Microeconomics principles and analysis phần 5 ppsx

242 CHAPTER 9. WELFARE
For proof of this, see Appendix C. So it appears that under certain circum-
stances we can pick the Pareto-e¢ cient allocation that we would like to see, and
then arrange that the economy is automatically moved to this allocation by the
process of competition –the competitive equilibrium “supports”the allocation.
But a few words of caution are appropriate here.
 First, in order to manipulate the economy in this way we need to have the
right property distribution
^
d as a starting point. But how do we arrange
for this distribution in the …rst place? If history has thrown up some other
property distribution d then perhaps it is possible to arrange transfers of
entitlements of property from one group of households to another before
production and trade takes place. These transfers are not base d on the
activities or choices of any of the agents in the economy –in the jargon they
are lump-sum transfers –and the political and administrative di¢ culties
associated with them should not be taken lightly (more of this in chapter
13, page 461). A principal di¢ culty is that of identifying who is entitled
to receive a transfer and who should be required to provide the resources.
Some resource endowments are intrinsically non-observable;
10
some are
intrinsically non transferable.
11
 Second, the conditions in Theorem 9.5 are fairly stringent. Reasonably
we could ask what guidance is available on e¢ cienc y grounds once we try
to accommodate real-world problems and di¢ culties. These di¢ culties
will involve either departures from the ideals of perfect competition or
relaxation of some of the assumptions that underpin the theorem. This
issues are addressed in section 9.3.2.
 Third, the discussion of e¢ ciency has b ee n conducted in a world of perfect

certainty. There are important issues raised by the mo de l of uncertainty
that we developed in chapter 8. These are handled in section 9.3.5.
9.3.2 Departures from e¢ ciency
The pair of theorems, 9.4 and 9.5, are undeniably attractive but, to be applicable
they clearly impose somewhat idealistic requirements. So, two things deserve
further consideration: (1) In situations where we have a private goods economy
with technology and preferences that satisfy the conditions of theorems 9.4 and
9.5, how does one quantify departures from the ideal? It may be useful to
have some guidance of this to have an idea of whether one imperfect state is
“better”or “worse”than another in e¢ ciency terms. (2) What if the underlying
assumptions about the private-goods economy were relaxed? What could we
then say about the conditions for an e¢ cient allocation? We deal with each of
these in turn.
10
(a) G ive an example of why this is so.
(b) Because of the problems of non observability policy makers often condition transfers on
individuals’ac tions, as with the income tax. Why will this giv e rise to e¢ ciency problems?
11
Again p rovide an exa mple.
9.3. PRINCIPLES FOR SOCIAL JUDGMENTS: EFFICIENCY 243
Waste
Consider the problem of quantifying ine¢ ciency. Suppose that we are in a
purely private-good economy. All the conditions for a competitive equilibrium
are present –which would involve prices p

and incomes y
1
; y
2
; ::: – but we

…nd that in fact one good (good 1) has its price …xed up above p

1
. Th is price-
wedge might be caused by a sales tax, for example, and it will in general distort
all other prices. Can we measure the loss that is induced by the price-wedge?
Let u s suppos e that we actually observe the consumer prices (p
1
; p
2
; :::; p
n
)
and producer prices (~p
1
; ~p
2
; :::; ~p
n
) such that
p
1
= ep
1
[1 + ];
p
2
= ~p
2
;

p
3
= ~p
3
::: ::: :::
p
n
= ~p
n
9
>
>
>
>
=
>
>
>
>
;
(9.16)
where  is the price-wedge imposed exogenously upon good 1. To make the
argument easier assume that all prices are positive and that all markets clear.
To measure waste we need a reference point. Since we have argued that under
the idealised conditions of a competitive equilibrium it seems natural to use
as the reference point the prices p

that would have prevailed in equilibrium.
Furthermore let
4p

i
:= p
i
 p

i
(9.17)
denote the deviations from the reference prices for each good i = 1; 2; :::; n.
Given that all consumers are maximising utility we must have that
MRS
h
ij
=
p
j
p
i
(9.18)
for all goods i and j, and all households h,
12
and
MRT
1j
=
p
j
p
1
[1 + ]
MRT

2j
=
p
j
p
2
::: ::: :::
MRT
nj
=
p
j
p
n
9
>
>
=
>
>
;
(9.19)
Now consider the net gain that person h would expe rience were one to go from
the reference allocation a

to the actual allocation a:
EV
h
= C
h

(p

; 
h
)  C
h
(p; 
h
)  [y
h
 y
h
] (9.20)
where C
h
is h’s cost function. Summing (9.20) over all h expresses the total loss
measured in the same units as income. Assume that all producer prices remain
12
Ske tch a diagram similar to Figure 9.5, but with a convex production possibility set, a nd
superimpose a set o f indi¤erence curves; use this to illustrate the conditions (9.18) and (9.19).
244 CHAPTER 9. WELFARE
constant –an implicit assumption of in…nite supply elasticities. Then (minus)
the aggregation over the consumers of the loss in equation (9.20) gives the total
measure of waste involved in the price distortion 4p thus:
13
(4p) :=
n
h
X
h=1


C
h
(p; 
h
)  C
h
(p  4p; 
h
)


n
X
i=1
R
i
4p
i

n
X
i=1
q
i
4p
i
(9.21)
We have (0) = 0 and Shephard’s Lemma implies:
x

h
i
= C
h
i
(p; 
h
) (9.22)
Using the materials’ balance condition and taking an approximation we then
get
14
(4p)  
1
2
n
X
i=1
n
X
j=1
n
h
X
h=1
@H
hi
(p; 
h
)
@p

j
4p
i
4p
j
(9.23)
where
@H
hi
(p;
h
)
@p
j
is the substitution e¤ect of a rise in the price of commodity j
on the demand for commodity i by household h – in other words the slope of
the compensated or Hicksian demand curve.
15
The interpretation of this can be based on the analysis of cost changes that
we developed for the …rm (page 33) and the consumer (page 91). The price
increase leads to an inc ome increase for someone (because of the e¤ect on sales
revenue) and the contribution to this from agent h’s consumption this is given
by the lightly shaded rectangle with dimension p
1
x
h
1
in Figure 9.6. However the
component of cost increase to agent h represented by the change in price 4p
1

is
represented by by the whole shaded area in Figure 9.6. The di¤erence between
the two represents the component of the waste generated by the price distortion
faced by person h directly from 4p
1
. It is illustrated in Figure 9.6 as () the
area of the heavily shaded “triangle”shape, app roximated by

1
2
4p
1
4x
h
1
where
4x
h
1
=
@H
h1
(p; 
h
)
@p
1
4p
1
Of course, one needs to take into account the other components of waste that

are generated from the induced price changes: the sum of the little triangles
such as that in Figure 9.6 gives the expression for loss (9.23).
16
13
Use equations (7.8), (7.9 ) and theorem 2.7 to show how (9. 21) follows from (9.20).
14
Show how to derive (9.23) using a Taylor approximation (see pag e 494).
15
Show how the the ex pressio n for waste must b e modi…ed if supply elasticities are less than
in…nite.
16
(a) Suppose there is only a single …rm producing good 1 that uses the market po wer it
enjoys to force up the price of good 1. If we neglect cross-price e¤ects and use consumer’s
surplus as an approximation to EV interpret the model as one of the wa ste that is attributable
to monopoly. [Hint: us e the equilibrium condition given in (3. 11s)]
(b) How is the waste related to the el asticity of demand for good 1?
9.3. PRINCIPLES FOR SOCIAL JUDGMENTS: EFFICIENCY 245
Figure 9.6: Component of e¢ ciency loss
This idea of quantifying waste gives us the basis for developing a coherent
analysis of economic policy that may be aimed at yielding welfare improvements
rather than shooting just at a welfare optimum. More of this in Chapter 13.
E¢ ciency and market “failures”
Now let us turn to the other main thing that needs consideration. This intro-
duces us to a class of economic problems that are sometimes –perhaps some what
unfairly –characterised as instances of “market failure.” It is perhaps better to
say that these are instances where unquali…ed reliance on the market mechanism
cannot be relied upon to produce an e¢ cient outcome. This is hardly aston-
ishing: the requirements for the “s upport” result in Theorem 9.5 may app ear
to be unacceptably strong. Relaxing these requirements raises two key issues.
1. The characterisation problem. Where the conditions for Theorem 9.4 are

violated the FOCs (9.12)– (9.13) are no longer valid. Furthermore, in
the presence of nonconvexities the FOCs are no longer su¢ cient to pin
down a unique allocation –see the two parts of the …gure where points on
di¤erent parts of on contour have the same MRS or MRT. So in all these
246 CHAPTER 9. WELFARE
cases the FOCs for the Pareto e¢ cient allocation need to be replaced or
supplemented in order to characterise an e¢ cient allocation.
2. The implementation problem. If the market mechanism cannot do the job
of supporting a particular allocation in this case, then what else might
work?
We shall discuss nonconvexities and the di¢ cult implementation issue further
in chapters 12 and 13. The characterisation issue where the conditions for
Theorem 9.4 are violated can be handled by a series of tweaks as follows in
sections 9.3.3 and 9.3.4.
9.3.3 Externalities
We have already seen the mechanics of externalities in a simple example of in-
teractions amongst …rms, discussed in chapter 3 (pages 55¤). Here we also need
to take into account a similar ph enome non of interactions amongst consumers.
We will handle each in turn under the labe ls production and consumption ex-
ternalities.
Production externalities
Unfortunately there are all too many practical examples of “negative”produc-
tion externalities–emissions into rivers, acid rain, tra¢ c congestion –where the
unregulated actions by one …rm signi…cantly a¤ects the cost function of other
…rms. So we shall focus on such detrimental interactions although virtually all
of the results can be easily reworked to deal with positive externalities too. We
can see the essential nature of the problem by considering a two-…rm example.
Suppose that q
1
1

the output of good-1 by …rm 1 a¤ects the technological possi-
bilities of other …rms: …rm 1 produces glue. Consider the position of …rm 2, a
restaurant. In the no-externality case we would normally write 
2
(q
2
)  0 to
characterise the net-output vectors q
2
that are technologically feasible. How-
ever, in view of the externality, …rm 1’s output (q
1
1
) will shift …rm 2’s production
function. If the externality is detrimental (the smell of glue does not enhance
enjoyment of the restaurant’s meals) then we have:
@
2
@q
1
1
> 0 (9.24)
Why? Consider a net output vector ^q
2
that was just feasible for …rm 2, before
…rm 1 increased its output; this means that – in terms of the …gure – the
relevant point lies on the boundary, so that 
2
(q
2

) = 0. Now suppose that …rm
1 increases its output q
1
1
: if the externality is strictly detrimental
17
, then this
must mean that ^q
2
–which had hitherto been just in the feasible set –must now
be infeasible (you have to use more electricity to run air conditioning). This in
17
Suppose …rms 1 and 2 experience diminishing returns to scale and generate negative
externa li ties: will production overall exhibit dimini shing returns to scale?
9.3. PRINCIPLES FOR SOCIAL JUDGMENTS: EFFICIENCY 247
Figure 9.7: The e¤ect of pollution on a victim’s production set
turn means that we now …nd 
2
(^q
2
) > 0 and that q
1
1
has shifted 
2
inwards:
in other words condition (9.24) holds –see Figure 9.7.
18
We could then appropriately de…ne the value, at the margin, of the damage
in‡icted upon …rm 2 by the externality generated by …rm 2. We could measure

this in terms of …rm 2’s output:

1

2
2
@
2
@q
1
1
(9.25)
where 
2
2
is the conventional di¤erential of …rm 2’s production function with
respect to its own output.
More generally, in the multi…rm case, we can represent an externality by
writing the production function for …rm g as:

g

q
g
; q
1
1
; q
2
1

; :::; q
g 1
1
; q
g +1
1
; :::

(9.26)
and if the externality generated by any of these …rms is potentially detrimental
we would have:
@
g
@q
f
i
> 0 (9.27)
Once again this means that if the detrimental externality (noxious emissions)
by other …rms were to increase, then …rm gs production p ossibilities are reduced
–see Figure 9.7.
18
Rework the analysis in equations (9.24) to (9.29) for a favourable externality.
248 CHAPTER 9. WELFARE
Figure 9.8: Production boundary and e¢ ciency with externalities
The general form of the marginal externality caused by …rm f when it pro-
duces good 1 (again evaluated in terms of good 2) may thus be written:
e
f
2l
:=

n
f
X
g =1
1

g
2
@
g
@q
f
i
(9.28)
We can then plug the production function with externalities into the problem
de…ning an e¢ cient allocation. We then …nd:
19

f
1

f
2
 e
f
2l
=

1


2
(9.29)
which can be expressed as:
ratio of
MRT – externality = shadow
prices
One implication of this is that market prices, that the …rm would use, do
not correspond to the “scarcity prices”of commodities in an e¢ cient allocation:
there is a “wedge” between them corresponding to the value of the marginal
externality.
20
This is illustrated in Figure 9.8. If the MRT were to equal just the
19
Substitute (9.26) into equation (9.8) and di¤erentiate to get this result.
20
Discuss how e quation (9.29) m ight be interpreted as a simple rule for setting a “polluter
pays” levy on output.
9.3. PRINCIPLES FOR SOCIAL JUDGMENTS: EFFICIENCY 249
ratio of scarcity prices the …rm would produce at point ^q
f
. If the scarcity prices
are adjusted by the marginal externality then we …nd the e¢ cient allocation at
point ~q
f
.
Consumption externalities
Consumption externalities can be handled in a similar manner, and the main
idea conveyed by means of a simple example. Alf is an asthmatic non-smoker
who is a¤ected by the actions of Bill a boorish smoker. To simplify the example
we use the device of bundling together all goods in the economy except one.

Let good 1 be tobacco, and good 2 the composite of everything else. Then, we
can write the utility function for Bill as U
b
(x
b
1
; x
b
2
) and for Alf as U
a
(x
a
1
; x
a
2
; x
b
1
).
The signs of the partial d erivatives of these functions are f airly obvious; in par-
ticular we may assume that @U
a
=@x
b
1
< 0, since Alf su¤ers when Bill consumes
commodity 1. But how awful is it for asthmatic Alf to be in boorish Bill’s
company? One way of capturing this is to try to measure Alf’s willingness to

pay to have the nuisance reduced –to get Bill to cut down on the tobacco. We
can do this by computing the amount that Alf would be prepared to sacri…ce in
order to get Bill to have one less cigarette; this is:
e
b
2l
:=
1
U
a
2
@U
a
@x
b
1
 0 (9.30)
where U
a
2
is Alf’s marginal utility derived from other goods de rived in the usual
way. From Alf’s point of view expression (9.30) is the marginal externality –or
the marginal damage –in‡icted through the consumption of good 1 by Bill the
boor. Translating this into our more general model of e¢ ciency with n goods
and n
h
households let us suppose the consumption of good 1 by any household
h potentially a¤ects the utility of some other household `, possibly as the result
of some side e¤ect. We would then write:


`
= U
`
(x
`
; x
1
1
; x
2
1
; :::; x
`1
1
; x
`+1
1
; :::) (9.31)
If the externality is detrimental by nature then we have
@U
`
@x
h
1
 0 (9.32)
for any two distinct households h and `. Analogous to (9.30) we may de…ne the
marginal externality imposed on others by household h as:
e
h
21

:=
n
h
X
`=1
1
U
`
2
@U
`
@x
h
1
(9.33)
Notice that the summation is required because we want to know the marginal
damage in‡icted on all parties, evaluated appropriately at the su¤erers’ mar-
ginal utility of other goods. When we take this relationship into account in the
250 CHAPTER 9. WELFARE
FOCs for e¢ ciency, we …nd the following:
21
U
h
1
U
h
2
+ e
h
21

=

1

2
: (9.34)
In other words we again have a simple relationship:
ratio of
MRS + externality = shadow
prices
Clearly, if there is a negative externality, then the marginal rate of substitution of
good 2 into good 1 will be greater than the price ratio in an e¢ cient allocation.
22
The interaction between …rms or between consumers leads to fairly straight-
forward extensions of the rules covering the characterisation of e¢ cient alloca-
tions. However, although the characterisation problem is relatively simple in
this case, the implementation problem may p rove to be intractable –even for
production externalities –in the absence of external intervention.
9.3.4 Public goods
The precise meaning of a public good is given in de…nition 9.4. So, if good 1 is
a pure public good it must be non-rival which requires that
x
h
1
= x
1
for all non-satiated households. It must also be non-excludable, which can be
interpreted as an extreme case of consumption externality: once provided there
is no means of charging for it.
Let us explore the e¢ ciency implications of non-rivalness. In fact we only

require a di¤erent form of aggregation in the e¢ ciency condition. Notice that in
this case if, for some household h we have x
h
1
< x
1
and yet U
h
1
> 0; then a Pareto-
superior allocation can be attained by allowing household h’s consumption of
good 1 to increase (as long as x
h
1
is strictly less than x
1
no additional resources
have to be used up to increase h’s consumption of this non-rival good, so we
might as well let household h increase its own utility since it will not thereby
reduce any one else’s utility). Therefore at the Pareto-e¢ cient allocation for each
household h, either x
h
1
= x
1
so that the household is consuming the non-rival
good to its maximum capacity, or x
h
1
< x

1
and U
h
1
= 0 so that the household
is consuming less than it could, but is satiated with the public good 1. Let us
assume that everyone is non-satiated;
23
each person must consume exactly the
same amount at a Pareto e¢ cient allocation. Thus we put x
h
1
= x
1
; h = 1; :::; n
h
21
Substitute (9.31) into equation (9.8) and di¤erentiate to get this result.
22
In a two-good model, show ho w condition (9.34) might be used to sugges t an an ap-
propriate tax on the good causing the exter nality, or an appropriate subs idy on th e other
good.
23
Derive the same condition assuming that the …rst h

ho useholds are non-satiated, and
the remaining n
h
 h


ho useholds are satiated.
9.3. PRINCIPLES FOR SOCIAL JUDGMENTS: EFFICIENCY 251
in the Lagrangean (9.8) as our new aggregation condition. Di¤erentiate the
Lagrangean with respect to x
1
and set it equal to zero:
n
h
X
h=1

h
U
h
1
(x
h
) = 
1
(9.35)
Now pick any other pure private good i that is being consumed in positive
amounts by everyone: from equation (9.35) we get
n
h
X
h=1
U
h
1
(x

h
)
U
h
i
(x
h
)
=

1

i
(9.36)
So we have established the result
Theorem 9.6 (E¢ ciency with public goods) In a Pareto e¢ cient state
without externalities for any pure private good i consumed by everyone and a
non-rival good 1 we have:
MRS
1
i1
+ MRS
2
i1
+ ::: + MRS
n
h
i1
=


1

i
= MRT
f
i1
; f = 1; :::; n
f
:
9
=
;
(9.37)
Figure 9.9 illustrates the two-good, two-person case in the case where pro-
duction is carried out by a single …rm. The top part of the diagram plots Alf’s
marginal rate of substitution of the private good (good 2) for the pu blic good
(good 1) as a function of the total supply of good 1. It is a graph of his willing-
ness to pay for additional units of the public good and it is downward sloping
on the assumption that Alf’s utility function is quasiconcave. The second part
of the diagram does the same job for Bill. At any level of provision of the public
good x
1
we can imagine asking ourselves “what is the total willingness to pay for
an extra unit of the public goo d ”(remember: because it is nonrival both parties
will bene…t from the extra unit). The graph of this total willingness to pay is
the downward sloping line in the b ottom part of the …gure (MRS
a
21
+MRS
b

21
);
the marginal cost of providing the public good is given by the graph of MRT
21
against x
1
; the intersection of these two curves gives the e¢ cient supply of public
goods x

1
.
9.3.5 Uncertainty
It is reasonably straightforward to apply the e¢ ciency concept in de…nition 9.2
to the case where an economy is characterised by uncertainty, following on the
analysis of section 8.6. The key issue is whether e¢ ciency is to be viewed before
or after the uncertain state-of-the-world is revealed (be careful not to confuse
the concept of a state-of-the -world ! 2  with that of a social state  2 ).
A standard approach is as follows. Consider a situation in which social states
are completely described by allocations. Take an allocation ^a in which the
consumption of household h under state-of-the-world ! is ^x
h
!
and the resulting
utility for household h is ^
h
, h = 1; 2; :::; n
h
.
252 CHAPTER 9. WELFARE
Figure 9.9: Conditions for e¢ cient provision of public goods

De…nition 9.5 An allocation ^a is ex-ante Pareto e¢ cient if it is feasible and
there is no other feasible allocation a with associated utility levels 
h
; h =
1; 2; :::; n
h
such that, for all h,

h
 ^
h
(9.38)
with strict inequality for at least one h.
This is a general approach. If we wish to impose the restriction that each
person or household’s utility conform to axioms 8.1 to 8.3 (page 185) that un-
derpin the von-Neumann-Morgenstern functional form of utility (Theorem 8.1)
then we may write

h
= E
h
u
h

x
h

=
X
!2


h
!
u
h

x
h
!

(9.39)
9.3. PRINCIPLES FOR SOCIAL JUDGMENTS: EFFICIENCY 253
where 
h
!
denotes the system of (subjective) probability weights used by house-
hold h, and E
h
denotes expectation with respect to this set of subjective prob-
abilities. Using (9.39) the condition (9.38) becomes
X
!2

h
!

u
h

x

h
!

 u
h

^x
h
!

 0: (9.40)
So ex-ante e¢ cien cy has the interpretation that there is no other allocation
which dominates it in terms of expected utility. However, it is also reasonable
to consider e¢ ciency only from the ex-post standp oint, after the state-of the-
world has been realised
De…nition 9.6 The allocation ^a is ex-post Pareto e¢ cient if there is no other
feasible allocation a with associated utility levels 
h
= u
h

x
h
!

; h = 1; 2; :::; n
h
such that, for all h, and all ! 2 
u
h


x
h
!

 u
h

^x
h
!

 0 (9.41)
with strict inequality for at least one h.
Comparing (9.40) and (9.41) we can see that the following must be true:
Theorem 9.7 (Ex-ante e¢ ciency) If there is no state-of-the-world which is
regarded by any household as impossible then any ex-ante Pareto-e¢ cient allo-
cation must also be ex-post Pareto-e¢ cient.
However, the reverse is not true; one can easily …nd social states that are
e¢ cient ex-post, but not ex-ante. Figure 9.10 illustrates this point. The axes
of the diagram viewed from origin O
a
are the same as in Figure 8.2 and give
consumption of a single commodity by person a in the two states-of-the-world;
the axes of the diagram viewed from origin O
b
do the same job for person b.
The points along the contract curve from O
a
to O

b
represent ex-ante-e¢ cient
allocations; p oints o¤ the curve are not e¢ cient ex ante: you can then increase
the expected utility of one person without reducing the expected utility of the
other. However any point in Figure 9.10 is ex-post e¢ cient: once the state-of-
the-world ! is known you can only increase the ex-post utility of one person by
reducing the consumption (and hence the utility) of the other.
Theorem 9.8 (Ex-ante e¢ ciency) If a competitive equilibrium exists in the
market for contingent goods then it is ex-ante Pareto e¢ cient.
The proof of this result is s traightforward, since the existence of all the
contingent markets permits one to extend the standard results on competitive
equilibrium and Pareto e¢ ciency just by rede…ning the particular markets in-
volved. Likewise we have:
Theorem 9.9 (Ex-ante support) If consumers are greedy and have concave-
contoured utility functions then any ex-ante Pareto-e¢ cient allocation with pos-
itive incomes for all consumers can be supported by a competitive equilibrium.
254 CHAPTER 9. WELFARE
Figure 9.10: Ex-ante and ex-post e¢ ciency
Again this follows by a simple extension of the results that we obtained in
the elementary model of e¢ ciency and equilibrium in the absence of uncertainty
in section 9.3 above. For the result to go through we require, in principle, lump-
sum transfers to be available in all states of the world !, and the existence of a
full set of n$ markets in contingent goods.
9.3.6 Extending the e¢ ciency idea
Let us reconsider the Pareto superiority criterion. Whilst it appears to have an
attractive interpretation in some welfare terms –“approve a switch from state

0
to state  if no-one is worse o¤ in  than he would have been in 
0

and at least
one person is strictly better o¤”–it is very limited as a general policy rule. A
principal reason for this is that it is so wretchedly indecisive. There are a lot of
pairs of possible social states which just cannot be compared using this criterion;
and, as it is quite di¢ cult to think up lots of real-life examples where there have
been demonstrable Pareto improvements, the Pareto-superiority criterion does
not strike one as overwhelmingly useful in practice.
Let us consider what might be done to make the Pareto superiority criterion
more discriminating and, perhaps, more useful as a criterion for making wel-
fare judgements. To do this, we convert the problem into a two-stage decision
process.
To …x ideas, consider the example of a government which has to decide
whether or not to build an airport, and assume that the airport is a “one-o¤”
9.3. PRINCIPLES FOR SOCIAL JUDGMENTS: EFFICIENCY 255
project – either one has an airport of given size and quality or one does not.
There is in fact a huge range of possible social states associated with this decision
even though there is only one type of airport which could be built: the reason
for this is that there are all sorts of ways in which the gains and losses arising
from the project may be distributed amongst the community. S o it may make
sense to consider (a) all the social states that could be obtained through a pure
redistribution (for example, by taxes and transfers) given that resources have
been committed to the airport; and (b) all the states of the world that could
be obtained (by similar methods) given that the airport is not built. In either
case we describe these other states (obtainable through redistribution) as b eing
accessible from the reference s tate. So the decision process is something like
this:
1. look …rst at the resource commitment that is involved in building the
airport;
2. then consid er the states you can generate from the outcome of step 1 by
a further rearrangement of incomes.

On a more general note –with many possible projects of di¤erent types and
sizes – the idea in step 1 is that the alternatives are mutually exclusive and
irreversible, and that in step 2 all the states can be reached from one another
by steps that are in principle reversible. Clearly the distinction between the two
may be somewhat arbitrary and is reminiscent of the distinction between the
“short” and “long run”. Nevertheless one can perhaps think of many practical
decisions where such a distinction could reasonably be drawn.
To see how we may use this to extend the Pareto superiority criterion let
 and 
0
be the two states under consideration (“airport” and “no airport”),
and let
^
() be the subset of  that is accessible from . Then consider the
following:
De…nition 9.7 The state  is potentially superior to 
0
if there exists 

2
^
()
such that 

is Pareto superior to 
0
.
The idea is this:  is potentially superior to 
0
if there is some other state,

accessible from , which is actually Pareto superior to 
0
. In the airp ort example,
the rule says: “building the airport (state ) is potentially superior to not
building the airport (state 
0
), even if some people actually lose out thereby,
if it can be shown that, once the airport is built, there is some hypothetical
income redistribution which (were it to be actually implemented) would mean
that everyone was at least as well o¤ as before and no one was worse o¤ (state


)”.
Again there are some obvious drawbacks to this criterion. One is on moral
grounds. The state  is counted as being superior to 
0
on the above conditions
even though the switch by income redistribution to 

never takes place. To some
people this will seem manifestly objectionable.
256 CHAPTER 9. WELFARE
Figure 9.11: 

is accessible from 
0
and 
0
is accessible from 


There is a second, powerful objection, this time on the grounds of logic.
In Figure 9.11 v (

) represents

v
a
(

) ; v
b
(

)

, the vector of utility payo¤s
corresponding to 

and 


0

is the utility-vector correspon ding to 
0
. The
set of utility vectors corresponding to states accessible from 

, and the set of
points corresponding to states accessible from 

0
have been sketched in. Clearly,
there are points in the set accessible from 

that lie to the north east of 
0
, and
so 
0
2
^
() and thus, 

is potentially superior to 
0
. However we could just as
easily turn the argument round, examine the points lying to the north and east
of 

and …nd that 

2
^
(
0
). We have the extraordinary conclusion that 

is
potentially superior to 
0

and 
0
is potentially superior to 

! The solution to
this problem that seems to suggest itself is to strengthen the de…nition so that
such apparent contradictions cannot occur. Consider the following:
De…nition 9.8 The state 

is unambiguously potentially sup erior to 
0
if 

is potentially superior to 
0
but 
0
is not potentially superior to 

.
Unfortunately this apparently more attractive criterion may be vacuous since
it could just restore the indecisiveness of the original Pareto superiority prin-
ciple, and it may also lead to intransitive rankings of social states. However,
despite the problem of “reversals”associated with the simple “potential su peri-
ority criterion”there is a useful interpretation of this concept in terms of the ag-
gregate “willingness to pay”expressed as
P
n
h
h=1

CV
h
, the sum of all households’
compensating variations. Here CV
h


0
! 

means () the cost of restoring
9.4. PRINCIPLES FOR SOCIAL JUDGMENTS: EQUITY 257
household h to state 
0
from state  and is positive if the move from 
0
to  is a
welfare gain for h. In the case where “accessibility”is de…ned as above in terms
of monetary transfers at given prices p, we have:
Theorem 9.10 (Potential superiority) A necessary and su¢ cient condition
for  to be potentiall y superior to 
0
is that
P
n
h
h=1
CV
h
(

0
! ) > 0.
Note that this uses a strong de…nition of accessibility: for weaker versions
it is usually the case that
P
n
h
h=1
CV
h
> 0 is necessary, but not su¢ cient for
potential superiority. The p ractical importance of Theorem 9.10 is considerable
in that potential Pareto superiority is used as the intellectual basis for the
applied welfare-economic technique of cost-bene…t analysis.
9.4 Principles for social judgments: equity
We turn to another broad general principle that might be considered as a “rea-
sonable” basis for welfare economics. However, rather than an extension of
Pareto e¢ ciency – as in the case of potential e¢ ciency – in this case it is a
separate criterion that complements principles of e¢ ciency, namely equity. We
will have two attempts at this.
9.4.1 Fairness
In our …rst attempt equity is interpreted as “fairness”. Furthermore “fairness”
is to be given a very speci…c interpretation that enables u s to carry through the
analysis with information about households that is no more speci…c than in our
treatment of e¢ ciency.
To assist in the exposition of this version of the equity principle it is conve-
nient to suppose that each so c ial state  is adequately described by the allocation
a of goods embedded in it. In particular let a particular household’s evaluation
of a social state depend only on x
h

, the consumption vector that household en-
joys: households are sel…sh. Under this restricted interpretation it is convenient
to use the conventional utility function U
h
(x
h
) as an index of household h’s
preferences.
De…nition 9.9 A consumption allocation [x] is fair if, for every pair of house-
holds h; ` = 1; 2; :::; n
h
, it is true that U
h
(x
h
)  U
h
(x
`
).
In other words an allocation is fair if it is such that no one in the community
wishes he had somebody else’s bundle instead of his own: fairness is the absence
of envy. This then yields:
24
24
Pro ve this. Hint: Note that the de…nition of competitiv e eq uilibrium implies that, for
every household h : U
h
(x
h

)  U
h
(x
h
) for all x
h
such that
P
i
p
i
x
h
i
 M
h
. Now consider
the utility h w ould enjoy wer e it to receive the consumption bundle of household h
0
under this
equilibrium, and note that all households face the same budget constraints.
258 CHAPTER 9. WELFARE
Theorem 9.11 (Fairness of competitive equili brium) If all households have
equal incomes then a competitive equilibrium is a fair allocation.
Of course not only would such an equal-income equilibrium be fair, it would
also be Pareto e¢ cient –which appears like a powerful endorsement.
However, just because it meets both the requirements of e¢ ciency and “no-
envy”fairness, one should not assume that such an equilibrium is ideal. Indeed
one has only to imagine two households one of which consists of several physi-
cally disabled people and the other composed of a single, able-bodied person, to

see that such an allocation of equal incomes, regardless of di¤erences between
households, is not very attractive. The “fairness” concept is, therefore, not of
itself overwhelmingly powerful or compelling.
9.4.2 Concern for inequality
It is reasonable to say that the fairness interpretation of the equity principles
is somewhat special, possibly even a touch arti…cial. Perhaps one might have
in mind that equity should involve the opposite of inequality – interpreted in
terms of the distribution of income, somehow de…ned, or the distribution of
utility. But here we reach a temporary check to our analytical progress. If one
is to interpret equity in this way one cannot avoid detailed utility comparison
between households, something that we have not yet introduced to the discussion
of this chapter. To make further progress we would …rst need to impose a lot
more structure on the welfare analysis.
9.5 The social-welfare function
Therefore let us look at what can be done using a third, more restrictive, ap-
proach to social welfare, one that underpins a lot of work in applied economics.
This will involve four main elements:
 a restriction on the range of issues on which each household’s preferences
is to count;
 an assumption that one either knows, or one may impute, the preferences
of households;
 a basis for comparing the levels and scales of utility of one household with
another;
 a function for aggregating the utilities enjoyed by (or imputed to) each
household.
To incorporate all these features we shall take a speci…c social welfare func-
tion:
W (U
1
(x

1
); U
2
(x
2
); :::; U
h
(x
h
); :::) (9.42)
9.5. THE SOCIAL-WELFARE FUNCTION 259
Notice that W is de…ned on the space of individual utilities –n ot on orderings,
as was the “constitution”function . We have further assumed that individual
utilities are determined by their own consumptions thus:

h
= U
h
(x
h
); h = 1; 2; :::; n
h
: (9.43)
Clearly we have a rule which assigns a welfare level (some number W ) to any
consumption allocation [x] by a two-stage process
[x]
(1)
! (
1
; 

2
; 
3
; :::)
(2)
! W: (9.44)
Where does W come from? Of the various answers that have be en attempted
in the social science literature over the last few decades I shall draw attention
to two in particular:
 Equal ignorance. Even though individuals may be perfectly informed
about what society actually looks like, society is supposed to form judg-
ments about alternative social states behind a “veil of ignorance.”It is as
though a representative individual were to make choices amongst alterna-
tive social states without knowing the identity that he or she would have
within the social state. If society chooses among distributions in the same
way that an individual makes choices amongst uncertain prospects then
it is appropriate to let W have the same basic structure as an individual
utility function under uncertainty (see Exercise 9.3).
 The PLUM principle: People Like Us Matter. Someone in the community
makes the decisions, and he/she/they impute their values to everybody
else. In practice this may mean that W is determined by the preferences
of a particular political interest group.
9.5.1 Welfare, national income and expenditure
To see the implications of assuming a well-de…ned soc ial-welfare function W as a
representation of social preferences, let us consider how the welfare level changes
when there is a small change in the allocation. Let each person’s consumption
of each commodity change by an amount dx
h
i
. We …nd

dW =
n
X
i=1
n
h
X
h=1
W
h
U
h
i
dx
h
i
(9.45)
Notice the following features which can be inferred from this simple relationship:
 the issues on which W ranks states are limited: each x
h
is assessed only
on the basis of h’s preferences for it.
 the cardinalisation of U
h
is important here, since we need to aggregate –
add up –the changes in utilities.
25
25
How will social welfare change if each U
h

is subjected to an arbitrary a¢ ne trans forma-
tion?
260 CHAPTER 9. WELFARE
 as a res ult a p erson’s “weight”depend s on both W
h
and U
h
i
–the impor-
tance of his utility to social welfare, and the marginal utility to him of
good i.
Suppose the government can choose the allocation [x
h
], subject to some
overall constraint
(x
1
; :::; x
n
)  0;
where x
i
:=
P
n
h
h=1
x
h
i

; i = 1; 2; ::; n

Clearly we will …nd that for any pair of goods that are being consumed by any
two households:
U
h
i
U
h
j
=
U
`
i
U
`
j
(9.46)
(This we knew anyway from our consideration of e¢ ciency problems.) However
we will also …nd the condition
W
h
U
h
i
= W
`
U
`
i

(9.47)
Why? Because if the cost of producing good i is the same whoever consumes it,
then we shall only be at a welfare maximum if W cannot be increased by some
small transfer of bread, butter or toothpaste from h to `.
Now let us examine the properties of the social welfare function (9.42) in
the case of a market economy. Each household h maximises its utilityU
h
(x
h
)
subject to a budget constraint
P
p
i
x
h
i
 y
h
where y
h
is the household’s income:
the argument can easily be extended to the case where y
h
is endogenously
determined. We may substitute from the demand functions for each household
back into its utility function to obtain the indirect utility function for each
household V
h
(p; y

h
).This then yields the social welfare function in terms of
prices and households’incomes:
W (V
1
(p; y
1
); V
2
(p; y
h
); :::; V
h
(p; y
h
); :::): (9.48)
Recall that for a consumer’s optimum purchases in a free market we have U
h
i
=

h
p
i
if good i is purchased in positive amounts where the term 
h
is the
marginal utility of money income for household h and equals @V
h
=@y

h
– see
(4.12) and page 88. So the social optimality condition (9.47) can be rewritten,
in the case of a market economy, as
W
h
V
h
y
= W
`
V
`
y
(9.49)
for any pair of households h and `. This is the e¤ect on social welfare of giving
one dollar to any household at the optimum; let us call it M. Hence it is
immediate that if there is some economic change a¤ecting individual incomes
(for example a change in natural resource endowments or in the technology),
the change in social welfare is
dW =
n
h
X
h=1
W
h
dU
h
=

n
h
X
h=1
W
h
V
h
y
dy
h
= M
n
h
X
h=1
dy
h
(9.50)
9.5. THE SOCIAL-WELFARE FUNCTION 261
The right-hand side of (9.50) is proportional to the change in national income
y
1
+ y
2
+ ::: + y
n
h
.
Now consider a change in the prices p leaving incomes y

h
unchanged. Dif-
ferentiating (9.48) we …nd that the e¤ect on social welfare is
n
h
X
h=1
W
h
"
n
X
i=1
V
h
i
dp
i
#
(9.51)
But, since each household is assumed to be maximising utility, (9.51) becomes
26

n
h
X
h=1
W
h
V

h
y
n
X
i=1
x
h
i
dp
i
= M
n
X
i=1
x

i
dp
i
(9.52)
This is simply M times the change in the cost of aggregate expenditure (by
all households on all go ods) as a result of the price changes. In a market
economy, aggregate expenditure equals n ational income; so we have e stablished
that, whatever the reason for the change in the soc ial state, the following result
holds:
Theorem 9.12 (National income) In the neighbourhood of a welfare opti-
mum, welfare changes are measured by changes in national income.
Unless we believe that somehow the distribution of resources just happens
to be ideal in every case we wish to examine this result is of limited appeal.
What of other cases?

9.5.2 Inequality and welfare loss
In our earlier discussion of equity as a general welfare principle (section 9.4) we
recognised that a method of comparing individual u tilities would be necessary in
order to introduce a meaningful criterion based on the common-sense notion of
inequality. The social-welfare function approach enables us to take the necessary
steps.
Clearly there is some form of loss that would result if households’ money
incomes were not “correctly”adjusted –according to the social welfare function
W –so as to be able to invoke Theorem 9.12. We can examine the nature of this
loss using an approach that is somewhat reminiscent of quantifying departures
from e¢ ciency –“waste” –discussed in section 9.3.2 above. To do this let us
make two further simplifying assumptions:
27
 all the V
h
are identical, and
 W is a s ymmetric and c oncave function.
26
Pro ve this . Hi nt: try using Roy’s identity.
27
Suppose the economy is composed of two types of households single i ndividuals, and
couples (who share their income). Show how t he results in this section can be established if
ho useholds are weighted by size and incomes adjusted to “per-person equivalents”.
262 CHAPTER 9. WELFARE
Given that all the households are assumed identical, the assumption of sym-
metry is a natural one: it implies that there is no signi…cance in the labelling of
individual households 1; :::; n
h
. The assumption that W is concave implies that
“society” – as represented by the social welfare function –is weakly averse to

an unequal distribution of income. Now national income is equal to the value
of all the resources in the community plus all the pro…ts made by …rms which,
in a market economy, can be written:
P
n
h
h=1
y
h
=
P
n
i=1
p
i
R
i
+
P
n
f
f=1

f
(p);
where

f
(p) =
P

n
i=1
p
i
q
f
i
(p)
9
=
;
(9.53)
So obviously national income will be …xed for a given price vector p and for
given resources and technology. So, in view of the concavity and symmetry of
W , we can see that for a given p, W would be maximised in a situation where
every household received has an equal share of national income; in other words
where everybody gets mean income Ey :=
P
h
y
h
=n
h
.
Consider now situations in which every household is not receiving an equal
share. A natural way of measuring the apparent loss attributable to the less-
than-ideal property distribution suggests itself in the light of chapter 8’s discus-
sion of the using the risk premium concept in the context of the expected-utility
model (see page 191). Consider the income that, were it to be given identically
to every household, would yield the same level of social welfare as the actual

incomes y
1
; y
2
; :::; y
n
h
. This income is clearly less than or equal to y, and the
di¤erence between the two can be regarded as a money measure of the shortfall
in social welfare attributable to the inequality of incomes.
De…nition 9.10 (a) The equally-distributed equivalent income  is a real num-
ber such that
W (V (; p); V (; p); :::) = W (V (y
1
; p); V (y
2
; p); :::): (9.54)
(b) The inequality index is 1 =y.
This is illustrated, for a two-household example, in Figure 9.12. The ray
through the origin is at 45

to the axes: any point on this represents a situation
of exact equality of income distribution. So, given an actual distribution of
income (y
a
; y
b
) represented by the point ^y we …nd the mean by drawing a
perpendicular from ^y to the 45


ray: this perpendicular meets the ray at Ey, the
point (y; y). The contour of the social welfare function W(V (p; y
a
), V (p; y
b
))
that passes through ^y is symmetric about the ray, and cuts the ray at point ,
with coordinates (; ). The more sharply curved is this contour, the greater
the inequality 1 =y.
You may have already spotted the similarity of these concepts to the concepts
of certainty equivalence and risk premium. This becomes even more evident if we
impose a further restriction on the social welfare function W . If we assume that
it is additively separable (analogous to the von Neumann Morgenstern utility
9.5. THE SOCIAL-WELFARE FUNCTION 263
Figure 9.12: The social-welfare function
function
28
) then, suppressing the price vector p (assumed to be held invariant)
we may rewrite (9.48) as
n
h
X
h=1
(y
h
) (9.55)
where  is an increasing, concave function of one variable. Then the equation
de…ning the equally-distributed equivalent income can be rewritten
 = 
1

1
n
h
n
h
X
h=1
(y
h
)
!
(9.56)
Recall that in the case of choice under uncertainty the curvature of the
function re‡ected the degree of risk aversion, an d hence the risk premium to be
imputed to any particular prospect. Likewise the curvature of  determines the
degree of “inequality aversion” that is implicit in the social welfare function:in
fact, assuming di¤erentiability of , we conventionally de…ne the following
De…nition 9.11 The (relative) inequality aversion of a social welfare function
is given by
(y) := y

y y
(y)

y
(y)
(9.57)
28
In this applicat ion what is the counter part to the probabilities used in the von Neumann-

Morgenstern ca se?
264 CHAPTER 9. WELFARE
Then, borrowing results from the theory of choice under uncertainty we may
immediately state
29
Theorem 9.13 (Concavity and inequality aversion) Let  and
^
 be in-
creasing, concave functions of one variable, such that
^
 is a concave transfor-
mation of . Then (a) ^  ;(b)
^
  .
So the greater is the inequality aversion implicit in the social welfare function,
the greater is the apparent loss attributable to any given unequal distribution
of income. It is important to emphasise that this is an apparent loss since there
is no reason to suppose that in practice it is legitimate to take total income as
given.
9.6 Summary
Moving from individual prefe rence s and decision-making –as in chapter 4 –to
preferences and decision-making for society is a challenge. Our three approaches
to social welfare show why this is so:
 The Arrow impossibility result (Theorem 9.1) is of fundamental impor-
tance in understanding why it is intrinsically so di¢ cult to express social
preferences as a general function of individual preferences.
 Of the principles for social judgment it is clear that Pareto e¢ ciency is
overwhelmingly important. It has a natural de…nition in models of perfect
certainty and can be extended without much di¢ culty to uncertainty.
In a pure private-good s economy the conditions for e¢ ciency are very

straightforward and can be ful…lled by a competitive equilibrium. But it
is di¢ cult to extend the notion of Pareto superiority (on which e¢ ciency
is based) to criteria that permit general application.
 The social-welfare function seems like a useful way of cutting through
the di¢ culties where gen eral principles appear indecisive. But where is it
supposed to come from? On what basis can we compare the utility levels
or utility scales of one person with another?
9.7 Reading notes
A good overview of the main issues in welfare economics is provided by Boadway
and Bruce (1984). On the “constitution” approach see Arrow (1951), Black
(1958) and the excellent paper by Vickrey (1960) ; for the basis of Theorem 9.2
see Black (1948).
The standard references on e¢ ciency with public goods are Samuelson (1954,
1955).
29
Pro ve this using the results from Chapter 8.
9.8. EXERCISES 265
Keenan and Snow (1999) summarise a variety of criteria for potential su-
periority and the relationship between them; the literature on the “reversals”
problems as soc iated with potential superiority was initiated by Kaldor (1939),
Hicks (1946) and Scitovsky (1941). The fairness discussion is based on an im-
portant contribution by Varian (1974).
Using the individual’s attitude to risk as the basis for a social welfare function
is attributable to Vickrey (1945) and Harsanyi (1955). On the social-welfare
interpretation of inequality and income distribution and its relationship to risk
aversion see Atkinson (1970). The developments of social-welfare criteria for
use in applied economics are reviewed in Harberger (1971) and Slesnick (1998).
9.8 Exercises
9.1 In a two-commodity exchange economy there are two large equal-sized groups
of traders. Each trader in group a has an endowment of 300 units of commodity

1; each person in group b has an endowment of 200 units of commodity 2. Each
a-type person has preferences given by the utility function
U
a
(x
a
) = x
a
1
x
a
2
and each b-type person’s utility can be written as
U
b
(x
b
) =
x
b
1
x
b
2
x
a
1
where x
h
i

means the consumption of good i by an h-type person.
1. Find the competitive equilibrium allocation
2. Explain why the competitive equilibrium is ine¢ cient.
3. Suggest a means whereby a benevolent government could achieved an e¢ -
cient allocation.
9.2 Consider a constitution  based on a system of rank-order voting whereby
the worst alternative gets 1 point, the next worst, 2, and so on, and the state
that is awarded the most points by the citizens is the one selected. Alf’s ranking
of social states changes during the week. Bill’s stays the same:
Monday: Tuesday:
Alf Bill Alf Bill
 
0
 
0

0
 
00


00

00

0

00
What is the social ordering on Monday? What is it on Tuesday? How does
this constitution violate Axiom 9.3?

266 CHAPTER 9. WELFARE
9.3 Consider an economy that consists of just three individuals, fa; b; cg and
four possible social states of the world. Each state of the world is characterised
by a monetary payo¤ y
h
thus:
a b c
 3 3 3

0
1 4 4

00
5 1 3

000
2 6 1
Suppose that person h has a utility function U
h
= log(y
h
).
1. Show that if individuals know the payo¤s that will accrue to them under
each state of the world, then majority voting will produce a cyclic decision
rule.
2. Show that the above conditions can rank unequal states over perfect equal-
ity.
3. Show that if people did not know which one of the identities fa; b; cg they
were to have before they vote, if they regard any one of these three identities
as equally likely and if they are concerned to maximise expected utility, then

majority voting will rank the states strictly in the order of the distribution
of the payo¤s.
4. A group of identical schoolchildren are to be endowed at lunch time with
an allocation of pie. When they look through the dining hall window in
the morning they can see the slices of pie lying on the plates: the only
problem is that no child knows which plate he or she will receive. Taking
the space of all possible pie distributions as a complete description of all
the possible social states for these schoolchildren, and assuming that ex
ante there are equal chances of any one child receiving any one of the
plates discuss how a von Neumann-Morgenstern utility function may be
used as a simple social-welfare function.
5. What determines the degree of inequality aversion of this social-welfare
function?
6. Consider the possible problems in using this approach as a general method
of specifying a social-welfare function.
9.4 Table 9.4 shows the preferences over three social states for two groups
of voters; the row marked “#” gives the number of voters with each set of
preferences; preferences are listed in row order, most preferred at the top.
1. Find the Condorcet winner (see footnote 3) among right-handed voters
only.