Economcs 21 Bentz X02 Topic3

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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS ECONOMICS 21

© Andreas Bentz page 1

Dartmouth College, Department of Economics: Economics 21, Summer Dartmouth College, Department of Economics: Economics 21, Summer Dartmouth College, Department of Economics: Economics 21, Summer ‘02‘02‘02

Optional Topic 3:Optional Topic 3:General EquilibriumGeneral Equilibrium

Economics 21, Summer 2002Andreas Bentz

Based Primarily on Varian, Ch. 29-31


ExchangeExchange

Wanna trade?





3

General Equilibrium: ExchangeGeneral Equilibrium: Exchange

Simplest setting: two consumers: person A, person B

two goods: x1, x2

pure exchange (no production)

In a pure exchange economy, a fixed amount of goodsis exchanged. Initially, every consumer is endowed with some of each good;

then they may engage in trade.

This allows us to study how prices change in response torelative scarcity.

How do we represent the possible allocations of thetwo goods between the two consumers? We can represent this in an Edgeworth box.

4

Exchange, cont’dExchange, cont’d

Some definitions:

an allocation X of goods:» bundle (x1

A, x2A) (person A); bundle (x1

B, x2B) (person B)

» This is any distribution of the two goods between the twoconsumers.

» Any allocation is feasible if the amount of good 1 thatperson A holds and the amount of good 1 that person Bholds add up to the total amount of good 1 in theeconomy, and similarly for good 2.

an endowment W (or, initial allocation) of goods:

» bundle (ω1A, ω2

A) (person A); bundle (ω1B, ω2

B) (person B)





5

Feasible AllocationsFeasible Allocations

All allocations in the Edgeworth box are feasible:

W

X

6

Edgeworth BoxEdgeworth Box

Definition: An allocation X is feasible if the totalamount of each good consumed is equal tothe total amount available:

x1A + x1

B = ω1

A + ω1B

x2A + x2

B = ω2

A + ω2B

Any allocation in the Edgeworth box isfeasible.

The initial endowment allocation (ω1A, ω2

A)(person A) and (ω1

B, ω2B) (person B)

determines the size of the Edgeworth box.





7

Edgeworth Box, cont’dEdgeworth Box, cont’d

Now we know how to illustrate all feasibleallocations in our two-consumer economy.

How do we represent preferences?

Each consumer has preferences over the twogoods.

Preferences are represented by indifferencecurves.

8

Building an Edgeworth BoxBuilding an Edgeworth Box

A’s quantity of good 1

A’ s q u an t i t y of g o o d 2

B’s quantity of good 1

B’ s q u an t i t y of g o o d 2





9

At allocation W (endowment), welfare gains for bothconsumers are possible:

Gains from ExchangeGains from Exchange

10

Pareto EfficiencyPareto Efficiency

At X, there are no further gains from trade: X is Pareto efficient.





11

Pareto Efficiency, cont’dPareto Efficiency, cont’d

Definition: Allocation X is a Paretoimprovement over allocation Y if:

every agent prefers (or is indifferent between) her consumption bundle under X to her bundle under Y;

that is: if for every agent allocation X is on a higher (or at least the same) indifference curve.

Definition: Allocation X is Pareto efficient if there is no other allocation that is a Pareto

improvement over X. The locus of all Pareto efficient allocations is the

contract curve.

12

Contract CurveContract Curve

The locus of all Pareto efficient allocations is thecontract curve. The contract curve joins all the tangencies between A’s and

B’s indifference curves.

A’s quantity of good 1

A’

s q u an t i t y of g o o d 2

B ’ s q u a n t i t y o f g o o d 1

B’sq

uantityofgood2

contract curve





13


The definitions are in terms of preferences. We want a criterion that tells us whether an

allocation is “good” in some sense.

Definition: According to the Pareto welfare

criterion, an allocation X is (socially) better

than Y if X is a Pareto improvement over Y.

What is attractive about this definition:» requires only a weak value judgement, and is powerful

and uncontentious;

» most other welfare criteria are contentious.

14


But: The Pareto criterion ranks allocations onlyincompletely. Example 1: If some agents “prefer allocation X to Y”, and

some agents “prefer Y to X”, the Pareto criterion cannot tell uswhich is better.

Example 2: Two Pareto efficient allocations cannot be

compared by the Pareto criterion. And: A Pareto efficient allocation may not have any

other nice properties. Example: Distribution: typically, an allocation where one

individual has everything and everyone else has nothing isPareto efficient.







17

The First TheoremThe First Theorem

Theorem: All competitive market equilibria (or,Walrasian equilibria) are Pareto efficient.

The First Fundamental Theorem of Welfare

Economics (Adam Smith’s “invisible hand”):

“[Every individual] generally, indeed, neither intendsto promote the public interest, nor knows how muchhe is promoting it. … he intends only his own gain,and he is in this, as in many other cases, led by aninvisible hand to promote an end which was no part

of his intention.”» [Smith A (1776) An Inquiry into the Nature and Causes of

the Wealth of Nations Book IV]

18

Alexander Pope, Essay on ManAlexander Pope, Essay on Man

On their own axis as the planets run,

Yet make at once their circle round the sun;

So two consistent motions act the soul;

And one regards itself and one the whole.

Thus God and Nature link’d the gen’ral frame,And bade self-love and social be the same.

Epistle III, An Essay on Man (1733)





19

First Theorem: DiscussionFirst Theorem: Discussion

Informational economy: agents only need toknow the prices they face. Then, the outcomeof market trade will be efficient.

In a two-agent world, this is not an excitingresult. But it holds for large numbers of agents:

a strong case for the market as an allocationmechanism.

20

The Second TheoremThe Second Theorem

Theorem: If preferences are convex, everyPareto efficient allocation can be achieved asthe equilibrium outcome of competitive markettrade.

“The Second Fundamental Theorem of Welfare

Economics”

Or: Given convexity of preferences, we can alwaysfind a set of prices that supports any Paretoefficient allocation as a market equilibrium for anappropriately chosen endowment allocation.





21

Second Theorem: IllustrationSecond Theorem: Illustration

22

Second Theorem: DiscussionSecond Theorem: Discussion

The second theorem is a theorem about theseparation of efficiency (a property of theallocation), and distribution.

Redistribution need not be concerned withefficiency:

We can pick any (Pareto efficient) allocation, andredistribute to an appropriate (not necessarilyPareto efficient) allocation. The market will thenachieve efficiency autonomously.





23

SecondSecond Th’mTh’m: Discussion, cont’d: Discussion, cont’d

Redistribution: all we need to do is: choose the allocation X we like (by some welfare criterion),

calculate the corresponding equilibrium prices,

redistribute endowments to anywhere along the (constructed) budgetline,

then, market trade will automatically achieve efficiency (by the first

theorem).


ProductionProduction

… more opportunities …





25

General Equilibrium: ProductionGeneral Equilibrium: Production

In the pure exchange model, the amounts of good 1 and good 2 in the economy weregiven.

We now study general equilibrium inproduction: how do producers decide howmuch (and using which input mix) to produce?

The quantities of the inputs capital (k) and labor (l)are given: how do firms produce output?

The Edgeworth (production) box contains allfeasible input combinations.

26

ProductionProduction

How much (and how) do firms produce?

Input allocation R is not productively efficient:production of both goods can be increased.

Firm 1’s quantity of l

F i r m 1 ’ s q u an t i t y of k

F i r m 2 ’ s q u a n t i t y o f l

Firm

2’squan

tityofk

contract curve

2’s isoquants:

quantity of

good 2

1’s isoquants:

quantity of

good 1

R





27

Production, cont’dProduction, cont’d

Recall from Topic 2 that profit-maximizing firms alwaysemploy inputs such that the ratio of marginal products(the slope of the isoquant) is equal to the ratio of inputprices.

For firm 1:

And for firm 2:

Since both pay the same input prices, so the isoquants are parallel.

r

w

MP

MPk

l

=

1

1

r

w

MP

MPk

l

=

2

2

MP

MPk

l

=

1

1

MP

MPk

l

2

2


Product MixProduct Mix

Are you being served?





29

Production Possibilities Frontier Production Possibilities Frontier

What are the combinations of outputs thiseconomy could (at best) produce (with givenamounts of inputs)?

For every quantity of good 2, what is the largestquantity of good 1 that can be produced (whenfactors are employed optimally)?

This gives us a schedule of an economy’sproduction possibilities: the different output

combinations the economy can maximallyproduce.

This is the Production Possibilities Frontier (PPF ).

30

PPF, cont’dPPF, cont’d

The contract curve in theEdgeworth (production)box tells us where it isnot possible to increaseproduction of one goodwithout reducingproduction of the other.

It has all the informationwe need for the PPF: Given any quantity of

good 2, what is themaximum that can beproduced of good 1?

Firm 1’s quantity of l

F i r m 1 ’ s q u an t i t y of k

F i r m 2 ’ s q u a n t i t y o f l

F

irm

2’squantityofk

1

2

PPF





31


As we move down the PPF, we gain more of good 1,but have to give up some of good 2. The absolute value of this ratio (the slope of the PPF) is the

marginal rate of transformation (MRT ).

32


What is MRT? As we gain one more unit of good 1, we need resources (k

and l) costing MC1.

How much of good 2 do we need to give up to “free up”enough to buy inputs worth MC1 (to produce this one unit of good 1)?

» If we produce one unit of good 2 less, we free up MC2.» If we produce 1 / MC2 units of good 2 less, we free up $1.

» If we produce MC1 / MC2 units of good 2 less, we free upMC1.

So MRT = MC1 / MC2. In a competitive market, this is equal top1 / p2.




33

PPF and ExchangePPF and Exchange

The economy is productively efficient (it produces on,not inside the PPF): which point on the PPF is chosen (the product mix)

determines the size of the Edgeworth (exchange) box.

34

EfficiencyEfficiency

MRT = MRS is efficient: Suppose MRT < MRS: we could have one more unit of good 1

for less of good 2 than how consumers are willing to substitute1 for 2: we could make consumers better off.

Efficiency in production: economy produces on

PPF

Efficiency in exchange: consumers consume on

contract curve

And: efficient productmix: MRT = p1 / p2 = MRS

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