Money-Metric Utility in Applied Welfare Analysis:

A Saddlepoint Rehabilitation∗

M. Ali Khan† and Edward E. Schlee‡

21 September 2018

Abstract

We attempt a rehabilitation of the money-metric, as proposed in Samuelson (1974),and, in its original incarnation, adduced by McKenzie (1957) as the minimum incomefunction. Our rehabilitation touches on both the partial equilibrium theory of the con-sumer, as well as its embedding in Walrasian general equilibrium analysis. It is articulatedthrough the vernacular of Uzawa’s (1958) saddle-point theorem of concave programming,and with a differentiable underpinning given to it by Debreu’s (1970, 1972) notions of aregular economy and of smooth preferences. We explore the viability of the aggregate ofthe individual money-metrics as a social welfare function, and relate our findings both tothe benefit function of Allais and the distance function of Malmquist-Shephard, as com-prehensively elaborated in Luenberger (1995) and Gorman (1970) respectively. From theperspective of applied welfare economics, we also connect the results to the fundamen-tal theorems of welfare economics, to Boadway’s (1974) paradox, and to Radner’s (1993)measure for cost-benefit in the small. Our analysis, by converting a constrained optimiza-tion problem to an unconstrained one, allows us to invoke lattice-theoretic structures, andthereby yield dramatically simpler comparative-static results that we explore in ongoingwork. (192 words)

Key words and phrases: Money-metric, saddlepoint inequalities, first fundamental theorem, benefit function,

distance function, cost-benefit analysis

JEL Classification Numbers: D110, C61, D610.

∗A first draft of this work was written during Khan’s visit to the Department of Economics at Arizona StateUniversity (ASU), March 1-6, 2017, and a second draft during Schlee’s visit to the Department of Economicsat Ryerson University as Distinguished Lecturer, May 1-5, 2017. Successive drafts were worked on duringKhan’s visit to ASU, October 28-30, 2017, and Schlee’s visit to JHU, March 12-15, 2018, and presentedat Ryerson, May 2017, the PET Conference, Paris, July 2017, ERMAS Conference, Cluj, Romania,August 2017, ASU, September 2017, Midwest Economic Theory Meetings, Dallas, October 2017 and at the2018 EWGET Conference, in Paris, July 2018. Both authors gratefully acknowledge the hospitality of therespective institutions. This version has benefited substantially from the encouragement and comments ofBeth Allen, Robert Becker, Luciano de Castro, Eddie Dekel, Peter Hammond, Michael Jerison, TsogbadralGalaabaatar, John Weymark, and Nicholas Yannelis.†Department of Economics, The Johns Hopkins University. E-mail akhan@jhu.edu‡Department of Economics, Arizona State University. E-mail edward.schlee@asu.edu

1

Contents

1 Introduction 3

2 Preliminaries: Notational and Conceptual 6

2.1 Money-metrics: The Background . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Example: Money-metric Utility and Quasi-linear Preferences . . . . . . . 7

2.3 The Non-concavity of Money-metric Utility . . . . . . . . . . . . . . . . . 9

2.4 The Money-metric Sum and Inequality . . . . . . . . . . . . . . . . . . . 10

2.5 Saddlepoints and the Kuhn-Tucker-Uzawa Theorem . . . . . . . . . . . . 12

3 Money-metrics, Saddlepoints and Concave Programming 13

4 Competitive Allocations and the Money-metric Sum 16

5 Money-metric Relations: Benefit and Distance functions 20

6 Money-metrics and Cost-Benefit Analysis in the Small 22

6.1 Radner’s Local Welfare Measure . . . . . . . . . . . . . . . . . . . . . . . 23

6.2 On Local Equivalences and Improvements . . . . . . . . . . . . . . . . . 23

7 A Technical Mopping-up 25

7.1 A Proof of Lemma 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

7.2 On the Differentiability of a Money-metric . . . . . . . . . . . . . . . . . 26

7.3 A Representation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 28

8 Concluding Remarks on Future Work 29

2

1 Introduction

A theorem of Blackorby and Donaldson (1988) asserts that the money-metric utility of

McKenzie (1957) and Samuelson (1974) is never a concave function of consumption ex-

cept for the knife-edge case of quasi-homothetic preferences that require each consumer’s

wealth expansion path to be a straight line.1 Since it is well-understood that the quasi-

concavity of an additive social welfare function necessitates, by the theorem of Debreu

and Koopmans (1982), that all but one of the individual utility representations be con-

cave, and that the quasi-concavity of the remaining representation be only necessary but

not sufficient, the authors draw out what they see to be the negative implications of the

construction for applied welfare economics.

We show that concavity cannot be guaranteed unless preferences are severely restricted.

This result vitiates the claim that money metrics are useful in applied welfare analysis

because they represent preferences exactly. [S]ince many plausible preference orderings

yield non-concave money metrics for all reference prices, we must conclude that social

welfare analysis based on money metrics is flawed.2

Indeed, the Fenchel-Finetti-Kannai example of a convex preferences relation that

does not have a concave representation is clearly relevant here.3 Furthermore, Gorman

(1959) had already put forward the objection that non-convexity of social preferences

over allocations leads to discontinuity in solutions, and Samuelson (1974) had openly

expressed his skepticism of the use of the money-metric in welfare economics.4 The du-

bious ethical implications stem from Gorman’s (1970) result5 that a social ordering Ω is

quasi-concave in individual allocations if the associated Bergson-Samuelson social wel-

fare function W of individual utility representations is quasi-concave, leading Blackorby

and Donaldson (1988) to write:

Quasiconcavity of Ω ensures that social judgements provide goods for everyone rather than

giving them exclusively to the few. Samuelson called this requirement “the foundation for

the economics of the good society.” Use of money metrics in a quasiconcave function W

may result in a social-evaluation function corresponding to Ω that is not quasiconcave.

1Preferences are quasi-homothetic if they generate, for some representation, an indirect utility function thatis affine in income of the form V (p, w) = a(p)+b(p)w, and they include homothetic and quasi-linear preferencesas special cases; see Blackorby and Shorrocks (1995) and Kreps (2013, Section 2.6) for formal definitions anddetails. For an alternative proof of the Blackorby-Davidson theorem based on the least concave representationof a preference relation, as in Debreu (1976), see Khan and Schlee (2017); also Section 2.3 below.

2See Blackorby and Donaldson (1988, Conclusion). On reference prices, see the comprehensive treatment inBlackorby, Laisney, and Schmachtenberg (1994).

3See Kannai (1977) and Kannai (1981) for the example and the antecedent background.4See the concluding paragraph of Khan and Schlee (2017), but note that unlike subsequent literature, Samuel-

son’s criticism is based on the very grounds of redistribution that Pareto optimality is criticized on; the professionwas not aware of the non-concavity of money-metrics until Blackorby and Donaldson (1988).

5See Gorman (1970), also Blackorby and Shorrocks (1995). For the implications of quasi-concavity of socialwelfare functions, as in the quote below, see (Chipman 1976), (Chipman 2006); and (Kemp 2009).

3

More recently, Decancq, Fleurbaey, and Schokkaert (2015, Chapter 2) remark on both

the fact and the causes of the withering away of the use of money metrics in theoretical

and empirical microeconomics.

[M]oney-metric utility ... has a somewhat surprising history. [It] had some impact on the

applied welfare economic literature during the eighties [but] lost popularity ... as authors

argued that it relied on an arbitrary choice of a reference values and could have nonegal-

itarian implications. Although it slowly disappeared from the applied welfare economics

literature, it was more or less independently developed within the social choice literature

in what is called the theory of fair allocation.6

This non-concavity property of the money-metric is elusive in that there is no under-

standing as to the ranges over which its concavity is violated; and indeed, Samuelson’s

conjecture that it is locally concave in the neighborhood of the bundle chosen at the

base price still awaits a proof or a refutation.7 This can underscore the sentiment that

the negative implications of the Blackorby-Davidson theorem are undeniable and con-

tinue to hold in force. This note contests this conclusion, and offers a rehabilitation.

In particular, we show that even when non-concave and even when they do not repre-

sent preferences, the sum of money-metrics has some natural welfare properties that are

worth bringing into salience.8

However, we first prepare the ground by laying out some antecedent theoretical lit-

erature and its empirical implementation. In terms of the theory, one way to read

an important slice of the last one-hundred years of applied welfare economics is as an

attempt to restore the simplicity of quasi-linearity in economies that are not quasi-

linear.9 The defining simplification of quasi-linearity is Marshall’s elusive “constancy

of the marginal utility of income” provided that the the commodity in which the pref-

erences are quasi-linear – usually taken to be the numeraire –is consumed in positive

quantities; see Samuelson (1942). Since the numeraire good enters utility linearly, one

difficulty is that its demand must be zero for some price-wealth pairs. An attraction

of the money-metric form lies precisely in its guarantee of the sought-after ‘constancy

of the marginal utility of money’ without the need to impose the strong assumption

of quasi-linear preferences, and furthermore, in the effortless way it handles corner so-

lutions and non-convexities, both in the consumption set and in the preferences; see

Section 2.2 below. It is thus that money-metric non-concavity has led to a conflicted

6For the influential papers of the eighties, the authors cite, among others, Deaton and Muellbauer (1980)and King (1983), but for the criticism, highlight only Blackborby and Donaldson’s (1988) influential paper. Forthe anti-egalitarian implications, see Donaldson (1992, p. 93).

7See Footnotes 4 and 5 in Khan and Schlee (2017), and Section 2.3 below in which we further elaborateSamuelson’s assertion.

8Anticipating somewhat, the reader is referred to Theorem 1, Remark 1 and Footnote 34 below.9For example, Willig (1976), Harberger (1971), McKenzie and Pearce (1976), Tirole (1988, pp. 7-13), Vives

(1999, chapter 3), and Hayashi (2017).

4

avoidance as well as its apologetic use, the former resulting in the exploration and for-

mulation of new welfare measures. Four such attempts are worth noting. Blackorby and

Donaldson (1987) develop a measure over price-wealth pairs called the welfare ratio that,

by construction, is linear in wealth and represents preferences over price-wealth pairs

if and only if preferences are homothetic.10 Hammond (1994, Section 3.1), after em-

phasizing the inegalitarian implications of the money-metrics sum, goes on to develop

a measure based on uniform poll taxes or subsidies. Bosmans, Decancq, and Ooghe

(2018) move away from additivity altogether and axiomatize a (generally) non-additive

welfare function based on individual money-metrics.11 Finally, acknowledging the non-

concavity, Chambers and Hayashi (2012, (p. 811)) axiomatize the use of the aggregate

money-metric for social choices involving risk by conceiving of it as the sum of von

Neumann-Morgenstern utility functions. They see their axiomatization as illuminating

the lack of any distributional content of the money-metric sum.

Approaches to the empirical estimation of money-metric welfare measures fall into

three groups: calculation of expenditure functions from parametrically estimated de-

mands; revealed preference; and first- or second-order approximations.12 Illustrating

the Decancq-Fleurbaey-Schokkaert sentiment quoted above, Diewert and Wales (1988,

p. 305) explicitly refuse to use the money-metric normalization altogether because of

the non-concavity considerations. Banks, Blundell, and Lewbel (1996, p. 1232-3) de-

velop and estimate a quadratic approximation of the money metric to calculate the

welfare cost of taxes, but do so only after acknowledging the Blackorby-Donalson ob-

jections. In their comprehensive project, Deaton and Zaidi (2002) compare the merits

of the money-metric sum and the welfare ratio of Blackorby and Donaldson (1987), but

write somewhat defensively that, despite the non-concavity, “Our own choice is to stick

with money metric utility” (p. 11). They develop aggregate consumption measures for

general welfare evaluations based on it.13

With this background, the motivation of this paper can be succinctly stated. First,

it is to articulate the observation that the non-concavity of the money-metric can be

circumvented by the remarkable fact that the classical consumer’s constrained maxi-

mization problem can be converted into an (unconstrained) saddlepoint problem without

any concavity assumptions when the money-metric is taken as the consumer’s objective,

10In the notation developed in the sequel, it equals w/e(p, u), the ratio of wealth and the minimum expenditureto reach utility u at prices p. The utility level is taken to be a poverty level of well-being.

11We discuss their rehabilitation in Section 2.412See King (1983) as an example of the first, Varian (1982), Knoblauch (1992) and Blundell, Browning, and

Crawford (2003) as examples of the second, and Banks, Blundell, and Lewbel (1996) and Deaton and Zaidi(2002) as examples of the third. Indeed, Varian (2012) explains that his original reason for pursuing revealedpreference was to estimate money-metric utilities.

13See also Deaton (1980) and Deaton (2003). The surveys by Blundell, Preston, and Walker (1994, p. 38) andSlesnick (1998, p. 1241) are explicit about non-concavity as an argument against the sum of money metrics.

5

whether or not it represents the consumer’s preferences. We can thereby show that any

competitive equilibrium allocation maximizes the sum of money-metric utilities at the

competitive price system. Second, it is to leverage this observation by exploiting the

sense in which the money-metric utility preserves aspects of concavity. It is thus that we

show how the sidelining of the Samuelson-McKenzie construction for can be effectively

blunted and brought into the service of applied welfare economics at the same conceptual

level as Pareto optimality.

Now to a more detailed reader’s guide. After a brief introduction to notational and

conceptual preliminaries in Section 2, the saddlepoint result is presented as Theorem

1 in Section 3. In Section 4, we relate the money-metric to general competitive analy-

sis, and present Theorem 2 that shows that competitive allocations maximize the sum

of money-metrics on the set of feasible allocations, and that the sum plus the added

term p(y − x) is maximized over the set of allocations (x, y) that are feasible or not.

Provided the money-metric represents consumer preferences, it thereby yields the first

welfare theorem as a simple corollary, an alternative proof that deserves consideration.

In Section 5, we connect this result to two alternative measures: (i) the benefit func-

tion originally due to Allais, and comprehensively elucidated in Luenberger (1995), and

(ii) the distance function originally due to Malmquist (1953) in the context of con-

sumer theory, and Shepard (1953) in the context of producer theory, and elaborated

in Gorman (1970). In Section 6, we turn to a synthetic overview of cost-benefit in the

“small,” and show that the derivative of the money-metric sum of equals seven other

local measures of welfare in a general equilibrium setting: an equivalence of welfare

measures associated with the names of Dupuit, Slutsky, Divisia, Allais-Luenberger and

Malmquist-Shepard, in addition to that of Radner.14 The equivalence fails for Debreu’s

coefficient of resource utilization, and for the measures devised by Blackorby-Donaldson

and Hammond in response to the non-concavity of the money-metric. Section 7 is tech-

nical, and can be skipped on a first reading, but in offering a representation theorem

and sufficient conditions for the differentiability of the money-metric, it is of substan-

tive importance. Finally, the concluding Section 8 consolidates the various strands, and

indicates directions for further analysis stemming from the fact that the conversion to

the unconstrained maximization problem allows the direct invocation of lattice-theoretic

methods for comparative-static analyses.

2 Preliminaries: Notational and Conceptual

This five-part section is primarily for the convenience of the reader: we supplement the

framework and notation by briefly reviewing pertinent facts about money-metric utilities

14This is laid out in Schlee (2013a), Schlee (2013b) and Schlee (2018).

6

and concave programming that dictate why the analysis presented in the sequel seemed

initially to be so unpromising for applied welfare analysis.

2.1 Money-metrics: The Background

We lay out the basic notation in this subsection. There are L > 1 commodities, and

the consumption set X ⊆ RL+ is non-empty and closed. The binary relation %⊆ X ×X

describing a consumer’s preferences is assumed to be complete, transitive, continuous,

and locally non-satiated. Note that neither %, nor the set over which it is defined, is

assumed to be convex, and, in particular, “discrete” commodities are allowed, as in

McKenzie (1957). For a price-wealth pair (p, w) ∈ RL+ × R+, the budget set and the

demand correspondence are

B(p, w) = x ∈ X | p · x ≤ w and d(p, w) = x ∈ B(p, w) |x % y, y ∈ B(p, w).

The money-metric M(·, ·) is the value function for the following wealth-minimization

problem15

M(x, p) = minx′%x

p · x′ and argmaxx′%xp · x′ ≡ h(p, x). (1)

We shall refer to the function h(·, ·) as the (Hicksian) compensated demand to distinguish

it from the (Marshallian) demands defined above. If the minimum does not exist, we

set M(x, p) equal to minus infinity.

In the classic textbook case ofX = RL+, the money-metric at price p assigns the wealth

level w to the indifference set passing through the point x = d(p, w). Since preferences

are locally non-satiated, this assignment of numbers to indifference sets represents %on X in this case; see Figure 2, but with the points a, b and x′ ignored. The money-

metric is related to the more conventional expenditure function, defined as e(p, u) =

minu(x′)≥u p · x′, by substituting for u the utility level u(x) attained by the commodity

bundle x. This is to say that M(x, p) = e(p, u(x)).

2.2 Example: Money-metric Utility and Quasi-linear Preferences

In this subsection, we calculate the money-metric for the familiar and historically-

important special case of quasi-linear preferences defined on two goods, X = R2+, and

15M(·, ·) is the minimum income function in McKenzie (1957), who emphasized it as a function of p for fixedx. Samuelson (1974, pp. 1272-1273) clearly ascribed the concept to McKenzie but saw it as a function of x for afixed p, writing that “It is not entirely new since, I believe, people have had a glimpse of it long before politicaleconomy became a scholarly discipline and took up mathematical symbolisms” (p. 1261). Given his emphasis,he changed the nomenclature of an expenditure or a minimum income function to a money-metric utility, and itis this change that recovers the inherent duality of the object and converts it into a powerful engine of analysis.

7

x1

p ・x = w0

p ・x = w1

u = w1

u = w0

x2

x'

ab

Figure 1: Monetary-labelled indifference curves

represented by the quasi-linear-in-good-2 form u(x1, x2) = φ(x1)+x2, where φ is strictly

concave and differentiable with φ′ > 0. The strict concavity of φ implies that preferences

are strictly convex. For this subsection, we set p2 = 1, so that good 2 is the numeraire.

For p1 > 0, define f(p1) to be the unique number (if any) that solves the Kuhn-Tucker

conditions for the consumer’s problem of choosing x1 ≥ 0 to maximize φ(x1) +w−p1x1,

i.e., by ignoring the non-negativity constraint on x2. These conditions are

φ′(f(p1))− p1f(p1) ≤ 0 f(p1) [φ′(f(p1))− p1f(p1)] = 0, andf(p1) ≥ 0.

If there is no solution to these conditions, set f(p1) =∞.

The demand for good 1 equals f(p1) if f(p1) ≤ w/p1 (that is, the constraint x2 ≥ 0

does not bind), and equals w/p1 otherwise. The indirect utility function for the consumer

is v(p1, w) = φ(f(p1)) − p1f(p1) + w if w ≥ p1f(p1) and φ(w/p1) otherwise. Note that

the marginal utility of wealth equals 1 if and only if the non-negativity constraint on

good 2 does not bind. For reference utility u in the range of u, the expenditure function

equals e(p1, u) = u− φ(f(p1)) + p1f(p1) if u ≥ φ(f(p1)) and equals p1φ−1(u) otherwise.

It follows that money-metric utility is given by

M(x, p) =

φ(x1) + x2 −

[φ(f(p1))− p1f(p1)

]if φ(x1) + x2 ≥ φ(f(p1))

p1φ−1(φ(x1) + x2) otherwise

(2)

In particular, for fixed p = (p1, 1), M(x, p) = φ(x1) + x2 up to an additive constant if

and only if the non-negativity constraint on x2 in the expenditure minimization problem

8

does not bind at p; otherwise the two functions diverge.16 Of course, since x2 enters the

representation linearly, the non-negativity constraint for x2 must bind for a region of

values for (p, u). Applications that hope to preserve the simplicity of quasi-linearity then,

often implicitly, restrict parameters to avoid corner solutions for the numeraire good.

And these parameters from the viewpoint of the consumer are endogenous equilibrium

variables from the viewpoint of the economy as a whole.17 A virtue of the money-metric

is that it preserves the constancy of the marginal utility of money without requiring

positive consumption of any particular good.

2.3 The Non-concavity of Money-metric Utility

In order to help the reader see the possible non-concavity of a money-metric, we outline

in this subsection an alternative proof of the Blackorby-Donaldson theorem based on

least-concave representation result of Debreu (1976); see Khan and Schlee (2017). Let

X = RL+. In this case there not only exists a representation u of preferences, but the

money metric itself is a representation; see Weymark (1985) or Section 7.3 below. For

any representation u, write money-metric utility as M(x, p) = eu(p, u(x)), where eu is

the expenditure function associated with u. There are two possible cases. Either there

is no concave representation of preferences, in which case we are done; or there is a

concave representation u. In the second case, the indirect utility is concave and strictly

increasing in w for each p.18 Since the expenditure function eu(p, ·) is the inverse of

the indirect utility for fixed p, eu(p, ·) is convex. So M(·, p) is a convex transformation

of any concave representation; which is to say that it is less concave than any concave

representation.19 By Debreu (1976), there is a least concave representation, u; that is,

if u is a concave representation, then u = T u for some concave, strictly increasing

function T : Range(u)→ R. So either M(·, p) is affinely related to u –that is M(x, p) =

a(p) + b(p)u(x) for b(p) > 0 – and hence the money-metric is also a least concave

representation— or it is not concave.

The literature is silent about the region of the consumption set over which the money-

metric is not concave, but Samuelson (1974, p. 1276) makes an intriguing assertion. He

16For example, if φ(x1) = lnx1, then M(x, p) = lnx1 + x2 + ln p1 + 1 if lnx1 + x2 ≥ − ln p1, and M(x, p) =p1x1 exp(x2) otherwise.

17One workaround is simply to drop the non-negativity constraint on the numeraire good altogether andassume its consumption can be any real number, as in Mas-colell, Whinston, and Green (1995, Chapter 3, p. 44and Section 10.C).

18If the consumer has expected utility preferences and we interpret u as a von Neumann-Morgentstern utility,this fact yields the conclusion that aversion for consumption risk implies aversion for wealth risk; see Kreps(2013, Proposition 6.16, p. 136).

19For two continuous real-valued functions f and g on a convex set D ⊆ Rn that represent the same binaryrelation %, g is less concave than f if there is a convex, and strictly increasing, real-valued function T onRange(f) that is convex.

9

writes that the money-metric M(x, p0) is “locally concave in all x that are near to [a

point demanded at p0] x0 (and which may as well be restricted to x’s that are at least

as good as x0).” However, But what he elucidates symbolically is the assertion that

∇M(x0, p0)·(x−x0) ≥M(x, p0)−M(x0, p0) for all x in a neighborhood of x0, that is, that

M(·, p0), is concave precisely the demanded point x0.20 Whereas the latter is certainly

true if money metric is differentiable at a demanded point, and the demanded point is

in the interior of the consumption set, Samuelson’s verbal assertion about concavity in

a neighborhood of a demand point remains an open question.21

2.4 The Money-metric Sum and Inequality

It is well to be clear as to the sense in which the non-concavity of the money-metric sum

leads to an inegalitarian outcome, and we turn to it in this subsection. We begin with an

example we think to be decisive for the use – misuse if one prefers – of the construction

in welfare analysis, and one that illuminates the relationship between our work and the

rehabilitation offered in Bosmans, Decancq, and Ooghe (2018), henceforth BDO.

Example 1 (Money-metrics and inequality). Consider an economy consisting of two

consumers with identical, strictly monotone, strictly convex preferences on X = RL+

that are not quasi-homothetic, and let M(·, ·) be their common continuous, non-concave

(at some p) money-metric. Then we know that it is not mid-point concave which is to

say,

M(12x+ 1

2y, p) < 1

2M(x, p) + 1

2M(y, p) for some x and y ∈ X. (3)

Let the aggregate endowment be x+ y. Then the sum of money metrics is higher at the

unequal allocation in which one consumer gets x and the other gets y than if they each

get (x + y)/2. Since the consumers have the same convex preferences, equal division is

Pareto optimal, and so one of the consumption bundles, say x, is worse than an equal

division of the aggregate resources.22

Our rehabilitation does nothing to rule out this example. We return to this point

in discussion of our Theorem 2, but here we just note that the particular inequalities

that the money-metric sum sanctions depend on the reference price; in particular, if we

consider a reference price p∗ one that supports equal division as a competitive equilibrium

20By “concave at the point x” for a differentiable function f we mean that ∇f(x) · (y − x) ≥ f(y)− f(x) forall y in some neighborhood of x.

21It may be worth mentioning that it was reflection on Samuelson’s assertions about local concavity of money-metric that lead us to think about what global properties, if any, of concave functions that it preserves, inde-pendently of differentiability and interiority assumptions, and thereby to Uzawa’s saddlepoint theorems.

22Note that for Equation 3, we appeal to Jensen (1906); for the continuity claim to Weymark (1985, Propo-sition 2): and it hardly needs saying, for the non-concavity claim to Blackorby and Donaldson (1988).

10

in this exchange economy, then the sum of those money-metrics would not increase with

a move to an unequal allocation.

The BDO rehabilitation, on the other hand, consists of a representation theorem for

a social preference relation, that is, a relation which specifies, for each possible profile

R of preferences in an economy, a binary relation %R over allocations. It asserts that

social preferences satisfy six axioms if and only if there is a reference price vector p and

a Schur-concave function23 W taking a profile of money-metrics at p into real numbers

such that the composition of W with the vector of money-metrics represents %R for each

preference profile R.24 The money-metric sum captures efficiency, the Schur-concave

welfare function, equity.

Since the sum of money metrics is a special case of the representation they prove,25

the BDO rehabilitation likewise does not rule out Example 1. Their novel axiom, the

one that they use to capture equity concerns, is referred to as the Efficiency-Preserving

Transfer Principle. To explain the axiom, consider an economy in which two consumers

have the same preferences, and an allocation in such an economy in which one consumer

is richer than the other: one has consumption y, the other consumption x and y > x.

The efficiency-preserving transfer principle asserts that if the richer consumer transfers

some consumption to the poorer consumer (but remains weakly richer), and the Scitovsky

set for these two consumers remain unchanged by the transfer, then the new allocation

is weakly preferred in the social order.26

How does Example 1 escape this axiom? One of two ways. Either the preferences of

the two consumers are quasi-homothetic, in which case their money-metrics are concave

and the example cannot arise; or they are not quasi-homothetic. If the latter, the

Scitovsky sets generally vary as the allocation of a fixed stock of goods changes, and

so the axiom does not apply.27 Indeed, Gorman (1953) proved that the Scitovsky set

is globally invariant to changes in the allocation of a stock of goods if and only if

preferences of consumers are quasi-homothetic with a common slope of their wealth-

expansion paths.28 If the Scitovsky set changes with a reallocation, as it typically does

23A function f on a nonempty convex set D ⊂ Rn is Schur-concave if f(x) ≥ f(y) whenever y majorizes x,that is, whenever

∑ki y[i] ≥

∑ki x[i] for k = 1, ..., n − 1, with equality for k = n, where y[n] ≥ y[n−1] ≥ y[1] and

x[n] ≥ x[n−1] ≥ x[1].24That is, social preferences over allocations at profile R are represented by UR(x1, ..., xI) =

W (M1(x1, p), ...,MI(xI , p)), where Mi is consumer i’s money metric in the preference profile R and xi is i’sconsumption.

25The function f(y) =∑yi is (weakly) Schur-concave.

26The Scitovsky set for a population of consumers (1, ..., I) at a reference allocation (x1, ..., xI) is the sum of at-least-as-good-as sets from this allocation. Formally, at an allocation (x1, ..., xI) it equals x′ ∈ X |x′ =

∑Ii=1 x

′i,

ui(x′i) ≥ ui(xi), i = 1, ..., I, where for each consumer i, ui represents that consumer’s preferences.

27In our view, a more descriptively-apt name for their axiom might have been Scitovsky-set-preserving transferprinciple.

28To be sure, Bosmans, Decancq, and Ooghe (2018) write clearly in their Footnote 14 that “This axiom doesnot cover all cases where the distributions before and after the transfer are both efficient (equal marginal rates

11

in Example 1, then the BDO axioms are silent about the desirability of an inequality-

producing transfer. As a final point of difference, the BDO rehabilitation is one in the

social choice register, as opposed to that of welfare economics. That is to say, BDO

consider axioms on how social rankings over allocations behave over difference profiles

of consumer preferences, rather than for a fixed society.29

2.5 Saddlepoints and the Kuhn-Tucker-Uzawa Theorem

In this subsection, we recall for the reader Uzawa’s (1968) saddlepoint theorem. Let ∅ 6=Z ⊆ Rn

+, and let f : Z → R and g : Z → Rm be functions, where n and m are positive

integers. Consider the constrained optimization problem maxz∈Zf(z) subject to g(z) ≤0, and the saddlepoint inequalities asserting the existence of z∗ ≥ 0 and λ∗ ≥ 0 such

that

L(z, λ∗) ≤ L(z∗, λ∗) ≤ L(z∗, λ)

for all z ∈ Z and λ ∈ Rm+ , and where L(z, λ) = f(z)− λ · g(z).

We can now present

Theorem (Uzawa). (i) If (z∗, λ∗) satisfies the saddlepoint inequalities, then z∗ solves

the optimization problem. (ii) If z∗ solves the optimization problem, Z is convex, f is

concave, g is convex, and there exists z ∈ Z such that g(z) << 0 (Slater’s constraint

qualification holds), then there exists λ∗ ≥ 0 such that (z∗, λ∗) satisfies the saddlepoint

inequalities.

If the saddlepoint inequalities hold, then z∗ maximizes f(z)− λ∗g(z) on Z ignoring the

constraint g(z) ≤ 0. A difficulty in applying claim (i) in Uzawa’s Theorem is to verify

that the saddlepoint inequalities hold. The deeper claim (ii) of Uzawa’s theorem gives

sufficient conditions for the saddlepoint to hold; it applies to the consumer’s problem

if preferences are representable by a concave function. There are two well-known im-

pediments for a successful application. First, the value of the multiplier, the marginal

utility of money, depends on the parameters of the problem, and thereby complicates

its use for comparative-static analysis. Second, it requires a concave representation of

preferences, and as already pointed out in the introduction, even if one imposes convex-

ity of preferences, some convex preference relations are not representable by a concave

function30 In any case, convexity of preferences is a strong assumption that precludes,

of substitution). Indeed, in some such cases the Scitovsky boundaries do not coincide, but rather intersect atthe societal bundle.”

29In particular, to have any bite, their efficiency-preserving transfer principal requires that the economy haveat least two consumers with the same preferences.

30See the references in Footnote 3. We remind the reader that concavity cannot be relaxed to quasi-concavityin Claim (ii) of Uzawa’s Theorem even if the constraint function g is linear. The Cobb-Douglas functionu(x1, x2) = x1x2 for the consumer’s problem is a counterexample.

12

X1

X0x1 + x2 = const

X2

X2

X1

Figure 2: The consumption set equals the vertical bars, so the consumption set is not convex.Assume that preferences are strongly monotone. If x2 x1 % x0, then x = x1 violates thelocal cheaper point at p0 = (1/2, 1/2). The point x2 is demanded at (p0,M(p0, x1)), but thereis no point in a small-enough neighborhood of x2 that is cheaper at those prices. We haveM(p0, x1) = M(p0, x2), even though x2 x1. The curves in the Figure are not indifferencecurves but meant as a visual aid.

for example, the discrete commodity setting that is allowed here.

3 Money-metrics, Saddlepoints and Concave Programming

From transitivity of %, it follows that

x % y implies that M(x, p) ≥M(y, p) (4)

since the constraint set for the income minimization problem at x is a subset of the

constraint set for that problem at y. From this fact it follows that if x′ ∈ d(p, w), then

x′ maximizes M(x, p) on the budget set B(p, w). In this section we are after something

bolder: that x′ maximizes M(x, p)− p · x on X ignoring the budget constraint.

Of course if M(·, p) represents %, then any maximizer x′ of M(·, p) on B(p, w) is

a demand. But without further restrictions on the consumption set, preferences, or

prices, a money metric does not in general represent preferences: in particular, we

can have x y, but M(x, p) = M(y, p).31 The possibility is depicted in Figure 2.

The consumption set equals the vertical bars, preferences are strongly monotone, and

x2 % x1 % x0. At the price p = (1/2, 1/2), M(x1, p) = M(x2, p). But nothing prevents

x2 x1. In that case M(·, p) does not represent the consumer’s preferences on X, and

x1 /∈ d(p, p · x1) even though it maximizes M(x, p) on the budget set B(p, w). We now

introduce a variant of the condition that McKenzie (1957) uses in this derivation of the

31See Weymark (1985) for an exceptionally clear discussion of money metrics as a preference representation.

13

Slutsky equation.32

Definition 1. A point x ∈ X satisfies the local cheaper point condition at p ∈ RL+

if, for every x′ ∈ d(p,M(p, x)) and every open neighborhood N of x′, there is a point

x′′ ∈ N ∩X such that p · x′′ < p · x′.

In Figure 2, if x2 x1 % x0, then x1 violates the local cheaper point condition since

x2 ∈ d(p, p · x1), but there is no point in X that is cheaper than it at price (1/2, 1/2).

Of course x2 itself violates the cheaper point condition as well, but this violation does

not pose a problem for us since x2 is itself demanded at price p and income p · x2:

x2 ∈ d(p, p · x2). The difficulty for x1 is that x1 /∈ d(p, p · x1).

The role of the local cheaper point condition is spelled out in the next lemma. With

it, compensated and uncompensated demands are equal; and the constraint x′ % x in

the wealth minimization problem (1) binds. We prove the Lemma in the Appendix.

Lemma 1. Fix (p, w) >> 0. If % is complete, transitive, continuous, and locally

nonsatiated and x ∈ X satisfies the local cheaper point assumption at p, then (a)

h(p, x) = d(p,M(p, x)); and (b) x′ ∼ x for any x′ ∈ h(p, x).

Of course in the classic case X = RL+, then the local cheaper point assumption holds

for every x 6= 0. Again, the point x = 0 poses no problem for us since it is itself

demanded at zero wealth at any price: 0 ∈ d(p, 0). We now turn to our main result for

a single consumer. It lays the foundation for a new agenda on the role of money metrics

in consumer theory.

Theorem 1 (Money-Metric Saddlepoint). Fix (p, w) ∈ RL+1++ . Define L(x, λ) =

M(p, x) + λ[w − p · x]. Consider the saddlepoint inequalities for (x∗, 1) with x∗ ∈ X:

L(x, 1) ≤ L(x∗, 1) ≤ L(x∗, λ) for every x ∈ X and λ ≥ 0. (5)

(a) If x∗ ∈ d(p, w), then the saddlepoint inequalities (5) hold.33

(b) If the saddlepoint inequalities (5) hold, and if x∗ satisfies the local cheaper-point

condition at p, then x∗ ∈ d(p, w) for w = p · x∗.

In either case, L(x∗, 1) = w.

32We warn the that the cheaper point assumption is being formulated in a slightly different way than inWalrasian general equilibrium theory where for each price, it is the existence of a cheaper point in the budgetset of each consumer that is being asserted.

33 The only properties we use to prove (a) are that % is complete and that budget balance holds at (p, w)(x′ ∈ d(p, w) implies p · x′ = w). In particular, we don’t use transitivity. Sonnenschein (1971) proves thatdemands exists if preferences are complete, continuous, and strictly convex, but not necessarily transitive.Shafer (1974) gives an alternative proof.

14

Note that we do not assume that preferences are convex, so M(·, p) need not even be

quasiconcave. Note too that part (a) holds even if the money metric does not represent

preferences at the reference price p, as in the example portrayed in Figure 2.34

Proof : (a) Let x∗ ∈ d(p, w). By local nonsatiation of preferences, p · x∗ = w, so

L(x∗, 1) ≤ L(x∗, λ) for every λ ≥ 0.

By the definition of M(p, x), it follows that

L(x, 1)− w = M(p, x)− p · x ≤ 0 (6)

for every x ∈ X, with equality if and only if x itself solves the income-minimization

problem at (p, x), that is, x ∈ h(p, x). We will prove that x∗ ∈ h(p, x∗). Since preferences

are complete, x∗ % x∗ and x∗ is itself feasible for the wealth-minimization problem at

x = x∗. Consider any y ∈ X with p · y < p · x∗. Since x∗ ∈ d(p, w), x∗ % y. By local

nonsatiation, y /∈ d(p, w), so x∗ y, and y is not feasible for the wealth-minimization

problem at (p, x∗). Since p · y < p · x∗ implies x∗ y, it follows that x∗ ∈ h(p, x∗) and

so L(x, 1) ≤ L(x∗, 1) for every x ∈ X and (5) holds.

(b) If the Saddlepoint inequalities (5) hold, then by Theorem 1 in Uzawa (1958),

x∗ maximizes M(p, ·) on B(p, w). If M(·, p) represents preferences on X—for example,

if X = RL+—then we would be done. Since we consider more general consumption

sets, we proceed using Lemma 1. Let x′ ∈ h(p, x∗), so by Lemma 1, x′ ∼ x∗ and

x′ ∈ d(p,M(p, x∗)). The inequalities L(x∗, 1) ≤ L(x∗, λ) for every λ ≥ 0 imply

that p · x∗ = w, so x∗ ∈ B(p, w). And the inequality L(x′, 1) ≤ L(x∗, 1) implies

that M(x′, p) − p · x′ ≤ M(x∗, p) − p · x∗. Since x′ ∼ x∗, M(x′, p) = M(x∗, p), so

L(x, 1) ≤ L(x∗, 1) reduces to w = p ·x∗ ≤ p ·x′. Since x∗ ∼ x′ % x for every x ∈ B(p, w)

and % is transitive, x∗ ∈ d(p, w). Again L(x∗, 1) = w.

We now specialize to the classic case of X = RL+ to illustrate a use of Theorem 1.

It asserts that the wealth expansion path at a price p equals the set of maximizers of

M(x, p) − p · x on X. It has the implication that we can replace the standard utility

maximization problem in classical consumer theory—maximize utility subject to a bud-

get constraint—with an unconstrained optimization problem using the money metric

representation.

Corollary 1. If X = RL+, then x′ ∈ d(p, w) for some w ≥ 0 if and only if x′ maximizes

M(x, p)− p · x on RL+.

The corollary puts consumer theory on the same footing as producer theory: maximizing

benefit minus cost on a consumption or production set, at least for the classic case

34Also see Footnote 8 above.

15

X = RL+. In the sequel to this paper we intend to use this formulation to simplify

dramatically the comparative statics of the consumer’s problem. A long-recognized

difficulty in applying the lattice-theoretic comparative statics methodology of Topkis

(1995) and Milgrom and Shannon (1994) to the consumer’s problem is that different

budget sets do not stand in the strong-set-order relation.35 The fact in Corollary 1

solves this problem by dispensing with the budget set altogether.

4 Competitive Allocations and the Money-metric Sum

We now extend the framework of Section 2 and to I consumers. This is to say that

each consumer i has a consumption set Xi ⊆ RL+ that is nonempty and closed; and a

preference relation %i⊆ Xi × Xi that is complete, transitive, continuous, and locally

nonsatiated. Define Mi(x, p) = minx′%ix p · x′ with hi(p, x) the set of solutions to his

problem for any p ∈ RL+. For (p, w) ∈ RL

+ × R+ and B(p, w) = x ∈ X | p · x ≤ w, we

let di(p, w) = x ∈ B(p, w) |x %i y for all y ∈ B(p, w).Let there be J ≥ 1 firms, with firm j endowed with a production set Yj ⊆ RL

satisfying 0 ∈ Yj (possibility of inaction). The aggregate production set is Y =∑

j Yj,

assumed to be closed. Let

πj(p) = maxyj∈Yj

p · yj

when it exists. Consumers are endowed with goods and ownership shares in firms.

Consumer i’s price-dependent wealth is wi(p) = p · ωi +∑

j θijπj(p), where ωi ∈ Xi is

consumer i’s endowment of goods and θij is i’s ownership share of firm j. We assume

that ω =∑

i ωi > 0.

Definition 2. Let A = ΠiXi × Πj Yj. An allocation is a point (x1, ..., xI , y1, ..., yJ) ∈A. An allocation (x1, ..., xI , y

1, ..., yJ) is feasible if∑i

xi ≤∑i

ωi +∑j

yj.

We will denote an allocation by (x,y). We denote aggregate consumption by x =∑i xi, aggregate production by y =

∑j yj, and the aggregate endowment by ω =

∑ωi.

Definition 3. A competitive equilibrium is a feasible allocation (x∗,y∗) and a price

vector p∗ ∈ RL+ such that

1. For every j = 1, ..., J , p∗ · y∗j ≥ p∗ · yj for every yj ∈ Yj;

2. For every i = 1, ..., I, x∗i %i xi for every xi ∈ B(p∗, wi(p∗)).

35See for example Quah (2007) and Mirman and Ruble (2008).

16

Note that feasibility in the definition of a competitive equilibrium includes market

clearing.

For each consumer i and p ∈ RL+, let X lcp

i (p) ⊆ Xi be the set of consumption plans

that satisfy the local cheaper point condition at p.

We now show that any competitive equilibrium allocation maximizes the sum of

money metric utilities on the set of feasible allocations—despite the non-concavity, and

even though the money metric need not represent preferences. Whatever normative ob-

jections one might make against the use of money-metrics in applied welfare economics,

if one makes the case for a Pareto optimal allocation that is supportable as a competitive

equilibrium, that allocation will maximize the sum of money metrics for some reference

price vector. Of course, what the sum cannot do is tell the analyst what a normatively

appealing price vector is, and Pareto Optimality is at best a necessary condition for dis-

tributive justice. The argument that we now present puts money-metrics on the same

level as Pareto Optimality as a criterion—at least for complete markets since the money

metric is defined for any reference price vector.36

In what follows we assume that competitive equilibrium prices are nonnegative. That

would follow, for example, if, for every good `, some consumer’s preferences were strictly

monotone in that good, no matter what the consumption levels of the remaining good,

or under free disposal. Here we simply suppose that equilibrium prices are nonnegative

without providing sufficient conditions for that supposition.

Theorem 2. Let Lp(x,y, µ) =∑

iMi(p, xi) + µ · [ω + y − x]. Consider the following

saddlepoint inequalities at (x∗,y∗, p∗) ∈ A× RL+:

Lp∗(x,y, p∗) ≤ Lp∗(x∗,y∗, p∗) ≤ Lp∗(x∗,y∗, µ) for every (x,y, µ) ∈ A× RL+. (7)

(a) If (x∗,y∗, p∗) is a competitive equilibrium, then the saddlepoint inequalities (7) hold.

It follows that (x∗,y∗) maximizes∑

iMi(xi, p∗) on the set of feasible allocations.

(b) If the saddlepoint inequalities (7) hold; and for every consumer i, x∗i ∈ Xlcpi (p∗);

then (x∗,y∗, p∗) is a competitive equilibrium for some distribution of endowments

and ownership shares.

Proof : Let (x∗,y∗, p∗) be a competitive equilibrium. Since for any µ ∈ RL+, µ · [ω +

y∗ − x∗] ≥ 0 and p∗ · [ω + y∗ − x∗] = 0, certainly Lp∗(x∗,y∗, p∗) ≤ Lp∗(x∗,y∗, µ) for

every µ ∈ RL+: Profit-maximization for each firm j at p = p∗ implies that p∗ · y∗ ≥ p∗ · y

36Our results go through under two kinds of market incompleteness. The first is that there simply aren’tmarkets for some goods, so consumers must consume their endowments of those goods. The second is thatevery consumer must consume the same physical good across a common set of states, locations, or dates. Thewelfare theorems must then be interpreted as taking these constraints on trade as given. On the inclusionof public goods and non-priced commodities as parameters of the money-metric, see Fleurbaey and Maniquet(2011) and Fleurbaey and Blanchet (2013).

17

for every y ∈ Y ; and for each consumer i, Mi(p∗, xi) − p∗ · xi ≤ 0 for every xi, with

equality at x∗i by Theorem 1. So Lp∗(x,y, p∗) ≤ Lp∗(x∗,y∗, p∗) = p∗ · (ω + y∗). Since

the saddlepoint inequalities (7) hold, it follows from Theorem 1 in Uzawa (1958) that

the competitive allocation maximizes the sum of money metrics on the set of feasible

allocations.

Now suppose that the saddlepoint inequalites (7) hold. That Lp∗(x∗,y∗, p∗) ≤Lp∗(x∗,y∗, µ) for every µ ∈ R+ implies that ω + y∗ − x∗ = 0, so the allocation (x∗,y∗)

is feasible. That Lp∗(x,y, p∗) ≤ Lp∗(x∗,y∗, p∗) for every (x,y) ∈ A immediately implies

that p∗ · y∗j ≥ p∗ · yj for every yj ∈ Yj for every firm j. If p∗(ω + y∗) = 0, then p∗x∗i = 0

for every consumer i and M(x∗i , p∗) = p∗ ·x∗i = 0 and the local cheaper point assumption

fails for each consumer i. Suppose p∗(ω + y∗) > 0 and consider consumer i. Let νi be

given by p∗ ·x∗i = νi(p∗ ·ω+p∗ ·y∗) and set w∗i = νi(p

∗ ·ω+p∗ ·y∗). (Since p∗(ω+y∗) > 0,

νi is well-defined.) Set consumer i’s endowment of goods equal νiω and i’s ownership

share of each firm equal to νi, so i’s wealth is w∗i . That x∗i ∈ di(p∗, w∗i ) now follows from

Theorem 1 (b).

Figure 2 helps illustrate the differences between parts (a) and (b). Suppose it

describes a single-consumer economy with endowment equal to x0 and that the line

x1 +x2 = const describes the set of final consumption bundles obtainable with the econ-

omy’s constant-returns technology. If x2 x1 x0 then unique competitive equilibrium

is p = (1/2, 1/2) and x = x2. The plan x2 maximizes the consumer’s money metric. The

point x2 is not covered by part (b) since it violates the local cheaper point assumption.

If however x1 x2 x0, then the equilibrium allocation is x1, which satisfies the local

cheaper point assumption and solves the saddlepoint inequalities at p = (1/2, 1/2). It is

also a competitive equilibrium, illustrating part (b).

We conclude this section with a series of remarks.

Remark 1 (CE allocations as unconstrained optima). The last sentence of Theorem

2(a) gives a version of the the title of this section. That a competitive equilibrium

allocation maximizes the sum of money metrics on the set of feasible allocations. The

saddlepoint inequalities (7) imply something more interesting: a competitive allocation

maximizes∑

iMi(xi, p)−p ·x+p ·y on the set A of allocations, feasible or not. Of course

choosing the production plan component of the allocation to maximize this objective just

amounts to profit maximization; the novelty is that the consumption allocation must

maximize∑

iMi(xi, p)− p · x if p is a competitive equilibrium price. Put another way:

if there is no maximizer of∑

iMi(xi, p)− p · x on the space of consumption allocations,

then p cannot be a competitive equilibrium price vector.

Related to this Remark, the next result is an immediate implication of Corollary 1

18

that we write out formally for completeness. The point is that we can solve for the

allocations demanded at price ignoring the budget constraint.

Corollary 2. Let Xi = RL+ for every consumer i, Then x∗ ∈ (d1(p, w1), ..., dI(p, wI)) for

some (w1, ..., wI) ≥ 0 if and only if x∗ maximizes∑

iMi(xi, p)− p · x on RLI+ .

Remark 2 (First Welfare Theorem). If for each consumer i, M(·, p∗) represents %i on

Xi, then the sum of money metrics is itself a welfare function. Theorem 2 (a) asserts that

any competitive allocation maximizes this sum at the corresponding competitive price.

So we obtain as a corollary to Theorem 2(a) the first welfare theorem. Saddlepoints are

a commonplace in discussing the first welfare theorem when consumers have concave

utility representations.37

Remark 3 (A Representative Consumer?). A natural question arises as to the sense

which the sum of money-metrics gives us a “representative consumer.” Remark 1 sug-

gests a limitation of this interpretation. A positive representative consumer refers to

a hypothetical consumer who owns all the economy’s wealth and has a preference re-

lation that generates the economy’s aggregate demand at every price-aggregate wealth

pair; maximizing the sum of money-metrics at a price-aggregate wealth pair generates

a set consisting of every vector of demands generated by any distribution of that ag-

gregate wealth (Corollary 2). The italicized words indicate the differences. An example

might help further illustrate the differences. Consider an exchange economy with two

consumers and two goods. Consumer i only cares for good i, the more the better. Con-

sumer i’s endowment is a fraction αi of the aggregate endowment ωi = αiω. (That is,

endowments are collinear.) This economy has an aggregate demand generated by Cobb-

Douglas preferences: U(x1, x2) = xα11 x

α22 , so clearly there is a positive representative

consumer for this economy. The representative consumer has homothetic preferences.

The economy, however, is clearly is not Gorman: the consumers do not have wealth

expansion paths that are straight lines with common slope. The sum of money-metric

utilities is M = p1x11 +p2x22 where xii is consumer i’s consumption of good i. Maximiz-

ing this sum subject to p1x1 + p2x2 ≤ w = p1ω1 + p2ω2 gives all aggregate consumption

plans that meet the budget constraint (with consumer i getting all of good i). The

“money-metric aggregate consumer” is indifferent between Pareto Optimal allocations

at a given aggregate wealth level (that “consumer’s” maximum value is simply aggregate

wealth at any such allocation).38

37Negishi (1960) is the classic reference for the differentiable case and Takayama and El-Hodiri (1968) for thegeneral case.

38To make the aggregate money-metric consumer look more like a representative consumer, one is temptedfirst to maximize the sum of money metric utilities with respect to an allocation x′ subject to the constraint thatthe aggregate consumption plan is no larger than some fixed vector (x1, ..., xL), and interpret the resulting valuefunction M(p;x1, ..., xL) as a representation of preferences of a representative consumer. Then maximize this

19

Remark 4 (Nontransitive consumers). Since Theorem 2(a) only uses the money metric

properties from Theorem 1(a), and the proof of Theorem 1(a) does not use transitivity;39

so the conclusion of Theorem 2(a) holds even if consumer preferences are not transitive.40

Remark 5 (Boadway Paradox). Consider two endowment distributions ωt = (ωt1, ..., ωtI)

for t = 0, 1 each associated with a competitive equilibrium (xt,yt, pt) with different

normalized prices. By Theorem 2, we have (a)∑

iMi(x0i , p

0) ≥∑

iMi(x1i , p

0) and (b)∑iMi(x

1i , p

1) ≥∑

iMi(x0i , p

1) with the possibility that both inequalities are strict. Recall

that the equivalent variation for a change from one price-wealth pair to another is the

wealth change at initial price that is utility-equivalent to the consumption plan at the

new price-wealth pair. Inequality (a) asserts that the sum of equivalent variations is

nonpositive from equilibrium 0 to 1; (b) asserts it is nonpositive from 1 to 0. That

both can be negative has come to be known as the Boadway paradox, found in Boadway

(1974).

Remark 6 (Sum of money metrics and price changes). Recall that one objection to using

the sum of money-metrics for welfare is that small changes in reference price vectors

might result in large changes in optimal allocations: since individual money metrics

are not in general concave, the sum is not generally even quasiconcave. In a smooth

economy with locally unique equilibria – for example the exchange economy in Debreu

(1970) with C1 demand functions satisfying a boundary condition41—we have this: as

endowments vary, equilibrium allocations vary continuously. Of course as endowments

vary, equilibrium prices vary. When prices vary in this endogenous way, allocations vary

continuously with them if utility is differentiable.

5 Money-metric Relations: Benefit and Distance functions

We now consider the relationship between money metric utility and two related func-

tions: the benefit function of Luenberger (1992a) and the distance function analyzed by

value function over affordable aggregate consumption plans. In the example, this maximized sum-of-money-metrics reduces just to M(p;x1, x2) = p1x1 + p2x2. Maximizing this subject to the aggregate budget setp1x1 + p2x2 ≤ w gives all affordable aggregate plans. With this procedure inefficient allocations are precludedby the “min” inequalities in the SP: in order for λ = 1 to minimize L(x, λ), it must be that x21 = x12 = 0.

39See footnote 33 to Theorem 1(a).40The sum of money metrics might a tool for welfare economics for non-standard theories of decision making,

as in Bernheim and Rangel (2009). See footnote 49 for an example.41In that setting, for all but a closed, Lebesgue-measure zero set of endowments, there are a finite number of

equilibria. Pick one of these endowments ω with a finite number of equilibria and one of these equilibria. Asendowments vary in a neighborhood of ω, equilibrium prices vary smoothly; see the Remark in Debreu (1970,p. 390).

20

Gorman (1970).42

We point out that versions of Theorems 1 and 2 hold for these functions. We refer

the reader to Luenberger (1992a) and Gorman (1970) for proofs of the properties of the

benefit and distance functions that we assert here.43

For xi ∈ Xi, ui ∈ Range(ui), g ∈ ∩Ii=1Xi (assuming the intersection is nonempty),

define bi(xi, ui), consumer i’s benefit function at (xi, u), to be the number β which solves

u(xi−βg) = ui if a solution exists, −∞ otherwise; define fi(xi, ui), consumer i’s distance

function at (xi, ui) to be the number γ which solves ui(xi/γ) = ui if a solution exists,

−∞ otherwise.44

To convert these numbers into money units, define Bi(xi, ui, p) = bi(xi, ui, p)p · g and

Fi(xi, ui, p) = fi(xi, ui)ei(p, ui).45 Parallel to Theorem 1, one can show that if x∗i ∈

di(p, w), then (x∗i , 1) solves the saddlepoint inequalities with either Bi(xi, Vi(p, w), p)

or Fi(xi, Vi(p, w), p) replacing the consumer’s money-metric utility in the Lagrangean.

As a corollary it follows that x∗i maximizes both Bi(xi, Vi(p, w), p) − p · xi and

Fi(xi, Vi(p, w), p) − p · xi on Xi—ignoring the budget constraint. An important lemma

in the proof of this fact spells out the relationship between the expenditure functions

and the benefit and distance functions:

ei(p, ui) = minxi∈Xi

p · xi/fi(xi, ui) = minxi∈Xi

p · xi − bi(xi, ui), (8)

where, for the first equality, prices are normalized so that p · g = 1.

Since we know of no proof of the fact, we sketch here the proof that if x∗i ∈ di(p, wi)then x∗i maximizes Fi(xi, Vi(p, w), p)−p·xi on Xi. Clearly p·x∗i = w and f(x∗i , V (p, wi)) =

1, so the objective equals 0 at x∗i . Substitute Vi(p, wi) into the utility argument on

both sides of the first equality in (8) to get w = minxi∈Xi(p · xi)/f(xi, Vi(p, wi)) ≤

p · x′/f(x′, Vi(p, wi)). For any affordable x′i, p · x′i ≤ w, so the preceding inequality

implies that f(x′i, Vi(p, wi))wi ≤ p · x′i, so the objective is at most 0 for any affordable

x′i ∈ Xi.

Define B(x, u) =∑

iBi(xi, ui) and F (x, u) =∑

i Fi(xi, ui), where u = (u1, ..., uI)

is a utility profile. Parallel to Theorem 2, if (x∗,y∗, p∗) is a competitive equilibrium,

then (x∗,y∗) maximizes B(x,u∗) + p∗ · y − p∗ · x and F (x,u∗) + p∗ · y − p∗ · x on A,

where u∗i = Vi(p∗, wi(p

∗)) for i = 1, ..., I. Luenberger (1992b) proves this last fact for

42The distance function was introduced in producer theory by Shepard (1953) and in consumer theory byMalmquist (1953); Deaton (1979) reviews some of the history of this function.

43Gorman (1970) is reprinted in Blackorby and Shorrocks (1995); proofs of the properties we use here can alsobe found in Deaton (1979). Also see Luenberger (1995). Finally, we refer the reader to Fuchs-Seliger (1990a)and Fuchs-Seliger (1990b) for a recent reconsideration of these measures in the context of the theory of revealedpreference.

44Gorman (1970) works in a setting for which the distance function is real-valued.45Luenberger (1992a) normalizes prices so that p · g = 1, so the extra multiplicative term drops out.

21

the aggregate benefit function.46 We can now present

Proposition 1. The conclusion of Theorem 2(a) continue to hold if each consumer i′s

money-metric Mi is replaced by Fi; or each consumer i money-metric is replaced by Bi.

For special cases Fi(xi, Vi(p, w), p) and Bi(xi, Vi(p, w), p) equal money-metric utility

Mi(xi, p) (sometimes up to an additive constant). For Xi = RL+, Fi(xi, Vi(p, w), p) =

Mi(xi, p) when i’s preferences are homothetic (the consumer’s wealth drops out of Fiwhen preferences are homothetic); and Bi(xi, Vi(p, w), p) = Mi(xi, p)+w when i’s prefer-

ences are quasi-linear with respect to some good ` and the reference bundle g equals the

`-th unit vector. Unlike the money-metric function, the benefit and distance functions

can be concave when preferences are not quasihomothetic: indeed, whenever a utility

representation is quasiconcave, then the associated benefit and distance functions must

be concave. These last facts imply that, like the money-metric function, the benefit

and distance functions need not represent preferences: as already mentioned, there are

convex preference relations not representable by a concave function (Kannai (1977)).47

The benefit and distance functions are concave when preferences are convex, and so they

avoid the inegalitarian implications of Example 2.4. They each also general fail property

(4) that the money metric satisfies. It is easy to show that the aggregate benefit and

distance functions are not generally Pareto consistent: a change can make either mea-

sure increase, but every person is worse off with the change. Such examples are easy to

construct even for quasihomothetic preferences.

6 Money-metrics and Cost-Benefit Analysis in the Small

We now consider how the sum of money-metrics behaves for small changes in an aggre-

gate production plan. In particular we show that the derivative of the sum of money

metrics equals a local welfare measure proposed by Radner (1993 [1978]). Since Radner

showed that a positive sign of this measure implies that a small change in the aggre-

gate production is a potential Pareto improvement, this result gives some support to

those using first-order approximations to money metric utility (for example, Deaton

(2003)). According, in the next section we add the assumption that, for each consumer

i, Mi(·, p) is differentiable in x at a particular point x0i in the interior of Xi satisfying

x0i ∈ di(p, p · x0

i ). In particular, if X = RL+, then we require that x0

i >> 0. In subsection

3, we give sufficient conditions on % for such differentiability to hold. This first-order

result46It is hard to believe that the aggregate distance function F (x, u∗) has not be used by someone, and that

the Proposition has not been proved for it , but we do not know a reference for it.47Compare with the first paragraph in Chambers, Chung, and Fare (1996).

22

6.1 Radner’s Local Welfare Measure

We now remind the reader of Radner’s (1993) local welfare measure. Let y : [0, 1]→ Y ,

where Y ⊆ RL is a connected, nonempty production set. Radner (1993) refers to the

scalar α as indexing projects. Suppose that y is differentiable at α = 0. The total

supply for project α is ω + y(α). A feasible consumption allocation of the total supply

ω+ y(α) is a point (x1, ..., xI) ∈ Xi such that∑

i xi = ω+ y(α). A feasible consumption

allocation x = (x1, ..., xI) is a valuation equilibrium relative to a price p ∈ RL++ if, for each

i = 1, ..., I, xi ∈ di(p, p · xi). We stress that the model is silent about how the aggregate

production plan is chosen; in particular it need not be part of a competitive equilibrium.

The valuation equilibrium assumption assures that the consumption allocation of a given

supply is Pareto optimal: the source of any inefficiency in the economy is the aggregate

production plan.

Radner (1993 [1978]) proposed this local measure of welfare,

p0 · y′(0), (9)

where the feasible allocation x(0) = (x1(0), ..., xI(0)) of the initial supply ω + y(0) is a

valuation equilibrium with respect to p0. In words, his measure is the change in the value

of the aggregate production plan measured by the initial valuation equilibrium prices.

He goes on to prove that, if (9) is positive, then then for some α > 0, the supply ω+y(α)

is a potential Pareto improvement over the supply ω+y(0) for every α ∈ (0, α): the total

supply of ω + y(α) can be allocated so that each consumer i’s prefers its consumption

in this allocation to xi(0).48 We next show that, under some conditions, the derivative

of the sum of money-metric utilities equals Radner’s measure.

6.2 On Local Equivalences and Improvements

In what follows let x(α) be a feasible consumption allocation of the supply ω + y(α).

Define M(α) =∑

iMi(p0, xi(α)), the sum of money-metric utilities. The next result

asserts that M′(0) equals Radner’s local welfare measure. Here int(Xi) refers to the

interior of Xi.

Proposition 2 (The sum of money metrics and Radner’s measure). Suppose that a)

x(·) is differentiable at α = 0; b) for i = 1, ..., I, x(0) ∈ int(ΠiXi); c) each Mi(p0, ·) is

differentiable in xi at xi = xi(0); and d) the allocation x(0) is a valuation equilibrium

relative to some p0 ∈ RL++. Then

M′(0) = p0 · y′(0)48If the measure is negative, the supply at all small-enough α is not a potential Pareto improvement over

allocation at α = 0.

23

We use this easy fact in the proof.

Lemma 2. Fix x0 ∈ int(Xi) and p0 ∈ mathbbRL++ with x0 ∈ di(p, p · x0). If Mi(p

0, ·) is

differentiable at x = x0, then

DxMi(x0i , p

0) = p0. (10)

Proof of Lemma 2: Since x0 ∈ di(p0, p0 · x0), x0 maximizes M(p0, x) − p0 · x on Xi

by Theorem 1(a). And since Mi(p0, ·) is differentiable at x0

i , the necessary Kuhn-Tucker

conditions must hold at xi = x0i . Since x0 ∈ int(Xi), the non-negativity constraints do

not bind, and (10) holds. .

Proof of Proposition 2: By Lemma 2, and a)-d), each Mi(p0, xi(α)) is differentiable

at α = 0 and the derivative equals p0 · x′i(0). Sum over all consumers and use feasibility

to find M′(0) =∑

i p0x′i(0) = p0 · y′(0).

The Proposition gives another sense in which the sum of money-metric utilities is not

a pathological welfare function. To make this sense more precise, we next show that, if

M′(0) > 0, then, for small-enough α > 0, the new supply can be allocated so that each

consumer i strictly prefers the new allocation to xi(0).49

Corollary 3 (Local Potential Pareto Improvements). Suppose that a) - d) of Proposition

2 holds andM′(0) > 0. Then for every α in some nonempty interval (0, α), the allocation

at α is a strict potential Pareto improvement over the allocation at α = 0.

Proof : Let xi(α) = xi(0) + 1I(y(α) − y(0)). For small enough α, xi(α) ∈ int(Xi).

For i = 1, , , I, let Mi(α) = Mi(p0, xi(α)) for all small-enough α. We have M′

i(0) =

p0y′(0)/I > 0, which implies Mi(α) > Mi(0) for all small-enough α. The conclusion

follows from (4). Radner (1993) proved the same conclusion holds, but under the stronger condition

that the supporting price is differentiable in the policy parameter α (Radner, 1993,

p. 136).

Under Radner’s additional condition that the equilibrium price is differentiable,

Schlee (2013a) shows that Radner’s local measure equals four other local welfare mea-

49We note that the conclusion holds even if we replace transitivity of preferences by strict convexity andpreferences appropriately smooth at the bundle demanded at p0. One notion of appropriate smoothness is this.Recall that Shafer (1974) proved the existence of a numerical representation ki : RL

+ × RL+ with the properties

that ki(x, y) = −ki(y, x) and x %i y if and only if ki(x, y) ≥ 0. If this function is differentiable in its first Larguments at xi = xi(0) when the second L arguments equal xi(0), then the conclusion of Proposition 2 goesthrough in this case, despite the failure of equation (4) in the absence of transitivity. Fountain (1981) pointsout the difficulties of welfare changes “in the large” for nontransitive consumers, but does not touch on localchanges.

24

sures, including (the derivative of) aggregate consumer’s surplus.50,51 Proposition 2 and

Corollary 3 extend that equivalence to the sum of money-metric utilities.

Corollary 4. Suppose that a) - d) of the Proposition hold. Suppose in addition that,

for a differentiable function p(·) : [0, 1] → RL+ that, for every α ∈ [0, 1] the allocation

(x1()α), ..., xI(α)) of the supply ω+ y(α) is a valuation equilibrium with respect to p(α).

ThenM′(0) equals the derivative of each of the five measures in the Proposition in Schlee

(2013a).

One can also show that the local equivalence extends to the aggregate benefit and

distance functions of Section 5. This follows since the conclusion of Lemma 2 extends

to these two functions: ∇xiBi(xi, Vi(p, w), p) = p = ∇xiFi(xi, Vi(p, w), p) whenever xi is

demanded at (p, w).

The results of this Section imposed differentiability of the money metric in the con-

sumption plan x. From the definition of the money metric, it is clear that differentiability

is an ordinal property, a property of preferences. From Lemma 2 it is also clear that the

inverse demand must be single-valued at any point of differentiability. We give sufficient

conditions on preferences for the money metric to be differentiable in x at a demanded

point in Section 7.2

7 A Technical Mopping-up

7.1 A Proof of Lemma 1

Let x′ ∈ h(p, x) and x′′ ∈ d(p,M(x, p)). Since p · x′ = M(p, x), x′′ % x′. By x′ % x and

transitivity, x′′ % x and x′′ ∈ h(p, x). Since x satisfies the cheaper point assumption,

there is a sequence xn in X with limit x′′ and p · xn < p · x′′ = M(p, x), so x xn for

every n. By continuity x % limxn, so x′′ % x′ % x % x′′ and both x′′ ∈ h(p, x) and

x′′ ∼ x′.50The three other measures are the Slutsky change in real income, the nominal Divisia price index, and

Debreu’s (1951) coefficient of resource utilization, normalized to be measured in terms of a numeraire good. AsSchlee (2013a) points out, the derivative of the five measures also equal the derivatives of the compensating andequivalent variations, bringing the total of locally equivalent measures to seven. Schlee (2018) points out thatthe local equivalence between Radner’s measure and the coefficient of resource utilization is false. The correctequivalence is between Radner’s measure and a measure related to the coefficient that takes as a reference pointa production plan that need not be a part of a Pareto optimal allocation, rather than the entire aggregateproduction set.

51In Schlee (2013a), the differentiability of the equilibrium price follows from the assumptions that the Paretooptimal allocation of the supply for each α is differentiable; and from monotonicity, strict quasiconcavity, anddifferentiability assumptions on utility consistent with smooth preferences in the sense analyzed by Debreu(1972).

25

7.2 On the Differentiability of a Money-metric

We give a set of sufficient conditions for Mi(p′, ·) to be differentiable in x at any x′ ∈

d(p′, p′ · x′).Let gi(x) = p ∈ R+ +L |

∑p` = 1 and x ∈ di(p, p · x), the normalized inverse

demand correspondence for consumer i.

Proposition 3 (Differentiability of the money metric). Suppose that X is convex. For

x0 ∈ di(p0, p0 · x0) with x0 ∈ int(Xi), Mi(p0, ·) is differentiable at x = x0 if

(a) %i is strictly convex;

(b) gi(x) is single-valued for every x in some open neighborhood Nb ⊆ X of x0;

(c) gi(x) is Lipschitz continuous at the point x0: namely, for some open neighborhood

Nc ⊆ X of x0, there is a real number K > 0 such that, for every x ∈ Nc

‖g(x0)− gi(x)(x)‖ ≤ K‖x0 − x‖

Before turning to a proof, we note that if gi(x) is single-valued and differentiable at

x0, then the conclusion follows. Clearly, strict convexity of preferences is not necessary.

If

u(x) =∑`

x`,

then M(p, x) = minp1, ..., pL∑

` x`, which is differentiable in x at every point in

the consumption set. By Lemma 2, uniqueness of the inverse demand obviously is

necessary.52 It remains to be seen how far Lipschitz continuity of the inverse demand

at a demand point can be relaxed to get differentiability of the money metric at a

demand point. In a follow-up paper on comparative statics with the money metric, we

confirm that if % has an increasing C1 utility representation with no critical point on the

consumption set X = RL++, then the money metric is monotone and C1 with no critical

point. In this case g is clearly continuous but not necessarily Lipschitz-continuous-at-a-

point.

We will use the following fact in the proof of Proposition 3.

Lemma 3. If %i is complete, transitive, continuous, locally nonsatiated, and strictly

convex, then the solution to the income minimization problem, hi(p, x), is continuous at

every (p, x) ∈ RL++ ×X.

52The case of homothetic preferences in which the homogenous-of-degree-1 representation is not differentiablemakes this point clear : the money metric takes the form u(x)b(p) where u is homogenous of degree 1. If sucha u is not differentiable at a demand point, then neither is the money-metric (e.g. Leontief preferences).

26

Proof of the Lemma: Under the assumptions on %i, there is a continuous represen-

tation ui of %i. Since %i is strictly convex, the solution hui(p, u) to the expenditure

minimization problem, minui(x)≥u p ·x, is single-valued. By standard arguments, hui(·, ·)is continuous (for example, Kreps, 2012, Proposition 10.3). Since the solution to the

income minimization problem, hi, satisfies hi(p, x) = hui(p, ui(x)) and ui is continuous,

the conclusion follows. .

Proof of Proposition 3: We omit the consumer i sub/superscript throughout. We

want to show that

lim‖x−x0‖→0

|M(p0, x0)−M(p0, x)− p0 · (x0 − x)|‖x− x0‖

= 0. (11)

From the Saddlepoint inequalities we have

M(p0, x0)−M(p0, x) ≥ p0 · (x0 − x), (12)

for every x ∈ X, so for any x ∈ X with x 6= x0

M(p0, x0)−M(p0, x)− p0 · (x0 − x) ≥ 0. (13)

Let N = Nb ∩ Nc. For x ∈ N , we have M(g(x), x) = g(x) · x and—recalling that g(x)

is a singleton on N—M(g(x), x0) < g(x) · x0 whenever x 6= x0. From these inequalities

find that

M(g(x), x0)−M(g(x), x) < g(x) · (x0 − x) (14)

which, adding and subtracting the same terms, is the same as

M(p0, x0)−M(p0, x)− p0 · (x0 − x) < (g(x)− p0) · (x0 − x)− Γ(x) (15)

where Γ(x) = [M(g(x), x0) −M(p0, x0)] + [M(p0, x) −M(g(x), x)]. Since % is strictly

convex, M(·, x) is differentiable in p, and

DpM(p0, x) = h(p0, x)

where h(p0, x) is unique solution to the income-minimization problem at (p0, x) (Kreps,

2012, Proposition 10.3). By the mean value theorem, for each x there is a number

α(x, y) ∈ [0, 1] such that, for y = x or y = x0)

M(p0, y)−M(g(x), y) = h(α(x, y)g(x) + (1− α(x, y))p0, y

)· (g(x)− p0) (16)

27

Let q(x, y) = α(x, y)g(x) + (1− α(x, y))p0 for y = x, x0. In this notation

[M(g(x), x0)−M(p0, x0)]+[M(p0, x)−M(g(x), x)] = [h(q(x, x0), x0)−h(q(x, x), x)]·(g(x)−p0)

(17)

Insert (17) into (15) to find

M(p0, x0)−M(p0, x)−p0 · (x0−x) < (g(x)−p0) · (x0−x+h(q(x, x0), x0)−h(q(x, x), x))

(18)

Combine (13) and (18) to find

|M(p0, x0)−M(p0, x)− p0 · (x0 − x)| ≤ Ω(x) (19)

where Ω(x) = (g(x) − p0) · (x0 − x + h(q(x, x0), x0) − h(q(x, x), x)) Since g(x0) = p0, g

is Lipschitz-continuous at the point x = x0, and by the Lemma h is continuous,

lim‖x−x0‖→0

Ω(x)

‖x− x0‖= 0, (20)

so (11) follows.

7.3 A Representation Theorem

We slightly extend the representation theorem in Khan and Schlee (2016). This theorem

differs from those in Weymark (1985) in that we do not impose convexity of the con-

sumption set. Let XD(p) ⊆ X be the set of consumption plans satisfying x ∈ d(p, p · x),

that is, that are demanded at (p, p · x).

Theorem 3. Let X ∈ RL+ be nonempty and closed. If %∈ X×X is complete, transitive,

continuous, and locally nonsatiated, then M(·, p) represents % on Xc(p) ∪XD(p).

Lemma 4. If x ∈ Xd(p), then the conclusion of Lemma 1 holds for (x, p); that is, (a)

d(p,M(p, x)) = h(p, x), and (b) if y ∈ d(p,M(x, p)) then y ∼ x.

Proof of Lemma: Suppose that y ∈ h(p, x), implying that p · y = M(p, x) and y % x;

it follows that y ∈ d(p,M(x, p)). Suppose that y ∈ d(p,M(x, p)), so y % x. By LNS,

p · y = M(x, p), so y ∈ h(p, x). Since x ∈ d(p,M(x, p)), it follows that y ∼ x.

We can now complete the proof of Theorem 3.

Proof of Theorem 3. Suppose that x % y. By transitivity, z ∈ X | z % y ⊆ z ∈X | z % y, so M(x, p) ≥M(y, p).

Now suppose that y x. By Lemmata 1 and 4, z′ ∈ d(p,M(z, p)) implies that z′ ∼ z

for z = x, y. So y′ ∼ y x ∼ x′. By transitivity, y′ x′. By LNS, M(y, p) > M(x, p).

28

8 Concluding Remarks on Future Work

We conclude this note with two remarks on future directions that are opened by the

rehabilitation of the money-metric pursued in this note: the first concerns the notion of

complementarity; and the second, comparative-static analysis, especially as it pertains

to the recent recasting of the theory of the consumer. Both draw on the version of the

Kuhn-Tucker-Uzawa theorem on non-linear programming presented here.

It is well-known that Samuelson (1974) used the money-metric to offer the fifth of

his six definitions of complementary commodities, and thereby connect to basic and

classical themes in consumer theory. It is this program that we carry forward in future

work. Thus, our next step is to proceed with this rehabilitation and reconsider the 5th

and 6th definitions of complementarity in Samuelson (1974), and relate it to other pro-

posed definitions in the literature. Of particular interest are the definitions suggested or

explored by Chipman (1977, 1980) and Kannai (1980), and more recently by Chambers,

Echenique, and Shmaya (2010).

Of equal, if not greater interest, is the use of the money-metric in the reconsidera-

tion of recent results in monotone comparative statics as obtained by Quah (2007) and

Mirman and Ruble (2008). A important tool to carry this out is Corollary 1, which

formulates the consumer’s problem as one subject only to non-negativity constraints

bypassing the budget constraint altogether. This point is important since one stumbling

block in applying lattice methods to the consumer’s problem is that different budget

sets do not stand in the strong-set order when endowed with the usual “greater than or

equal to” partial order ≥.

29

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