12. General equilibrium: An exchange economy Varian, Chapter 30.

12. General equilibrium:An exchange economy

Varian, Chapter 30

The simplest market

• Two people, or agents– Agent A and Agent B

• Two goods– Good x and Good y

• Agents interact by exchanging or trading goods• There is no production of either good

Questions we’re interested in

• Will unregulated exchange lead to “good” outcomes?– Under what conditions?

• How does trade occur?– Bargaining?– Using prices?

• Can an outsider (e.g., government) intervene to improve things?

Endowments

Person A

x

y

Ay

Endowment

• Endowments

A has (Ax, A

y)

B has (Bx, B

y)

Ax (A

x+Bx)

(Ay+B

y)• Any possiblebundle ofgoods for A isin this space

Preferences

Person A

Ay

EndowmentA’s indifferencecurves

Ax

x

y

(Ax+B

x)

(Ay+B

y)

The simplest market

• Allocation: A pair of consumption bundles– (xA, yA), (xB, yB)

• Feasible allocation: pair of consumption bundles that add up to total endowment– (xA, yA)+(xB, yB)=(A

x, Ay)+(B

x, By)

The Edgeworth Box: endowments

Person A x

y

Person BBx

Ay

Endowment

By

• Endowments

A has (Ax, A

y)

B has (Bx, B

y)

Ax

• Any feasibleallocation ofgoods amongthe agents isa point in thisbox

The Edgeworth Box: preferences

Person A x

y

Person BBx

Ay

Endowment

By

A’s indifferencecurves

B’s indifferencecurves

Ax

Trade in the Edgeworth Box

Person A x

y

Person BBx

Ay

Endowment

By

A’s indifferencecurves

B’s indifferencecurves

Ax

Points to whichA would be willingto trade

Trade in the Edgeworth Box

Person A x

y

Person BBx

Ay

Endowment

By

A’s indifferencecurves

B’s indifferencecurves

Ax

Points to whichB would be willingto trade

Mutually beneficial trade

Person A x

y

Person BBx

Ay

Endowment

By

A’s indifferencecurves

B’s indifferencecurves

Ax

If trade is voluntarythey should end upsomewhere in here

What does the result look like?

Person A x

y

Person BBx

Ay

Endowment

By

Ax

At thereare no moregains fromtrade

Describing potential outcomes

• Pareto set, or contract curve: The set of all points that could be the outcome of a bargain – Depends on – Can only make one individual better off by

making the other worse off

The Pareto Set

Person A x

y

Person BBx

Ay

Endowment

By

Ax

Allocation if A hasall the bargainingpower

Allocation if B hasall the bargainingpower

Pareto set

Potential bargaining outcomesfrom endowment

Example: Pareto set

• Consumer A:– (A

x, Ay)=(5,10)

– u(xA, yA)=1/3 ln(xA) + 2/3 ln(yA)

• Consumer B:– (B

x, By)=(10,5)

– u(xB, yB)=1/2 ln(xB) + 1/2 ln(yB)

• Find the contract curve

The Pareto Set

Person Ax

yPerson B

Bx

Ay

EndowmentB

y

Ax

Pareto set

155

15

10 5

10

Can we narrow down outcomes?

• So far we’ve said nothing about the mechanism or process by which people trade

• We’ve found that agents should get to the contract curve…..

• …..but we’re not sure what point they’ll reach on that curve

• If trading is via prices, this indeterminacy can be resolved

What about prices?

• A price-based process– Set py=1– Try a value of px=p– Gives slope of budget line for both consumers– Find gross demands of both consumers– Vary p until demand is a feasible allocation

• Excess supply of x– Lower p

• Excess demand of x– Raise p

Gross demand A

Person A x

y

Person BBx

Ay

Endowment

By

Ax

Gross demand B

Person A x

y

Person BBx

Ay

Endowment

By

Ax

Market equilibrium

Person A x

y

Person BBx

Ay

Endowment

By

Ax

At these prices, excessdemand and excesssupply are both zero

Prices and excess supply

Person A x

y

Person BBx

Ay

Endowment

By

Ax

Budget constraint

Amount of x thatB wants to buy

Amount of x thatA wants to sell

Excess supply of x

Prices and excess demand

Person A x

y

Person BBx

Ay

Endowment

By

Ax

Budget constraint

Amount of y thatB wants to sell

Amount ofy that A wants tobuy

Excessdemand for y

A’s price offer curve

Person Ax

y

Person BBx

Ay

Endowment

By

Ax

Agent A’s priceoffer curve

B’s price offer curve

Person A x

y

Person BBx

Ay

Endowment

By

Ax

Agent B’s priceoffer curve

Using price offer curves to find the market equilibrium

Person A x

y

Person BBx

Ay

Endowment

By

Ax

Agent B’s priceoffer curve

Agent A’s priceoffer curve

Example: Price offer curves

• Consumer A:– (A

x, Ay)=(5,10)

– u(xA, yA)=1/3 ln(xA) + 2/3 ln(yA)

• Consumer B:– (B

x, By)=(10,5)

– u(xB, yB)=1/2 ln(xB) + 1/2 ln(yB)

• Find the IOCs and the equilibrium allocation and price.

The Pareto Set

Person Ax

yPerson B

Bx

Ay B

y

Ax

Pareto set

155

15

10 5

10

EndowmentPOC A

POC B

6.43

8.57

9 6

The First Fundamental Theorem of Welfare Economics

• The First Fundamental Theorem of Welfare Economics– All equilibria resulting from a competitive

market are Pareto efficient– There are no gains from trade available from

the result of a price-based exchange

• There may be other undesirable properties of a market outcome

The First Fundamental Theorem of Welfare Economics

Person A x

y

Person BBx

Ay

Endowment

By

Ax

The market equilibriumallocation is on the contractcurve, so is Pareto efficient

Contract curve

The Second Fundamental Theorem of Welfare Economics• The Second Fundamental Theorem of

Welfare Economics– If preferences are convex, any Pareto efficient

allocation can be reached by a competitive market, with the correct redistribution of the endowments

– Using redistribution then allowing price-based trade, a planner can choose any PE allocation

The Second Fundamental Theorem of Welfare Economics

Person A x

y

Person BBx

Ay

Endowment

By

Ax

Can the market get usto any Pareto efficientallocation we want?

Pareto efficientallocations

Redistribute goodx from B to A

Answer: Yes, withconvex preferencesand redistribution

Trade

Policy implications

• How to help the poor– Give them money!– Price controls, quantity controls, etc lead to

inefficient outcomes