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