Transcript No Slide Title

Prerequisites
Almost essential
Welfare: Basics
Frank Cowell: Microeconomics
December 2006
Welfare: Efficiency
MICROECONOMICS
Principles and Analysis
Frank Cowell
Welfare Principles
Frank Cowell: Microeconomics
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Try to find general principles for running the economy.
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We have already slipped in one notion of “desirability”
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“Technical Efficiency”
…applied to the firm
What about a similar criterion for the whole economy?
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May be more fruitful than the constitution approach.
A generalised version of efficiency.
Subsumes technical efficiency?
And what about other desirable principles?
Overview...
Welfare:
efficiency
Frank Cowell: Microeconomics
Pareto
efficiency
Fundamental
criterion for
judging economic
systems
CE and PE
Extending
efficiency
A short agenda
Frank Cowell: Microeconomics
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Describe states of the economy in welfare terms...
Use this analysis to define efficiency.
Apply efficiency analysis to an economy
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Use standard multi-agent model
Same as analysed in earlier presentations.
Apply efficiency concept to uncertainty
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distinguish ex-ante and ex-post cases
The essential concepts
Frank Cowell: Microeconomics
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Social state
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Pareto superiority
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Describes economy completely.
For example, an allocation
At least as much utility for all;
Strictly greater utility for some
Pareto efficiency
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Uses concept of Pareto superiority.
Also needs a definition of feasibility...
A definition of efficiency
Frank Cowell: Microeconomics
Link to
Blocking
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The basis for evaluating social
states: vh(q)
the utility level enjoyed by
person h under social state q
A social state q is Pareto
superior to state q' if:
1. For all h: vh(q)  vh(q')
2. For some h: vh(q) > vh(q')
Note the similarity with the
concept of blocking by a
coalition
A social state q is Pareto
efficient if:
1. It is feasible
2. No other feasible state
is Pareto superior to q
“feasibility” could be
determined in terms of the
usual economic criteria
Take the case of
the typical GE
model.
Derive the utility possibility set
Frank Cowell: Microeconomics
 From the attainable set...
(x1a, x2a)
(x1b, x2b)
A
 ...take an allocation
 Evaluate utility for each agent
 Repeat to get the utility
possibility set
A
ua=Ua(x1a, x2a)
ub=Ub(x1b, x2b)
U
Utility possibility set – detail
Frank Cowell: Microeconomics
ub
 Utility levels of each person.
 U-possibility derived from A
 Fix all but one at some utility
level then max U of that person
 Repeat for other persons and
U-levels to get PE points
rescale this for
a different
cardinalisation
Efficient points
on boundary of U
U
constant ub
an efficient
point
But not all
boundary points
are efficient
ua
Find the
corresponding
efficient allocation
Finding an efficient allocation (1)
Frank Cowell: Microeconomics
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Use essentially the same method.
Do this for the case where
 all goods are purely private
 households 2,…,nh are on fixed utility levels uh
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Then problem is to maximise U1(x1) subject to:
 Uh(xh) ≥ uh, h = 2, …, nh
f f
 F (q ) ≤ 0, f = 1, …, nf
 xi ≤ qi + Ri , i= 1, …, n
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Use all this to form a Lagrangean in the usual way…
Finding an efficient allocation (2)
Frank Cowell: Microeconomics
max L( [x ], [q],
Lagrange multiplier for
l, each
m, k)
:=constraint
utility
Lagrange multiplier
for
1
1
h
h) 
U (x ) +

l
[U
(x
each firm’s
h h technology
Lagrange
multiplier for
f
f
 f mf Fmaterials
(q )balance, good i
+ i ki[qi + Ri  xi]
where
xi = h xih
qi = f qi f
uh]
FOCs
Frank Cowell: Microeconomics
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Differentiate Lagrangean w.r.t xih. If xih is positive at
the optimum then:
lhUih(xh) = ki
Likewise for good j:
lhUjh(xh) = kj
Differentiate Lagrangean w.r.t qif. If qif is nonzero at
the optimum then:
mfFif(qf) = ki
Likewise for good j:
mfFjf(qf) = kj
Interpreting the FOC
Frank Cowell: Microeconomics
From the FOCs for any
household h and goods i and j:

Uih(xh)
for every household:
MRS = shadow price ratio
ki
———— = —
Ujh (xh) kj
From the FOCs for any firm f
and goods i and j:
Fif(qf) ki
———— = —
Fjf (qf) kj

for every firm:
MRT = shadow price ratio
familiar
example
Efficiency in an Exchange Economy
Frank Cowell: Microeconomics
x1b
x2a
b
O
 Alf’s indifference curves
 Bill’s indifference curves
 The contract curve
 The set of
efficient
allocations is
the contract
curve.
Allocations where
MRS12a = MRS12b
Includes cases
where Alf or Bill
is very poor.
Oa
x2b
x1a
Apply to
uncertainty
Frank Cowell: Microeconomics
Efficiency in an Exchange Economy:
Uncertainty
b
xRED
a
xBLUE
 Alf's indifference curves
over prospects
Ob
 Bill's indifference curves
 The contract curve
 The set of
ex-ante
efficient
allocations is
the contract
curve.
Allocations where
a
b
MRSREDBLUE = MRSREDBLUE
Oa
Ex-post
efficient
allocations
b cover whole
xBLUE
diagram
a
xRED
Overview...
Welfare:
efficiency
Frank Cowell: Microeconomics
Pareto
efficiency
Relationship
between
competitive
equilibrium and
efficient allocations
CE and PE
Extending
efficiency
Efficiency and equilibrium
Frank Cowell: Microeconomics
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The Edgeworth Box diagrams are suggestive.
Points on the contract curve…
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So, examine connection of market with efficiency:
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…are efficient
…could be CE allocations
Will the equilibrium be efficient?
Can we use the market to implement any efficient outcome?
Or are there cases where the market “fails”?
Focus on two important theorems
Welfare theorem 1
Frank Cowell: Microeconomics
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Assume a competitive equilibrium.
What is its efficiency property?
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THEOREM: if all consumers are greedy and there are
no externalities then a competitive equilibrium is
efficient.
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Explanation:
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If they are not greedy, there may be no incentive to trade.
If there are externalities the market takes no account of
spillovers.
Welfare theorem 2
Frank Cowell: Microeconomics
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Pick any Pareto-efficient allocation
Can we find a property distribution d so that this
allocation is a CE for d?
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THEOREM: if, in addition to conditions for theorem 1,
there are no nonconvexities then an arbitrary PE
allocation be supported by a competitive equilibrium.
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Explanation:
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If “lump sum” transfers are possible then we can arbitrarily
change the initial property distribution.
If there are nonconvexities the equilibrium price signals could
take us away from the efficient allocation...
Supporting a PE allocation
Frank Cowell: Microeconomics
x1b
 The contract
Ob curve
 An efficient allocation
 Supporting price ratio = MRS
 The property distribution

x2a [R]
 A lump-sum transfer
 Support the
Allocations where
MRS12a = MRS12b
^
 [x]
p1
allocation by a
CE.
p2
 This needs an

adjustment of
the initial
endowment.
Beware: lump-
Oa
x2b
x1a
sum transfers
may be tricky to
implement
Supporting a PE allocation
(production)
Frank Cowell: Microeconomics
 Firm f’s technology set
q2f
 f’s net output in the efficient
allocation
 Supporting price ratio = MRT

f
^
q
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f’s net output in the
allocation is profitmaximising for f.
p1

p2
q1f
0
what if preferences
and production
possibilities were
different?
Household h makes “wrong”
choice
Frank Cowell: Microeconomics
x2h
 Household h ’s utility function
violates second theorem
 Suppose we want to allocate
this consumption bundle to h.
 Introduce prices
~
xh 
 h's choice given this budget

^xh 
p1

p2
0
x1h
Nonconvexity
leads to “market
failure”
Firm f makes “wrong” choice
Frank Cowell: Microeconomics
q2f
 Firm f ’s production function
violates second theorem
 Suppose we want to allocate
this net output to f.
 Introduce prices.
 f's choice at those prices
f
^
q


p1

p2
0

~
qf
q1f
Another example
of “market failure”
Frank Cowell: Microeconomics
Competitive Failure and
Efficiency
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1.
The market “failures” raise two classes of problem that
lead to two distinct types of analysis:
The characterisation problem
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2.
The implementation problem
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Description of modified efficiency conditions…
…when conditions for the welfare theorems are not met.
Design of a mechanism to achieve the allocation.
These two types of problem pervade nearly all of
modern micro economics.
It’s important to keep them distinct in one’s mind.
Overview...
Welfare:
efficiency
Frank Cowell: Microeconomics
Pareto
efficiency
Attempt to
generalise the
concept of Pareto
superiority
CE and PE
Extending
efficiency
Why extend the efficiency
concept?
Frank Cowell: Microeconomics
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Pareto efficiency is “indecisive.”
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Pareto improvements may be elusive.
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What about the infinity of PE allocations along the contract
curve?
Beware the politician who insists that everybody can be
made better off
Other concepts may command support.
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"potential" efficiency
fairness
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…
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Indecisiveness of PE
Frank Cowell: Microeconomics
ub
 Construct utility-possibility
set as previously
 Two efficient points
 Points superior to q
 Points superior to q'.
Boundary points
cannot be compared
on efficiency grounds
v(q)

 q and q' cannot be
compared on efficiency
grounds
 v(q)
U
ua
A way forward?
“Potential” Pareto superiority
Frank Cowell: Microeconomics
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Define q to be potentially superior to q' if :
There is a q* which is actually Pareto superior to q'
q* is “accessible” from q.
To make use of this concept we need to define
accessibility.
This can be done using a tool that we already have from
the theory of the consumer.
The idea of accessibility
Frank Cowell: Microeconomics
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Usually “q* accessible from q” means that income total
in q* is no greater than in q.
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“if society can afford q then it can certainly afford q* ”
This can be interpreted as a “compensation rule”
The gainers in the move from q' to q...
...get enough income to be able to compensate the
losers of q'  q.
This can be expressed in terms of standard individual
welfare criteria
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the CV for each person
A result on potential superiority
Frank Cowell: Microeconomics
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Use the terminology of individual welfare
CVh(q'  q) is the monetary value the welfare change
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for person h…
…of a change from state q' to state q
…valued in terms of the prices at q
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CVh > 0 means a welfare gain; CVh < 0 a welfare loss.
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THEOREM: a necessary and sufficient condition for q
to be potentially superior to q' is
Sh CVh(q'  q) > 0.
But can we really base welfare economics on the
compensating variation...?
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Applying potential superiority
Frank Cowell: Microeconomics
ub
 q is not superior to q' and q'
is not superior to q.
 q* is superior to q.
 Points accessible from q'.
v(q*)
 There could be lots
of points accessible by
lump-sum transfers.

v(q)

 q' is potentially
superior to q
 v(q)
U
ua
Difficulties
Problems with accessibility
Frank Cowell: Microeconomics
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What prices should be used in the evaluation.
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We speak only of potential income gains.
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compensation is not actually paid.
does this matter?
If no income transfer takes place…
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those in q?
those in q' ?
so that there are no consumer responses to it
can we accurately evaluate gains and losses?
But this isn’t the biggest problem...
Re-examine potential superiority
Frank Cowell: Microeconomics
 The process from q to q', as
before.
ub
 The process in reverse from
q' to q.
points accessible
from q
 Combine the two.
q is potentially
superior to q and …
v(q)

points accessible
from q
 v(q)
ua
q is potentially
superior to q!
Extending efficiency: assessment
Frank Cowell: Microeconomics
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The above is the basis of Cost-Benefit Analysis
It relies on notion of “accessibility”
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a curious concept involving notional transfers?
more than one possible definition
“Compensation” is not actually paid
If equilibrium prices differ substantially, the rule may
produce contradictions.
What next?
Frank Cowell: Microeconomics
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This has formed part of the second approach to the
question “how should the economy be run?”
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The approach covers widely used general principles
Efficiency
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Neat. Simple. But perhaps limited
Potential efficiency
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The first was “the constitution”.
Persuasive but perhaps dangerous economics/politics
A natural way forward:
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Examine other general principles
Consider problems with applying the efficiency concept
Go to the third approach: a full welfare function.