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General Equilibrium: Basics

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### General Equilibrium: Basics

Almost essential

A Simple Economy

Useful, but optional

Firm: Optimisation

Consumer Optimisation

MICROECONOMICS

Principles and Analysis

Frank Cowell

Note: the detail in slides marked “ * ” can only be seen if you run the slideshow

Limitations of Crusoe model

- The Crusoe story takes us only part way to a treatment of general equilibrium:
- there's only one economic actor…
- …so there can be no interaction

- Prices are either exogenous (from the mainland? the world? Mars?) or hypothetical
- But there are important lessons we can learn:
- integration of consumption and production sectors
- decentralising role of prices

When we use something straight from Crusoe we will mark it with this logo

Onward from Crusoe…

- This is where we generalise the Crusoe model
- We need a model that will incorporate:
- many actors in the economy…
- …and the possibility of their interaction
- the endogenisation of prices in the economy

- But what do we mean by an “economy”…?
- We need this in order to give meaning to “equilibrium”

Overview…

General Equilibrium: Basics

The economy and allocations

The components of the general equilibrium problem

Incomes

Equilibrium

The components

- At a guess we can model the economy in terms of:
- Resources
- People
- Firms

- Specifically the model is based on assumptions about:
- Resource stocks
- Preferences
- Technology

- (In addition –for later – we will need a description of the rules of the game)

What is an economy?

- Resources (stocks)

R1 , R2 ,…

n of these

- Households (preferences)

nh of these

U1, U2 ,…

- Firms (technologies)

nfof these

F1, F2,…

An allocation

A competitive allocation consists of:

Note the shorthand notation for a collection

- A collection of bundles (one for each of the nh households)
- A collection of net-output vectors (one for each of the nf firms)

utility-maximising

⋌

[x] := [x1, x2, x3,… ]

profit-maximising

⋌

[q] := [q1, q2, q3,… ]

- A set of prices (used by households and firms)

p := (p1, p2, …, pn)

How a competitive allocation works

- Implication of firm f’s profit maximisation

- Firms' behavioural responses map prices into net outputs

p qf(p)

{ , f=1,2,…,nf}

- Implication of household's utility maximisation

- Households’ behavioural responses map prices and incomes into demands

p, yhxh(p)

{ } { , h=1,2,…,nh}

- The competitive allocation

just a minute! Where do these incomes come from??

An important model component

An important missing item

- For a consumer in isolation it may be reasonable to assume an exogenous income
- Derived elsewhere in the economy

- Here the model involves all consumers in a closed economy
- There is no “elsewhere”

- Incomes have to be modelled explicitly
- We can learn from the “simple economy” presentation

Overview…

General Equilibrium: Basics

The economy and allocations

A key role for the price system

Incomes

Equilibrium

Modelling income

- What can Crusoe teach us?
- Consider where his “income” came from
- Ownership rights of everything on the island

- But here we have many persons and many firms
- So we need to proceed carefully
- We need to assume a system of ownership rights

What does household h possess?

- Resources

Rih 0,

i=1,…,n

R1h, R2h, …

0 Vfh 1,

f =1,…,nf

V1h, V2h, …

- Shares in firms’ profits

introduce prices

The components of h’s income

Incomes- Resources

- Shares in firms

Rents

- Net outputs

- Prices

look more closely at the role of prices

Profits

The fundamental role of prices

- Net output of i by firm f depends on prices p:
- qif = qif(p)

- Supply of net outputs

- Thus profits depend on prices:
- n
- P f(p):= S piqif(p)
- i=1

- Again writing profits as price-weighted sum of net outputs

indirectly

directly

- So incomes can be written as:
- n nf
- yh= S piRih+SVfhPf(p)
- i=1 f=1

- Income = resource rents + profits

Holding by h of shares in f

Holding by h of resource i

- Income depends on prices:
- yh= yh(p)

- yh(•) depends on ownership rights thathpossesses

Prices in a competitive allocation

- The allocation as a collection of responses

p qf(p)

{ , f=1,2,…,nf}

- Put the price-income relation into household responses

- Gives a simplified relationship for households

p, yhxh(p)

{ } { , h=1,2,…,nh}

p

- Summarise the relationship

yh= yh(p)

[q(p)]

Let's look at the whole process

p

[x(p)]

share ownership

R1a, R2a, …

V1a, V2a, …

R1b, R2b, …

V1b, V2b, …

…

…

The price mechanism*- System takes as given the property distribution

- Property distribution consists of two collections

- Prices then determine incomes

- Prices and incomes determine net outputs and consumptions

a

d

- Brief summary…

[y]

[q(p)]

[x(p)]

prices

allocation

distribution

Overview…

General Equilibrium: Basics

The economy and allocations

Specification and examples

Incomes

Equilibrium

What is an equilibrium?

We just copy and slightly modify our earlier work

- What kind of allocation is an equilibrium?
- Again we can learn from previous presentations:
- Must be utility-maximising (consumption)…
- …profit-maximising (production)…
- …and satisfy materials balance (the facts of life)

- We can do this for the many-person, many-firm case

Competitive equilibrium: basics

- For each h,maximise

- Households maximise utility, given prices and incomes

Uh(xh), subject to

n

Spi xih yh

i=1

- Firms maximise profits, given prices

- For each f,maximise

- For all goods the materials balance must hold

n

S pi qif, subject toFf(qf ) 0

i=1

- For each i:

xi£ qi + Ri

what determines these aggregates?

aggregate consumption of good i

aggregate stock of goodi

aggregate net output of good i

Consumption and net output

- “Obvious” way to aggregate consumption of good i?
- nh
- xi=S xih
- h=1

- Appropriate if i is a rival good
- Additional resources needed for each additional person consuming a unit of i

Sum over households

- Opposite case: a nonrival good
- Examples: TV, national defence…

- An alternative way to aggregate:
- xi= max {xih}
- h

- Aggregation of net output:
- nf
- qi:= Sqif
- f=1

- if all qf are feasible will q be feasible?
- Yes if there are no externalities
- Counterexample: production with congestion…

By definition

To make life simple:

- Assume incomes are determined privately
- All goods are “rival” commodities
- There are no externalities

Competitive equilibrium: summary

- It must be a competitive allocation

- A set of prices p
- Everyone maximises at those prices p

- The materials balance condition must hold

- Demand cannot exceed supply:
- x ≤q + R

An example

- Exchange economy (no production)
- Simple, standard structure
- 2 traders (Alf, Bill)
- 2 Goods:

Alf__

Bill__

- resource endowment

(R1a, R2a)

(R1b, R2b)

- consumption

(x1a, x2a)

(x1b, x2b)

- utility

Ua(x1a, x2a)

Ub(x1b, x2b)

diagrammatic approach

Alf’s optimisation problem

- Resource endowment

Increasing preference

- Prices and budget constraint

x2a

- Preferences

- Equilibrium

- Ra

R2a

- x*a

- Budget constraint is
- 22
- Spi xia≤Spi Ria
- i=1i=1

- Alf sells some endowment of 2 for good 1 by trading with Bill

Oa

x1a

R1a

Bill’s optimisation problem

- Resource endowment

Increasing preference

- Prices and budget constraint

x2b

- Preferences

- Equilibrium

- x*b

- Budget constraint is
- 22
- Spi xib≤Spi Rib
- i=1i=1

- Rb

R2b

- Bill, of course, sells good 1 in exchange for 2

Ob

R1b

x1b

R1b

x1b

Ob

x2a

R2b

R2a

x2b

Oa

x1a

R1a

Combine the two problems- Bill’s problem (flipped)

- Superimpose Alf’s problem

Incomes from the distribution…

- Price-taking trade moves agents from endowment point…

- …to the competitive equilibrium allocation

- [R]

…match expenditures in the allocation

- The role of prices

- [x*]

- This is the Edgeworth box
- Width: R1a + R1b
- Height: R2a + R2b

Alf and Bill as a microcosm

The Crusoe equilibrium story translates to a many-person economy

Role of prices in allocations and equilibrium is crucial

Equilibrium depends on distribution of endowments

Main features are in the model of Alf and Bill

But, why do these guys just accept the going prices…?

See General Equilibrium: Price-Taking

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