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Prerequisites. Almost essential A Simple Economy Useful, but optional Firm: Optimisation Consumer Optimistion. General Equilibrium: Basics. MICROECONOMICS Principles and Analysis Frank Cowell . November 2006 . Limitations of Crusoe model.

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general equilibrium basics

Prerequisites

Almost essential

A Simple Economy

Useful, but optional

Firm: Optimisation

Consumer Optimistion

General Equilibrium: Basics

MICROECONOMICS

Principles and Analysis

Frank Cowell

November 2006

limitations of crusoe model
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
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
Overview...

General Equilibrium: Basics

The economy and allocations

The components of the general equilibrium problem.

Incomes

Equilibrium

the components
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
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
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
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, yhxh(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
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.
overview1
Overview...

General Equilibrium: Basics

The economy and allocations

A key role for the price system.

Incomes

Equilibrium

modelling income
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
What does household h possess?

Rih  0,

i =1,…,n.

  • Resources

R1h, R2h, ...

0  Vfh  1,

f =1,…,nf.

V1h, V2h, ...

  • Shares in firms’ profits

introduce prices

incomes

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
The fundamental role of prices

firm f, good i.

n-vector of prices

  • Net outputs depend on prices:

qif = qif(p)

  • Supply of net outputs
  • Thus profits depend on prices:
  • n
  • Pf(p):=S pi qif(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 pi Rih+SVfh Pf(p)
  • i=1 f=1

Holding by h of resource i

Holding by h of shares in f

  • Incomes = resource rents + profits
  • Note that the function yh(•) depends on the ownership rights that h possesses
  • Incomes depend on prices:

yh= yh(p)

prices in a competitive allocation
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, yhxh(p)

{ } { , h=1,2,...,nh}

p

  • Summarise the relationship

yh= yh(p)

[q(p)]

Let's look at the whole process

p

[x(p)]

the price mechanism

resource distribution

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

overview2
Overview...

General Equilibrium: Basics

The economy and allocations

Specification and examples

Incomes

Equilibrium

what is an equilibrium
What is an equilibrium?
  • 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.

We just copy and slightly modify our earlier work

competitive equilibrium basics
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 all goods the materials balance must hold
  • For each f,maximise

n

Spi qif, subject toFf(qf ) 0

i=1

aggregate stock of good i.

  • For each i:

what determines these aggregates?

xi£ qi + Ri

aggregate consumption of good i.

aggregate net output of good i.

consumption and net output
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
  • Full additional resources are needed for each additional person consuming a unit of good i.

Sum over households

  • An alternative way to aggregate:
  • xi=max {xih }
  • h
  • Opposite case: a nonrival good
  • Examples: TV, national defence...
  • Aggregation of net output:
  • nf
  • qi:= S qif
  • f=1
  • if all the qf are feasible will q be feasible?
  • Yes if there are no externalities
  • Counterexample: production with congestion...

By definition

to make life simple
To make life simple:
  • Assume incomes are determined privately.
  • All goods are “rival” commodities.
  • There are no externalities.
competitive equilibrium summary
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
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
Alf’s optimisation problem
  • Resource endowment

Increasing preference

  • Prices and budget constraint

x2a

  • Preferences
  • Equilibrium
  • Ra

R2a

  • x*a
  • Budget constraint is

22

S pi xia≤S pi Ria

i=1 i=1

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

Oa

x1a

R1a

bill s optimisation problem
Bill’s optimisation problem
  • Resource endowment

Increasing preference

  • Prices and budget constraint

x2b

  • Preferences
  • Equilibrium
  • x*b
  • Budget constraint is

22

S pi xib≤S pi Rib

i=1 i=1

  • Rb

R2b

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

Ob

R1b

x1b

combine the two problems

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
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.