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Lecture1

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Lecture 1: General Equilibrium

Mauricio Romero

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Introduction

Pure Exchange Economies

Pareto efficiency

Edgeworth Box


Introduction

Previous classes

.., Consumers behavior ( decision theory) was often analyzed separately from firm behavior (producer theory)

.., When analyzed together, each market was viewed in isolation

Previous classes

.., Consumers behavior ( decision theory) was often analyzed separately from firm behavior (producer theory)

.., When analyzed together, each market was viewed in isolation

.., But markets are often intertwined

Previous classes

.,. Consumers behavior ( decision theory) was often analyzed

separately from firm behavior (producer theory)

.,. When analyzed together, each market was viewed in isolation

.,. But markets are often intertwined

II> Transportation: Uber/ metro/ ecobici / car

II> Wages across sectors

11> Fruits

11> Beer and tacos

Example - Fruits

.,. Suppose that apple and bananas are substitutes

Apples Bananas

p p

ss ss

DD DD

~-------Q ~---------+Q

Example - Fruits

.,. Suppose that apple and bananas are substitutes

.,. Supply curve for apples shifts out

Apples

p

ss , SS '

DD

~---------;Q

Example - Fruits

.., Suppose that apple and bananas are substitutes

.., Supply curve for apples shifts out

.., DD for bananas decreases (exogenous)

Apples Bananas

p p

ss - ss· A'l' i

DD

Q (

'aQ ~

Example - Fruits




.., DD for apples decreases (exogenous) - maybe a little

Apples

p

ss SS '

- BB·

Bananas

p

ss

DD

' DD'

Q

~

ss

DD

~---~----+Q ' DD'

~---------tQ

Example - Fruits




.., DD for apples decreases (exogenous) - maybe a lot

Apples Bananas

p p

ss , SS '

DD

- DD '

Q

~

ss

DD

- DD '

Q

~

-Q.~2'

-Q.~2'

Example - Fruits

.., What happens if apple and bananas are

A tour down memory lane

.., Leon Walras started it all (1834-1910)

.., First to use mathematical tools in economics

.., Supply and demand curves as solutions to a maximization problem

.., Started th

.., Walras was ultimately afte uestions (is the market economy good?) \."--""E,a_,.-;

.., But first, he tackled positive questions (is there an

equilibrium? is it unique?)

.., Made a lot of progress. In particular came up with "Walras Law": Sum of the values of excess demands across all markets

must equal zero always


.., Vi lfredo Pareto was Wa I ras student ( 1848-1923)

.., Abandoned utilitarianism (i.e., utility functions)

.., Embraced "preferences"

.., Utility functions only have ordinal content

.., Comparing "utils" across individuals is meaningless

.., (Pareto) optimum/ efficiency: Achieved if we can't make someone better-off without making someone worst-off


1J>, Francis Edgeworth (1845 - 1926) ~

.,. Introduced indifference curves

.,. Was the first to ask: Where will voluntary exchange lead to?

.,. He conjecture his result was aligned with Walras' result


.,. Existence

.,. Showed it was Pareto efficient

1J>, Two Nobel prizes (Arrow - 1Q72 and Debreu - 1974) - ~ ~


Introduction


Pareto efficiency

Edgeworth Box




.., How are goods distributed among consumers?

.., What incentives are there to exchange goods? What institutions mediate the exchange?

.., Is there a distribution of goods that leaves everyone satisfied and there aren't any incentives to deviate?


.., What are the properties of such an equilibrium?

.., Is it unique?

.., Is it stable?

.., Is it efficient?


.,. Assume there are ----~ ...

.,_ L goods,£ = {1 , ... , L} ~

.,_ Each consumer i is characterized by a utility function lJ)

.,_ Each consumer can consume goods i\J E i!;J

.,_ Each consumer has an initial endowment of Q E IR~ .

.,_ Each consumer is characterized by the pair~

.,_ Assume the utility functions represent neoclassic preferences ______.

Utility functions and neoclassic preferences

.,. A brief reminder


.,. A brief reminder

.,. Utility functions are ordinal not cardinal


... A brief reminder

... Utility functions are ordinal not cardinal

... They are used to represent preferences



... Utility functions are ordinal not cardinal


... If x >-- ; y then u;(x) > u;(y)

11> If f is any increasing function then f(u;(x)) > f(u;(y))

11> Hence f(ut)) also represents >-- ;

11> u;(x) > u;(y) mea ns something, but u;(x) - u;(y) does not

... Neoclassic preferences are well behaved



... Utility functions are ordinal not cardinal ~


11> (?', >--G)hen u;(x) > u;(y)

II> If f is any increasing function then f(u;(x)) > f(u;(y))

11> Hence f(ut)) also represents >-- ;

11> u;(x) > u;(y) means something, but u;(x) - u;(y) does not

... Neoclassic preferences are well behaved

11> They can be represented by a utility function

11> They are weakly monotonic

II> They are quasi-concave


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Ca~owb Definition (Exchange economy) /

A pure exchange economy 1s £ = {!) ( u1 , w 1) iEI) where I is the

set of agents, u1 is a representation of consumer i 's preferences and wi is consumer i 's initial endowment.

'-----...r,\>J( tt~

.., Let = tQ be the total ~ndowment of the economy. f.,:.. L 1= 1 4 v-lfJ ttl.,. ~"p 1,.f, l't

.., An ~ of resources is denoted by0= (x1,x2 , ... ,{0

whe,c~lW\4 ~tJ ~~ fQ'.:


Definition (Feasible allocation)

The set ~ allocation F of an economy

£ = Q, ( ui, wi) iEI) is defined by: j17 t,.. :;;;.:__g_.. ~ =:c. vJ E, Iv

I I

F = { x = (x1,x2, ... , x 1): xi E IRt, I>i C).l 1= 1 ~ f

4tt


Introduction


Pareto efficiency

Edgeworth Box


Pareto efficiency

Pareto efficiency

Let £ be an economy. A feasibl allocation of resources x = (x1, x2 , ... , x 1) is Pareto efficient if there isn 't anotht,r_ feasible

al_location x =: (x1, x2 , ... , x1) such that for everY. aJ:~ · • .• ':JZ] ;:::~ (x'2 and for at least one agent i*, u' (x' l,j u' (x' ).

Pareto efficiency

We say that x Pareto

( Ai Ai f'>r ( i i) u; ~ 'l:Ju; x1 , ... , xL

and there is at least one consumer j for which

Uj(x{ , ... , xi_ G}uj(x{ , ... , x{_).

Thinking about Pareto efficiency

vtJ 0.?, NADA-- Lo

Vof--1' tvil

--lo~


.., If x is a Pareto efficient feasible allocation, does it mean that x Pareto dominates all other feasible allocations?

.., If there are two allocations 0 nc@s it always the case that one Pareto dominates the other?

.., For Pareto efficiency, the initial endowments only matter in

the sense that they determined the total endowment of the economy

.., Social planner should strive to achieve Pareto efficiency at the very leastl


.., If x is a Pareto efficient feasible allocation, does it mean that

x Pareto dominates all other feasible allocations?

.., If there are two allocations (x and y) is it always the case that

one Pareto dominates the other?

.., For Pareto efficiency, the initial endowments only matter in the sense that they determined the total endowment of the economy

... ive ieve Pareto efficiency at the owever, she may have other concerns such as


.., The set of all Pareto allocations is known as the contract

curve

........._

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Introduction


Pareto efficiency

Edgeworth Box


Edgeworth Box 4 2,. ~ I~

Edgeworth Box

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Edgeworth Box

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The Edgeworth box

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Edgeworth Box

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