PARTIAL EQUILIBRIUM Positive Analysis

[See Chap 12 ]

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.

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.

1 Model • We are interested in market 1.

– Price is denoted by p 1, or p. – Firms/Consumers face same price (law of one price). – Firms/Consumers are price takers.

Model • There are J agents who demand good 1.

– Agent j has income m j

– Utility u j(x 1,…,x N)

– Prices {p 1,…,p N}, with {p 2,…,p N} exogenous.

• There are K firms who supply good 1.

– Firm k has technology f k(z 1,…,z M)

– Input prices {r 1,…,r M} exogenous.

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.

2 MARKET DEMAND

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.

Market Demand

• Assume there are two goods, x1 and x2.

• Agent j’s Marshallian demand for x1 is j x1 (p1,p2,mj) • The market demand is the sum of individual Marshallian demands: J = = j Market demand X 1 ( p1, p2 ;m1,..., mJ ) ∑ x1 ( p1, p2 , m j ) i=1

3 Market Demand To derive the market demand curve, we sum the quantities demanded at every price

p1 p1 Individual A’s Individual B’s p1 Market demand demand curve demand curve curve

A B x1 x1 X1

x’ x1 x’’ x1 x’+x’’ x1

A B x1 + x2 = X1

Example

• There are 1000 identical consumers each with Marshallian demand: j x1 (p1,p2,mj) = 10 - 0.1p

• The market demand function is given by

1000 1000 = j = − = − X1 ∑x1 ∑10 0.1p 10 ,000 100 p j=1 j =1

MARKET SUPPLY

4 Market Supply

• The market supply curve is given by the sum of individual firms’ supply curves:

K = = k Market supply Q ∑ q (p,r 1, r2 ) k =1

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. 14

Short-Run Market Supply Curve To derive the market supply curve, we sum the quantities supplied at every price

Firm A’s P supply curve P P qB qA Firm B’s Market supply Q supply curve curve P

qA’ quantity qB’ quantity qA’+ qB’’ Quantity

qA+ qB = Q

5 Example

• There are 100 identical firms each with the supply curve

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

• The market supply function is given by

100 100 = = 10 p = 1000 p Q ∑qk ∑ k=1 k =1 3 3

SHORT -RUN EQUILIBRIUM

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. 18

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.

Equilibrium Price Determination

The interaction between Price market demand and market S supply determines the equilibrium price

Q1 Quantity 20

Example: Demand Side • 15 Agents

– All have utility u(x 1,x 2)=x 1x2 – 10 have income m=10 – 5 have income m=5 • Demand j j x1 = m /2p 1 • Market demand

X1 = 200/2p 1 = 100/p 1

7 Example: Supply Side • 9 Firms 1/3 1/3 – All have technology f(z 1,z 2)=(z 1-1) (z 2-1) • Cost curve 1/2 3/2 c(q) = 2(r 1r2) q + (r 1+r 2)

In short-run no shutdown so c(0)=r 1+r 2. 2 • Profit max supply: q*(p,r 1,r 2) = p /9r 1r2 2 • Market supply: Q(p,r 1,r 2) = p /r 1r2 • Equilibrium price: 1/3 p* = (100r 1r2) 22

Market Reaction to a Shift in Demand

Start at equilibrium. Price S

Q1 Quantity 23

Market Reaction to a Shift in Demand If many buyers experience an increase in their demands, Price the market demand curve S will shift to the right

P2 Equilibrium price and equilibrium quantity will P1 D’ both rise

Q1 Q2 Quantity 24

8 Market Reaction to a Shift in Demand

If the market price rises, Price MC firms will increase their level of output

P2 This is the short-run supply response to an P 1 increase in market price

Q1 Q2 Quantity 25

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 26

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.

9 Shifts in Supply Small increase in price, Large increase in price, large drop in quantity small drop in quantity

Price S’ Price S’ S S

P’ P’ P P

D D

Q’ Q QuantityQ’Q Quantity Elastic Demand Inelastic Demand 28

Shifts in Demand Small increase in price, Large increase in price, large rise in quantity small rise in quantity

Price Price S

P’ P’ P P

D’ D’ D D

Q Q’ QuantityQQ’ Quantity Elastic Supply Inelastic Supply 29

LONG -RUN EQUILIBRIUM

10 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.

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.

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.

11 Long-Run Equilibrium

This is a long-run equilibrium for this industry P = MC = AC Price Price MC S

q1 Quantity Q1 Quantity34 A Typical Firm Total Market

Example continued • Individual analysis 1/2 3/2 – Firm’s cost: c(q) = 2(r 1r2) q + (r 1+r 2) 1/2 1/2 -1 – Hence AC(q) = 2(r 1r2) q + (r 1+r 2)q -1/3 2/3 – AC is minimized at q* = (r 1r2) (r 1+r 2) 1/3 1/3 – Price is p* = AC(q*) = 3(r 1r2) (r 1+r 2)

• Market analysis ⅓ -1/3 -1/3 – Demand X(p*) = 100/p* = 33 (r 1r2) (r 1+r 2) ⅓ -1 – Number of firms = X(p*)/q* = 33 (r 1+r 2)

• The math here is a bit messy. Let r 1=r 2=1. 35

PUTTING IT ALL TOGETHER

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. 37

Increase in Demand

This is a long-run equilibrium for this industry P = MC = AC Price Price MC S

q1 Quantity Q1 Quantity38 A Typical Firm 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

P1’

P1 D’ D

q1 Quantity Q1 Quantity39 A Typical Firm Total Market

13 Increase in Demand

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

Economic profit > 0 Price Price MC S

P1 D’ D

q1 q2 Quantity Q1 Q2 Quantity40 A Typical Firm Total Market

Increase in Demand

• In the long run, new firms enter the industry

Economic profit will return to 0 Price MC Price S S’

P1 D’ D

q1 Quantity Q1 Q3 Quantity41 A Typical Firm Total Market

Increase in Demand

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

Price Price MC S S’

P1 LS D’ D

q1 Quantity Q1 Q3 Quantity42 A Typical Firm Total Market

14 Example • Market demand: X(p) = 1500 – 50p

• Cost function c(q) = 100 + q 2/4

• 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.

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. 44

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.

15 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. 46

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.