Transcript General Equilibrium: Basics

Prerequisites
Almost essential
A Simple Economy
Frank Cowell: Microeconomics
Useful, but optional
Firm: Optimisation
Consumer Optimistion
November 2006
General Equilibrium: Basics
MICROECONOMICS
Principles and Analysis
Frank Cowell
Limitations of Crusoe model
Frank Cowell: Microeconomics
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The Crusoe story takes us only part way to a
treatment of general equilibrium:
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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:
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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...
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This is where we generalise the Crusoe model.
We need a model that will incorporate:
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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...
Frank Cowell: Microeconomics
General Equilibrium:
Basics
The economy
and allocations
The components
of the general
equilibrium
problem.
Incomes
Equilibrium
The components
Frank Cowell: Microeconomics
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At a guess we can model the economy in terms of:
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Specifically the model is based on assumptions
about:
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Resources
People
Firms
Resource stocks
Preferences
Technology
(In addition –for later – we will need a description
of the rules of the game)
What is an economy?
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Resources
(stocks)
R1 , R2 ,...
n of these
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Households
(preferences)
U1, U2 ,...
nh of these
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Firms
(technologies)
F1, F2 ,...
nf of these
An allocation
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A competitive allocation consists of:
Note the shorthand notation
for a collection
utility-maximising
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A collection of⋌bundles (one for
each of the nh households)
[x] := [x1, x2, x3,... ]
profit-maximising
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A collection of⋌net-output vectors
1, q2, q3,... ]
[q]
:=
[q
(one for each of the nf firms)
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A set of prices (used by
households and firms)
p := (p1, p2, ..., pn)
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How a competitive allocation
works
 Implication of firm f’s profit
maximisation
p
 {qf(p) , f=1,2,...,nf }
p, {yh } {xh(p) , h=1,2,...,nh }
 Firms' behavioural responses
map prices into net outputs
 Implication of household's
utility maximisation
 Households’ behavioural
responses map prices and
incomes into demands
 The competitive allocation
just a minute! Where do these
incomes come from??
An important
model
component
An important missing item
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For a consumer in isolation it may be reasonable
to assume an exogenous income.
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Here the model involves all consumers in a closed
economy.
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Derived elsewhere in the economy.
There is no “elsewhere.”
Incomes have to be modelled explicitly.
We can learn from the “simple economy”
presentation.
Overview...
Frank Cowell: Microeconomics
General Equilibrium:
Basics
The economy
and allocations
A key role for the
price system.
Incomes
Equilibrium
Modelling income
Frank Cowell: Microeconomics
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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?
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Resources
R1h, R2h, ...
Rih  0,
i =1,…,n.
0  Vfh  1,
f =1,…,nf.
h, V h, ...
V
 Shares in firms’ 1
2
profits
introduce
prices
Incomes
Frank Cowell: Microeconomics
Resources
 Shares in firms
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Rents
Net outputs
 Prices
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Profits
look more
closely at the
role of prices
The fundamental role of prices
Frank Cowell: Microeconomics
firm f,
n-vector
i.
ofdepend
prices
Net good
outputs
on prices:
 Supply of net outputs
qif = qif(p)
Thus profits depend onindirectly
prices:
n
directly
P f(p):= S pi qif(p)
 Again writing profits as priceweighted sum of net outputs
i=1
Holding by hcan be writtenHolding
So incomes
as: byhIncomes = resource rents +
of resource
n i
nf
of shares in f
yh = S pi Rih + S Vfh P f(p)
i=1
profits
f=1
Incomes depend on prices :
yh = yh(p)
 Note that the function yh(•)
depends on the ownership
rights that h possesses
Prices in a competitive allocation
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 The allocation as a collection of
responses
p
 {qf(p) , f=1,2,...,nf }
p,p{yh } {xh(p) , h=1,2,...,nh }
 Put the price-income relation
into household responses
 Gives a simplified
relationship for households
 Summarise the relationship
yh = yh(p)
p 
[q(p)]
[x(p)]
Let's look at the
whole process
The price mechanism
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resource distribution
share
a ownership
a
 System takes as given the
property distribution
R1 , R2 , ...
V1a, V2a, ...
R1b, R2b, ...
V1b, V2b, ...
...
...
d
distribution
 Property distribution consists
of two collections
 Prices then determine incomes
 Prices and incomes determine
net outputs and consumptions
 Brief summary...
a
[y]
prices
allocation
[q(p)]
[x(p)]
Overview...
Frank Cowell: Microeconomics
General Equilibrium:
Basics
The economy
and allocations
Specification and
examples
Incomes
Equilibrium
What is an equilibrium?
Frank Cowell: Microeconomics
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What kind of allocation is an equilibrium?
Again we can learn from previous
presentations:
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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
Frank Cowell: Microeconomics
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For each h, maximise
n
Uh(xh), subject to S pi xih  yh
i=1
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For each f, maximise
 Households maximise utility,
given prices and incomes
 Firms maximise profits, given
prices
 For all goods the materials
balance must hold
n
S pi qif, subject to Ff(qf )  0
i=1
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For each i:
aggregate stock
of good i.
xi  qi + Ri
aggregate
consumption
of good i.
aggregate net
output of good i.
what determines
these
aggregates?
Consumption and net output
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“Obvious” way to aggregate  Appropriate if i is a rival good
overi?
 Full additional resources are
consumption of Sum
good
households
nh
xi = S xih
needed for each additional person
consuming a unit of good i.
h=1
An alternative way to
aggregate:
 Opposite case: a nonrival good
 Examples: TV, national defence...
xi = max {xih }
h
By definition
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...
To make life simple:
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Assume incomes are determined privately.
 All goods are “rival” commodities.
 There are no externalities.
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Competitive equilibrium: summary
Frank Cowell: Microeconomics
It must be a competitive
allocation
 A set of prices p
 Everyone maximises at
The materials balance
condition must hold
 Demand cannot exceed
those prices p
supply:
xq+R
An example
Frank Cowell: Microeconomics
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Exchange economy (no production)
Simple, standard structure
2 traders (Alf, Bill)
2 Goods:
Alf__
Bill__
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resource endowment
(R1a, R2a)
(R1b, R2b)
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consumption
(x1a, x2a)
(x1b, x2b)
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utility
Ua(x1a, x2a)
Ub(x1b, x2b)
diagrammatic
approach
Alf’s optimisation problem
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 Resource endowment
 Prices and budget constraint
x2a
R2
a
 Preferences
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 Equilibrium
Ra
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x*a
 Budget constraint is
2
2
i=1
i=1
S pi xia ≤ S pi Ria
Oa
R1 a
x1a
 Alf sells some
endowment of 2 for
good 1 by trading with
Bill
Bill’s optimisation problem
Frank Cowell: Microeconomics
 Resource endowment
 Prices and budget constraint
x2b
 Preferences
 Equilibrium
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R2b
x*b
 Budget constraint is
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2
2
i=1
i=1
S pi xib ≤ S pi Rib
Rb
 Bill, of course, sells
good 1 in exchange for 2
Ob
R1b
x1b
Combine the two problems
Frank Cowell: Microeconomics
x1b
R1 b
Incomes from the
distribution…
x2a
R2
a
Ob
[R]
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 Bill’s problem (flipped)
 Superimpose Alf’s problem.
 Price-taking trade moves
agents from endowment point…
R2
b
…match expenditures
in the allocation
 [x*]
...to the competitive equilibrium
allocation
 The role of prices
 This is the Edgeworth box.
 Width: R1a + R1b
 Height: R2a + R2b.
x2b
Oa
R1a
x1a
Alf and Bill as a microcosm
Frank Cowell: Microeconomics
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The Crusoe equilibrium story translates to a manyperson 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.