Discussion Paper No. 698

THE MEASUREMENT OFSOCIAL WELFARE

Atsushi Tsuneki

September 2007

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

The Measurement of Social Welfare

Atsushi Tsuneki*
Institute of Social and Economic Research, Osaka University

JEL classification: H20; D61; D63
Keywords: Social Welfare, Deadweight Loss, Measurement, Second OrderApproximation*Corresponding Author. Institute of Social and Economic Research, Osaka
University, 6-1, Mihogaoka, Ibaraki, Osaka, 567-0047 Japan.
Tel.:+81-6-6879-8562; Fax: 81-6-6878-2766;e-mail: tsuneki@iser.osaka-u.ac.jp

Abstract
This paper derives a basic formula for the measure of social welfare, a second
order approximation to the difference of the value of the Bergson-Samuelsonsocial welfare function between the socially optimal resource allocation and theone in the present suboptimal economy. We discuss pros and cons of our approachfor the measurement of welfare compared with other approaches to applied
welfare economics, especially the measurement of deadweight loss.

1

Introduction
The purpose of this paper is to present our basic formula for the measure ofsocial welfare and discuss its pros and cons compared with other approaches toapplied welfare economics.

Up to now, approaches for measuring the economic

performance of the society chiefly concentrated to the measurement of inefficiency
such as deadweight loss.

In some circumstances, however, it becomes essential

to evaluate the equity aspect of the economic policy at the same time as efficiency.
We present a methodology for evaluating the social welfare directly instead ofevaluating the efficiency loss to answer the question.
We introduce our model in the next section.

In section 3, our point iselaborated by taking a second order approximation to the linear social welfare
function when tax distortions prevail. In section 4, the economic implications of
our approach are discussed, especially in comparison with literatures on themeasurement of deadweight loss, which include Hotelling (1938), Hicks (1942),
Allais (1943, 1977), Boiteux (1951), Debreu (1951, 1954), Harberger (1964), andDiewert (1981, 1984, 1985).

Section 5 briefly concludes.

2. The Model
Let us set up the model of our economy. There are H consumers having
quasi-concave utility functions f h ( x h ), h = 1, K , H defined over a translatedh )T is a consumption vector of goods 1, K, N byorthant Q h where x h ≡ ( x1h , K , x N

the h -th consumer. The initial endowment vector of the h -th consumer is given
by x h , h = 1, K, H .
production

There are K firms and firm k produces y k using thepossibilities

set

S k , k = 1, K, K ,
2

which

are

subsetsoftheN -dimensional nonnegative orthant and satisfies standard regularity properties.

We define the expenditure function:1m h ( p, u h ) ≡ min h { pT x h : f h ( x h ) ≥ u h } .
x(1)where p 〉〉 0 N . See Diewert (1982; 554) for the regularity properties that must be
satisfied by the functions m h .
The profit function π k , k = 1, K, K is defined as

π k ( p) ≡ max x { p T y : y ∈ S k }, k = 1, K , K .

(2)

The regularity properties of the profit function are summarized in Diewert (1982;
580-1).
We assume below that the production possibilities sets S k , k = 1, K, K areconvex. We also assume for simplicity that markets are complete and that the only
source of distortions is indirect taxes levied on consumers. Extensions of theseassumptions are not difficult.

See Diewert (1983, 1986) and Tsuneki (1987,

1995).
Generally, the desirable properties of the ordering of social states are
summarized in the Bergson-Samuelsonian social welfare function (BSSWF
hereafter).

(See Samuelson (1956) for a discussion of the BSSWF and its

properties listed below.)

We first assume that the underlying social ordering iscompatible with the Pareto partial ordering (i.e., if all individual utilities increase,1x 〉〉 0 N means that each element of the vector x is strictly positive, x ≥ 0 N means thateach element of x is nonnegative, and x > 0 N means x ≥ 0 N but x ≠ 0 N .
superscript T means transpose.
3

Athen so does social welfare) so that the resulting BSSWF becomes Pareto-inclusive.
Suppose also that the evaluation of social states is individualistic (or welfarist);i.e., the utility vector u prevailing at the state is the only information used in theevaluation.

Also suppose that the evaluation takes the form of a continuous

ordering of utility vectors.

Then, Debreu’s (1959; 56) representation theorem isapplied to get the BSSWF, S (u ) .

Pareto-inclusiveness implies that S ismonotone increasing in u .
We introduce the value of BSSWF at the socially optimal allocation and its
value at the present tax-distorted equilibrium.

We call the numerical differenceof the value of these two resource allocations BSSW measure of welfare. For thenumerical evaluation of the difference of the value of BSSWF to be meaningful,
we assume below that each individual’s utility is measurable on a cardinal scalewith a unique zero and that utility functions are identical. Furthermore, we
assume that utilities are interpersonally comparable.2
In deriving the formula for BSSW measure, we first characterize the socially
optimal resource allocation in the form of mathematical programming as follows:(3)

Max h k {S ( f 1 ( x1 ), f 2 ( x 2 ),K, f H ( x H )) s.t.
x ,yH+x h ; y k ∈ S k , k = 1,K K } .
h =1

∑h =1
H

xh ≤

∑k =1 y ktheutilityK∑

Hereafterwestrengthen

theassumptiononfunctionsf h ( x h ), h = 1, K , H , so that they are concave following Negishi (1960).
Furthermore we assume that S is concave with respect to u by introducing an
2 This kind of assumption is not at all exceptional in the study of public economics andpractical policy analysis. See Feldstein(1976; 79). For a more detailed discussion for justifying
the assumption of identical utility functions among individuals, see also Mirrlees(1982).
4

explicit value judgment for equity by the social planner.
At this point, we use the concept of the overspending function B which isdefined as

B ( q, p, u ) ≡∑ m ( q, u ) − ∑ q x
H

h =1

H

h

T

h

h =1

h

−

∑ π ( p) .
K

k

k =1

In Appendix I, B is restated with its economic interpretation and its usefulproperties are summarized. Using the Uzawa (1958; 34) - Karlin (1959; 201)
Saddle-point Theorem using the definition (1), (2) and (A.1), we can rewrite (3) as
follows3 (the calculation is in Appendix II):(4)

Max u Min p ≥ 0 N {S (u ) − B( p, p, u )} .

We assume that (i) ( u 0 , p 0 ) solves (4), (ii) the first order conditions for (4) holdwith equality so that p 0 〉〉 0 N , (iii) B is twice continuously differentiable at the
0 + B 0 is negative definite. From assumptions (i) and (ii),
optimum, and (iv) Bqq
pp

we find the first order conditions for (4) are:(5)

∇ u S (u ) = ∇ u B( p, p, u ) ,(6)− ∇ p B ( p , p, u ) − ∇ q B ( p , p, u ) = 0 ,where ∇ u S (u ) ≡ [∂S / ∂u1 , ∂S / ∂u 2 , K , ∂S / ∂u H ]T .
Condition (6) is the equality of demand and supply at the optimum while (5) is therule to equate the marginal social importance of each person to the inverse of his
marginal utility of income (see Negishi (1960)).4

Note that the solution depends

on ∇ u S (u ) implies that a distributive value judgment of the social planner is
3 We also assume that the Slater constraint qualification condition applies in this economy;i.e., we require that a feasible solution for (3) exists that satisfies the first N inequality
constraints strictly.
4 With appropriate lump-sum transfers across households, the budget constraints ofindividuals are satisfied and the government budget constraint is implied by them and (6).
5

explicitly introduced.
In principle, it is possible to compute socially optimal resource allocation andits associated S (u 0 ) so that we can compute the BSSF measure globally.

The

numerical general equilibrium approach by Shoven and Whalley (1972, 1973,1977) which computes equilibria directly corresponding to various tax and
expenditure policies should be helpful in this process.

However, an obvious

drawback of the numerical general equilibrium approach is that we must haveinformation on global functional forms of utility and production functions.
Furthermore, in the numerical general equilibrium approach, very restrictivefunctional forms are adopted to make global computation possible, and theserestrictions are easily rejected in econometric tests using more general functional
forms (see Jorgenson (1984; 140)).
In contrast, local approximation approach we adopt in the sections that follow,
only local information of these functions is required, and it does not involve any
numerical computations that are more complicated than a single matrix inversion
whereas the global computation of the social optimum often causes substantial
numerical difficulties.

Therefore in this paper, we concentrate on the study ofapproximate measures of welfare.

Specifically, we derive a second orderapproximation to the BSSW measure.

3.

Second Order Approximation
To compute the second order approximation to the BSSW measure around the

optimal equilibrium, we first construct a z -equilibrium:(7)∇ u B( p( z ) + tz , p( z ), u ( z )) = ∇ u S (u ( z )) + λz ;6− ∇ q B( p ( z ) + tz , p( z ), u ( z )) − ∇ p B ( p ( z )) + tz , p ( z ), u ( z )) = 0 .

(8)When z = 0 , (7) and (8) coincide with the first order conditions for the maximumof social welfare (5) and (6), if we define u (0) ≡ u 0 and p(0) ≡ p 0 .

When z = 1 ,(8) is a set of equations to show the market clearing conditions at the tax-distorted
equilibrium, if u (1) ≡ u1 and p (1) ≡ p1 are the values prevailing at the observed
distorted equilibrium.

If we assume that the level of lump-sum transfers fromthe government to consumers is appropriately chosen, there exist budget
constraints for consumers compatible with (7) and (8).

(8) and these budget

constraints imply the budget balance of the government.

When z = 1 , (7)quantifies the ‘equity’ distortions at the observed equilibrium; i.e., −λ h shows
the difference between the marginal social importance of the h -th person and theinverse of his marginal utility of income.

It must be noted that both ∇ u S (u )and the marginal utility of income are not invariant to a monotone transformationof f h ( x h ) .

However, they are adjusted proportionally so that (5) is valid.

We

must also adjust λh proportionally to h ’s marginal utility of income and∇ u S (u ) so that (7) is valid.
Now differentiate (7) and (8) with respect to z ;(9)⎡z , Bz ⎤⎡B z− S uu
u
z
'
(
)
uu
uq
⎡
⎤
⎢ −λ
⎥
− ⎢
=
⎢ z
z
z ⎥ ⎢⎣ p' ( z )⎥⎦ ⎢ z⎢⎣ Bqq t⎢⎣ Bqu , Bqq + B pp ⎥⎦z ⎤+ Buq t ⎥⎥⎥⎦2 B( p ( z ) + tz , p ( z ), u ( z )), S z ≡ ∇ 2 S (u ( z )) for i, j = q, p, u .
where Bijz ≡ ∇ ij
ij
ij

Note thatz =0
Bup
H × N . Assumptions (iii) and (iv) guarantee, via the Implicit Function

Theorem, that once continuously differentiable functions u (z )satisfying (9) exist at z close to 0.

and

p(z )

Premultiplying ⎡0TH , p( z )T ⎤ to both sides of
⎢⎣
⎥⎦

7

(9) and using property (iii) of the overspending function in Appendix I, we show in
Appendix III that∑h =1∇u m h ( p( z) + tz, uh ( z))uh′ ( z)H(10)

is derived.

z ( p ' ( z ) + t ) + t T B z u ' ( z )⎤,= z ⎡t T Bqq
qu⎢⎣⎥⎦
Evaluating (10) at z = 0 , we get∑h =1∇u m h ( p 0 , uh0 )uh′ (0) = 0 .
H(11)

We next differentiate (10) with respect to z , and evaluate at z = 0 to obtain(12)

0 u ' (0) +u ' (0)T Buu∑h =1∇u m h ( p 0 , uh0 )uh′′ (0)H

0 ( p ′(0) + t ) − p ′(0)T B 0 u ′(0).
= t T Bqqqu
Premultiplying (9) evaluated at z = 0 by [0TH , p ′(0)T ] and adding the resulting
identity to (12), we haveu ' (0) T Buu0 u ' (0) + ∑h =1 ∇ u m h ( p 0 , u h0 )u h′′ (0)
H

(13)

0
0
= p ′(0) T B pp
p ′(0) + [ p ′(0) + t ]T Bqq[ p ′(0) + t ].

A second order Taylor approximation to the BSSW measure LBSSW at z = 0 is
as follows:(14)

L BSSW ≡ S (u (0)) − S (u (1)) = − {∇ u S (u 0 ) T u ′(0)+ 1 / 2[u ′(0) T S uu0 u ′(0) + ∇ u S (u 0 ) T u ′′(0)]}.

Substituting (5) into (11), we have ∇ u S (u 0 )T u ′(0) = 0.

Substituting it and (5),(13) into (14), we finally have
(15)

0
0
LBSSW ≅ − 1 / 2{ p ′(0) T B pp
p ′(0) + [ p ′(0) + t ]T Bqq[ p ′(0) + t ]+ u ′(0) T [ S uu0 − Buu0 ]u ′(0)} ≥ 0,

where the inequality comes from the concavity of B with respect to p and q , the0 , which is implied by the concavity of the utility
positive semidefiniteness of Buu
0 implied by the concavity of thefunctions, the negative semidefiniteness of S uu
8

BSSWF.

4. Implications
The remarkable advantage of the approximation approach relative to thenumerical general equilibrium approach is that it can be implemented from thederivatives of the overspending function evaluated at the optimum equilibrium, sothat we need not know global functional forms for utility and production functions.
As an important corollary of this fact, our approximate measure can be derived
from any set of flexible functional forms using local information around theequilibrium.

However, as long as we must know the derivatives at the optimum

as in (15), we must actually know the optimal prices so that we must compute theoptimum or we must depend on some ‘guessing’ process about the values at the
optimum.

Harberger(1964)

suggested

replacingthese(unobservable)

derivatives by those which are evaluated at the observed distorted equilibrium,
since they can be calculated using data prevailing at the observed equilibrium.
We now compare the empirical implementability of L BSSW in (15) and (9) andthe usual approximate deadweight loss measure represented by Debreu (1954),
Harberger (1964) and especially Diewert (1985).

Despite their apparent

similarities, their meanings are completely different.

First, (15) includes a pureequity term u ′(0) T [ S uu0 − Buu0 ]u ′(0) which does not exist in the measurement ofdeadweight loss.

Second, p ′(0) and u ′(0) calculated from (9) are also affectedby the pure equity term λ .
In (15), we need both the substitution matrices B1pp , B1qq and income effectmatrices B1qu , tax rates t and the distributional distortion parameters λ so that9the informational requirements are much higher than the usual measurement ofdeadweight loss.

Though it is possible to calculate B1pp and B1qq from local

information on ordinary demand curves and supply curves at the distorted
equilibrium, we have to know the marginal utility of income ∇ u B( p1 + t , p1 , u1 )to compute λ from (7) or B1qu from ordinary demand curves.

As the marginalutility of income is not an operational concept without knowing its original utility
function, in light of these observations, we must conclude that the approximate
BSSW measure lacks empirical operationality without a knowledge of the originalutility functions whereas the measurement of deadweight loss is free from thisproblem.
It is chiefly for this reason that standard literature has concentrated to thecomputation of deadweight loss instead of BSSWF.

Needless to say, however, theinformational advantage of using the deadweight does not mean that it is asuperior measure to the BSSW measure of welfare. As long as we can measure
the difference between the first order weight of a BSSWF and the inverse of themarginal utility of income, the same type of analysis in the measurement ofdeadweight loss can be carried out using a BSSWF.

As we have stressed thus far,adopting a BSSWF is superior to measuring deadweight loss for the evaluation
based on the BSSWF can capture both efficiency of equity aspect of the economy.
The only equity-concerned welfare measure we know is the one originated byBoiteux (1951), which was taken up again recently by Diewert (1981, 1984, 1985).
Boiteux’x measure takes the sum of Hicksian (1942; 128) equivalent variations
where its reference prices vector is chosen from the viewpoint of normative equity.
Unfortunately, this reference prices vector is difficult to compute.
10

In addition,the second order approximation of the Boiteux measure requires the informationon the hypothetical income vector at the optimum which is difficult to obtain (seeDiewert (1984; 36)).

In contrast, we can avoid this problem by explicitly

assuming the form of utility functions and the comparability of utilities acrossindividuals as value judgment of the social planner.

In this sense, our approach

to measure the value of BSSWF directly seems much more natural way toevaluate the performance of the economy including both equity and efficiency.

5. Conclusion
Present analysis has derived our basic formula for the measure of social
welfare. Although it demands more information for its measurement than usual
analysis of the deadweight loss, it has an important significance because it canevaluate both efficiency and equity aspect of the economy. We hope that our study
serves for the future policy analysis.

11Appendices
Appendix I:

The Properties of the Overspending Function.

An overspending function, introduced into economics by Bhagwati, Brecher
and Hatta (1983; 608) summarizes the general equilibrium relations of an
economy within one equation.

It may be interpreted as the aggregate net

expenditure of consumers facing prices q minus the aggregate profits of firms
facing prices p .

It inherits many useful properties of expenditure functions andprofit functions which are exhibited in Diewert (1982).

We collect several

important properties for later use.
An overspending function is defined by(A.1)

B (q, p, u ) ≡ ∑h =1 m h (q, u h ) − ∑h =1 q T x h − ∑k =1 π k ( p ) .
H

HKIt has the following properties.
(i)

B is concave with respect to p and q .

(ii)

If B(q, p, u ) is once continuously differentiable with respect to q and p
at (q, p, u ) , then ∇ q B (q, p, u ) is the aggregate net consumption vectorand − ∇ p B(q, p, u ) is the aggregate net production vector.

(iii) The following identities are valid for any (q, p, u ) if B is twice
continuously differentiable at (q, p, u ) :(A.2)

q T Bqq = 0TN ,(A.3)

q T Bqu = (∇ u B)T = (∂m1 (q, u1 ) / ∂u1 , K , ∂m H (q, u H ) / ∂u H ) ,(A.4)

p T B pp = 0TN ,

2 B (q, p, u ) for i, j = q, p, u .
where Bij ≡ ∇ ij
Property (i) follows from the fact that an expenditure function is concave withrespect to prices and a profit function is convex with respect to prices.
12

Property(ii) is a straightforward consequence of Hotelling’s (1932; 594) lemma and theHicks (1946; 331)-Shephard (1953; 11) lemma.

Property (iii) is a consequence ofthe linear homogeneity of an expenditure function and a profit function withrespect to prices.

Appendix II
In this Appendix, we show that (3) and (4) are equivalent given concave utility
functions and convex production sets, provided the Slater constraint qualification
holds.

In (3), S (u ) is assumed to be concave, the set {x h : f h ( x h ) ≥ u h } is convex

from the concavity off h ( x h ), S k is also assumed to be convex and theinequalities are linear.

Therefore, (3) is a concave programming problem and the

Uzawa-Karlin Saddle Point Theorem is applicable.

Rewrite (3) as :max u , x h , y k min p ≥0 N {S (u ) + p T [∑k =1 y k + ∑h =1 x h − ∑h =1 x h ] :
K

H

H

(A.5)
f h ( x h ) ≥ u h , h = 1, K , H ; y k ∈ S k , k = 1, K , K },≡ max u min p ≥ 0 N {S (u ) +

∑h =1
H

pT x h ++(A.6)≡ max u min p ≥ 0 {S (u ) +Nusing the definitions (1) and (2).

∑h =1[max − pT x h : f h ( x h ) ≥ uh ]H

∑k =1max pT y k : y k ∈ S k ]}K

∑h =1 pT x h + ∑k =1π k ( p) − ∑h =1m h ( p, uh )}HKHFrom the definition (A.1), we have (4).

Appendix III

13

In this Appendix, we derive (10).

Premultiplying both sides of (9) by[0TH , p( z )T ] , we have:(A.7)

z u ′( z ) − p ( z )T [ B z + B z ] p ′( z ) =− p ( z )T Bqu
qq
pp

z t.
p ( z )T Bqq

From (A.2), (A.3) and (A.4) evaluated at ( p, q, u ) = ( p( z ) + tz , p( z ), u ( z ) ), we havez = − zt T B z , p ( z )T B z = (∇ B)T − zt T B z and p ( z )T B z = 0T .
p ( z )T Bqq
qq
qu
u
qu
pp
N

Substituting these equations into (A.7), we have (10).

14Acknowledgement: The author thanks Professors W. Erwin Diewert and John
Weymark for their valuable suggestions and support in writing the article.
Discussion with Hideki Konishi and the suggestions by Hiroshi Yoshikawa werealso very helpful. The remaining errors are strictly my own responsibility.

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18