Dupuit-marshall consumer's surplus, utility, and revealed preference

JOURNAL

OF ECONOMIC

Notes,

THEORY

20, 26O-270 (1979)

Comments,

and

Letters

to the

Dupuit-Marshall Consumer’s Surplus, and Revealed Preference

Editor

Utility,

E. E. ZAJAC Bell Laboratories,

Murray

Hill, New Jersey 07974

Received January 26, 1977; revised February 13, 1979

1.

INTRODUCTION

The single-commodity Dupuit-Marshall consumer’s surplus is defined as the area between the consumer’s market (Marshallian) demand curve and a price horizontal. In applications the Dupuit-Marshall surplus would appear to be more useful than Hick’s compensating or equivalent variations, since the latter are given by areas under more-difficult-to-measure compensated (Hicksian) demand curves. However, whereas the compensating and equivalent variations readily generalize to the multicommodity case as pathindependent line integrals, the multicommodity generalization of the DupuitMarshall surplus is well-known to be path-dependent (see for example, Hotelling [4], Silberberg [I 11, Willig [12, 131, and Chipman and Moore [3]). The path dependence of the Dupuit-Marshall surplus immediately casts doubt on its validity as an indicator of a consumer’s preferences. For example, starting at a price vector, PA, consider price changes along two different paths, say Paths 1 and 2, to a final price vector, PB. With different values of the Dupuit-Marshall surplus along the two paths, one can pick a looped traverse, from PA to PB along Path 1 and return to PA along Path 2, or the reverse, for which the net Dupuit-Marshall surplus is positive. If this surplus is supposed to indicate preference levels, one would have to conclude that one prefers the price vector PA and the associated demanded commodity bundle to itself. In fact, the path dependence would imply the existence of a “happiness pump”; after each traverse to the starting price one is happier, and infinite happiness is approached as one traverses the loop indefinitely. Two developments in the literature have evolved to cope with the path dependence infirmity. In one development, Willig [12, 131 has pioneered the use of the Dupuit-Marshall surplus as an approximation to the path independent Hicksian compensating and equivalent variations. He has shown that surplus will over a wide range of circumstances, the Dupuit-Marshall 260 OO22-0531/79/02026O-11$02.00/O Copyright AlI rights

0 1979 by Academic Press, Inc. of reproduction in any form reserved.

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numerically vary only slightly from path to path between two endpoints and along any particular path will closely approximate both the compensating and equivalent variations. In another development, several authors (e.g., Silberberg [ll], Rader [8], Chipman and Moore [3]) have pointed out that the Dupuit-Marshall surplus is path independent when the consumer’s preferences are homothetic and have investigated its properties in that special case. Both these developments are important and useful, and help justify the use of the Dupuit-Marshall surplus in applications. Nevertheless, they still leave open a number of issues. For example, one observes that naively treating the Dupuit-Marshall surplus like a utility function in theoretical work often leads to a correct analysis, in spite of its path dependence and the “happiness pump” example. This would imply that the Dupuit-Marshall surplus is somehow linked to a valid utility function. The purpose of this paper is to point out an explicit linkage and, further, to clarify when the Dupuit-Marshall surplus can and cannot validly be used. Briefly, the situation is akin to that in revealed preference theory. The “happiness pump” example shows that binary Dupuit-Marshall surplus comparisons in terms of endpoints of integrals in price space cannot unambigously delineate the consumer’s preference field. This is parallel to the observation that binary revealed preference comparisons (“weak axiom”) of commodity bundles are also insufficient to delineate the preference field. Houthakker’s [5] insight was to replace the binary comparison with a monotone chain of commodity bundle comparisons as the starting point (“strong axiom”) of revealed preference theory. This allowed revealed preferences to infer the entire preference field. Similarly, we find that restricting attention to integration paths along which the Dupuit-Matshall surplus varies monotonically is the key to relating it unambiguously to consumer’s preferences. Put in other terms, a path-independent form of surplus, such as the equivalent variation or the Dupuit-Marshall surplus for homothetic preferences, allows the imputation of a utility function over the entire preference field, while path-dependence generally precludes using the Dupuit-Marshall surplus to impute an entire utility function. However, Section 2 of the paper shows that the restriction to a path along which the Dupuit-Marshall surplus varies monotonically gives one an ordinal preference indicator along that path. This indicator is an analog, in continuous form, to a monotone chain of revealed preferences. The above analogy is most direct in terms of Sakai’s [9] theory of revealed favorabilities, i.e., comparisons of price vectors given the consumer is spending his budget and maximizing preferences. (See also Chipman and Moore’s [3] linkage of consumer’s surplus to “indirect” preference relations in price-income space). However, as Chipman and Moore point out, an

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alternative to the usual definition in terms of price paths is a definition of the Dupuit-Marshall surplus as a line integral in terms of paths in commodity space. This definition of Dupuit-Marshall surplus then directly links to standard revealed preference theory. Thus, restriction to monotone variations of the Dupuit--Marshall surplus results in explicit linkage to a utility function and to revealed preference. Further, Section 3 of the paper shows that if the Dupuit-Marshall surplus either strictly increases or decreases along a path, it imputes a utility index whose marginal utility of income is unity everywhere along the path. However, this unitary value of the marginal utility of income adheres only to the path of integration. Finally, Section 4 shows that the Dupuit-Marshall surplus is everywhere tangent to some utility function if the variation of the DupuitMarshall integral is defined in a certain, precise way. This variation does not depend on the direction in commodity or price space along which the surplus is defined, and it allows one to define a gradient function which corresponds to naive differentiation of the Dupuit-Marshall surplus with respect to one of its limits of integration. The existence of this gradient in turn resolves the paradox mentioned above. Namely, it explains why the naive use of the Dupuit-Marshall surplus often results in the correct values of prices or I commodities for a constrained utility optimization.

2. RELATIONSHIP

OF THE DUPUIT-MARSHALL

SURPLUS TO UTILITY

Since we are interested in the principal connections between the DupuitMarshall surplus and utility, we shall assume smooth utility functions. In particular, we first assume that direct utility functions, U(X) (X = (x1 , xZ ,..., x,) = commodity vector), satisfy Katzner’s [6] (p. 38) smoothness assumptions: (1) U(X) is twice continuously differentiable in the open positive orthant, E, and continuous where finite in the closed positive orthant J!?;(2) au/ax, > 0 for all X in E; (3) U is strictly quasi concave;l (4) If for X’ and X” in E, U(X’) = U(X”) and X’ > 0, then X” > 0. Further, when necessary for sufficient smoothness we assume: (5) the bordered Hessian of U is everywhere nonvanishing. Adopting these assumptions allows us to use theorems given in Katzner’s book or facilitates the proofs of the paper’s propositions. Let the Dupuit surplus be denoted by D(P, Pf, mi): D(P~, Pf, /vi) = /-” X(P, mi) . dP,

where P = (pl, pz ,..., p,),

is the price vector,

X(P, mi) = (x,(P, mi),

1 As in [6], X > 0 means x, > 0, all i, while X :-- 0, means xi > 0, all i.

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xz(P, mi),..., xn(P, ,mi)) is the market demand vector, the superscripts i andf refer to initial and,final states, X * dP denotes inner product, and the integration is along some piecewise smooth price path I’ between P” and Pf.

Definition (1) is seen to be the multiproduct generalization of the usual single-product definition of consumer’s surplus. It is convenient to parameterize the integration path, say r, by setting w

= MS),

P2VL

P&h

O
P(0) = Pi, P( 1) = Pf.

That is, as s varies continuously from zero to unity, the price vector sweeps out the path, r, starting at the initial vector, Pi and ending at the final vector, Pf. Applications often explore the change in D as the lower-limit price in Eq. (1) is varied. Hence, we rewrite the Dupuit integral (1) in terms of an integral computed along r and with a variable lower limit:2 D(P(s), Pf, mi) =

l X(P(u), mi) * P da, ss

(2)

where J’ = dP(s)/ds (throughout (-) denotes d( )/ds). Now for any indirect utility index L(P, m), by Roy’s lemma ([6] p. 60) and the fact that ti = 0 for m = mi, we have Ii = (aL/BP) 9i) + (aL/am)k = -X(s)X . P = X(s)D,

(3)

where aL/aP is the gradient of L with respect to prices. Since h(s), the marginal utility of income, is known to be positive ([6], p. 44) we have: B = i/h(s).

(4)

If as one moves along r from Pi to Pf (0 < 5 < l), L(s) is monotonically nondecreasing, then, by Eq. (4), D(s) will also be monotonically nondecreasing. Likewise, if D(s) is monotonically nondecreasing for 0 < s < 1, so will be L(s). We summarize these properties of D as an ordinal preference indicator as: PROPOSITION 1. Along path r, D(s) is monotone nondecreasing (nonincreasing) in s, 0 < s < 1, i&T the consumer’spreferences are monotone nondecreasing(nonincreasing) in s, it is constant for all s ifl r is on an isopreference surface; and points of B = 0 coincide with points oft = 0.

Furthermore, for any two price vectors, P” and Pf, and a path of price changes, r, consider the trace in the p1 , p2 plane of the isoutility surface 2 This conforms to the usual usage in the single-product case where consumer’s surplus is the area between the market demand curve and a price horizontal, and increases in surplus correspond to lowering of the price horizontal.

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through Pi. Follow this trace from (pIi, pza) until pIi changes to plf, and pzi changes to the value, say jz, which is necessary to maintain the constant utility level.3 Next move along the isoutility surface’s trace in the (pB , p3) plane so that pz changes from j2 to pzf and p3 from p3i to say j3 . A chain of such isoutility changes brings one to the vector P = (plf, p2f,..., pf,_, , j,J. A final change from P to Pf involves only a change of pn from j,, to pnf. But by Roy’s lemma aL/ap, = -Ax,, and since h and X, are both positive, this last price change is monotone in utility and, by Proposition 1, monotone in D. Hence, we have* PROPOSITION

1’.

constant-incomepath

Between any two price vectors there aln,ays exists a qf price changesalong which D varies monotonically.

We have followed the common practice in the theoretical literature of defining D in terms of the price vector as the variable of integration. However, for the smooth utility fields we have assumed there exists a continuous, singlevalued inverse demand vector, say P(X, mi), for a constant income. At demand X and prices P(X, mi), the consumer consumes his entire income at maximum satisfaction.5 We can imagine a path traced out in commodity space, say, Q, as X is varied from an initial Xi to a final Xf. This will generate the image path, r, between Pi = P(J?, mi) and Pf = (Xf, m”) in price space. Integration by parts and the identity P . A’ .+: mi gives D(s) = j=l X(P(u), mi) . PC/CT= - 3” P(,Y(,), mi) . J? LICS,

(5)

5

s

where the first integral is along the path I’and the second along a. Furthermore, differentiation of the second integral gives D = P . ST,

while differentiation

(6)

of the direct utility function, ri = (au/ax)*R

= hp.-Y

U(X), gives = AD,

(7)

From (7) it is clear that Propositions 1 and 1’ have duals, valid in commodity space. To generate the dual propositions one computes D from Eq. (5) along a 3 More explicitly, Hence,

along the isoutility 8, = -

&:

trace dpJdp,

=

(xlu,lx,)dp,

+ ~2”.

-- (aL/3p,)/(Q’2p2)

=

-xI,‘xt

4 I am indebted to R. D. Willig for pointing out Proposition 1’. 5 Katzner [6] normalizes the inverse demand function with respect to the nth price ([6], p. 44). However, it is obvious that one can equally well normalize with respect to income, viz., let p&k’, mf) = (m%LY/&i)/(Xl/LLY) * 2’. Then P . X = nz” and p&Y, rn<) is continuous and single-valued.

.

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path of commodity bundles and substitutes “Q” for “r” in Proposition 1 and “commodity” for “price” in Proposition 1’. The dual, commodity bundle forms of Propositions 1 and 1’ are in turn variants of a well-known result in revealed preference theory. Namely, a chain of monotonically inferior revealed preference comparisons links commodity bundles Xa and Xb if and only if X” is preferred to Xb (see for example, [6], Theorem 6.2-8). Propositions 1 and 1’ express this result for a continuous chain of comparisons under the smoothness of utility functions assumptions made at the outset and for the special case of constant income.

3. D Ihmms

A UTILITYINDEX

WITHMARGINALUTILITY

OF INCOMEOF UNITY

An attractive feature to early writers on the Dupuit-Marshall surplus was that it seemed to measure utility changes in terms of units of income, as, for example “that maximum sacrifice (in income) that each consumer would be prepared to make, in order to be able to purchase the good” (Dupuit, as quoted by Burns [2]). However, if utility is to be validly measured in income units, the marginal change in utility with respect to change in income should be a constant (because of utility’s ordinality, this constant can be considered to be unity). Indeed, the importance in the application of consumer’s surplus of the constancy of the marginal utility of income was stressed by writers like Marshall, and the implications of a constant marginal utility of income have occupied the attention of much of the extensive consumer’s surplus literature (for recent discussions see [2], [II]). In particular, Samuelson’s [lo] fundamental paper showed that a constant marginal utility of income, say independent of prices when income is fixed, implies unitary income elasticities of demand and homogeneous demand functions of degree minus one in prices (which further implies homothetic underlying preference-see [7], p. 394). This result would seem to say that viewing the Dupuit-Marshall surplus as a measure of utility in income units is justified only for one particular form of consumer’s preferences [l 11. However, if one restricts attention to paths in commodity or price-income space where D, L, or U are strictly monotone, the Dupuit-Marshall surplus can in fact be viewed as measuring utility in income units, regardless of the form of the consumer’s preference field. For then, D coincides with a utility index which along the path of D’s de$nition has a marginal income of unity. Thus, even though the Dupuit-Marshall surplus cannot be expected to measure utility in income units everywhere in commodity or price-income space, it does give such a measure along a strictly monotone path. This result falls out of a more general theorem which is itself of interest. As a preliminary to proving the more general theorem we introduce the notion of a complete, strictly monotonic path:

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DEFINITION. A path in price-income or commodity space which intersects all preference levels and along which preferences vary strictly monotonically is a complete, strictly monotone path (CSMP). Obviously, a CSMP extends to infinite values of prices or income or to infinite amounts of quantities. Also, the (P, m) vector that corresponds to a budget plane for which X is the utility maximizing commodity bundle is not unique. This nonuniqueness means that a CSMP in commodity space maps into many CSMP’s in price-income space. However, we restrict attention, as in the previous section, to the constant-income images in price-income space of commodity space CSMP’s. This normalization then makes one-to-one the mapping from commodity to price-income CSMP’s, and we have:

PROPOSITION 2. For a consumer at constant income, to any given CSMP there correspondsa unique (except for an additive constant) utility labelling of the preference field with the property that the marginal utility of income is unity along the given CSMP.

Proof. The proof proceeds by first constructing the required labelling by a monotone transformation of an arbitrary labelling and then by proving that except for an additive constant, the required labelling is unique. Consider first any given CSMP in price-income space and take any utility function L. Because of the CSMP’s completeness and strict monotonicity, there is a one-to-one mapping of values of L onto points along the CSMP. Consequantly, we can write the values of h = aLlam encountered as one traverses the CSMP as a function of L: h = h(L). Let L = 4(L), where # is defined by d#/aL = I/X(L). Since h is positive, $ is monotone in L, and L is a proper indirect utility function. But, since x = &T/am = (d#/dL)(aL/&n) = 1, E has the required unitary marginal utility of income. Furthermore, instead of using L to label points along the CSMP, use some parameterization in terms of s,o
BAn intuitive notion of the labelling process in Proposition 2 follows from the differential form of Eq. (3): dL = A( 4%‘. dP + dm). Imagine marching in small discrete steps in the direction of increasing utilities along a CSMP and at each step going through a two-stage process. At the starting point say one is on an isopreference surface labelled L, In Stage

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Now suppose that D is strictly monotone increasing or decreasing along r in price-income space. Then, by Proposition 1, preferences will also be strictly monotonically increasing or decreasing along l? Clearly I’ can then be embedded in a CSMP. For example, if Pi is at a higher preference level than Pf, one can form a path consisting of a ray from the origin in price-income space to (Pi, mi), followed by r, and then followed by the continuation of the ray from the origin through (Pf, mi). From, BL/BP = -AX, X > 0, X > 0, preferences along the rays are strictly monotonically decreasing, and this path is a CSMP.’ An L with aEpm = 1 which is generated by this CSMP will obviously have aL/arn = 1 along r. Furthermore, by Eq. (3), this implies that a = E. Since z is unique except for an additive constant, it can be chosen to coincide with D along r. Likewise, suppose D is strictly monotone increasing or decreasing along the path Sz in commodity space. We can consider Q’s (constant-income) image in price-income space and repeat the above argument. Hence, we have shown: PROPOSITION 2’. rf along rea, LW(U( s1) or D(s) is strictly monotone in s, then there exists a utility function E(P, m)(U(X)) which along (1) D’s numerical values, and (2) a marginal utility of income of unity.

r(sz)has:

4. D IS TANGENT

TO AN INDIRECT UTILITY

INDEX

Suppose one naively treats D as if it were path independent. From Eq. (1) one would then formally conclude that D’s “gradient” with respect to variation of its lower limit is given by aD(P, Pf, mi)/aP = -X(P, mi). However, a gradient of D is not well defined without specification of how the integration path r varies as P varies, so the use of D’s “gradient” is suspect. For example, path dependence means that a search for a maximum value of D under a constraint on prices will generally be fruitless. For suppose D’s maximum value, say D, , occurred at point PA. Then, as in the “happiness pump” of Section 1, integration along a suitable loop back to PA may increase D to a 1, instead of moving to the next, higher preference level along the CSMP, depart from the CSMP and reach the higher preference level by setting dP = 0 and increasing only m. If dm is the required increase, set dL = dm, that is, assign the value LO+ dm to this next, higher preference level. Since dP = 0, we have from dL = X(-X. dP + dm) that dL = hdm. But dL = dm, and thus this labelling has X = 1 at the starting point. In Stage 2, go back to the starting point and proceed along the CSMP to the preference level just labelled. Starting there, repeat the two stages for the next step, and so on. 7 In the interest of simplicity, a CSMP has been defined to be a continuous path. Clearly, Proposition 2 works for a piecewise continuous path as well, since one can compute .I% on the continuous segments and articulate the E values at points of discontinuity by adjusting additive constants.

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value greater than D, . So the naive use of this “gradient” in constrained optimization problems would appear to lead to a nonsensical search for a nonexistent maximand. On the other hand, Proposition 2 shows that along a r for which D is strictly monotone, D has the numerical values of a utility index, L, for which X = at/am is unity and hence for which ai;jaP = -X. This would suggest that the above naive computation of D’s “gradient” might be justified. This is indeed the case if the variation of D and r with D’s lower limit is suitably defined. Further, by this definition D as a function of its lower limit is tangent to an indirect utility function, a result which does not depend on D’s monotonicity along its path.

P*, J

Pf r

P

Pi

c

0

FIG. I. r’ is the traverse along the straight-line along r to Pf.

P,

segment from P to P and thence

Consider an arbitrary point P and path r with endpoints Pi, Pf, where I is not necessarily a path along which D is strictly monotone. We associate with I’ a new path, r’. This is the traverse from P along a straight line to Pi and thence along p to Pf (see Fig. 1). Let D*(P, Pi, r) be the value of the Dupuit-Marshall surplus along I”. D* is uniquely defined for every P and is thus a proper scalar function of P. From Eq. (1) and the mean value theorem, D*(P, Pi, T) - D*(Pi, Pl, T) = -X(p, mi) . (P - Pi) where P, component by component, lies between P and Pi. Hence, we immediately have: PROPOSITION

3. At P = Pi, the gradient of D*(P, Pi, r) is given by aD*(P, Pi, r)laP Ipzpi = --(Pi,

mi),

and is independentof r.

Thus, although D depends on the integration sense defined above, does not.

path, r, its gradient in the

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Now for any L(P, m) let X* equal aLlam evaluated at Pi, mi, and let L*(P, m) = (L(P, d/h*)

+ K,

(8)

where K is a constant. Then from Roy’s lemma

aL* = --(Pi, - ap p,pi m=mi

mi),

Further, chose the constant K so that D*(Pi, Pi, I’) = L*(Pi, mi). Then by Proposition 3, we have PROPOSITION 4. At P = Pi, D*(P, Pi, r) is tangent to L*(P, mi).

Proposition 4 is independent of integration paths of D. However, different choices of Pi lead to different values of the scale factor, X*, and hence to different definitions over preferences of L *. Thus D* is tangent to a different utility function at each Pi. Finally, suppose one is searching for a P at which are satisfied the necessary conditions for a constrained maximum of an indirect utility index. More precisely, suppose the constraints are g”(P) d 0, k = l,..., K. Consider g(P,n)

= D*(P, Pi, r) - 1 /Vg”(P),

where (1 = (hl, AZ,..., hK) is a vector of Lagrange multipliers. For brevity, denote by 9p and 2G$the gradients of 9 with respect to the components of P and (1 respectively. From Proposition 4 immediately follows: PROPOSITION 5. SupposeP satisfies the Kuhn-Tucker necessaryconditions for the problem: maxp L(P, m”) such that g”(P) ,< 0. Define L* by Eq. (8) with h” = aLlam evaluated at P = P, m = mi. Then (P, Al^) satisfy the Kuhn-Tucker relations:

(9) tr they satisfy the Kuhn-Tucker necessary conditions for max, L*(P, m”) such that g”(P) < 0.

the problem:

Since L* is a monotone transformation of L, Proposition 5 shows that the naive assumption that D’s gradient is --X yields the values, p, that satisfy the necessary conditions for a constrained optimum.* Again, we note without * Only D*‘s gradient enters in Eqs. (9). Thus, simply assuming that --X is the gradient with resmt to prices of an indirect utility function, without reference to consumer’s surplus, also yields the P values. 642/20/2-10

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E. E. ZAJAC

proof that duals to Propositions 4 and 5 are generated when P's are replaced by X’s and L's by Uk9

ACKNOWLEDGMENTS I would like to thank several colleagues for helpful comments on various drafts of this paper, including E. E. Bailey, J. C. Panzar, H. 0. Pollak, 1. H. Rohlfs, W. W. Sharkey, and W. E. Taylor. I would also like to thank the referee and editor for suggestions for shortening of proofs and, in particular, for raising the question of the relationship of some of the paper’s propositions to known revealed preference results. Finally, I am especially indebted to R. D. Willig, not only for encouraging me to persist in this investigation, but also for insisting on detailed supporting arguments for “intuitively obvious” assertions. His insistence greatly helped the precision and clarity of the exposition. Needless to say, final responsibility for errors is mine.

REFERENCES 1. W. J. BAUMOL AND D. F. BRADFORD, Optimal departures from marginal cost pricing, Amer. Econ. Rev. 50 (1970), 265-283. 2. M. E. BURNS, A note on the concept and measure of consumer’s surplus, Amer. Econ. Rev. 43 (1973), 335-344. 3. J. C. CHIPMAN AND J. C. MOORE, The scope of consumer’s surplus arguments, in “Evolution, Welfare, and Time in Economics” (A. M. Tang, F. M. Westfield, and J. S. Worley, Eds.), Lexington Books, 1977. 4. H. HOTELLING, The general welfare in relation to problems of taxation and railway and utility rates, Econometrica 6 (1938), 242-269. 5. H. HOUTHAKKER, Revealed preference and the utility function, Economica 17 (1950), 242-269.

6. D. W. KATZNER, “Static Demand Theory,” Macmillan Co., New York, 1970. 7. L. J. LAU, Duality and the structure of utility functions, J. Econ. Theory 1 (1969), 374-396. 8. T. RADER, Equivalence of consumer surplus, the divisia index of output, and Eisenberg’s addilog social utility, J. Econ. Theory 13 (1976), 58-66. 9. Y. SAKAI, Revealed favorability, indirect utility, and direct utility, J. Econ. Theory 14 (1977), 113-129. 10. P. SAMUELSON, On the constancy of the marginal utility of income, in “Studies in Mathematical Economics and Econometrics: In Memory of Henry Schultz” (0. Lange, F. McIntyre, and T. 0. Yntema, Eds.), pp. 75-91, Univ. of Chicago Press, 1942. 11. E. SILBERBERG,Duality and the many consumer’s surpIuses, Amer. Econ. Rev. 62 (1972), 942-951. 12. R. D. WILLIG, “Welfare Analysis of Policies Affecting Prices and Products,” Memorandum No. 153, Center for Economic Growth, Stanford University, 1973. 13. R. D. WILLIG, Consumer’s surplus without apology, Amer. Econ. Rev. 66 (1976), 589-597.

9 An example of the use of D in the sense of Propositions 5 and 4 is Baumol and Bradford 111.