GE Excess Demand & Prices

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Prerequisites
Almost essential
General equilibrium: Basics
Useful, but optional
General Equilibrium: Price Taking
GENERAL EQUILIBRIUM: EXCESS
DEMAND AND THE RÔLE OF PRICES
MICROECONOMICS
Principles and Analysis
Frank Cowell
Note: the detail in slides marked “ * ” can
only be seen if you run the slideshow
March 2012
Frank Cowell: GE Excess Demand & Prices
1
Some unsettled questions
 Under what circumstances can we be sure that an
equilibrium exists?
 Will the economy somehow “tend” to this
equilibrium?
 And will this determine the price system for us?
 We will address these using the standard model of a
general-equilibrium system
 To do this we need just one more new concept
March 2012
Frank Cowell: GE Excess Demand & Prices
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Overview…
General Equilibrium:
Excess Demand+
Definition and
properties
Excess Demand
Functions
Equilibrium
Issues
Prices and
Decentralisation
March 2012
Frank Cowell: GE Excess Demand & Prices
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Ingredients of the excess demand function
 Aggregate demands (the sum of individual households'
demands)
 Aggregate net-outputs (the sum of individual firms' net outputs)
 Resources
 Incomes determined by prices
check this out
March 2012
Frank Cowell: GE Excess Demand & Prices
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Aggregate consumption, net output
 From household’s demand function  Because incomes depend
on prices
xih = Dih(p, yh)
= Dih(p, yh(p) )
 So demands are just functions of p  xih(•) depends on holdings of
xih = xih(p)
resources and shares
 If all goods are private (rival) then  “Rival”: extra consumers require
aggregate demands can be written: additional resources. Same as
“consumer: aggregation”
xi(p) = Sh xih(p)
 From firm’s supply of net output
qif = qif(p)
 Aggregate:
qi = Sf qif(p)
March 2012
 standard supply functions/
demand for inputs
valid if there are no externalities.
graphical
As in “Firm and the market”)
summary
Frank Cowell: GE Excess Demand & Prices
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Derivation of xi(p)
 Alf’s demand curve for good 1
 Bill’s demand curve for good 1
 Pick any price
 Sum of consumers’ demand
 Repeat to get the market demand curve
p1
Alf
March 2012
p1
p1
x1a
Bill
x1b
x1
The Market
Frank Cowell: GE Excess Demand & Prices
6
Derivation of qi(p)
 Supply curve firm 1 (from MC)
 Supply curve firm 2
 Pick any price
 Sum of individual firms’ supply
 Repeat…
 The market supply curve
p
p
p


q1
low-cost
firm
March 2012
q1+q2
q2
high-cost
firm
both firms
Frank Cowell: GE Excess Demand & Prices
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Subtract q and R from x to get E:
p1
p1
Demand
Supply
q1
x1
p1
Resource
stock
p1
R1
net output of i
aggregated over f
demand for i
aggregated over h
Ei(p) := xi(p) – qi(p) – Ri
1
Resource
stock of i
E1
March 2012
Frank Cowell: GE Excess Demand & Prices
8
Equilibrium in terms of Excess Demand
Equilibrium is characterised by a
price vector p*  0 such that:
 For every good i:
The materials balance condition
(dressed up a bit)
Ei(p*) 0
 For each good i that has a
positive price in equilibrium
(i.e. if pi* > 0):
Ei(p*) = 0
March 2012
If this is violated, then somebody,
somewhere isn't maximising…
You can only have excess supply
of a good in equilibrium if the
price of that good is 0
Frank Cowell: GE Excess Demand & Prices
9
Using E to find the equilibrium
 Five steps to the equilibrium allocation
From technology compute firms’ net output functions
and profits
2.
From property rights compute household incomes and
thus household demands
3.
Aggregate the xs and qs and use x, q, R to compute E
4.
Find p* as a solution to the system of E functions
5.
Plug p* into demand functions and net output functions
to get the allocation
 But this begs some questions about step 4
1.
March 2012
Frank Cowell: GE Excess Demand & Prices
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Issues in equilibrium analysis
 Existence
• Is there any such p*?
 Uniqueness
• Is there only one p*?
 Stability
• Will p “tend to” p*?
 For answers we use some fundamental properties of E
March 2012
Frank Cowell: GE Excess Demand & Prices
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Two fundamental properties…
 Walras’ Law. For any price p:
You only have to work
with n-1 (rather than n) equations
n
pi Ei(p) = 0
S
i=1
Hint #1: think about the "adding-up"
property of demand functions…
 Homogeneity of degree 0. For any
price p and any t > 0 :
You can normalise the prices by
any positive number
Ei(tp) = Ei(p)
Hint #2: think about the
homogeneity property of
demand functions…
Can you explain why they are true?
Reminder: these hold for any competitive allocation,
not just equilibrium
March 2012
Frank Cowell: GE Excess Demand & Prices
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Price normalisation*
 We may need to convert from n numbers p1, p2,…pn to n1
relative prices
 The precise method is essentially arbitrary
 The choice of method depends on the purpose of your model
 It can be done in a variety of ways:
You could divide by
n
pi
S
i=1
ppapnuméraire
n
labour
MarsBar
to give a
neat
oftheory
n-1
standard
value
system
“Marxian”
Mars
bar
theory
of
ofvalue
value
set
of set
prices
thatprices
sum
to 1
March 2012
 This method might seem weird
 But it has a nice property
 The set of all normalised prices
is convex and compact
Frank Cowell: GE Excess Demand & Prices
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Normalised prices, n=2
 The set of normalised prices
p2

 The price vector (0,75, 0.25)
(0,1)
J={p: p0, p1+p2 = 1}
(0, 0.25)
•
(0.75, 0)

(1,0)
March 2012
p1
Frank Cowell: GE Excess Demand & Prices
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Normalised prices, n=3
p3
 The set of normalised prices

 The price vector (0,5, 0.25, 0.25)
(0,0,1)
J={p: p0, p1+p2+p3 = 1}
(0, 0, 0.25)
•

(0,1,0)

(1,0,0)
(0, 0.25 , 0)
0
(0.5, 0, 0)
p1
March 2012
Frank Cowell: GE Excess Demand & Prices
15
Overview…
General Equilibrium:
Excess Demand+
Is there any p*?
Excess Demand
Functions
Equilibrium
Issues
•Existence
•Uniqueness
•Stability
Prices and
Decentralisation
March 2012
Frank Cowell: GE Excess Demand & Prices
16
Approach to the existence problem
 Imagine a rule that moves prices in the direction of excess
demand:
• “if Ei >0, increase pi”
• “if Ei <0 and pi >0, decrease pi”
• An example of this under “stability” below
 This rule uses the E-functions to map the set of prices into itself
 An equilibrium exists if this map has a “fixed point”
• a p* that is mapped into itself?
 To find the conditions for this, use normalised prices
• pJ
• J is a compact, convex set
 We can examine this in the special case n = 2
• In this case normalisation implies that p2  1  p1
March 2012
Frank Cowell: GE Excess Demand & Prices
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Why?
Existence of equilibrium? *
Why boundedness below?
As p2  0, by normalisation, p11
 ED diagram, normalised prices
 Excess demand function with
well-defined equilibrium price
 Case with discontinuous E
 Case where excess demand
for good2 is unbounded below
As p2  0 if E2 is bounded below then
p2E2  0
By Walras’ Law, this implies p1E1  0
as p11
So if E2 is bounded below then E1
can’t be everywhere positive
1
p1
Excess
supply
good 2 is
free here

p1*
good 1 is
free here

0
March 2012
Excess
demand
 E-functions are:
continuous,
bounded below
 No equilibrium
price where E
crosses the axis
E1
E never
crosses the axis
Frank Cowell: GE Excess Demand & Prices
18
Existence: a basic result

An equilibrium price vector must exist if:
excess demand functions are continuous and
2. bounded from below
• (“continuity” can be weakened to “upper-hemi-continuity”)
1.

Boundedness is no big deal
• Can you have infinite excess supply…?

However continuity might be tricky
• Let's put it on hold
• We examine it under “the rôle of prices”
March 2012
Frank Cowell: GE Excess Demand & Prices
19
Overview…
General Equilibrium:
Excess Demand+
Is there just one p*?
Excess Demand
Functions
Equilibrium
Issues
•Existence
•Uniqueness
•Stability
Prices and
Decentralisation
March 2012
Frank Cowell: GE Excess Demand & Prices
20
The uniqueness problem
 Multiple equilibria imply multiple allocations, at normalised
prices…
 …with reference to a given property distribution
 Will not arise if the E-functions satisfy WARP
 If WARP is not satisfied this can lead to some startling
behaviour…
let's see
March 2012
Frank Cowell: GE Excess Demand & Prices
21
Multiple equilibria
 Three equilibrium prices
 Suppose there were more of resource 1
 Now take some of resource 1 away
1
p1 

single equilibrium
jumps to here!!

three equilibria
degenerate to one!


0
March 2012
E1
Frank Cowell: GE Excess Demand & Prices
22
Overview…
General Equilibrium:
Excess Demand+
Will the system tend to p*?
Excess Demand
Functions
Equilibrium
Issues
•Existence
•Uniqueness
•Stability
Prices and
Decentralisation
March 2012
Frank Cowell: GE Excess Demand & Prices
23
Stability analysis
 We can model stability similar to physical sciences
 We need…
• A definition of equilibrium
• A process
• Initial conditions
 Main question is to identify these in economic terms
Simple
example
March 2012
Frank Cowell: GE Excess Demand & Prices
24
A stable equilibrium
Stable:
Equilibrium:
If we apply a small shock
Status
quo isadjustment
left
the built-in
undisturbed
by gravity
process (gravity)
restores
the status quo
March 2012
Frank Cowell: GE Excess Demand & Prices
25
An unstable equilibrium *
Equilibrium:
Unstable:
This
If weactually
apply afulfils
small the
shock
definition
the built-in adjustment
process (gravity) moves us
But…
away from the status quo
March 2012
Frank Cowell: GE Excess Demand & Prices
26
“Gravity” in the CE model
 Imagine there is an auctioneer to announce prices, and to adjust
if necessary
 If good i is in excess demand, increase its price
 If good i is in excess supply, decrease its price (if it hasn't
already reached zero)
 Nobody trades till the auctioneer has finished
March 2012
Frank Cowell: GE Excess Demand & Prices
27
“Gravity” in the CE model: the auctioneer using
tâtonnement*
Announce p
individual dd & ss
individual
individual
dd & ssdd
Adjust p
Adjust p
individual
& ss
dd & ss
Adjust p
…once we’re at
equilibrium we
trade
Equilibrium?
Equilibrium?
Equilibrium
?
Equilibrium?
March 2012
Evaluate
Evaluate
excess dd
Evaluate
excess dd
excess dd
Evaluate excess dd
Frank Cowell: GE Excess Demand & Prices
28
Adjustment and stability
 Adjust prices according to sign of Ei:
• If Ei > 0 then increase pi
• If Ei < 0 and pi > 0 then decrease pi
 A linear tâtonnement adjustment mechanism:
 Define distance d between p(t) and equilibrium p*
 Given WARP, d falls with t under tâtonnement
Two examples:
with/without
WARP
March 2012
Frank Cowell: GE Excess Demand & Prices
29
Globally stable…
 Start with a very high price
1
 Yields excess supply
 Under tâtonnement price falls
p1(0)
Excess
supply
•p *
1
 Start instead with a low price
 Yields excess demand
Excess
demand
 Under tâtonnement price rises
 If E satisfies
p1
WARP then
the system must
converge…
p1(0)
E1
0
E1(0)
March 2012
E1(0)
Frank Cowell: GE Excess Demand & Prices
30
Not globally stable… *
 Start with a very high price
1
 Start again with very low price
p1
•
Excess
supply
 …now try a (slightly) low price
 …now try a (slightly) high price
Locally
Stable
 Check the “middle” crossing
Excess
demand
 Here WARP
does not hold
Unstable
March 2012
•
•0
 Two locally
stable
equilibria
Also locally
stable
 One unstable
E1
Frank Cowell: GE Excess Demand & Prices
31
Overview…
General Equilibrium:
Excess Demand+
The separation
theorem and the role
of large numbers
Excess Demand
Functions
Equilibrium
Issues
Prices and
Decentralisation
March 2012
Frank Cowell: GE Excess Demand & Prices
32
Decentralisation
 Recall the important result on decentralisation
• discussed in the case of Crusoe’s island
 The counterpart is true for this multi-person world
 Requires assumptions about convexity of two sets,
defined at the aggregate level:
A := {x: x q+R, F(q) 
• the “better-than” set: B(x*) := {Shxh: Uh(xh )Uh(x*h ) }
• the “attainable set”:
 To see the power of the result here…
• use an “averaging” argument
• previously used in lectures on the firm
March 2012
Frank Cowell: GE Excess Demand & Prices
33
Decentralisation again *
 The attainable set
 The “Better-than-x* ” set
 The price line
 Decentralisation
x2
 A = {x: x  q+R, F(q)0}
 x*
 B = {Shxh: Uh(xh)  Uh(x*h)}
B
 x* maximises income over A
p1

p2
 x* minimises expenditure
over B
A
0
March 2012
x1
Frank Cowell: GE Excess Demand & Prices
34
Problems with prices
 Either non-convex technology (increasing returns or
other indivisibilities) for some firms, or…
 …non-convexity of B-set (non-concave- contoured
preferences) for some households…
 …may imply discontinuous excess demand function
and so…
 …absence of equilibrium
 But if there are large numbers of agents everything
may be OK
two
examples
March 2012
Frank Cowell: GE Excess Demand & Prices
35
A non-convex technology*
output
One unit of input
produces exactly
one of output
B
 q'
 The case with 1 firm
 Rescaled case of 2 firms,
 … 4 , 8 , 16
 Limit of the averaging process
 The “Better-than” set
• q*
 “separating” prices and equilibrium
 Limiting attainable set is
convex

q°
input
 Equilibrium q* is sustained by
a mixture of firms at q° and q'
A
March 2012
Frank Cowell: GE Excess Demand & Prices
36
Non-convex preferences*
 The case with 1 person
 Rescaled case of 2 persons,
x2
 A continuum of consumers
 The attainable set
No equilibrium here
 “separating” prices and equilibrium
 x'
• x*
 Limiting better-than set is convex
B
 Equilibrium x* is sustained by a
mixture of consumers at x° and x'
 x°
A
x1
March 2012
Frank Cowell: GE Excess Demand & Prices
37
Summary
Review
 Excess demand functions are handy tools for getting
results
Review
 Continuity and boundedness ensure existence of
equilibrium
Review
 WARP ensures uniqueness and stability
Review
 But requirements of continuity may be demanding
March 2012
Frank Cowell: GE Excess Demand & Prices
38

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