

A US

A US A US A US

US A US A A USUS AMR MR P P P P

Intuitively, the more price elastic the demand, the lower is the price charged.

2 2

,

2 2

max (20 ) (30 ) 10( ) 20 30 10 10

10 20

FOC: 10 2 0 * 5 * $15

20 2 0 * 10 * $20

US A

A A US US A US A A US US A USQ Q

A A US US

A A A

A

US US US

US

Q Q Q Q Q Q Q Q Q Q Q Q

Q Q Q Q

Q Q PQ

Q Q PQ

15( 1) 3 more elastic, lower price

5

20( 1) 2 less elastic, higher price

10

A

US

dQ P

dP Q

dQ P

dP Q

Also 1 1 1 1

(1 ) 20(1 ) 10 15(1 ) (1 )2 3

US US A AUS AMR P P MR

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)

144

In general, a two-market monopolist which can separate the markets will solve the following

problem.

,max ( ) ( ) ( )

A B

A A B B A BQ Q

TR Q TR Q C Q Q

FOC:

0

0

A

A

B

B

MR MCQ

MR MCQ

A BMR MR 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 A B

F

P’

P* G MC

D

Q

Q’ Q* MR

In a regular monopolist, the price is P*

and the quantity is Q*. In doing so, the

consumers can enjoy a CS of ABP*

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*.

145

Perfect price discrimination

P

total amount

of money charged MC

D

Q

Q*=QC

0

max ( ) ( )

Q

QP x dx C Q

FOC: ( ) '( ) 0d

P Q C Q P MCdQ

Two-part tariff P

membership fee

MC

P*

D

Q

Q*=QC

The monopolist will charge a

lump-sum equal to the total

shaded area under the demand

curve for a quantity equal to QC.

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.

Note that in this case the monopolist will sell

the competitive output QC.

146

Example: Let 2: 30 and ( )D P Q TC Q Q 2( ) 2

Profit-maximizing condition for a perfect competitive market:

30 2 * 10 * 20

TC Q Q MC Q

P MC

Q Q Q P

P

S

30

E

20 Membership fee = CS= ($30 20)(10)

$502

D

Q

10

Example:

Grocery Membership fee

Telephone, gas and electricity Rent for the equipment

Amusement park Entrance fee

147

Multi-plant monopoly

1 2

1 2 1 1 2 2,

1

1

2

2

1 2

max ( ) ( ) ( )

FOC: 0

0

Q QTR Q Q TC Q TC Q

MR MCQ

MR MCQ

MR MC MC

Example:

1 2

2 2 2 2 2 2

1 2 1 2 1 2 1 2 1 1 2 2 1 2,

2 2

1 2 1 2 1 2

1 2

1

1 2

2

max [20 ( )]( ) 2 20 20 2 2

20 20 2 3 2

FOC: 20 4 2 0 (1)

20 2 6 0

Q QQ Q Q Q Q Q Q Q Q Q Q Q Q Q

Q Q Q Q Q Q

Q QQ

Q QQ

1 2

2 2 1

(2)

(2) x 2 40 4 12 0 (3)

(3) (1) 20 10 0 * 2 * 4

Q Q

Q Q Q

148

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

( )1

2P t x

1(

1)

2P t x

V

1

V P

t

0z 1x 1

2z 2x 1z

1

1 1 1

1 1( )2 2

V PP t x V x

t

and 1

1 2 2

1 1( )

2 2

V PP t x V x

t

1 :P price of the good

:t transportation cost

149

Since the road is 1 mile long, hence there are 12

V PN

t

consumers, hence the total demand for the

monopolist’s product given that it operates just 1 retail outlet is 1 1

2( ,1) ( )

NQ P V P

t .

# of retail outlet

Note that the demand function is a decreasing function of P . When P , consumers who are

further away from the shop will buy the product. As a matter of fact, if 2

tP V , then all consumers

will buy the product. Define ( ,1)2

tp N V .

c : the monopolist’s per unit cost.

F : the setup costs for each retail outlet

Profit function facing the monopolist which is selling to the whole market:

( ,1) [ ( ,1) ]2

tN N p N c F N V c F

2 outlets

P

V

0z 1

4

1

2z

3

4 1z

( ,2)4

tp N V ; ( ,2) 2

4

tN N V c F

2

11

2

1

2

2

2

V Px

t

V P

t

tV P

tP V

2

2 2 2

1 1( )

4 4

V PP t x V x

t

1 1 1

2 4 4 4

V P V P tP V

t t

150

3 outlets

P

V

0z 1

6

1

2z

5

6 1z

( ,3)6

tp N V ; ( ,3) 3

6

tN N V c F

n outlets

( , )2

tp N n V

n ; ( , )

2

tN n N V c nF

n

max ( , )2n

tN n N V c nF

n

FOC: 2

20

2 22

Nt Nt NtF n n

F Fn

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.]