Market demand curves and Dupuit–Marshall consumers’
surpluses: a general equilibrium analysis
a b ,*
Charles Blackorby , David Donaldson
a
University of British Columbia and GREQAM, Vancouver, Canada b
Department of Economics, University of British Columbia, 997 –1873 East Mall, Vancouver,
Canada BC V6T 1Z1
Received 30 October 1997; received in revised form 1 April 1998; accepted 15 May 1998
Abstract
This paper investigates a claim of Hicks that the area to the left of a market demand curve—calculated by assuming that other markets are in equilibrium—is equal to the aggregate overall Dupuit–Marshall consumers’ surplus together with producers’ surplus in other markets, and extends it to general equilibrium in an economy with a competitive private sector. The usefulness of the result is questioned by considering path independence and the possibility of consistency with a Bergson–Samuelson social-welfare function. In the only interesting possibility,
´
there must be no income effects on non-numeraire goods for each person. 1999 Elsevier Science B.V. All rights reserved.
Keywords: Consumer’s surplus; General equilibrium
[P]olitical economy, being concerned only with wealth, can take account of the intensity of a wish only
through its monetary expression. Political economy only bakes bread for those who can buy it, and leaves to social economy the care of supplying it to those with nothing of value to give in exchange. Dupuit
(1969) p. 262
This involves the consideration that a pound’s worth of satisfaction to an ordinary poor man is a much greater thing than a pound’s worth of satisfaction to an ordinary rich man . . . In earlier generations, many statesmen, and even some economists, neglected to make adequate allowance for considerations of this class, especially when constructing schemes of taxation; and their words or deeds seemed to imply a want of sympathy with the sufferings of the poor, though more often they were simply due to lack of thought. Marshall (1920) p. 108
*Corresponding author. Tel.:11-604-224-0933; fax:11-604-822-5915; e-mail: [email protected] 0165-4896 / 99 / $ – see front matter 1999 Elsevier Science B.V. All rights reserved.
1. Introduction
In consumer’s surplus analysis, an interesting argument, originally due to Hicks (1946a), (1946b), is used to justify the use of a kind of partial-equilibrium analysis. It is most clearly stated in Sugden and Williams (1978). Suppose that an economy consists of an undistorted competitive private sector and a government sector that produces marketed goods. If the government increases its production of one good, price is changed in that market, but price and quantity changes in other markets are also induced. The argument claims that the area to the left of the market demand curve in the original market is equal to the sum of Dupuit–Marshall consumers’ surpluses in all markets plus producers’ surpluses in the other markets. The market demand curve is the one that obtains when all other markets clear on the transition path (see Section 2 for a discussion). Adding this surplus to the change in government-sector profit provides a cost-benefit test.
We show that this argument is true in general equilibrium. As in Hicks and Sugden / Williams, price changes are generated by changing a public-sector input-output vector. In principle, all prices in the economy could change. The area to the left of the market demand curve(s) (translated into a general-equilibrium setting) is equal to the sum across all consumers of the sum of Dupuit–Marshall surpluses and producers’ surpluses in all markets. In general equilibrium, the Dupuit–Marshall surpluses include changes in consumers’ (lump-sum or full) incomes. In addition, we show that project profitability at the simple average of before-project and after-project prices approximates the cost-benefit test.
Having established this result, we turn to its interpretation and usefulness. It is well known that, when more than one price changes, Dupuit–Marshall surpluses suffer from a path-dependency problem (Silberberg, 1972; Chipman and Moore, 1976). That is, the value of the surplus can be different depending on the path of integration. Accordingly, we investigate the possibilities for path independence. In addition, we ask whether the aggregate surplus is consistent with a Bergson–Samuelson social-welfare function. If there is only one consumer, this requires the surplus to accord with his or her well-being. We study four cases which differ in the way prices and incomes are normalized. First, we allow prices and incomes to be unrestricted, and show that it is not possible to have path independence. As a consequence, it is impossible for the surplus to be consistent with a single consumer’s preferences or, in the many-consumer case, with a social-welfare function.
The second case uses normalized prices—prices divided by aggregate income. In the single-consumer case, path independence and consistency with individual preferences requires homotheticity (Silberberg, 1972; Chipman and Moore, 1976). In the many-consumer case, we show that there must be an aggregate many-consumer with homothetic preferences. This requires individuals to have parallel quasi-homothetic preferences with a restriction across consumers that ensures a homothetic aggregate.
The third normalization requires prices to sum to one. As in the first case, this leads to an impossibility.
´
The fourth and last normalization sets the price of a numeraire good to one. This ´
Thus, there are two circumstances in which Hicks’s result has normative significance. But these two possibilities are not encouraging. We live in many-consumer economies and we cannot reasonably expect to find preferences that support an aggregate consumer,
´
nor can we bargain on the absence of income effects on all but the numeraire good for all consumers. In addition, these two possibilities force the evaluator to be indifferent to income inequality, a standard but ethically unattractive feature of most cost-benefit analysis.
2. Partial equilibrium
Before presenting our main general equilibrium result, we discuss the problem in partial equilibrium. Following Sugden and Williams (1978), suppose that the govern-ment operates a rail service between a city and one of its suburbs, as illustrated in Fig. 1.
b a
A project is planned which will reduce the price of rail trips from p to p (b stands forr r
‘before’, a stands for ‘after’). Cheaper transportation results in increased demand in the (competitive) suburban rental housing market, shifting the demand curve to the right—
b a b
from D ( p , p ) to D ( p , p ). This results in an increase in the price of housing from ph r h h r h h
a b
to p which, in turn, shifts the demand curve for rail trips to the left—from D ( p , p ) toh r r h a
D ( p , p ). The initial equilibrium is at C and K, and the equilibrium after the change isr r h
at E and I.
Dupuit–Marshall consumers’ surpluses can be measured in many different ways. If
c
ˆ
consumers’ surplus in the rail market is calculated first, it is sr5ACFD. Once p hasr a
changed, the relevant housing demand curve is D ( p , p ) and consumers’ surplus in theh r h c
ˆ
housing market is sh5 2GILJ. Producers’ surplus (the increase in landlords’ profit) in
p
the housing market is sh5GIKJ, and consumers’ surplus plus producers’ surplus in housing is
c p c c p
ˆ ˆ ˆ
s 1s :h5sr1sh1sh5ACFD2GILJ1GIKJ5ACFD2ILK. (1) Alternatively, the consumers’ surplus in housing can be computed first, and, in this
c b
˜
case, it is sh5 2GHKJ, the area to the left of the original demand curve D ( p , p ).h r h a
The relevant demand curve for the rail-market surplus becomes D ( p , p ), andr r h c
˜
consumers’ surplus is sr5ABED. Using this method, consumers’ surplus plus
producers’ surplus in housing is
c p c c c
˜ ˜ ˜
s 1s :h5sr1sh1sh5ABED2GHKJ1GIKJ5ABED1HIK. (2) A third method for calculating the surpluses can be obtained by considering a price path which reduces the price of rail trips, keeping the housing market in equilibrium. When this is done, price and quantity in the rail market move along the market demand
M
curve D ( p ). The claim made by Hicks (1946a), (1946b) and by Sugden and Williamsr r
(1978) is that the area ACED to the left of the market demand curve is equal to the overall consumers’ surplus plus producers’ surplus in the housing market. This can be seen intuitively by thinking of the change in p as a sequence of very small changes.r
Fig. 1. Market demand curves and Dupuit–Marshall consumers’ surpluses.
In general, these three numbers are not the same; in order for them to coincide, the
1
consumers’ surpluses must be path independent, a requirement that is not often satisfied. If path independence is satisfied, the consumers’ surplus is well defined for all price paths.
To prove the result mathematically, we define the Dupuit–Marshall consumers’
b b a a
surplus as a line integral along a continuous price path from ( p , p ) to ( p , p ) given byr h r h
1
b b
the functions P : [0,1]∞5 and P : [0,1]∞5 with ( p , p )5(P (0),P (0)) and
r h r h r h
a a
( p , p )r h 5(P (1),P (1)). Thenr h a a
( p , p )r h c
s 5 2
E
[D ( p , p )dpr r h r1D ( p , p )dp ]h r h h b b( p , p )r h 1
5 2
E
[D (P (t),P (t))dP (t)r r h r 1D (P (t),P (t))dP (t)].h r h h (3)0
If the functions P and P are differentiable, then (3) can be written asr h 1
c
9 9
s 5 2
E
[D (P (t),P (t))P (t)r r h r 1D (P (t),P (t))P (t)]dt.h r h h (4)0
Producers’ surplus in the housing market is
a
ph 1
p
sh5
E
S ( p )dph h h5E
S (P (t))dP (t),h h h (5)b 0
ph
where S is the housing supply function. It follows that the sum of overall consumers’h
surplus and producers’ surplus in the housing market is given by
1 1
c p
s 1sh5 2
E
D (P (t),P (t))dP (t)r r h r 1E
[2D (P (t),P (t))h r h 1S (P (t))]dP (t).h h h (6)0 0
M M M M M
Along the market equilibrium path, D (P (t),P (t))h r h 5S (P (t)), where P (t) and P (t)h h r h c
are equilibrium or market prices and, defining s as the overall consumers’ surplus alongm
the market-equilibrium path,
1
c p M M M
sm1sh5 2
E
D (P (t),P (t))dP (t)r r h r 0a
pr
M
5:2
E
D ( p )dpr r r5ACED. (7)b
pr
A cost-benefit test can be performed by adding the consumers’ surplus ACED to the change in government profit that results from the project.
2
A path-independent example is provided by the demand functions
2
300 600
]]] ]]]
D ( p , p )r r h 5p 12p , D ( p , p )h r h 5p 12p , (8)
r h r h
and the supply function
S ( p )h h 520, (9)
b a b
with pr510 and pr55. The decrease in p induces an increase in p from pr h h510 to
a
ph512.5.
Then, computing consumers’ surplus in the rail market first,
5 12.5 12.5
300 600
c p
ˆ ]] ]]]
s 1sh5 2
E
p 120dpr2E
512p dph1E
20 dphr h
10 10 10
10 12.5 12.5
5[300 ln( pr120)]5 2[300 ln(512p )]h 10 1[20p ]h 10
30 30
] ]
5300 ln 2300 ln 150550. (10)
25 25
When consumers’ surplus in housing is calculated first,
12.5 5 12.5
600 300
c p
˜ ]]] ]]
s 1sh5 2
E
1012p dph2E
p 125dpr1E
20 dphh r
10 10 10
12.5 10 12.5
5 2[300 ln(1012p )]h 10 1[300 ln( pr125)]5 1[20p ]h 10
35 35
] ]
5 2300 ln 1300 ln 150550. (11)
30 30
For any value of p , equilibrium in the housing market requiresr
600
]]] 520, (12)
pr12ph
so
pr ]
ph5152 2, (13)
and the market demand curve is given by
300
M
]]]]
D ( p )r r 5p 1302p 510. (14)
r r
It follows that
5
c p 10
sm1sh5 2
E
10 dpr5[10p ]r 5 550, (15)10
3
A path-dependent example is provided by the demand functions
5ph 30025pr
] ]]]
D ( p , p )r r h 5 p 15, D ( p , p )h r h 5 p 25, (16)
r h
and
S( p )h 520, (17)
b a b a
with pr510 and pr55. p increases from ph h510 to ph511. Calculating the rail-market surpluses first,
5 11 11
b a
5ph 30025pr
c p
ˆ ] ]]]
s 1sh5 2
E
S
p 15 dpD
r2E
S
p 25 dpD
h1E
20 dphr h
10 10 10
5 11 11
50 275
] ]
5 2
E
S
p 15 dpD
r2E
S
p 25 dpD
h1E
20 dphr h
10 10 10
559.66221.21120558.45. (18)
If the housing-market surplus is computed first,
11 5 11
b a
30025pr 5ph
c p
˜ ]]] ]
s 1sh5 2
E
S
p 25 dpD
h2E
S
p 15 dpD
r1E
20 dphh r
10 10 10
11 5 11
250 55
] ]
5 2
E
S
p 25 dpD
h2E
S
p 15 dpD
r1E
20 dphh r
10 10 10
5 218.83163.12120564.29. (19)
Equilibrium in the housing market requires, for any value of p ,r
30025pr
]]] 25520, (20)
ph
so
pr ]
ph5122 5, (21)
and the market demand curve is given by
60
M
]
D ( p )r r 5 p 14. (22)
r
The area to the left of the market demand curve is
3
These demand functions can be rationalized by the utility function U(r,h)55 ln(r25)1h with income equal
5
60
c p ] 10
sm1sh5 2
E
S
p 14 dpD
r5[60 ln pr14p ]r 5 561.59. (23)r 10
Note that all three answers are different because of path dependence.
There is an additional important difficulty with the above result. It is that, in general,
p
the demand curves will shift as prices change because producers’ surplus s and theh
change in net revenue from the project are changes in income. Consequently, the above argument requires there to be no income effects in both markets. If there are only two markets, this cannot be true because budget constraints must be satisfied. Consequently, if all prices and incomes change, the consumers’ surplus calculated above cannot be correct.
3. General equilibrium: the basic result
Suppose that an economy consists of two sectors—a competitive private sector and a
m
public sector. Only private goods are produced, m$2 of them; p[5 is the price
11 vector.
m
The private-sector technology is described by the profit functionP:5 ∞5 .P is
11 1
homogeneous of degree one, and we assume that it is differentially strongly convex (Diewert et al., 1981). We employ the usual convention that inputs are measured
4
negatively. Using Hotelling’s theorem,Pj( p), if positive, is the private-sector supply of good j. IfPj( p) is negative, it is equal to minus the demand for an input. There are H$1
h m11 h
individuals, and V :5 ∞5 is person h’s indirect utility function. Each V is
11
decreasing in p, increasing in y, homogeneous of degree zero, and we assume that each
5 h
is differentially strongly quasi-convex in p for each y. D is person h’s demand functionj
for good j, h51, . . . ,H, j51, . . . ,m, and, using Roy’s theorem, is given by
h h
V ( p, y )j h h
]]]
D ( p, y )j 5 2 h h . (24)
V ( p, y )y
m
The government sector’s input–output vector is g[5 . It may be constrained to lie in a production set but, because we use consumers’ surplus analysis to search for social improvements, we do not assume that it is chosen optimally. Profit from government operations is p?g and it is paid out to consumers as lump-sum transfers (taxed from
h m H h
them if negative). Endowments are v [5 , h51, . . . ,H, and v 5o v is the
1 h51
h H h h
aggregate endowment. r is person h’s profit share, and oh51r 51. Similarly, s is
H h
person h’s share of government profit withoh51s 51. The (lump-sum or full) income
h
of person h is y and it is equal to
h h h h
y 5r P( p)1p?v 1 s p?g. (25)
4
Given that the economy is assumed to be competitive the assumption of one firm entails no loss of generality. 5
1 H H h
y5( y , . . . , y ) is the economy’s income vector and y5oh51y is aggregate income,
with
H h
y5
O
y 5P( p)1p?v 1p?g. (26)h51
Equilibrium requires, for each j51, . . . ,m,
H
h h
D ( p, y):j 5
O
D ( p, y )j 5Pj( p)1v 1j g .j (27)h51
b
We consider a public project that moves the public input–output vector g from g
a
(before) to g (after). A path of integration is described by the continuous functions
m b a m b
G: [0,1]∞5 , with g 5G(0) and g 5G(1), and P: [0,1]∞5 , with p 5P(0) and
a b a b a
p 5P(1). Equilibrium prices corresponding to g and g are p and p . The income
1 H
vector along the paths G and P is Y(t)5(Y (t), . . . ,Y (t)), t[[0,1], and is given by
b a
(25) evaluated at g5G(t) and p5P(t) with y 5Y(0) and y 5Y(1). Aggregate income
H h b a
along the paths is Y(t)5oh51Y (t) with Y(0)5y and Y(1)5y , and is given by (26)
evaluated at g5G(t) and p5P(t).
The Hicksian partial-equilibrium claim discussed in Section 2 above can be demon-strated in this general equilibrium model. The aggregate Dupuit–Marshall consumers’ surplus, allowing for income change, is
a a ( p , y )
m
sba5
E
F
2O
D ( p, y)dpj j1dyG
j51 b b ( p , y ) 1
m
5
E
F
2O
D (P(t),Y(t))dP (t)j j 1dY(t) .G
(28)j51 0
1 H
It is equal to the sum of the individual consumer’s surpluses s , . . . ,s , andba ba a a
( p , y )
H H m
h h h h
sba5
O
sba5O E
F
2O
D ( p, y )dpj j1dyG
h51 h51 j51 b b ( p , y ) 1
H m
h h h
5
OE
F
2O
D (P(t),Y (t))dP (t)j j 1dY (t) .G
(29)h51 j51 0
Theorem 1 shows that this consumers’ surplus can be computed in another way in an undistorted competitive economy: one in which all agents are price takers, producer and consumer prices are equal, and all taxes and / or transfers are lump-sum. The general equilibrium analogue to the market-demand-curve surplus of (7) is computed along a particular path where all markets are in equilibrium. Along a market-equilibrium path,
M M 1M H M
H
M M h M hM
D (P (t),Y (t))j 5
O
D (P (t),Yj (t))h51 M
5Pj(P (t))1v 1j G (t),j (30)
j51, . . . ,m, for all G(t), t[[0,1]. It is possible for these paths to be different even if
b a
the end points g and g are the same because the pathshG(t)ut[[0,1]jcan be different,
t[(0,1).
Theorem 1. For any market-equilibrium price path, the aggregate Dupuit–Marshall
consumers’ surplus (28) is equal to
1
H m
h M a a b b
sba5
O
sba5 2EO
G (t)dP (t)j j 1[ p ?g 2p ?g ] (31)h51 j51 0
or, equivalently,
1
H m
h M
sba5
O
sba5EO
P (t)dG (t).j j (32)h51 j51 0
Proof. Along the market-equilibrium path, using (29) and (30) and noting that Y(t)5
H h b a
oh51Y (t), Y(0)5y , and Y(1)5y ,
1
H m
h M M M M
sba5
O
sba5E
F
2O
D (P (t),Y (t))dP (t)j j 1dY (t)G
h51 j51 0
1 1 1
m m m
M M M M a b
5 2
EO
Pj(P (t))dP (t)j 2EO
vjdP (t)j 2EO
G (t)dP (t)j j 1[ y 2y ]j51 j51 j51
0 0 0
1
m
a b a b M a b
5 2[P( p )2P( p )]2[ p ?v 2p ?v]2
EO
G (t)dP (t)j j 1[ y 2y ]. (33)j51 0
Using (26), the change in income is
a b a b a b a a b b
y 2y 5[P( p )2P( p )]1[ p ?v 2p ?v]1[ p ?g 2p ?g ], (34)
and (33) becomes
1
m
M a a b b
sba5 2
EO
G (t)dP (t)j j 1[ p ?g 2p ?g ], (35)j51 0
which is (31).
1
m
a a b b M a a b b
sba5[ p ?g 2p ?g ]1
EO
P (t)dG (t)j j 2[ p ?g 2p ?g ], (36)j51 0
which yields (32). j
It is straightforward to interpret the results of Theorem 1. Eq. (31) is the general equilibrium analogue of (7), the partial-equilibrium result. Because p?g is
government-a a b b
sector profit, [ p ?g 2p ?g ] is the change in profit. It results in a change in
consumers’ incomes. The first term is easiest to interpret if the government sector produces a single good ( j51) with a change in its price but no change in the prices of government inputs (other prices may change). In that case, the surplus is simply
1
M a a b b
2
E
G (t)dP (t)1 1 1[ p ?g 2p ?g ]. (37)0
[image:11.612.58.389.79.136.2]The first term corresponds to the area to the left of the market demand curve—ACED in Fig. 1—and it should be added to the change in government-sector profit. The summation sign in (31) indicates that, in each market in which the government participates, consumers’ surpluses should be computed for each output or input whose price changes. It is true, of course, that there may be private-sector involvement in some or all of the goods (including inputs) that correspond to the non-zero elements of g. Eq. (31) indicates that this can be accounted for completely through the market equilibrium
M
path hP (t)ut[[0,1]j.
Eq. (32) is an area under market demand curves. Suppose that some prices
b a
¯
( j[^#h1, . . . ,mj) are unaffected by the change in g, then pj 5pj5p for thosej
a b
¯
values of j, and the corresponding terms in (32) are oj[^p [ gj j2g ], the change inj
government-sector profit. If ^5h1, . . . ,mj, this is the usual cost-benefit test in an undistorted competitive economy. In the case where all prices can change, it is possible to approximate the path hP(t), t[[0,1]j linearly with
P (t)j 5a G (t)j j 1b ,j (38)
b a
j51, . . . ,m, P(0)5p , P(1)5p . In this case, (32) becomes
b a
p 1p a b
F
]]G
sba5 2 ?[ g 2g ]. (39)
b a
This is the change in government-sector profit computed at the average of p and p , before and after prices. Such averaging, based on partial-equilibrium arguments, is common in cost-benefit analysis and (39) provides a general-equilibrium justification.
4. Interpretation
market-demand-curve surplus is an indicator of welfare change. To do that, we investigate it for all possible paths hG(t)ut[[0,1]j and the corresponding paths hp5
b a
P(t)ut[[0,1]j. We do not restrict the end points g 5G(0) and g 5G(1) and we
assume, throughout the section, that individual incomes can move independently along
6 m
market-equilibrium paths. It is not true that all possible price vectors p[5 are in
11 some path. The results presented below do not depend on this, however. They are local results and hold for feasible prices and incomes.
Because the demand functions are homogeneous of degree zero and because the market-clearing equations only determine relative prices, many different normalizations are possible: dividing by aggregate income, holding income equal to a constant, dividing by one price, and letting the sum of prices be constant. Of course, these normalizations are mutually exclusive and, as we shall see, each leads to a different result. Each subsection below is devoted to one of them.
We use the following result from Courant (1936) on path independence of line integrals of the form
a
x 1
K K
k k
EO
f (x)dxk5EO
f (X(t))dX (t),k (40)k51 k51
b 0
x
K K b
x5(x , . . . ,x )5(X (t), . . . ,X (t))5X(t)[5 and X: [0,1]∞5 , with X(0)5x and
1 K 1 K
a
X(1)5x . If the integral in (40) is path independent, then there must be a potential
a b
function F such that it is equal to F(x )2F(x ), and it must, therefore, be true that
K K
≠F(x) k
]]
dF(x)5
O
≠x dxk5O
f (x)dxk (41)k
k51 k51
for all (dx , . . . ,dx ). This requires1 K
≠F(x) k
]] 5≠ f (x), (42)
xk
k51, . . . ,K, and, because
2 2
≠ F(x) ≠ F(x)
]]≠ 5]], (43)
xk≠xl ≠xl≠xk
k l
≠f (x) ≠f (x)
]]≠ 5]] (44)
xl ≠xk
for all x.
6
4.1. No normalization
In this subsection, we normalize neither prices nor income and show that it is impossible for the consumers’ surplus s , described by (31) or (32), to be pathba
independent for either a one- or a many-consumer economy. Theorem 2 was proved by Chipman and Moore (1976) for one consumer.
Theorem 2. For any H$1, there are no well-behaved preferences such that s , givenba
by (28) or (29), is path independent.
Proof. Using (27), (28) and (29), the surplus can be rewritten as
a a ( p , y )
m H H
h h h
sba5
E
F
2OO
D ( p, y )dpj j1O
dyG
. (45)j51h51 h51 b b
( p , y )
Path independence requires
H
≠ h h ≠(1)
]h
O
D ( p, y )j 5]]≠p 50, (46)≠y h51 j
or
≠ h h
]hD ( p, y )j 50 (47)
≠y
for all h and all j which violates the regularity conditions on preferences. j
Path independence does not necessarily imply, however, that the aggregate consum-ers’ surplus sbais consistent with a social-welfare function. We say that, for any H$1,
sbais consistent with a Bergson –Samuelson social-welfare function if and only if there
H
exists an increasing function W :5 ∞5 such that
a b 1a H a 1b H b
sba$0↔W(u )$W(u )↔W(u , . . . ,u )$W(u , . . . ,u ) (48)
b 1b H b
for all alternatives B and A and all paths between them. u 5(u , . . . ,u ) and
a 1a H a 7
u 5(u , . . . ,u ) are the utility vectors in B and A. If there is a single consumer, (48) requires the surplus to be consistent with preferences. If sba is consistent with a social-welfare function, it must be path independent (see Lemma 1 in Appendix A). This observation together with Theorem 2 is sufficient for Corollary 2.1.
Corollary 2.1. For any H$1, there are no well-behaved preferences such that sba is consistent with a Bergson –Samuelson social-welfare function.
7
This result suggests that, without normalizations of price(s) or income, the result of Theorem 1 is nothing more than an intellectual curiosity.
4.2. Normalized prices
“
In general equilibrium, only relative prices are determined and, if p is an equilibrium
“
price vector, so islp for alll .0. We consider first a one-consumer economy in which prices along the equilibrium path can be divided by income without disturbing
N 8
equilibrium to create the normalized price path hP (t)ut[[0,1]j, where
P (t)1 P (t)m
N N N N
]] ]]
p 5P (t)5(P (t), . . . ,P (t)):1 m 5
S
Y(t), . . . , Y(t)D
. (49) In this case the consumers’ surplus becomesNa
p
m
N N N
sba5 2
EO
D ( p ,1)dp .j j (50)j51 Nb
p
Theorem 3. In a one-consumer economy, the consumer’s surplus in normalized prices
N
s , given byba (50), is path independent if and only if preferences are homothetic; that is,
* * N
u5V( p, y)5V(a( p)y)5V(a( p )) (51)
*
m11 m
for all ( p, y)[5 , wherea:5 ∞5 is homogeneous of degree minus one and V
11 11
is increasing. In this case,
N Na Nb
sba5lna( p )2lna( p )
a a b b
5ln(a( p )y )2ln(a( p )y )
21 21
* a * b
5ln V (u )2ln V (u ). (52)
Proof. Path independence and (50) imply
N N
≠D ( p ,1)j ≠D ( p ,1)k
]]]N 5]]]N (53)
≠pk ≠pj
N m
for all j,k[h1, . . . ,mjand all p [5 . For any y.0,
11
N ≠D ( p, y)j ≠D ( p ,1) 1j
]]]≠p 5]]] ]N y (54)
≠p
k k
for all j,k51, . . . ,m, and (53) implies
8
≠D ( p, y)j ≠D ( p, y)k
]]]≠ 5]]] (55)
pk ≠pj
m11
for all j,k[h1, . . . ,mjand all ( p, y)[5 . Therefore, the cross-partial derivatives of
11
the uncompensated demand functions are symmetric, which implies homotheticity and (51) (Silberberg, 1972; Chipman and Moore, 1976).a is homogeneous of degree minus
h
one because each V is homogeneous of degree zero.
Given (51),
N aj( p )
N ]]
D ( p ,1)j 5 2 N , (56)
a( p )
N
j51, . . . ,m. In this case, sba becomes
Na
p
N
m a
( p )
j
N ]] N
sba5
EO
N dpja( p )
j51 Nb
p
Na
p
N 5
E
d lna( p )Nb
p
Na Nb
5ln(a( p ))2ln(a( p ))
a a b b
5ln(a( p )y )2ln(a( p )y )
21 21
* a * b
5ln V (u )2ln V (u ). j (57)
N
sbais an exact index of welfare change if and only if (51) is satisfied with normalized prices which implies path independence. Accordingly, we have
Corollary 3.1. In a one-consumer economy, the consumer’s surplus in normalized
N
prices s , given byba (50), is an exact index of welfare change if and only if preferences
are homothetic and(51) is satisfied.
Although Theorem 3 and its corollary provide a possibility, it is a very restrictive one. Path independence requires symmetry of the cross-derivatives of the uncompensated demand functions, which implies homotheticity. Consequently, the utility function is given by (51). Normalizing prices is equivalent to setting income equal to one,
m m
2oj51D ( p,1)dpj j5oj51aj( p) /a( p)dpj5d lna( p), and the logarithmic structure of (52) results.
homothetic or quasi-homothetic and parallel. In the latter case, preferences must aggregate to overall homotheticity.
˜
The income vector y is defined by
1 H
y y
S
D
˜ ] ]
y5 , . . . , (58)
Y Y
H h
˜
and, becauseoh51y 51,
h 2 H
˜ ˜ ˜ ˜
y5
S
12O
y ,y , . . . ,yD
. (59)h±1
The aggregate consumers’ surplus is written as
Na˜Na ( p , y )
m H H
y h N h N h
˜ ˜
sba5
E
F
2OO
D ( p ,y )dpj j 1O
dyG
j51h51 h51 Nb˜Nb
( p , y ) Na˜Na ( p , y )
m H
h N h N
˜
5
E
F
2OO
D ( p ,y )dpj jG
. (60)j51h51 Nb˜Nb
( p , y )
y
Theorem 4. For H$2, the consumers’ surplus in normalized prices s , given by (60),ba
is path independent if and only if
h
*
h h h h
V ( p, y )5V (a( p)y 1b ( p)), (61)
m h m
where a:5 ∞5 is homogeneous of degree minus one, b :5 ∞5 is
homoge-11 *h 11
9
neous of degree zero, and V is increasing, h51, . . . ,H. In addition,
h
O
b ( p)5constant. (62)h y
The surplus sba can be written as
y Na Nb
sba5lna( p )2lna( p )
a a b b
5ln(a( p )Y )2ln(a( p )Y ).
H h 1 h
˜ ˜ ˜
Proof. Because oh51dy 50, dy 5 2oh±1dy , and (60) can be rewritten as
9
Na˜a ( p , y )
m
h N h 1 N h N h h
˜ ˜ ˜ ˜
E
F
2O O
F
D ( p ,y )j 1DjS
p ,12O
yDG
dpj 1O
dy 2O
dyG
.j51 h±1 h±1 h±1 h±1 Nb˜b
( p , y )
(64)
Path independence requires, for g±1,
≠ h N h 1 N h
˜ ˜
]≠˜g
F
O
D ( p ,y )j 1DjS
p ,12O
yDG
50 (65)y h±1 h±1
or that
≠ g N g 1 N h
˜ ˜
]≠˜g
F
D ( p ,y )j 1DjS
p ,12O
yDG
50. (66)y h±1
This in turn implies that
≠ g N g ≠ h N h
˜ ˜
]≠˜g[D ( p ,y )]j 5]h[D ( p ,y )]j (67)
y ≠y˜
for all g and h. Consequently,
g g h h
D ( p, y )Yjy 5D ( p, y )Yjy (68) or
g g h h
D ( p, y )jy 5D ( p, y ),jy (69)
so that there must exist an aggregate consumer. It follows that the individual indirect utility functions can be written as in (61) (Gorman, 1953, 1961) with homogeneity
1 H 1 H
properties of a andb , . . . ,b implied by homogeneity of degree zero of V , . . . ,V .
˚
The aggregate consumer’s utility function is V, with
H
* h
˚V( p,Y)5V
S
a( p)Y1O
b ( p) .D
(70)h51
y
The consumers’ surplus sba is the surplus for the aggregate consumer (in normalized prices). Path independence requires homotheticity by Theorem 3, and (62) results. The surplus is calculated as in Theorem 3. j
y
If sbais consistent with a social-welfare function (48), path independence is implied. Corollary 4.1 follows.
y
Corollary 4.1. The consumers’ surplus in normalized prices sba is consistent with a Bergson –Samuelson social-welfare function if and only if it is path-independent and
(61) and (62) are satisfied, with
H 21 H
h
* h h
W(u)5
O
V (u )5a( p)S D
O
y 1constant. (71)Corollary 4.1 indicates that the evaluator must be indifferent to inequality in the distribution of income. The result of Theorem 4 and its corollary is connected to results on social evaluation using the sum of compensating or equivalent variations in Blackorby and Donaldson (1985). Consistent aggregation is possible only when an aggregate consumer exists. As in the result above, evaluators must be indifferent to inequality in the distribution of income.
Eq. (61) requires each consumer to have quasi-homothetic preferences. Although
h h h h
¯
there is a representation of the preferences of household h (V ( p, y )5a( p)y 1b ( p)) in which the marginal utility of income is independent of the level of income, it is not a constant: becausea is homogeneous of degree minus one, it must depend on at least one price. In addition, this property of the marginal utility of income is not shared by all of the (ordinally equivalent) representations of preferences.
4.3. Prices normalized to add to one
In this subsection, we normalize prices to add to one. This means that income and each price must be divided by the sum of unadjusted prices along the path of integration.
h
ˇ ˇ
Writing the resulting prices and income as p and y , with
h
p h y
]]] ]]]
ˇ ˇ
p5
O
m and y 5 m , (72)p
O
pi51 i i51 i
S
the consumers’ surplus is equal to s , which is given byba a a
ˇ ˇ
( p , y )
m H
S h
ˇ ˇ ˇ ˇ
sba5
E
F
2O
D (p,y )dpj j1O
dyG
. (73)j51 h51 b b
ˇ ˇ
( p , y )
Theorem 5. For any H$1, with prices and incomes normalized so that prices sum to S
one, there are no well-behaved preferences such that s , given by (73), is pathba
independent.
m
˜
ˇ ˇ ˇ
Proof. Because oi51dpi50, writing p5(p , . . . ,p ),2 m
a a
˜ ˇ
( p , y )
m H
S h
ˇ ˇ ˇ ˇ ˇ ˇ
sba5
E O
F
[D (p,y )1 2D (p,y )]dpj j1O
dyG
. (74)j52 h51
b b
˜ ˇ
( p , y )
Path independence requires
≠ ≠
]hD (p,y )1 ˇ ˇ 2]hD (p,y )j ˇ ˇ 50, (75)
ˇ ˇ
≠y ≠y
≠ h h ≠ h h
]hD ( p, y )1 5]hD ( p, y )j (76)
≠y ≠y
for all h and all j±1. The individual budget constraints imply that
m
≠ h h
]
O
pj hD ( p, y )j 51, (77)≠y
j51
and, therefore, using (76),
≠ h h 1
]hD ( p, y )j 5]]]m . (78)
≠y
O
pi51 i
Integrating,
h
y
h h ]]] hj
D ( p, y )j 5
O
m 1f ( p) (79)p
i51 i hj
for some functionf . (79) and Roy’s theorem imply that
h
y
h h h h
]]]
˚
u 5V ( p, y )5
O
m 1c ( p), (80)p
i51 i h
˚
where c is homogeneous of degree zero and 5 means ‘is ordinally equivalent to’.
Consequently, there is an aggregate consumer. The demand functions can be written as
h m
y
h h h h h
]]] ˇ ˇ
D ( p, y )j 5 m 2
S D
O
pi cj( p)5y 2cj(p ), (81)i51
O
i51pih h H h
wherecj is the jth partial derivative ofc . Defining c( p)5oh51c ( p),
H h
ˇ ˇ ˇ ˇ
D (p,y )j 5
O
y 2cj(p ). (82)h51
It follows that
a a
ˇ ˇ
( p , y )
m H
S h
ˇ ˇ ˇ ˇ
sba5 2
E O
[c1(p )2cj(p )]dpj1O
dy , (83)j52 h51 b b
ˇ ˇ
( p , y )
and path independence requires, for all j,k±1,
ˇ ˇ ˇ ˇ
c1j(p )2ckj(p )5c1k(p )2cjk(p ), (84)
and, therefore,
ˇ ˇ
c1j(p )5c1k(p ). (85)
ˇ
m
ˇ ˇ ˇ ˇ
c1(p )5d(p )1
S D
O
pi 1e(p ).1 (86)i52
m
ˇ
Because oi51pi51,
ˇ ˇ ˇ ˇ
c1(p )5d(p )(11 2p )1 1e(p ),1 (87)
ˇ
which is a function of p1 alone. Because c is homogeneous of degree zero, c1 is homogeneous of degree minus one, and it follows that
l1 ]
ˇ
c1(p )5 ˇp , (88)
1
wherel1 is an arbitrary constant.
Although good one was chosen in the argument above to be the residual commodity, any good can play that role and, therefore, (88) generalizes to all goods with
lj
]
ˇ
cj(p )5 ˇp , (89)
j
j51, . . . ,m. Because cj is homogeneous of degree minus one,
m c ˇ
(p ) l l
j j j
]]] ]]]] ]
ˇ
cj( p)5cj
SS D D
O
pi p 5 m 5 m 5 p (90)j i51
O
ps
O
p pd
ˇi51 i i51 i j
for all j and, integrating,
m
c( p)5
O
ljln pj1g, (91)j51
m
whereg is a constant. Because c is homogeneous of degree zero, oj51l 5j 0.
Now consider the case m52. There is an aggregate consumer with preferences
represented by
Y
]]
˚
V( p,Y )5p 1p 1lln p12lln p .2 (92)
1 2
Ifl 50, our curvature assumption is violated. Without loss of generality, letl .0. The expenditure function corresponding to V is E, with
E(u, p)5( p11p )(u2 2lln p11lln p ).2 (93) Given this,
l( p12p )2 ]]]
E (u, p)11 5 2 2 , (94)
( p )1
and our assumptions require it to be negative for all (u, p). But, because it is positive when p2.p , an impossibility results.1 j
S
Samuelson social-welfare function because that requires the integral in (73) to be path independent. This observation proves
S
Corollary 5.1. For any H$1, there are no well-behaved preferences such that sba is consistent with a Bergson –Samuelson social-welfare function.
´
4.4. A numeraire good
´
It is also possible to choose some good as a numeraire, normalizing its price to one. If
h h
¯ ¯
good k is chosen, prices and income become pj5p /p , jj k 51, . . . ,m, and y 5y /p .k
1 H
¯ ¯ ¯
Letting y5(y , . . . ,y )5y /p , consumers’ surplus isk
a a
¯ ¯
( p2k, y )
H
k h
¯ ¯ ¯ ¯ ¯
sba5
E
F
2O
D (p,y )dpj j1O
dyG
, (95)j±k h51 b b
¯ ¯
( p2k, y )
¯ ¯
where p2k is p without its kth element. Theorem 6 characterizes preferences that allow
k
¯sba to be path independent.
´
Theorem 6. For any H$1, if good k[h1, . . . ,mj is chosen as numeraire, the
k
¯
consumers’ surplus s , given by (95), is path independent if and only if the individualba
indirect utility functions can be written as
h h h y 1c ( p)
*
h h
]]]]
V ( p, y )5V
S
D
, (96)pk
h
*
h
wherec is homogeneous of degree one and V is increasing, h51, . . . ,H. The surplus
is given by
H H H
k kh h a h a h b h b
¯ ¯ ¯ ¯ ¯ ¯
sba5
O
sba5O
(y 1c (p ))2O
(y 1c (p ))h51 h51 h51
H ha h a H hb h b
y 1c ( p ) y 1c ( p )
]]]] ]]]]
5
O
S
aD
2O
S
bD
. (97)p p
h51 k h51 k
Proof. The surplus can be written as
a a
¯ ¯
( p2k, y )
H H
k h h
¯ ¯ ¯ ¯ ¯
sba5
E
F
2O O
D (p,y )dpj j1O
dyG
, (98)j±k h51 h51 b b
¯ ¯
( p2k, y )
and path independence requires
H
≠ h ≠ g h ≠(1)
]≠¯g
O
D (p,y )j ¯ ¯ 5]¯gD (p,y )j ¯ ¯ 5]]≠p¯ 50 (99)y h51 ≠y j
¯ ¯
≠ h h
]hD ( p, y )j 50, (100)
≠y
h
because D is homogeneous of degree zero. By Hotelling’s theorem,j
≠
h h h h h
]
D ( p, y )j 5≠pE (V ( p, y ), p), (101)
j
h h
where E is the expenditure function corresponding to V . Eqs. (100) and (101) imply
h h
E (u , p)ju 50 (102)
or, equivalently,
h h
E (u , p)uj 50 (103)
h
for all (u , p) and all j±k. Integrating, and taking account of the fact that (103) holds for all j±k and that the expenditure function and its utility derivative are homogeneous of degree one in prices,
h h ¯h h
¯
E (u ,p )u 5f (u , p )k h h
¯
5p f (u ,1)k h h
5:p f (u )k (104)
h
for some function f . Integrating again yields
h h h h h
E (u , p)5pkf (u )2c ( p) (105)
h h h h h
for functions f andc . Setting E (u , p)5y and inverting yields
h h h y 1c ( p)
*
h h
]]]]
V ( p, y )5V
S
D
, (106)pk
h h
which is (96). For each h, c must be homogeneous of degree one because V is
homogeneous of degree zero. Given (106),
h h h
¯ ¯ ¯
D (p,y )j 5 2cj(p ), (107)
j±k. Then, the surplus of consumer h is given by
a h a ¯ ¯
( p2k, y )
kh h h
¯ ¯ ¯ ¯
sba5
E O
F
cj(p )dpj1dyG
j±k
b h b ¯ ¯
a h a ¯ ¯
( p2k, y )
h h
¯ ¯
5
E
[dc (p )1dy ]b h b ¯ ¯
( p2k, y )
h a h a h b h b
¯ ¯ ¯ ¯
5[y 1c (p )]2[y 1c (p )]. (108)
k H kh
¯ ¯
Because sba5oh51s , the aggregate surplus is given by (97).ba j It follows immediately from Theorem 6 that
k
¯
Corollary 6.1. For any H$1, the consumers’ surplus sba is consistent with a Bergson –Samuelson social-welfare function if and only if (96) holds and the
social-welfare function W is given by
H 21 h
* h
˚
W(u)5
O
V (u ). (109)h51
´
The indirect utility functions in (96) have no income effects for non-numeraire goods. Because of this, the aggregate Dupuit–Marshall consumers’ surplus is the same as the aggregate of the Hicksian compensating and equivalent variations. In addition, as in
h h h
˜
Section 4.2, there is a representation of each household’s preferences (V ( p, y )5( y 1
h
c ( p)) /p ) such that the marginal utility of income is independent of the level ofk
income. But it is not constant because it depends on p . This property is not shared by allk
of the representations of households’ preferences. It is true, of course, that the Dupuit–Marshall consumers’ surpluses depend on household demand functions and, because of this, are unaffected by applications of increasing transformations to utility functions. Consequently, any attempt to characterize necessary or sufficient conditions for consistent aggregation in terms of marginal utilities of income is both misleading and meaningless.
´
Each choice of numeraire good results in different preferences and a different social-welfare function. Because consumers must have different preferences for different
´
choices of numeraire, this result is unsatisfactory. In each case, however, the evaluator must be indifferent to inequality in the distribution of income.
5. Conclusion
We have extended the claim of Hicks that the aggregate Dupuit–Marshall consumers’ surplus associated with the market demand curve(s) is equal to the consumers’ surplus in all markets together with producers’ surpluses in other markets in partial and general equilibrium in an undistorted economy. This attractive result can be used to justify a cost-benefit test which simply computes project profitability at the average of before-and after-project prices.
consistent with a Bergson–Samuelson social-welfare function. Possibilities arise only ´
when normalized prices (prices divided by aggregate income) or a numeraire are employed. In the first case, an aggregate consumer with homothetic preferences is required. In the second, there is an aggregate consumer, income effects for each person
´
are confined to the numeraire, and preferences must be different for different choices of
´ ´
numeraire. Because of this, a project may pass the cost-benefit test using one numeraire and fail it using another. But preferences are supposed to be primitives in economic models; choosing them to accommodate an arbitrary price normalization is not in the spirit of the cost-benefit enterprise. All of these results suggest that the Dupuit–Marshall consumers’ surplus methodology that pervades cost-benefit analysis is theoretically unsatisfactory.
Acknowledgements
Financial support through a grant from the Social Sciences and Humanities Research Council of Canada is gratefully acknowledged. We are indebted to a referee and an Associate Editor for helpful comments. Earlier versions of the paper were presented at Dalhousie University, the University of Nottingham, and the University of British Columbia. We thank the participants for their comments.
Appendix A
Lemma 1. The line integral a
x
m k
EO
f (x)dxk (A.1)k51 b x
satisfies
a x
m
k a b
EO
f (x)dxk$0↔F(x )$F(x ) (A.2)k51 b x
m
for all continuous paths in 6#5 and some function F :6 ∞5 if and only if it is
path-independent.
b a ¯ ˆ ˆ
Proof. Consider, for any x , x [6, any two paths C and C. Define C9as the reverse of
a b ˜ b a b
ˆ ¯ ˆ
C (from x to x ) and C as the path from x to x along C and back to x along C9. Eq. (A.2) implies that the line integral must be zero on any closed path (from any x to itself). Therefore,
m m m
k k k
EO
f (x)dxk5EO
f (x)dxk1EO
f (x)dxk k51 k51 k51¯
˜ C Cˆ9
m m
k k
5
EO
f (x)dxk2EO
f (x)dxk50. (A.3)k51 k51
¯ ˆ
C C
Consequently,
m m
k k
EO
f (x)dxk5EO
f (x)dx ,k (A.4)k51 k51
¯ ˆ
C C
and the integral is path-independent.
b a
If the integral is path-independent, for all paths between x and x ,
a
x
m
k a b
EO
f (x)dxk5G(x )2G(x ) (A.5)k51 b
x
for some function G:6 ∞5 and (A.2) is satisfied. j
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´ ˆ ´
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