Prerequisites
Almost essential
A Simple Economy
Useful, but optional
Firm: Optimisation
Consumer Optimisation
GENERAL EQUILIBRIUM: BASICS
MICROECONOMICS
Principles and Analysis
Frank Cowell
Note: the detail in slides marked “ * ” can
only be seen if you run the slideshow
July 2015
Frank Cowell: General Equilibrium Basics
1
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
July 2015
Frank Cowell: General Equilibrium Basics
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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”
July 2015
Frank Cowell: General Equilibrium Basics
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Overview
General Equilibrium:
Basics
The components of
the general equilibrium
problem
The economy
and allocations
Incomes
Equilibrium
July 2015
Frank Cowell: General Equilibrium Basics
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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)
July 2015
Frank Cowell: General Equilibrium Basics
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What is an economy?
 Resources (stocks) R , R ,…
1
2
n of these
 Households
(preferences)
U1, U2 ,…
nh of these
 Firms
(technologies)
F1, F2 ,…
nf of these
July 2015
Frank Cowell: General Equilibrium Basics
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An allocation
A competitive allocation consists of:
shorthand notation
for a collection
utility-maximising
 A collection of⋌bundles (one for
each of the nh households)
[x] := [x1, x2, x3,… ]
profit-maximising
 A collection of⋌net-output vectors
(one for each of the nf firms)
[q] := [q1, q2, q3,… ]
 A set of prices (used by
households and firms)
p := (p1, p2, …, pn)
July 2015
Frank Cowell: General Equilibrium Basics
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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 }
where do these
incomes come
from?
July 2015
 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
An important
model
component
Frank Cowell: General Equilibrium Basics
8
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
July 2015
Frank Cowell: General Equilibrium Basics
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Overview
General Equilibrium:
Basics
A key role for the
price system
The economy
and allocations
Incomes
Equilibrium
July 2015
Frank Cowell: General Equilibrium Basics
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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
July 2015
Frank Cowell: General Equilibrium Basics
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What does household h possess?
 Resources
R1h, R2h, …
Rih  0,
i =1,…,n
0  Vfh  1,
f =1,…,nf
 Shares in firms’
profits
V1h, V2h, …
introduce
prices
July 2015
Frank Cowell: General Equilibrium Basics
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Incomes
 Resources
 Shares in firms
Rents
 Net outputs
 Prices
Profits
July 2015
look more
closely at the
role of prices
Frank Cowell: General Equilibrium Basics
13
The fundamental role of prices
 Net output of i by firm f depends  Supply of net outputs
on prices p:
qif = qif(p)
 Thus profits depend on prices:
 Again writing profits as priceP
n
f(p):=
S
i=1
weighted sum of net outputs
pi qif(p)
directly
indirectly
 So incomes can be written as:
n
nf
i=1
f=1
yh = S pi Rih + S Vfh P f(p)
Holding by h
of resource i
Holding by h
of shares in f
 Income depends on prices :
yh = yh(p)
July 2015
 Income = resource rents + profits
 yh(•) depends on ownership
rights that h possesses
Frank Cowell: General Equilibrium Basics
14
Prices in a competitive allocation
 The allocation as a collection of
responses
 {qf(p) , f=1,2,…,nf }
p
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 
July 2015
[q(p)]
[x(p)]
Let's look at the
whole process
Frank Cowell: General Equilibrium Basics
15
The price mechanism*
resource distribution
share
a ownership
a
 System takes as given the
property distribution
R1 , R2 , …
V 1a, V 2a, …
R1b, R2b, …
V 1b, V 2b , …
…
…
d
distribution
July 2015
 Property distribution consists
of two collections
 Prices then determine incomes
 Prices and incomes determine
net outputs and consumptions
 Brief summary…
a
[y]
prices
[q(p)]
[x(p)]
allocation
Frank Cowell: General Equilibrium Basics
16
Overview
General Equilibrium:
Basics
Specification and
examples
The economy
and allocations
Incomes
Equilibrium
July 2015
Frank Cowell: General Equilibrium Basics
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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
July 2015
Frank Cowell: General Equilibrium Basics
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Competitive equilibrium: basics
 For each h, maximise
n
Uh(xh), subject to S pi xih  yh
i=1
 Households maximise utility, given
prices and incomes
 Firms maximise profits, given prices
 For all goods the materials balance
must hold
 For each f, maximise
n
S pi qif, subject to Ff(qf )  0
i=1

For each i:
xi  qi + Ri
aggregate
consumption
of good i
aggregate stock of good i
aggregate net
output of good i
July 2015
what determines
these
aggregates?
Frank Cowell: General Equilibrium Basics
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Consumption and net output
 “Obvious” way to aggregate  Appropriate if i is a rival good
 Additional resources needed for each
consumption of good i?
additional person consuming a unit of i
nh
xi = S xih
h=1
Sum over
households
 An alternative way to
aggregate:
 Opposite case: a nonrival good
 Examples: TV, national defence…
xi = max {xih }
h
 Aggregation of net output:  if all qf are feasible will q be feasible?
By definition
nf
qi := S qif
f=1
July 2015
 Yes if there are no externalities
 Counterexample: production with
congestion…
Frank Cowell: General Equilibrium Basics
20
To make life simple:
 Assume incomes are determined privately
 All goods are “rival” commodities
 There are no externalities
July 2015
Frank Cowell: General Equilibrium Basics
21
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
July 2015
Frank Cowell: General Equilibrium Basics
22
An example
 Exchange economy (no production)
 Simple, standard structure
 2 traders (Alf, Bill)
 2 Goods:
Alf__
Bill__

resource endowment (R1a, R2a)

consumption
(x1a, x2a)
(x1b, x2b)

utility
Ua(x1a, x2a)
Ub(x1b, x2b)
(R1b, R2b)
diagrammatic
approach
July 2015
Frank Cowell: General Equilibrium Basics
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Alf’s optimisation problem
 Resource endowment
 Prices and budget constraint
x2a
R2
a
 Preferences

 Equilibrium
Ra

x*a
 Budget constraint is
2
2
i=1
i=1
S pi xia ≤ S pi Ria
 Alf sells some endowment of 2
for good 1 by trading with Bill
Oa
July 2015
R1a
x1a
Frank Cowell: General Equilibrium Basics
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Bill’s optimisation problem
 Resource endowment
 Prices and budget constraint
x2b
 Preferences
 Equilibrium

R2b
x*b
 Budget constraint is

2
2
i=1
i=1
S pi xib ≤ S pi Rib
Rb
 Bill, of course, sells
good 1 in exchange for 2
Ob
July 2015
R1b
x1b
Frank Cowell: General Equilibrium Basics
25
Combine the two problems
x1b
R1b
Incomes from the
distribution…
x2a
R2
a
Ob
[R]

 Bill’s problem (flipped)
 Superimpose Alf’s problem
 Price-taking trade moves
agents from endowment point…
b
R
2
…match expenditures in
…to the competitive equilibrium
allocation
the allocation
 The role of prices
 [x*]
 This is the Edgeworth box
 Width: R1a + R1b
 Height: R2a + R2b
x2b
Oa
July 2015
R1a
x1a
Frank Cowell: General Equilibrium Basics
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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
July 2015
Frank Cowell: General Equilibrium Basics
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