partial equilibrium positive analysis...31 long-run analysis • in the long run, firms can enter...

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1

PARTIAL EQUILIBRIUM

Positive Analysis

[See Chap 12 ]

2

Equilibrium

• How are prices determined?

• Partial equilibrium

– Look at one market.

– In equilibrium, supply equals demand.

– Prices in all other markets are fixed.

• General Equilibrium

– Look at all markets at once.

– Consider interactions.

3

Example: Car Market• What happens if the Chinese Govt builds more

roads?

• Partial equilibrium

– Demand for cars rises.

– Quantity of cars rises.

– Price of cars rises.

• General equilibrium

– Price of inputs rises. Increases car costs.

– Value of car firms rises. Shareholders richer and

buy more cars.

4

Model

• We are interested in market 1.

– Price is denoted by p1, or p.

– Firms/Consumers face same price (law

of one price).

– Firms/Consumers are price takers.

5

Model

• There are J agents who demand good 1.

– Agent j has income mj

– Utility uj(x1,…,xN)

– Prices {p1,…,pN}, with {p2,…,pN} exogenous.

• There are K firms who supply good 1.

– Firm k has technology fk(z1,…,zM)

– Input prices {r1,…,rM} exogenous.

6

Competitive Market

• To understand how the market functions we

consider consumers’ and firms’ decisions.

• The consumers’ decisions are summarized by

the market demand function.

• The firms’ decisions are summarized by the

market supply function.

7

MARKET DEMAND

8

Market Demand

• Market demand is the quantity demanded by

all consumers as a function of the price of the

good.

– Hold constant price of the other goods.

– Hold constant agents’ incomes.

9

Market Demand

• Assume there are two goods, x1 and x2.

• Agent j’s Marshallian demand for x1 is

x1j(p1,p2,mj)

• The market demand is the sum of

individual Marshallian demands:

J

i

j

j

J mppxmmpp1

2111211 ),,( ),...,;,(X demandMarket

10

Market Demand

x1 x1x1

p1p1p1

x’ x’’

p1

To derive the market demand curve, we sum the

quantities demanded at every price

x1A

Individual A’s

demand curve

x1B

Individual B’s

demand curveMarket demand

curve

x’+x’’

X1

x1A + x2

B = X1

11

Example

• There are 1000 identical consumers each

with Marshallian demand:

x1j(p1,p2,mj) = 10 - 0.1p

• The market demand function is given by

p,p.xXj j

j 1000001010101000

1

1000

1

11

12

MARKET SUPPLY

13

Market Supply

• The market supply curve is given by the

sum of individual firms’ supply curves:

K

k

k )r(p,rqQ1

21, supply Market

14

Long- and Short- Run

• The market supply differs depending on the time period considered.

• Short run

– Market supply is sum of the quantity supplied by existing firms.

– No new firms can enter the industry.

• Long run

– Market supply is sum of the quantity supplied by the existing and entering firms.

– New firms may enter the industry.

15

Short-Run Market Supply Curve

quantity Quantityquantity

PPP

qA’ qB’

P

To derive the market supply curve, we sum the

quantities supplied at every price

qA

Firm A’s

supply curve qB

Firm B’s

supply curve

Market supply

curve

qA’+qB’’

Q

qA+ qB = Q

16

Example

• There are 100 identical firms each with

the supply curve

qk (p,r1,r2) = 10p/3

• The market supply function is given by

3

1000

3

10100

1

100

1

ppqQ

k k

k

17

SHORT-RUN EQUILIBRIUM

18

Equilibrium Price and Quantity

• The equilibrium price is the price at

which quantity demanded equals

quantity supplied.

• An equilibrium price, p*, solves

• The equilibrium quantity is the quantity

demanded and supplied at the

equilibrium price.

),,(),...,;,( 21

*

12

* rrpQmmppX J

19

Short-Run Equilibrium

• In the short-run equilibrium, the number

of firms in an industry is fixed.

• These firms are able to adjust the

quantity they are producing by altering

the levels of the variable inputs they

employ.

20

Equilibrium Price Determination

Quantity

Price

S

D

Q1

P1

The interaction between

market demand and market

supply determines the

equilibrium price

21

Example: Demand Side

• 15 Agents

– All have utility u(x1,x2)=x1x2

– 10 have income m=10

– 5 have income m=5

• Demand

x1j = mj/2p1

• Market demand

X1 = 200/2p1 = 100/p1

22

Example: Supply Side

• 9 Firms

– All have technology f(z1,z2)=(z1-1)1/3(z2-1)1/3

• Cost curve

c(q) = 2(r1r2)1/2q3/2 + (r1+r2)

In short-run no shutdown so c(0)=r1+r2.

• Profit max supply: q*(p,r1,r2) = p2/9r1r2

• Market supply: Q(p,r1,r2) = p2/r1r2

• Equilibrium price:

p* = (100r1r2)1/3

23

Market Reaction to aShift in Demand

Quantity

Price

S

D

Q1

P1

Start at equilibrium.

24

Market Reaction to aShift in Demand

Quantity

Price

S

D

Q1

P1

Q2

P2 Equilibrium price and

equilibrium quantity will

both rise

If many buyers experience

an increase in their demands,

the market demand curve

will shift to the right

D’

25

Market Reaction to aShift in Demand

Quantity

Price

MC

Q1

P1

This is the short-run

supply response to an

increase in market price

Q2

P2

If the market price rises,

firms will increase their

level of output

AC

26

Shifts in Supply and Demand Curves

• Demand curves shift because

– incomes change

– prices of substitutes or complements change

– preferences change

• Supply curves shift because

– input prices change

– technology changes

– the number of producers change

27

Shifts in Supply and Demand Curves

• When either a supply curve or a

demand curve shift, the equilibrium

price and quantity will change

• The relative magnitudes of these

changes depends on the elasticities of

market demand and supply.

28

Shifts in Supply

Quantity Quantity

PricePrice

S

S’

S

S’

D

D

P

P

Q

P’

Q’

P’

QQ’

Elastic Demand Inelastic Demand

Small increase in price,

large drop in quantity

Large increase in price,

small drop in quantity

29

Shifts in Demand

Quantity Quantity

PricePrice

S

S

D D

P P

Q

P’

Q’

P’

Q Q’

Elastic Supply Inelastic Supply

Small increase in price,

large rise in quantity

Large increase in price,

small rise in quantity

D’ D’

30

LONG-RUN EQUILIBRIUM

31

Long-Run Analysis

• In the long run, firms can enter and leave the

market.

– Assume there are many potential entrants.

– All firms are identical.

• New firms enter if profits are positive:

– Entry causes the short-run industry supply curve to

shift outward;

– Market price and profits fall;

– The process continues until profits are zero.

32

Long-Run Analysis

• Existing firms exit if profits are negative:

– Exit of firms causes the short-run industry supply

curve to shift inward;

– Market price and profits rise;

– The process continues until profits are zero.

• A perfectly competitive market is in long-run

equilibrium if there are no incentives for firms

to enter or leave the industry.

33

Long-Run Equilibrium

• Firms are profit maximising

– Hence p = MC

• Firms make zero profits

– Hence p = AC

• Hence we have AC = MC

– Firms operate at minimum average cost.

34

Long-Run Equilibrium

A Typical Firm Total MarketQuantity Quantity

MC

AC

S

D

q1

P1

Q1

This is a long-run equilibrium for this industry

P = MC = ACPrice Price

35

Example continued

• Individual analysis

– Firm’s cost: c(q) = 2(r1r2)1/2q3/2 + (r1+r2)

– Hence AC(q) = 2(r1r2)1/2q1/2 + (r1+r2)q

-1

– AC is minimized at q* = (r1r2)-1/3(r1+r2)

2/3

– Price is p* = AC(q*) = 3(r1r2)1/3(r1+r2)

1/3

• Market analysis

– Demand X(p*) = 100/p* = 33⅓(r1r2)-1/3(r1+r2)

-1/3

– Number of firms = X(p*)/q* = 33⅓(r1+r2)-1

• The math here is a bit messy. Let r1=r2=1.

36

PUTTING IT ALL TOGETHER

37

Increase in Demand

• Initially firms produce at minimum AC.

• Suppose demand rises.

• Very short run - output fixed. – Market price rises.

• Short run – firms vary output, but no entry– Each firm increases output

– Total output rises, and price falls.

• Long run – new firms enter.– Each firm produces at min AC

– Total quantity rises, and price falls.

38

Increase in Demand

A Typical Firm Total MarketQuantity Quantity

MC

AC

S

D

q1

P1

Q1

This is a long-run equilibrium for this industry

P = MC = ACPrice Price

39

Increase in Demand

A Typical Firm Total Market

q1 Quantity Quantity

MC

AC

S

D

P1

Q1

P1’

Market price rises to P1’

• Suppose that market demand rises to D’

• In the very short run, output is fixed

D’

Price Price

40

Increase in Demand

A Typical Firm Total Market

q1 Quantity Quantity

MC

AC

S

D

P1

Q1

D’

P2

Economic profit > 0

Q2

• In the short run, each firm increases output to q2

q2

Price Price

41

Increase in Demand

A Typical Firm Total Market

q1 Quantity Quantity

MC

AC

S

D

P1

Q1

D’

Economic profit will return to 0

Q3

• In the long run, new firms enter the industry

S’

PricePrice

42

Increase in Demand

A Typical Firm Total Market

q1 Quantity Quantity

MC

AC

S

D

P1

Q1

D’

Q3

S’

• The long-run supply curve is a horizontal line at p1

LS

Price Price

43

Example

• Market demand: X(p) = 1500 – 50p

• Cost function c(q) = 100 + q2/4

• Long run equilibrium

– AC(q)=100q-1+q/4

– AC(q) is minimized at q*=20.

– This yields p* = AC(q*) = 10.

– Industry output: X(p*) = 1000.

– Number of firms = 1000/20 = 50.

44

Example cont.• Market demand falls: X(p)=1200 – 50p

• Very short run. Output fixed.

– Output is Q(p)=1000, so price p = 4.

• Short run. No entry/exit.

– Firm’s supply function q*(p) = 2p.

– Market supply: Q(p) = 50q*(p) = 100p.

– New equilibrium price p = 8.

• Long run. Firms exit.

– Price rises to p* = 10. Firm’s output q* = 20.

– Demand X(p*)=700.

– No. of firms = X(p*)/q* = 700/20 = 35.

45

Long-Run Supply

• We have assumed that one firm’s costs are

independent of the number of firms.

– Long run supply curve is flat.

• Long-run supply curve may be increasing.

– prices of scarce inputs may rise

– new firms may have worse cost functions.

• Long-run supply curve may be decreasing.

– News firms attract larger pool of labor.

• See Chap 12 for analysis of these cases.

46

Existence of Equilibrium• Does there exist a p* where X(p*)=Q(p*)?

• Such a p* exists if

– Preferences are convex → demand continuous.

– Cost is convex → supply is continuous.

• Problem: Non-convex costs.

– Fixed cost can cause supply function to jump.

– May jump over demand.

• Solution: Aggregation

– At p’, agent wants 5 units.

– At p’, firm indifferent between suppying 0 and 10.

– No problem if have 10 agents and 5 firms.

47

Uniqueness of Equilibrium• Is the equilibrium price p* unique?

– Could there be multiple equilibrium prices?

• Supply curve is upward sloping

– Law of supply.

• Demand curve is usually downward sloping

– Unless Giffen Good.

• Together, these imply uniqueness.

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