General Equilibrium (without Production) or Exchange ...econ.ucsb.edu/~grossman/teaching/Econ100B_Winter... · General vs. Partial Equilibrium The Edgeworth Box The Contract Curve

General vs. Partial Equilibrium The Edgeworth Box The Contract Curve The Core

General Equilibrium (without Production)or

Exchange(Chapter 31)


General Equilibrium

• Events in one market have effects on other markets (spillovers)

• Demand for x depends upon prices of complements,substitutes; income

• Supply of x depends upon factor prices

• Previously, we’ve taken these as given– doing partialequilibrium analysis

• But its important to understand interdependence of markets–general equilibrium analysis

Partial equilibrium analysis says that competitive markets yieldefficient outcomes—is this still true in general equilibrium?


General Equilibrium

Our approach:

• Simple environment—the entire economy• 2 kinds of goods• 2 people

• Focus on exchange• Abstract away from production of new goods• Give people endowments• Specify preferences• Allow them to trade

• Make predictions about behavior of utility-maximizers

• Evaluate welfare


Endowment Economy

• Consumers A and B; goods 1 and 2

• Endowments: ωA = (ωA1 , ω

A2 ) and ωB = (ωB

1 , ωB2 )

• Example: ωA = (6, 4) and ωB = (2, 2)

• This means total endowment of good 1 isωA1 + ωB

1 = 6 + 2 = 8 and of good 2 is ωA2 + ωB

2 = 4 + 2 = 6


Allocations

• Endowment represents where people start, but through trade,their allocations may change

• General allocation or consumption: xA = (xA1 , xA2 ) and

xB = (xB1 , xB2 )

• (xA, xB) is feasible if it uses at most the aggregateendowment:

xA1 + xB1 ≤ ωA1 + ωB

1 and xA2 + xB2 ≤ ωA2 + ωB

2

• Helpful graphical tool: Edgeworth Box

• Allows us to simply depict all feasible allocations


Edgeworth Box

Edgeworth Box

A( , )6 4

B( , )2 2

OA

OB

6

8

4

6

2

2

Theendowmentallocation

The Endowment Allocation


Edgeworth Box

Edgeworth Box

Feasible Reallocations

OA

OB

1 1A B

xA2

2

2

A

B

xA1

xB1

xB2


How do we think about equilibrium?

• In partial equilibrium analysis:• Treat each good separately• Find p and q that equate supply and demand

• But this is general equilibrium analysis: where do supply anddemand come from?

• A and B can trade with each other

• For everything to be balanced, the amount that A gives uphas to equal amount that B receives (for each good, and viceversa)

• In other words Supply = Demand for each good

• This will determine prices for each good

• How do we find supply and demand curves?

• Go back to utility maximization problem

• Need to specify preferences to do this


Utility maximization

• Preferences are given

• Given prices for each good, endowment bundle serves asincome

• Can write down budget constraint

p1x1 + p2x2 ≤ p1ω1 + p2ω2

• Solve utility maximization problem

• Gives you optimal allocation, as a function of price ratio

• x∗1 − ω1 > 0 means person demands more of good 1

• x∗1 − ω1 < 0 means person is willing to supply good 1

• Key question: what prices will make it so that A demandexactly as much of each good as B supplies?


Consumer A’s Preferences

Preferences of A

Adding Preferences to the Box

2A

1A

xA2

xA1

More

preferred

For consumer A.

OA


Consumer B’s Preferences

Preferences of B


xB2

xB1

More

preferred

For consumer B.

OB

2B

1B


Consumer B’s Preferences

Preferences of B


2B

1B

xB1

xB2

More

preferred

For consumer B. OB


Putting Them Both Together

Putting both of them together

Edgeworth’s Box

2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB


Pareto-improving allocations

• Given a particular allocation, a Pareto-improving allocationimproves the welfare of at least one consumer withoutreducing the welfare of another.

• How do we depict Pareto-improving allocations in theEdgeworth box?


Pareto-Improving Allocations

Putting both of them together

Edgeworth’s Box

2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB


Pareto-Improving Allocations

Pareto-improving allocations

Pareto-Improvements

2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB

The set of Pareto-improving allocations


Pareto-optimal allocations

• An allocation is Pareto-optimal if it is feasible and there isother feasible allocation that is a Pareto-improvement over it.

• In other words, there is no was to make anyone better offwithout making someone worse off.

• The set of all Pareto-optimal allocations is called the contractcurve.


A Pareto-Optimal Allocation

A Pareto-Optimal Allocation

Pareto-Optimality

A is strictly better offbut B is strictly worse

off

B is strictly betteroff but A is strictlyworse off

Both Aand B are

worseoff

Both A andB are worseoff


The Set of Pareto-Optimal Allocations

Pareto-Optimal Allocations

Pareto-Optimality

2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB

All the allocations marked bya are Pareto-optimal.

The contract curve


Pareto-optimal Allocations

• From the figures, we can see that an allocation at which theindifference curves of the two consumers are tangent must bePareto-optimal

• Tangency implies they have the same slope

• What is the slope of an indifference curve? The Marginal rateof substitution (MRS)!

• Condition for Pareto-optimality:

MRSA =

∂uA(xA1 ,xA2 )

∂xA1∂uA(xA1 ,x

A2 )

∂xA2

=

∂uB(xB1 ,xB2 )

∂xB1∂uB(xB1 ,xB2 )

∂xB2

= MRSB

• We also require feasibility:

xA1 + xB1 = ωA1 + ωB

1 and xA2 + xB2 = ωA2 + ωB

2


ExampleIdentifying Pareto-optimal allocations• Recall total endoments: ωA

1 + ωB1 = 6 + 2 = 8 and

ωA2 + ωB

2 = 4 + 2 = 6• Let uA(xA1 , x

A2 ) = ln(xA1 ) + 2 ln(xA2 ) and

uB(xB1 , xB2 ) = ln(xB1 ) + 2 ln(xB2 ).

• MRS of consumer A:

MRSA =

1xA12xA2

=xA2

2xA1

• Similarly,

MRSB =

1xB12xB2

=xB2

2xB1

• So a Pareto-optimum is a feasible allocation for which

xA22xA1

=xB2

2xB1


ExampleFinding all Pareto-optimal allocations (deriving the contract curve)

• We can simplify tangency condition to:

xA2xA1

=xB2xB1

• Recall endowment/feasibility requirement:

xA1 + xB1 = 8 and xA2 + xB2 = 6

• Re-write tangency condition, substituting xB1 = 8− xA1 andxB2 = 6− xA2 :

xA2xA1

=6− xA28− xA1

or

xA2 =3

4xA1

• This is the equation of the contract curve.• In this case, it’s just the diagonal of the rectangle


Core Allocations

How can we narrow down our prediction about the outcomeresulting from trade?

• We’ve derived the contract curve: xA2 = 34x

A1

• Note that clicker options D and E are both on this curve.

• However, Pareto Optimal, but given her endowment,(ωB = (2, 2), Person B would never agree to trade toallocation E = (6,4.5,2,1.5).

• We need to restrict attention to Pareto-optimal allocationsthat are Pareto-improvements over the initial endowment.

• These allocations are called core allocations, or the core—theset of all PO allocations that are welfare improving for bothconsumers relative to their own endowments

• An allocation will be in the core if it is feasible and it is notblocked by any consumer


Core Allocations

Core Allocations

The Core

2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB

Pareto-optimal trades blockedby B

Pareto-optimal trades blockedby A


The Core

Core Allocations

The Core

2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB

Pareto-optimal trades not blockedby A or B are the core.

Documents

General Equilibrium (without Production) or Exchange ...econ.ucsb.edu/~grossman/teaching/Econ100B_Winter... · General vs. Partial Equilibrium The Edgeworth Box The Contract Curve