Download PARTIAL EQUILIBRIUM Positive Analysis Equilibrium Example: Car

PARTIAL EQUILIBRIUM
Positive Analysis
[See Chap 12 ]
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Equilibrium
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How are prices determined?
Partial equilibrium
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–
–
•
Look at one market.
In equilibrium, supply equals demand.
Prices in all other markets are fixed.
General Equilibrium
–
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Look at all markets at once.
Consider interactions.
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Example: Car Market
•
What happens if the Chinese Govt builds more
roads?
Partial equilibrium
•
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•
Demand for cars rises.
Quantity of cars rises.
Price of cars rises.
General equilibrium
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Price of inputs rises. Increases car costs.
Value of car firms rises. Shareholders richer and
buy more cars.
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1
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.
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Model
•
There are J agents who demand good 1.
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–
–
•
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.
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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.
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2
MARKET DEMAND
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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.
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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
Market demand = X1 ( p1 , p2 ; m1 ,..., m J ) = ∑ x1j ( p1 , p2 , m j )
i =1
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3
Market Demand
To derive the market demand curve, we sum the
quantities demanded at every price
p1
p1
Individual A’s
demand curve
Individual B’s
demand curve
p1
Market demand
curve
p1
x1A
x’
X1
x1B
x’’
x1
x’+x’’
x1
x1
x1A + x2B = X1
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Example
• There are 1000 identical consumers each
with Marshallian demand:
x1j(p1,p2,mj) = 10 - 0.1p
• The market demand function is given by
1000
1000
X1 = ∑ x1j = ∑10 − 0.1p = 10,000−100p
j =1
j =1
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MARKET SUPPLY
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4
Market Supply
• The market supply curve is given by the
sum of individual firms’ supply curves:
K
Market supply = Q = ∑ q k (p,r1 , r2 )
k =1
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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.
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– New firms may enter the industry.
Short-Run Market Supply Curve
To derive the market supply curve, we sum the
quantities supplied at every price
P
Firm A’s
supply curve
P
P
qB
qA
Firm B’s
supply curve
Market supply
curve
Q
P
qA’
quantity
qB’
quantity
qA’+qB’’ Quantity
qA+ qB = Q
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5
Example
• There are 100 identical firms each with
the supply curve
qk (p,r1,r2) = 10p/3
• The market supply function is given by
100
100
Q = ∑ qk = ∑
k =1
k =1
10 p 1000 p
=
3
3
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SHORT-RUN EQUILIBRIUM
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Equilibrium Price and Quantity
• The equilibrium price is the price at
which quantity demanded equals
quantity supplied.
• An equilibrium price, p*, solves
X ( p * , p2 ; m1 ,..., mJ ) = Q( p* , r1 , r2 )
• The equilibrium quantity is the quantity
demanded and supplied at the
equilibrium price.
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6
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.
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Equilibrium Price
Determination
The interaction between
market demand and market
supply determines the
equilibrium price
Price
S
P1
D
Q1
Quantity
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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
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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
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Market Reaction to a
Shift in Demand
Start at equilibrium.
Price
S
P1
D
Quantity
Q1
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Market Reaction to a
Shift in Demand
If many buyers experience
an increase in their demands,
the market demand curve
will shift to the right
Price
S
P2
P1
D’
Equilibrium price and
equilibrium quantity will
both rise
D
Q1
Q2
Quantity
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8
Market Reaction to a
Shift in Demand
If the market price rises,
firms will increase their
level of output
Price
MC
AC
P2
P1
Q1
Q2
This is the short-run
supply response to an
increase in market price
Quantity
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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
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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.
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Shifts in Supply
Small increase in price,
large drop in quantity
Price
Large increase in price,
small drop in quantity
Price
S’
S’
S
S
P’
P
P’
P
D
D
Q’
Q’ Q
Quantity
Q
Elastic Demand
Quantity
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Inelastic Demand
Shifts in Demand
Small increase in price,
large rise in quantity
Large increase in price,
small rise in quantity
Price
Price
S
S
P’
P’
P
P
D’
D’
D
Q
Q’
Elastic Supply
D
Quantity
Q’
Quantity
Inelastic Supply
Q
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LONG-RUN EQUILIBRIUM
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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.
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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.
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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.
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Long-Run Equilibrium
This is a long-run equilibrium for this industry
Price
P = MC = AC
Price
MC
S
AC
P1
D
q1
Quantity
A Typical Firm
Q1
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Quantity
Total Market
Example continued
• Individual analysis
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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.
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PUTTING IT ALL TOGETHER
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12
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.
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Increase in Demand
This is a long-run equilibrium for this industry
Price
P = MC = AC
Price
MC
S
AC
P1
D
q1
Quantity
A Typical Firm
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Quantity
Q1
Total Market
Increase in Demand
• Suppose that market demand rises to D’
• In the very short run, output is fixed
Price
Price
Market price rises to P1’
MC
S
AC
P1’
P1
D’
D
q1
A Typical Firm
Quantity
Q1
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Quantity
Total Market
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Increase in Demand
• In the short run, each firm increases output to q2
Price
Price
Economic profit > 0
MC
S
AC
P2
P1
D’
D
q1
q2
Quantity
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Quantity
Q1 Q2
A Typical Firm
Total Market
Increase in Demand
• In the long run, new firms enter the industry
Economic profit will return to 0
Price
Price
MC
S
S’
AC
P1
D’
D
q1
Quantity
A Typical Firm
Q1
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Quantity
Q3
Total Market
Increase in Demand
• The long-run supply curve is a horizontal line at p1
Price
Price
MC
S
S’
AC
P1
LS
D’
D
q1
A Typical Firm
Quantity
Q1
Q3
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Quantity
Total Market
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Example
• Market demand: X(p) = 1500 – 50p
• Cost function c(q) = 100 + q2/4
• Long run equilibrium
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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.
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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.
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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.
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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.
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– No problem if have 10 agents and 5 firms.
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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