DP2017-31

Behavioral Heterogeneity:
Pareto Distributions of Homothetic
Preference Scales and Aggregate
Expenditures Income Elasticities
Jean-Michel GRAND MONT

November 15, 2017

Behavioral Heterogeneity : Pareto
Distributions of Homothetic Preference
Scales and Aggregate Expenditures
Income Elasticities ∗J.M. Grandmont (1)September, 2017

* I have benefited from conversations with Frédérique Bec, Xavier d’Haultfoeuille,
Francis Kramarz and Laurent Linnemer. I am very grateful to Fanda TRAORE
for her efficient typing. As usual, I am sole responsible for any remaining
shortcomings of the paper.

1

ICEF, Department of Economics, University Ca’ Foscari di Venezia at San Giobbe,
Italy, CREST (EXCESS, UMR CNRS 9194), University of Paris-Saclay, France, and RIEB
Fellow, University of Kobe, Japan.

1Behavioral Heterogeneity : Pareto Distributions of Homothetic
Preference Scales and Aggregate Expenditures Income Elasticities
Abstract
We evaluate the income elasticity of the aggregate budget share spent on
a sub-group of commodities, in a competitive framework, by a continuum of
agents having the same income, but heterogeneous behavior described by an"homothetic preferences scaling factor" having a bounded Pareto distribution
in the population. If individual budget share increases globally significantlyin the limit from low to large incomes, aggregate budget share is locally increasing with medium range incomes when the logarithm of the heterogeneity
factor has an increasing (exponential) density with a large support. Aggregate income elasticity converges to that exponential density parameter when
its support becomes infinitely large. Symmetric results hold in the decreasing case. Applications are made to market expenditures, wealth effects on
portfolio choice with many risky assets, concave expenditures, that are compatible with standard (expected) utility maximization or other "behavioral"
decision making processes.
Keywords : Behavioral heterogeneity, aggregation, preference scales,aggregate income elasticity, power law, Pareto distribution, exponential distribution, market demand, wealth effect on aggregate portfolio choices.

1

1

Introduction

This work seeks to make theoretical progress on the analysis of macroeconomic aggregates by relying significantly on a distributional approach, i.e.
on specified properties of distributions of heterogeneous individual preferences characteristics. Such an approach has been initiated in particular by
W. Hildenbrand (1983, 1994) and the author (1987, 1992). It aims at coping with the observed behavioral diversity of economic agents, which has
been well documented empirically for long in households demand analysis(see Barten (1964, 1977), Calvet and Comon (2003), Deaton and Muellbauer(1980), Jorgenson and Selsnik (1987), Lewbel (1997), Lewbel and Pendakur(2017), Matzkin (2007), McFadden (2001), Muellbauer (1977, 1980), Prais
and Houthakker (1971)), or in households heterogeneous portfolio choices(see Calvet et alii (2007, 2009, 2014), Chan and Kogan (2002), Curcuru etalii (2010)): for useful recent surveys, see Blundell and Stoker (2005), Lewbel (2007). A distinguishing feature of the approach adopted here is that wetake as a reguirement that the introduced heterogeneity should be compatible
with individual agents behaving "rationally", i.e. through (expected) utility
maximization, as well as with various kinds of "behavioral" biases. This isof course to be contrasted with a large part of (micro or macro) economic
theorizing, that is often based more or less implicitly (and misleadingly) on a
"representative agent" approach, in which all agents are supposed to behave
similarly, whether in markets for goods and services, or in their portfolio
choices in finance.
The distributional approach employed in this work, as in previous contributions of the author on this topic (1987, 1992, 1993), seeks to accountfor behavioral heterogeneity through "preference scaling factors" (the word
"preference" being there as a reminder of our requirement of individual behaviors being compatible with "rationality"). These are essentially (possible commodity specific) versions of the well known "households equivalence
scales", due to Prais and Houthakker (1971), Barten (1964, 1977), that havebeen subsequently much used in econometric studies (Blundell and Lewbel(1991), Deaton and Muellbauer (1980), Jorgenson and Selsnik (1987), Lewbel (1997), Lewbel and Pendakur (2008), Muellbauer (1977, 1980)). Such astructure was also used long ago in general equilibrium analysis to show thatsuitably dispersed distributions on the space of consumers’ characteristics
led to a nice "smoothing" of competitive aggregate demand (Mas-Colell and2

Neuefind (1977), E. Dierker, H. Dierker and Trockel (1984)). Specifically,
one considers a collection (continuum) of agents who have the same income
and the same individual (possibly (expected) utility maximizing) "base" behavior, but who behave actually differently, i.e. as if the prices they facewere multiplied by these (possibly commodity specific) heterogeneity scaling factors. Previous analysis showed that when the distributions of suchcommodity specific scaling factors (actually, of their logarithms), were "flatenough", aggregate expenditures became less sensitive to prices (Grandmont(1992), Quah (1997, 2001), Giraud and Maret (2002)). Such aggregate insensitivity property has been further shown to occur as well for non-parametricdefinitions of "behavioral heterogeneity" (see Kneip (1999), Jerison (1999),
Hildenbrand and Kniep (2005) and their references). These "behavioral heterogeneity" concepts were however mostly defined directly as properties ofthe distributions of individual demand functions themselves, without any
explicit guarantee that there were compatible with standard individual "rational" utility maximization, in the spirit of the early "irrational" approachof Becker (1962).
The aim of this paper is to show that one can actually get sharper aggregate properties by exploiting further the convenient parametric nature of
the distributions of such heterogeneous preference scaling factors. We focus
here on the simple case where such scaling factors are "homothetic", i.e. thesame for all commodities exchanged in the market, as in Grandmont (1987),
Quah (1997). Such a formulation is indeed particularly adapted to an application we wish to make, i.e. portfolio choices in financial markets, if only
to be compatible with standard individual decision making specifications,
including VNM expected utility maximization, ambiguity, loss aversion, or
whatever. We shall show that focusing, for the distributions of such homothetic scaling factors, on power laws, which do have important applicationsin economics and finance (Gabaix (2009), generates fruitful implications also
here. We consider accordingly a continuum of agents who have the same income and the same "base" behavior, but whose actual behavior is indexed bysuch a multiplicative heterogeneity scaling factor, that is distributed among
the agents following a bounded Pareto distribution. We focus on the income
elasticity of the aggregate budget share spent by these agents on a sub-groupof commodities. We derive then sharp results on the aggregate income elasticity from rather weak limiting properties of individual "base" behavior forquite low and large income levels, and by relying significantly on the spec3

ification of a Pareto distribution for the scaling factor. While our analysis
is compatible with standard "rational" (expected) utility maximization overthe entire income range, as mentioned earlier, it is compatible as well withany other "behavioral", more or less "irrational" departures in the medium
income range, as we do not require any specific properties of the individual "base" behavior there. In the "increasing" case, one requires that theindividual "base" budget share spent on the subgroup of commodities, orassets, under consideration, is very small for low incomes, and quite large forhigh incomes. Then if the logarithm of the homothetic scaling factor has anincreasing (exponential) density with a large support, the aggregate budget
share is locally increasing for medium range incomes. Further, the aggregate
income elasticity converges to the (positive) parameter of this exponential
density when its support becomes infinitely large. Symmetric results arevalid in the "decreasing" case.
A detailed application is made to expenditures on a subgroup of commodities in a competitive framework. There, low budget shares allocated tothis subgroup for low incomes can be viewed as resulting from low marginal
rates of substitution for these commodities at low consumption levels. And
conversely, large budget shares spent on the subgroup for large incomes, areresulting from large marginal rates of substitution in favor of these commodities at large consumption levels. Another detailed application is madeto aggregate portfolio choice in financial markets involving several risky securities and a single riskfree asset. There, a low budget share invested in
the riskless asset for low incomes, at the individual level, is then the consequence of very large degrees of relative risk aversion (or of loss aversion) inthat income range. Symmetrically, a very low degree of risk aversion (i.e. risk
neutrality) for large incomes implies a vanishing investment, at the individuallevel, in the risk free asset. Our approach enables us then to get the aggregate budget share invested in the riskfree asset to be locally decreasing withmedium range wealth levels. Such a property has been known to be valid at
the individual level since the contribution of Arrow (1970), when individual
relative degrees of risk aversion are decreasing everywhere with income, provided however that there is a single risky asset to be invested in. By contrast,
when there are several risky securities as here, such a property has been wellknown to be impossible to obtain at the individual level, without imposing
extremely particular specifications, e.g. of the agent’s VNM expected utility(Cass and Stiglitz (1970, 1972), Hart (1974)). Our results show that going4

to the aggregate level does improve significantly the perspective. Detailed
applications to higher order derivatives, i.e. concavity or convexity of aggregate expenditures, are also presented, a topic that has attracted a significant
attenion in the literature (see Caroll and Kimball (1996)).
The paper is organized as follows. We present the basic framework and assumptions in section 2. We state the core formal results on aggregate income
elasticities in section 3 when the homothetic heterogeneity scaling factor has
a bounded Pareto distribution, i.e. when its logarithm has an exponential
density. The case where that density is uniform is considered in section 3.1,
and monotonically increasing or decreasing in section 3.2. Detailed applications to market demand are given in section 4.1, to aggregate portfolio
choice in section 4.2, to concavity of aggregate expenditures in section 4.3.
We conclude briefly in section 5. Proofs are gathered in an appendix.

2

Framework

We study aggregate expenditures for a particular group of commodities(consumption goods, services, physical or financial assets) of a collection(continuum) of agents in a given period, in a competitive set up. Theircommon "base" behavior is described by an individual expenditure functionw∗ (p, β), where p = (p1 , . . . , pn )  0 is a vector of positive market prices
for all commodities exchanged in the period under consideration and β > 0
is income. It satisfies 0 5 w∗ (p, β) 5 β and is homogenous of degree 1.
The corresponding individual budget share function s∗ (p, β) = w∗ (p, β)/βlies accordingly in [0,1] and is homogenous of degree 0. Both expenditure
functions are continuously differentiable in (p, β)  0 up to any requiredorder.
The agents’ actual behavior is governed by these expenditure functions
up to a "behavioral heterogeneity" scaling factor λ, an arbitrary positive real
number. An agent corresponding to the scaling factor λ has "tastes" for allcommodities that are homothetically divided by λ : she behaves accordingly"as if" the market price vector were λp instead of p. We shall work foranalytical convenience with the heterogeneity parameter α = log λ. If theagent’s income is b > 0, the resulting expenditures functions are given by5w(α, p, b) = w∗ (eα p, b) = eα w∗ (p, e−α b),s(α, p, b) = s∗ (eα p, b) = s∗ (p, e−α b).

(2.1)
(2.2)

There is a continuum of such individual agents, normalized to 1, thatis described by a probability distribution over the behavioral heterogeneity
parameter, generated by a density function g(α) = 0. It is concentrated
on an interval [α0 , α1 ] with α0 < α1 , i.e. g(α) = 0 for α 6∈ [α0 , α1 ], andcontinuously differentiable up to any required order on that interval. Under
the assumption that all these agents face the same current price system p  0
and have the same current income b > 0, aggregate expenditures and budget
shares for the group of commodities under consideration, are given by
Z α1
W (p, b) =w(α, p, b)g(α)dα,
Z α1
S(p, b) = W (p, b)/b =s(α, p, b)g(α)dα.

(2.3)

α0

(2.4)

α0

Our aim is to analyze how these aggregate expenditures vary with income, namely to evaluate ∂W (p, b)/∂b and ∂S(p, b)/∂b, in relation with theproperties of individual expenditure functions and of the behavioral heterogeneity scaling factor density distribution g(α). It is clear from (2.2) that∂s∂sb (α, p, b) = − (α, p, b), so that by differentiation of (2.4)
∂b
∂α
Z α1
∂S
∂s
b (p, b) = −(α, p, b)g(α)dα.
(2.5)
∂b
α0 ∂α
An elementary integration by parts yields accordingly
Lemma 1. Aggregate expenditures’ income derivatives satisfyb∂S(p, b) =
∂b

Z α1
α0

s(α, p, b)g 0 (α)dα − [g(α)s(α, p, b)]αα10
(2.6)

where

[g(α)s(α, p, b)]αα10

∗

−α1

∗

−α0

= g(α1 )s (p, eb) − g(α0 )s (p, e∂W∂S(p, b) = S(p, b) + b (p, b).
∂b
∂b

6

b),
(2.7)

The above suggests a potentially fruitful research strategy. Aggregate expenditures’ income derivatives, or elasticities, depend on two interacting features : a)the relative sizes of individual "base" budget shares for low and large incomes atthe endpoints of the distribution, namely s∗ (p, e−α1 b) compared to s∗ (p, e−α0 b),
together with b) the shape of the scaling parameter density, through its derivativeg 0 (α) and its values at the endpoints of the distribution, g(α0 ) compared to g(α1 ).
As a quick simple example, one is sure to get a locally increasing aggregate budget
share, i.e. b∂S(p, b)/∂b > 0, if the scaling parameter density is non-decreasing,g 0 (α) = 0 (in which case g(α1 ) = g(α0 )), whenever the individual budget shares∗ (p, β) is much higher for large incomes β than for low ones, so that,g(α0 )s∗ (p, e−α0 b) − g(α1 )s∗ (p, e−α1 b) > 0.

(2.8)

In such a configuration, a significant increase of individual base behavior "in thelarge" (for large incomes β compared to low ones) translates into a locally increasing behavior for intermediate incomes b through a smoothing effect of the
distribution of the behavioral heterogeneity scaling parameter. In the sequel, weshall rely on the two following alternative configurations where base individual
budget shares increase (assumption (H.0)), or decrease (assumption (H.1)), moreor less significantly when comparing large incomes β to low ones.
(H.0) (Increasing individual budget share "in the large") Given p  0, there exist
0 < β0∗ < β1∗ and 0 5 s∗0 < s∗1 5 1 such that s∗ (p, β) 5 s∗0 for 0 < β 5 β0∗ ands∗1 5 s∗ (p, β) for β1∗ 5 β.
(H.1) (Decreasing individual budget share "in the large") Given p  0, there exist
0 < β0∗ < β1∗ and 1 = s∗0 > s∗1 = 0 such that s∗ (p, β) = s∗0 for 0 < β 5 β0∗ ands∗1 = s∗ (p, β) for β1∗ 5 β.
In order to ensure the possibility that these increasing or decreasing properties ofindividual behavior, "in the large", may induce corresponding local monotonicity
properties in the aggregate, we consider distributions of the scaling parameter thathave a large enough support. In addition to either (H.0) or (H.1), we shall assume
throughout(H) Given β0∗ and β1∗ in either (H.0) or (H.1), the support of the distribution of
the heterogeneity scaling parameter α is large, so that eα0 β1∗ < eα1 β0∗ . The agents’common income b lies in the open interval (eα0 β1∗ , eα1 β0∗ ), so that the scaled incomese−α b at the endpoints of the density distribution g(α) satisfy7e−α1 b < β0∗ and β1∗ < e−α0 b.

(2.9)

This strategy will enable us to get approximate relative evaluations of the individual "base" budget shares s∗ (p, e−α b) at the endpoints of the density distribution
α0 , α1 , as for instance in (2.8), by direct comparison of s∗ (p, β0∗ ) and s∗ (p, β1∗ ). As
for the distribution of the behavioral heterogeneity parameter itself, we shall focus
on monotone distributions, and actually bounded exponential distributions for theindexing parameter α = log λ, which corresponds to the case of bounded Pareto
distributions for the heterogeneity scaling factor λ itself. That will enable us toderive sharp results on the income elasticities of aggregate expenditures for large
heterogeneity distributions, i.e. when |α1 − α0 | tends to +∞.

Exponential distributions of α = log λ3

We consider bounded exponential distributions for the indexing parameter α,i.e. g 0 (α) = εg(α) for α in [α0 , α1 ] in which caseg(α) = γ0 eαε for α ∈ [α0 , α1 ], g(α) = 0 for α 6∈ [α0 , α1 ], with
γ0 = ε/(eα1 ε − eα0 ε ) when ε 6= 0, and γ0 = 1/(α1 − α0 ) when ε = 0. (3.1)
Equivalently, the heterogeneity scaling factor λ = eα follows a bounded Pareto
distribution on [λ0 , λ1 ], where λ0 = eα0 , λ1 = eα1 , with a density f (λ) given byf (λ) = γ0 λε−1 for λ ∈ [λ0 , λ1 ], f (λ) = 0 for λ 6∈ [λ0 , λ1 ] with also here
γ0 = ε/(λε1 − λε0 ) when ε 6= 0 and γ0 = 1/(log λ1 − log λ0 ) when ε = 0.

3.1

Uniform distributions (ε = 0)

In this benchmark configuration, we get immediately from (2.6)∂Ss∗ (p, e−α0 b) − s∗ (β, e−α1 b)(p, b) =(3.2)∂b
α1 − α0
It is then clear that one gets a locally increasing aggregate budget share, i.e.
∂Sb (p, b) > 0, when individual budget shares s∗ (p, β) are increasing "in the large",
∂b
b

8

∂S
(p, b) < 0 under∂b
assumptions (H.1) and (H). On the other hand, if the support of the distribution,i.e. |α1 − α0 |, becomes increasingly large, given β0∗ , β1∗ , aggregate budget shares becomes less sensitive to income variations. One can in fact show that the aggregate
budget share income elasticity converges then uniformly to ε = 0.

i.e. under assumptions (H.0) and (H). Symmetrically, one gets b

Proposition 1. Under assumption (H), given p  0 and the agents’ common
income eα0 β1∗ < b < eα1 β0∗ ,∂Sa) under assumption (H.0), one has b (p, b) > 0. Furthermore∂b∂Sb (β, b)
1
5 ∗
(3.3)
0 < ∂bS(p, b)s1 (log(b/β1∗ ) − α0 )converges to 0 when α0 tends to −∞, uniformly on compact income subintervals[bm , bM ] of (eα0 β1∗ , eα1 β0∗ ).
∂Sb) Under assumption (H.1), one has b (p, b) < 0. Furthermore∂b∂Sb (p, b)
1
5 ∂b
<0
(3.4)
s∗0 (log(b/β0∗ ) − α1 )S(p, b)converges to 0 when α1 tends to +∞, uniformly on compact subintervals [bm , bM ]of (eα0 β1∗ , eα1 β0∗ ).
A proof is given in the appendix. The implications for aggregate expenditures
W (p, b) are immediate through (2.7). Details are left to the reader.

3.2

Strictly monotonic exponential distributions

We focus now on exponential distributions of the heterogeneity parameter, g 0 (α) =εg(α) for α in [α0 , α1 ] with ε 6= 0. We get from (2.6) and (3.1)∂S(p, b) = εS(p, b) + γ0 [eα0 ε s∗ (p, e−α0 b) − eα1 ε s∗ (p, e−α1 b)]∂bwith γ0 = ε/(eα1 ε − eα0 ε ).

b

(3.5)

When ε > 0, it is straightforward to verify here, as above in our heuristic
discussion of (2.8), that the aggregate budget share is locally increasing, in fact∂Sb (p, b) − εS(p, b) > 0, if individual budget shares s∗ (p, β) increase strongly∂b

9enough when comparing low incomes β 5 β0∗ to large incomes β = β1∗ , as inassumption (H.0). One can in fact show the much stronger result that the aggre∂S
gate income elasticity b (p, b)/S(p, b) actually converges to ε > 0, uniformly on∂bcompact income intervals, when the support [α0 , α1 ] of the heterogeneity density
becomes infinitely large, if individual budget shares s∗ (p, β) converge fast enoughto 0 (faster than β ε ) when β 5 β0∗ goes to 0.
∂S(p, b) − εS(p, b) < 0∂b
if individual budget shares s∗ (p, β) decrease strongly enough when comparing low
incomes β 5 β0∗ to large incomes β = β1∗ , as in assumption (H.1). Here also,∂S
one can show that the aggregate income elasticity b (β, b)/S(p, b) converges to∂b
ε < 0, uniformly on compact income intervals, when the support [α0 , α1 ] of theheterogeneity density becomes infinitely large, if individual budget shares s∗ (p, β)go fast enough to 0 when β = β1∗ increases without bound.
Symmetrically, when ε < 0, one verifies also easily that b

Proposition 2. Under assumption (H), given p  0 and the agents’ common
income eα0 β1∗ < b < eα1 β0∗ ,∂Sa) If ε > 0, under assumption (H.0), one has b (p, b) > εS(p, b) > 0 if∂b[log s∗1 − log s∗0 ]/(α1 − α0 ) > ε.
(3.6)
∂S
The aggregate income elasticity b (p, b)/S(p, b) converges uniformly to ε when∂b
α0 → −∞, α1 → +∞, if
0 5 s∗ (p, β) 5 aβ γε for all β 5 β0∗ , with a = 0, γ > 1.
(3.7)
∂Sb) If ε < 0, under assumption (H.1), one has b (p, b) < εS(p, b) < 0 if∂b[log s∗1 − log s∗0 ]/(α1 − α0 ) < ε.
(3.8)∂SThe aggregate income elasticity b (p, b)/S(p, b) converges uniformly to ε when∂b
α0 → −∞, α1 → +∞, if
0 5 s∗ (p, β) 5 aβ γε for all β = β1∗ , with a = 0, γ > 1.

(3.9)

A proof is given in the appendix. The above has immediate implications for
aggregate expenditures, through (2.7) and (3.5).
Corollary 1. Under the assumptions of Proposition 2, income derivatives ofaggregate expenditures are given by10

∂W(p, b) = (1 + ε)S(p, b) + [g(α0 )s∗ (p, e−α0 b) − g(α1 )s∗ (p, e−α1 b)].
∂b

(3.10)∂W(p, b) > (1 + ε)W (p, b) > 0. Undera) When ε > 0, (H.0) and (3.6) imply b∂b∂W(3.7), b(p, b)/W (p, b) converges uniformly to 1 + ε when α0 → −∞, α1 → +∞.
∂b∂Wb) When ε < 0, (H.1) and (3.8) imply b(p, b) < (1 + ε)W (p, b), while∂b∂Wb(p, b)/W (p, b) converges uniformly to 1 + ε when α0 → −∞, α1 → +∞ under∂b(3.9). Aggregate expenditures are then decreasing in income if 1 + ε < 0.
It may be noted that when ε > 0, the claimed results are valid in the limiting,but nevertheless relevant, configuration where s∗ (p, β) = 0 for all 0 < β 5 β0∗ ,which corresponds to the case where log s∗0 = −∞ in (3.6) and a = 0 in (3.7). In
that case, convergence of the aggregate income elasticity to ε > 0 obtains when
α0 → −∞, even when α1 > log(b/β0∗ ) remains bounded above. When s∗0 > 0 andthus a > 0 in (3.7), one needs both α0 → −∞ and α1 → +∞. The distribution overthe heterogeneity parameter becomes then more concentrated on large α, hence thescaled incomes e−α b on low levels. Both the aggregate budget share S(p, b) and its
income derivative become small, while the aggregate income elasticity gets closer
to ε > 0. An analogous symmetric argument applies to the case ε < 0.

4

Applications

We apply next the above results and methods to a few specific frameworks :aggregate market demand, portfolio choice, and concave aggregate expenditures.

4.1

Aggregate market demand

We consider the case where "base" behavior is the result of maximizing individual
preferences, or utility U (x), over commodity bundles x = (x1 , . . . , xn ) = 0 for low
and large incomes. Given the commodity price vector p = (p1 , . . . , pn )  0, "base"expenditures for each commodity h, wh∗ (p, β), s∗h (p, β), are the result of maximizing
U (x) subject to p.x = β for β < βm and for βM < β, where 0 < βm < βM
are given thresholds. We assume standard conditions (strict monotonicity andconcavity, smoothness) ensuring in particular interior solutions xh > 0 within thisadmissible income range, that can be characterized by standard FOC relations

11

equating marginal rates of substitution of any pair of commodities h, k with thecorresponding price ratios ph /pk . Expenditure functions wh (α, p, b) and sh (α, p, b)
corresponding to the heterogeneity scaling factor λ = eα , as in (2.1), (2.2), arethen obtained by maximizing U (x/λ) = U (e−α x) subject to p.x = b, whenevere−α b < βm or βM < e−α b. Other (possibly non utility maximizing) behaviors areof course compatible with our analysis for medium range incomes βm 5 β 5 βM .
We wish to see under which conditions on the utility function U (x) one
can get the configurations described in Propositions 1,2 above. For the simplicityof the exposition, we focus on the "increasing" case (H.0) and ε = 0. Similararguments extend easily to the other case (H.1) and ε 5 0. Intuitively, one should
get the individual "base" budget share s∗ (p, β) of a particular group of commodities
h = (1, . . . , i) to be increasing in the "large" as in (H.0) if the marginal rates of∂U∂Usubstitution(x)/(x) between commodities h = 1, . . . , i in that group and∂xh∂xk
those outside k = i+1, . . . , n, become low for small income commodity bundles, andhigh for large income commodity bundles, in a sense we wish to make precise. And
similarly for limiting conditions such as (3.7) for incomes going to 0 in Proposition
2 for the case ε > 0.
A two commodities CES example
Given p = (p1 , . . . , pn )  0, let base expenditures be the result of maximizing
the CES utility function
U (x1 , x2 ) = a1

1
x1−ρ
x1−ρ2
1
+ a2 2, ρ1 > 0, ρ2 > 0
1 − ρ11 − ρ2(4.1)

with a1 > 0, a2 > 0, under the budget constraint p.x = β for low incomes 0 < β <βm and large ones βm < βM < β. The optimum base budget share of commodity
1, s = p1 x1 /β = s∗ (p, β) is then the unique solution of the FOCs∗ (p, β)ρ1
= β ρ2 −ρ1 d[1 − s∗ (p, β)]ρ2(4.2)

with d = a1 pρ11 −1 /a2 pρ22 −1 . When ρ2 > ρ1 , s∗ (p, β) is increasing within these income
ranges β < βm , βM < β. Then (H.0) applies : the right hand side of (4.2) a) goes to
0 (s∗ (p, β) decreases to 0) when β < βm goes down to 0, and b) goes to +∞ (s∗ (p, β)goes up to 1) when βM < β tends to +∞. Therefore, for any arbitrary thresholds
0 < s∗0 < s∗1 < 1, there exist β0∗ low enough and β1∗ large enough such that (H.0)
applies.
Then Proposition 1 applies for uniform distributions (ε = 0). For increasing

12

exponential densities ε > 0, Proposition 2 applies as well provided that ρ2 issignificantly greater than ρ1 . Indeed one gets from (4.2) for any β 5 β0∗s∗ (p, β) 5 β (ρ2 −ρ1 )/ρ1 d1/ρ1 ,(4.3)

which is bounded above by aβ γε for some γ > 1 as in (3.7) (provided that, without
loss of generality, β0∗ < 1) if (ρ2 − ρ1 )/ρ1 = γε > ε.
As an illustration of the impact of large supports of the behavioral heterogeneity
parameter density g(α), one may note that the limit value ε > 0 in Proposition∂S
2 for the aggregate income elasticity b (p, b)/S(p, b), can be somewhat different∂bfrom the underlying individual base income elasticity of s∗ (p, β) for low incomes :the latter goes, from (4.2), to (ρ2 − ρ1 )/ρ1 > ε when β goes to 0. This occurs
despite the fact that the distributions considered in Proposition 2 give increasing
weight on large heterogeneity parameters α, hence on low income e−α b, when ε > 0.
Multicommodities markets
The core economic mechanism, in the foregoing two commodities CES example,
rests on a simple property of the marginal rate of substitution M RS12 (x1 , x2 ) =∂U∂U(x1 , x2 )/(x1 , x2 ) when expenditures are fixed proportions of variable in∂x1∂x2
come β, x1 = µ1 β, x2 = µ2 β, i.e. µ1 = s/p1 , µ2 = (1 − s)/p2 where commodity 10 sbudget share s is fixed. If ρ2 > ρ1
M RS12 (µ1 β, µ2 β) = β ρ2 −ρ1 (a1 µρ22 /a2 µρ11 )(4.4)

goes to 0 when β < βm goes to 0, to +∞ when βM < β goes up to +∞. Thenfor low incomes, one needs to lower commodity 10 s consumption (decrease s) toreestablish the optimum equality of the marginal rate of substitution M RS12 withthe given price ratio p1 /p2 , and a similar symmetric argument applies for large
incomes. Hence the (H.0) pattern that arises in such a case. We consider now asimilar mechanism in a multicommodity context.
We use a simple two steps procedure. Every commodity bundle x = (x1 , . . . , xn ) =0 is split in two groups x = (x1 , x2 ) with x1 = (x1 , . . . , xi ), x2 = (xi+1 , . . . , xn ).
And similarly for the price vector p = (p1 , . . . , pn )  0, i.e. p = (p1 , p2 ) with p1 =
(p1 , . . . , pi ), p2 = (pi+1 , . . . , pn ). The base expenditures resulting from maximizing
U (x) subject to p.x = β for β < βm and βM < β, we focus on the budget share alloiX∗
cated to the first group of commodities, which is noted s (p, β) =s∗h (p, β). Let
h=1β1 , β2 be a preliminary spending allocation for the two groups, with β1 + β2 < βm13

or βM < β1 + β2 , and define the corresponding "indirect" utility by the constrained
maximization
V (p, β1 , β2 ) = M ax U (x1 , x2 )/p1 .x1 = β1 , p2 .x2 = β2 .

(4.5)

Under standard regularity conditions on U (x) (strict monotonicity and concavity,smoothness, interior solutions), the "indirect" utility V (p, β1 , β2 ) will display thesame regularity properties. Further, the original optimum allocation of total income
β among the two groups will be the result of maximizing V (p, β1 , β2 ) under the
total budget constraint β1 + β2 = β. The corresponding optimum condition is then∂V∂V(p, β1 , β2 ) =(p, β1 , β2 ).
∂β1
∂β2

(4.6)

The interpretation of this two steps procedure follows standard lines. The first stage(4.5) allocates expenditures so as to equalize, within each group of commodities,
marginal utilities of additional spending∂V(p, β1 , β2 ) =
∂β1

1 ∂U
(x),
ph ∂xhh = 1, . . . , i,∂V(p, β1 , β2 ) =
∂β2

1 ∂U
(x),
pk ∂xkk = i + 1, . . . , n,under the budget constraints p1 .x1 = β1 , p2 .x2 = β2 . These marginal utilities arethen equalized, accross the two groups, in the second stage (4.6), so as to reachthe original optimum allocation obtained by maximizing directly U (x) subject to
p.x = β. The outcome of (4.6), under the budget constraint β1 + β2 = β, yields
then the optimum budget share, β1 /β = s∗ (p, β), β2 /β = 1 − s∗ (p, β).
On expects that, similarly to our two commodities CES example, the (H.O)
pattern and the conditions of Propositions 1,2 with ε = 0, will obtain if the "indirect" utility’s marginal rate of substitution
M RS12 (p, β1 , β2 ) =∂V∂V(p, β1 , β2 )/(p, β1 , β2 )
∂β1
∂β2

(4.7)

becomes low for small incomes β1 + β2 < βm and large for high incomes βM <β1 + β2 , along lines analogous to (4.4).
Proposition 3. Let the indirect utility function V (p, β1 , β2 ) be defined as in(4.5).
1) Assume that for any fixed budget share 0 < s < 1 of the first group ofcommodities, the marginal rate of substitution M RS12 (p, β1 , β2 ) in (4.7), with β1 =14sβ, β2 = (1 − s)β, tends : a) to 0 when β < βm goes down to 0, b) to +∞ whenβM < β increases without bound. Then for any thresholds 0 < s∗0 < s∗1 < 1, there
exist 0 < β0∗ < βm < βM < β1∗ , with β0∗ small enough and β1∗ large enough, suchthat (H.0) is satisfied. Then Proposition 1 applies when ε = 0, as well as the firstpart of Proposition 2. a) when ε > 0, under condition (3.6).
2) Assume in addition, when ε > 0
M RS12 (p, aβ γε+1 , β) → 0 when β 5 β0∗ tends to 0

(4.8)

for some a > 0, γ > 1. Then for β0∗ small enough, condition (3.7) is satisfied andthe whole Proposition 2.a) applies.
The argument of the proof is straightforward. Details are given in the appendix. A
similar symmetric argument can be easily designed to generate the (H.1) pattern,
to be applied to decreasing densities of the heterogeneity parameter α, i.e. ε 5 0.

4.2

Aggregate portfolio choices

We consider the aggregate market behavior of a centinuum of agents who have an
identical income b to invest in several risky securities and a single riskfree asset. The
issue we investigate is : under which conditions the aggregate amount invested in
the risky assets is an increasing proportion of their income. Or equivalently, when
the aggregate amount invested in the riskfree asset is a decreasing proportion of
their income. It has been well known since the work of Arrow (1970, Ch.3), thatin the case of a single risky security and of a single expected utility maximizing
investor, such a behavior would obtain if the agent’s relative degree of risk aversion
was a decreasing function of her wealth. But it has been also known since thenthat such an investment behavior was no longer true in the case of several risky
securities (Cass and Stiglitz (1970, 1972), Hart (1974)). Without mentioning theimpact of modern "behavorial" departures from expected utility maximization...
We show that swithching to an aggregate viewpoint with heterogeneous agents
allows to make the above portfolio property to be valid even with many risky assets
when there are no personal outside income insurance motives (as in the standard
frameworks quoted above), and when the available risky securities still imply incomplete markets. We require individual "base" behavior, as measured by thebudget shares function s∗h (p, β) of every asset, to be the result of consistent expected utility maximization only for quite low incomes β < βm , and quite large
ones βM < β, while allowing for any "behavioral" departure (loss aversion, ambiguity or whatever...) in the medium income range βm 5 β 5 βM . Variationsof relative degrees of risk aversion are needed only "in the large", i.e. we assume15

high relative degrees of risk aversion for small incomes β, and a very low one (riskneutrality) for quite large incomes β, while anything can happen in medium income ranges. The proportion of aggregate investment in the riskless security isthen shown to be decreasing with the investors’ (identical) wealth b for large heterogeneous populations as in Propositions 1,2 above, in the (H.1) pattern, with
ε 5 0.
Formally, there are n risky securities available today. A unit of any such asset
h = 1, . . . , n yields dhj > 0 units of income tomorrow, that may depend on theoccurrence of some event j = 1, . . . , J. Its price today is qh > 0, the corresponding
gross rates of return being Rhj = dhj /qh > 0. There is a single riskfree asset, a unitof which generates a sure unit of income tomorrow. Its current unit price is q0 > 0,
the corresponding sure gross rate of return being R0 = 1/q0 > 0. The current asset
prices vector is noted p = (q0 , q1 , . . . , qn )  0. We assume that every portfolio of
the risky securities involves some possible loss by comparison to the riskfree asset.
(Incomplete asset markets) The exists ∆Rm < 0 such that for every λh = 0nXwithλh = 1, one has for some j = 1, . . . , J,
h=1nX

λh Rhj − R0 < ∆Rm < 0.

(4.9)

h=1

The "base" behavior of individual investors is described by the non-negative
proportions of the variable income β > 0 invested in the riskless asset, s =nXsh = 1.
s∗ (p, β) = 0, and in the risky ones, sh = s∗h (p, β) = 0, with s +h=1

As in section 2, we consider a continuum of agents with a common income b,
whose actual portfolio choices s(α, p, b), sh (α, p, b), are affected by a behavioral
heterogeneity scaling factor λ = eα , as in (2.2), and look at the proportion of theiraggregate investments in the various assets S(p, b), Sh (p, b), defined as in (2.4). We
investigate then, within the framework of section 3, conditions ensuring that theaggregate budget share invested in the riskless asset, S(p, b), decreases with incomeb. To do so, we need to focus on the case where the individual "base" budget shareof the riskfree security, s∗ (p, β), decreases "in the large", as in the (H.1) pattern.
We assume that for low incomes, 0 < β < βm , and large ones, βM < β, where0 < βm < βM are given thresholds, individual "base" choices of a portfolio s =s∗ (p, β), sh = s∗h (p, β), are driven by the maximization of a standard VNM expectedJX
utility E[u(yj )] =πj u(yj ), where πj > 0 are the subjective probabilities of
j=1

16

occurrence of each state j, withJX

πj = 1, subject to s +

j=1

n
X

sh = 1. The incomes

h=1

generated by such a portfolio, are given by, for each state j = 1, . . . , J (there is no
exogenous risky income, so no outside income insurance motive).
yj = (sR0 +

n
X

n
X
sh Rhj )β = R0 β +(Rhj − R0 )sh β.

h=1(4.10)

h=1

On the other hand, the corresponding budget shares s(α, p, b), sh (α, p, b) associatedto the heterogeneity scaling factor λ = eα , as in (2.2), can be viewed as resulting
from the maximization of the expected utility E[u(e−α yj )] under the same constraints in similar income ranges. We make the standard "rationality" assumption
for all low income ranges yj < ym and large ones yM < yj generated by arbitrary
portfolios (s, sh ) as in (4.10) with β < βm and βM < β respectively :(U) The VNM utility function u(y) is, on the relevant income ranges y < ym
and yM < y, continuous, twice continously differentiable for y > 0, increasing(u0 (y) > 0, with limy→0 u0 (y) = +∞), strictly concave (u00 (y) < 0).
As a counterpart of (4.9), the following assumption states that risky securities
are nevertheless financially attractive.
There is at least one risky asset h such that
E[Rs − R0 ] =JX

πj (Rhj − R0 ) > 0.

(4.11)

j=1

Proposition 4. Let the assumptions (U), (4.9) and (4.11) hold.
1) Assume that investors are risk neutral for high incomes :There is y1∗ > yM such that ρ(y) = −yu00 (y)/u0 (y) = 0 for y > y1∗ .

(4.12)

Then there exists β1∗ > βM such that s∗ (p, β) = 0 for all β = β1∗ .
2) Assume that investors are very much risk averse for low incomes :ρ(y) → +∞ when y → 0.

(4.13)

Then for any 0 < s∗0 < 1, there exists 0 < β0∗ < βm such that s∗ (p, β) > s∗0 for allβ 5 β0∗ .

17

3) Under assumptions (4.12), (4.13), the (H.1) pattern applies with 1 > s∗0 >= 0 as above. For any exponential distribution of the heterogeneity scaling
parameter α and any investors’ common income b satisfying (H), the aggregate
budget share invested in the riskless security, S(p, b), is decreasing in b when ε 5 0.
Its elasticity with respect to b tends to ε when α1 → +∞.

s∗1

A proof is given in the appendix. As a concluding remark, it may be worth reemphasizing that the standard "rationality" expected utility assumption was usedhere only for low and large incomes, as in (U). And this only to assume formallyrisk neutrality for large incomes, as in (4.12), or a relative degree of risk aversion
that tends to +∞ (not necessarily monotonically !) for low incomes as in (4.13). Asa matter of fact, a cursory look at the argument in the proof suggests strongly thatthe driving mechanism to derive the above results 1) and 2) (and hence the whole
Proposition from Propositions 1.b), 2.b)) is the investors’ degree of loss aversion,that is absent for high incomes and becomes infinite for low incomes. On the other
hand, the whole analysis is compatible with any "behavioral rationality departure"in the medium range incomes.

4.3

Concave/convex aggregate expenditures

We extend the results of section 3 to higher order derivatives, namely concavity/convexity of aggregate budget shares S(p, b) and expenditures W (p, b) withrespect to income b. We maintain throughout our assumption (3.1) of exponential
distributions for the heterogeneity parameter α = log λ (i.e. Pareto distributionsof the scaling factor λ). It is convenient to reformulate the income derivative of
aggregate expenditure (3.10) as∂W(p, b) = W (p, b)(1 + ε) − γ0 [eαε w(α, p, b)]αα10 ,∂bwith the maintained notation [f (α)]αα10 = f (α1 ) − f (α0 ).
b(4.14)

Lemma 2. Differentiation with respect to income b, of (3.5) and (4.14), yields∗∂2S∂Sα(ε−1) ∂s
(p,
b)
=
(ε
−
1)
(p,
b)
−
γ
[e
(p, e−α b)]αα10 ,
0
∂b2
∂b
∂β
∂2W
∂W
∂w∗
b 2 (p, b) = ε(p, b) − γ0 [eαε(p, e−α b)]αα10 .
∂b
∂b
∂β

b

(4.15)(4.16)In what follows, as in section 3, we let assumption (H) hold, while the pricesystem p  0 and the agents’ common income eα0 β1∗ < b < eα1 β0∗ are given.

18

We keep assumption (H.0), or (H.1), that enabled us to evaluate the sign and∂Smagnitude offrom a comparison of individual budget shares s∗ (p, β) for low and∂blarge incomes, i.e. for β ≤ β0∗ versus β1∗ ≤ β for some appropriate thresholds β0∗ <β1∗ . Our strategy is also here to compare the income derivatives of individual "base"∂s∗∂w∗
budget shares(p, β), or expenditures(p, β), for low and large incomes∂β∂ββ. We focus on budget shares through assumptions (D.0), or (D.1) below. The∂s∗
individual "base" income derivative(p, β) will play accordingly here, through∂βthese conditions, a role similar to that of s∗ (p, β) in section 3 through (H.0), (H.1).
The main technical difference being that, by construction, the budget share s∗ (p, β)
is bounded since it lies in [0,1], whereas the income derivative needs not.
(D.0) (Increasing individual budget share’s income derivative "in the large")
Given p  0 and β0∗ < β1∗ as in (H.0) or (H.1), there exist σ0∗ < σ1∗ such that∂s∗∂s∗(p, β) 5 σ0∗ for 0 < β < β0∗ , while σ1∗ 5(p, β) for β1∗ 5 β.
∂β∂β(D.1) (Decreasing individual budget share’s income derivative "in the large")
Given p  0 and β0∗ < β1∗ as in (H.0) or (H.1), there exist σ0∗ > σ1∗ such that∂s∗∂s∗(p, β) = σ0∗ for 0 < β 5 β0∗ , while σ1∗ =(p, β) for β1∗ 5 β.
∂β∂β
We focus essentially on one configuration in the main text here, where the aggregate budget share S(p, b) is locally decreasing and convex in income (Proposition 5
below, under assumptions (H.1), (D.0) and a decreasing exponential heterogeneity
distribution g(α), with ε 5 0). Such a configuration seems indeed of specific potential interest, as it can generate aggregate expenditures W (p, b) that are increasing
and concave (Corollary 2 below, with −1 < ε < 0), i.e. an aggregate (multicommodities) "concave consumption function", a topic that has attracted a significant
attention (see, e.g. Caroll and Kimball (1996), Gourinchas and Parker (2002)).
Two other interesting configurations are considered in the appendix, where theaggregate budget share S(p, b) is 1) locally increasing and convex (Proposition 6,
under assumptions (H.0), (D.0) and a significantly increasing exponential heterogeneity distribution, with ε = 1) and 2) locally increasing and concave (Proposition
7, under assumptions (H.0), (D.1) and a moderately increasing exponential heterogeneity distribution with 0 5 ε 5 1).
Proposition 5. (Locally decreasing and convex aggregate budget share)
Let (H.1) hold, as well as (D.0) with σ0∗ < σ1∗ 5 0, and consider a decreasing
exponential heterogeneity distribution with ε 5 0.

19

1) One has b∂S∂2S(p, b) > (ε − 1) (p, b) if σ1∗ is close enough to 0, i.e.
2∂b∂b[log |σ1∗ | − log |σ0∗ |]/(α1 − α0 ) < ε − 1.

The aggregate budget share is then locally strictly decreasing,(4.17)∂S(p, b) < 0, and∂b∂2S
strictly convex, b 2 (p, b) > 0, if either ε = 0, or under condition (3.8) when∂b
ε < 0.
2) Assume that the individual "base" budget share is non-increasing with in∂s∗
∂2S
∂S
come,
(p, β) 5 0, for all β > 0. Then the aggregate income elasticity, b 2 (p, b)/ (p, b),
∂β
∂b
∂b
∂s∗
converges uniformly to ε − 1 5 −1 when α0 → −∞, α1 → +∞, if |(p, β)|∂βa) goes fast enough to 0 when β tends to +∞, and b) is bounded above or does
not diverge to +∞ too fast when β goes down to 0 :∂s∗(p, β)| 5 a1 β ν1 (ε−1) with a1 = 0, ν1 > 1,∂β∂s∗b) for 0 5 β0∗ , |(p, β)| 5 a0 β ν0 (ε−1) with a0 > 0, 0 5 ν0 < 1.
∂βa) for β = β1∗ , |(4.18)(4.19)

A detailed proof is given in the appendix. On the other hand, it is immediate to ver∂2S∂S
ify that when the aggregate budget share income elasticity b 2 (p, b)/ (p, b) con∂b
∂b
∂2W
∂W
verges to δ, the corresponding income elasticity for aggregate expenditure b 2 (p, b)/(p, b)∂b∂b
converge to 1 + δ. One should expect accordingly that a moderately decreasing exponential heterogeneity distribution with −1 < ε < 0, should lead to an increasing
and concave (again, multidimentional) aggregate expenditure.
Corollary 2. (Locally increasing and concave aggregate expenditure)
Let the assumptions of Proposition 5 above and of Corollary 1 at the end ofsection 3 hold, with −1 < ε < 0, and let α0 → −∞, α1 → +∞.
∂W(p, b)/W (p, b) →a) Under condition (3.9), the aggregate income elasticity b∂b
1 + ε > 0.
∂2W∂Wb) The aggregate income elasticity b 2 (p, b)/(p, b) → ε < 0.
∂b
∂b

20

5

Conclusion

The analysis presented in this paper shows that exploiting further the parametric structure of heterogeneity preference scaling factors, here by assuming Pareto
distributions, may be a significantly fruitful research avenue to get sharp monotonicity properties of macroeconomic aggregates. This allowed us to go well beyond
earlier results about possible "insensitivity" properties of aggregate expenditures
obtained with less specific notions of "behavioral heterogeneity", recalled in the introduction. In particular, the application made here to the study of wealth effects
on aggregate portfolio choices, shows that this approach enables us to get neataggregate monotonicity properties that are usually hard, or even impossible, to getat the individual level from standard microeconomic (expected utility maximizing,
ambiguity,...) theory in financial markets.
The theoretical analysis of this paper focussed on a single group of agents
displaying heterogeneous behavior generated from a single individual "base" expenditure function and a single homothetic heterogeneity scaling factor Pareto distribution. Clearly, it can be easily extended to encompass other possible sources ofheterogeneity, e.g. by considering several groups of such individuals, these groupsnX
i = 1, . . . , n having relative sizes µi > 0 with
µi = 1, and being endowed withi=1

individual incomes bi > 0, individual "base" expenditure functions and heterogeneity factor distributions that are possibly different. For a given income distribution
with b = Σi µi bi , the elasticity of the aggregate budget share with respect to aggregate income b, can be easily deduced from the results of this paper if each group fits
our framework (of course, its relation with each group’s aggregate elasticity withrespect its income bi will have to take into account covariance, as noted e.g. by
Paluch, Kneip and Hildenbrand (2012)). Further, the impact of a (local) change ofthe income distribution, given aggregate income, can also be easily derived withinsuch an extended framework. Such an extended model is then able to cope witha wide variety of observable behavioral heterogeneities, that have been so much
documented in the empirical literature, as reviewed in the introduction, in various
frameworks (goods and services, portfolio choices, ...).
Preliminary research work suggests also that the methodological approach ofthis paper, which focussed as a starting point on aggregate income elasticities, canpresumably also be applied fruitfully to price elasticities, by using commodity specific scaling factors, as in Grandmont (1992, 1993). This may open the possibility
to extend the analysis to general equilibrium frameworks, with perfect of imperfectcompetition.

21

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Quah, J. (1997), "The Law of Demand when Income is Price Dependent",
Econometrica, 65, 1421-1142.
Quah, J. (2001), "Demand is Heterogeneous in Grandmont’s Model", Nuffield
College, Working Paper, W12.

25

Appendix
1. Proof of Proposition 1a) Under (H.0), (3.2) implies∂S(p, b) = [s∗1 − s∗0 ]/(α1 − α0 ) > 0.
∂bbMoreover S(p, b) = s∗1 (log( ∗ )−α0 )/(α1 −α0 ) > 0. Combining this inequality with
β1(3.2) yields immediately (3.3), hence the result.
bb) Under (H.1), (3.2) implies∂S(p, b) 5 [s∗1 − s∗0 ]/(α1 − α0 ) < 0.
∂bbMoreover S(p, b) = s∗0 (α1 −log( ∗ ))/(α1 −α0 ) > 0. Combining this inequality with
β0(3.2) yields immediately (3.4), hence the result. Q.E.D.
b

2. Proof of Proposition 2a) Case ε > 0. Under (H.0), (3.5) implies∂S(p, b) − εS(p, b) = γ0 [eα0 ε s∗1 − eα1 ε s∗0 ].
∂b
Since γ0 > 0, the right hand side will be positive if and only if (3.6) is satisfied(including the limiting case where s∗0 = 0, or log s∗0 = −∞).
b

When the support [α0 , α1 ] becomes large, with α0 < log(b/β1∗ ), log(b/β0∗ ) < α1 ,a lower bound for S(p, b) is given by
Z log(b/β ∗ )1S(p, b) =s∗1 γ0 eεα dα

α0

= s∗1

γ0 b ε[( ) − eα0 ε ] > 0.
ε β1∗

(A.1)

One gets then from (3.5)
b∂S(p, b)|eα0 ε s∗ (p, e−α0 b) − eα1 ε s∗ (p, e−α1 b)|
∂b
−ε 5ε
b
S(p, b)s∗1 [( ∗ )ε − eα0 ε ]
β1

26

(A.2)

Since s∗ (p, e−α0 b) = s∗1 > 0, one needs to have α0 → −∞ to make the right handside of (A.2) go to 0. This condition is actually sufficient when s∗0 = 0, hences∗ (p, e−α1 b) = 0, even when α1 > log(b/β0∗ ) remains bounded above. Under the
more general condition (3.7), where one may have a > 0, one getsb∂S(p, b)
eα0 ε + aeα1 ε(1−γ) bγε
∂b
−ε 5ε
b
S(p, b)s∗1 [( ∗ )ε − eα0 ε ]
β1

(A.3)

which tends uniformly to 0 when α0 → −∞ and since γ > 1, when α1 → +∞.
b) Case ε < 0. Under (H.1), (3.5) implies∂S(p, b) − εS(p, b) 5 γ0 [eα0 ε s∗1 − eα1 ε s∗0 ].
∂b
Since γ0 > 0, the right hand side will be negative if and only if (3.8) is satisfied(including the limiting case where s∗1 = 0, or log s∗1 = −∞).
When the support [α0 , α1 ] becomes large, with α0 < log(b/β1∗ ), log(b/β0∗ ) < α1 ,a lower bound for S(p, b) is given byb

Z α1S(p, b) =log(b/β0∗ )= s∗0s∗0 γ0 eεα dα

γ 0 α1 εb[e− ( ∗ )ε ] > 0.
ε
β0

(A.4)

One gets then from (3.5)
b∂S(p, b)|eα1 ε s∗ (p, e−α1 b) − eα0 ε s∗ (p, e−α0 b)|∂b− ε 5 |ε|.
bS(p, b)s∗0 [( ∗ )ε − eα1 ε ]
β0

(A.5)

Since s∗ (p, e−α1 b) = s∗0 > 0, one needs to have α1 → +∞ to make the righthand side of (A.5) go to 0. This condition is sufficient when s∗1 = 0, hences∗ (p, e−α0 b) = 0, even when α0 remains bounded away from −∞. Under the
more general condition (3.9), where one may have a > 0, one getsb∂S(p, b)
eα1 ε + aeα0 ε(1−γ) bγε∂b− ε 5 |ε|bS(p, b)s∗0 [( ∗ )ε − eα1 ε ]
β0

27

(A.6)

which tends uniformly to 0 when α1 → +∞ and, since ε(1 − γ) > 0, when α0 →−∞. Q.E.D.
3. Proof of Proposition 3
We consider the standard case where M RS12 (p, β1 , β2 ) in (4.7) is a decreasing
function of β1 that tends to +∞ when β1 goes to 0, and an increasing functionof β2 that goes down to 0 when β2 tends to 0, in the small income configuration
β1 + β2 < βm . Symmetrically M RS12 (p, β1 , β2 ) tends to 0 when βM < β1 goes up
to +∞, and tends to +∞ when βM < β2 increases without bound. For β < βmand βM < β, the optimum individual base budget share s∗ (p, β) of the first groupof commodities is then the unique solution, given β, of
M RS12 (p, sβ, (1 − s)β) = 1,

(A.7)

where the left hand side of (A.7) is a decreasing function of s that tends to +∞when s goes down to 0, and tends to 0 when s goes up to 1.
The assumption made in Proposition 3.1) states that M RS12 (p, sβ, (1 − s)β) is
an increasing function of β, given s, that tends to 0 when β goes down to 0. Thus
for any 0 < s∗0 < 1 fixed, there exists β0∗ small enough such that M RS12 (p, s∗0 β, (1−s∗0 )β) is uniformly small, hence less than 1, for all β 5 β0∗ . Then one needs to lower
s to reestablish the optimum of (A.7), i.e. s∗ (p, β) < s∗0 for all β 5 β0∗ . A symmetricargument applies for an arbitrary 1 > s∗1 > s∗0 and β1∗ > βM , so that s∗1 < s∗ (p, β),as in (H.0).
Thus Proposition 1 applies for the case ε = 0. When ε > 0, Proposition 2. a)applies as well if condition (3.6) is satisfied. Assume now (4.8) for some a > 0 and
γ > 1. One has for β 5 β0∗ going to 0
M RS12 (p, aβ γε+1 , (1 − aγε )β) < M RS12 (p, aβ γε+1 , β) → 0.

(A.8)

So if β0∗ is chosen small enough, the above is small, hence less than 1, which implies
that s∗ (p, β) < aβ γε , i.e. (3.7), for all β 5 β0∗ . Thus Proposition 2. a) applies fully
for the case ε > 0. Q.E.D.
4. Proof of Proposition 41) Under (4.12), let β1∗ > βM be large enough so that all incomes generated byarbitrary portfolios as in (4.10) satisfy yj > y1∗ . Then for any β = β1∗ , if the "base"budget share invested in the riskfree security at the optimum portfolio (s, (sh )), is
positive, s = s∗ (p, β) > 0, one should have E[(Rhj − R0 )u0 (yj )] 5 0 for every risky

28

asset h, with equality whenever sh > 0, where (yj ) is the resulting optimum income
stream (4.10). That would not be possible for the risky security h mentioned in(4.11), since risk neutrality means that u0 (yj ) is a constant independent of yj . Thuss∗ (p, β) = 0 for all β = β1∗ .
2) Let 0 < s∗0 < 1, an arbitrary "large" budget share invested in the riskfree
asset, be fixed. Consider an arbitrarily small income level 0 < β < βm . Let (sh (β))
be the budget shares of risky securities that maximize expected utility E[u(yj )]nX∗with s = s0 andsh (β) = 1 − s∗0 . Let the resulting income stream yj∗ (β) be given
h=1

by (4.10) :yj∗ (β) = R0 β +nXsh (β)β(Rhj − R0 ) > 0.

(A.9)

h=1

From the corresponding FOC, marginal expected utilities E[Rhj u0 (yj∗ (β))] > 0
for additional investment in risky assets h must all be equalized for assets withsh (β) > 0, while marginal expected utilities must be lower for those other assets kwith sk (β) = 0.
Consider now the optimum portfolio budget shares when the riskless security
proportion s is free to vary. The necessary and sufficient condition for s∗ (p, β) > s∗0is that marginal expected utility of investing in the riskfree asset exceeds that ofinvesting in the risky securities, when these are evaluated at the optimum portfolio(sh (β)) obtained when s is fixed at s∗0 :E[nXsh (β)β(Rhj − R0 )u0 (yj∗ (β))] < 0.

(A.10)

h=s

Taking an exact first order Taylor development of u0 (yj∗ (β)) at R0 β and using (A.9)
generates the equivalent inequality
E[(yj∗ (β) − R0 β)u0 (R0 β)] < E[(yj∗ (β) − R0 β)2 ρ(θj (β))u0 (θj (β))/θj (β)]
where the income θj (β) > 0 lies in [R0 β, yj∗ (β)]. Dividing both sides by βu0 (R0 β) >nX
0 and introducing the notation ∆Rj (β) =sh (β)(Rhj − R0 ), leads to the equivh=1

alent inequality
E[∆Rj (β)] < E[(∆Rj (β))229

β u0 (θj (β))ρ(θj (β))].
θj (β) u0 (R0 β)(A.11)

We now make β > 0 tend to 0 and prove that the right hand side of (A.11)
tends to +∞, which implies the existence of a small income level β0∗ > 0 such
n
X
∗
∗
∗
that s (p, β > s0 for all β 5 β0 , as claimed. Sincesh (β) = 1 − s∗0 > 0,
the income θj (β) > 0 lies in the interval [R0 β, (

n
X

h=1

sh (β)Rhj + s∗0 R0 )β] and tends

h=1

thus to 0, but the ratio β/θj (β) remains bounded away from 0. On the other
hand, (4.9) implies the existence of a state j (that may depend on β) such that
∆Rj (β)(1 − s∗0 ) < ∆Rm < 0. For that state j, one has yj∗ (β) < R0 β, henceθj (β) 5 R0 β in which case u0 (θj (β))/u0 (R0 β) = 1. If π > 0 is the minimum of allstate probabilities, the right hand side of (A.11) is larger thanπ[∆Rm /(1 − s∗0 )]2

βρ(θj (β)).
θj (β)

Since θj (β) goes to 0, ρ(θj (β)) tends to +∞ by (4.13), so the above goes to +∞when β goes to 0, which proves 2).
3) The claims made in 3) of Proposition 4 follow immediately, from Proposition
1. b) when ε = 0 and Proposition 2. b) when ε < 0. Q.E.D.
5. Proof of Proposition 51) Under (D.0), (4.5) implies

b

h
i
∂2S
∂S
α0 (ε−1) ∗α1 (ε−1) ∗
(p,
b)
−
(ε
−
1)
(p,
b)
=
γ
e
σ
−
e
σ
0
1
0 .
∂b2
∂b

(A.12)

If σ0∗ < σ1∗ 5 0, since γ0 > 0, the right hand side is positive if and only if (4.17)
holds (including the limiting case where σ1∗ = 0, or log |σ1∗ | = −∞). To complete∂S
the proof, one needs to prove that(p, b) < 0. Under (H.1), this is true if ε = 0∂b(Proposition 1), and under condition (3.8) when ε < 0 (Proposition 1), and undercondition (3.8) when ε < 0 (Proposition 2.b).
2) Let [α0 , α1 ] become large, with α0 < log(b/β1∗ ), log(b/β0∗ ) < α1 . Under the∂s∗assumption(p, β) 5 0 for all β > 0, a lower bound for
∂β
Z α1
∂s∗
∂S
(p, b) =(p, e−α b) e−α g(α)dα(A.13)∂b∂β
α0
is given by30

∂S(p, b)∂b

Z α1=log(b/β0∗ )= |σ0∗ |γ0 |σ0∗ |eα(ε−1) dαiγ0 h α1 (ε−1)
e− (b/β0∗ )ε−1 .
ε−1(A.14)One gets then from (4.15)b∂2S(p, b)ε−1∂b2− (ε − 1) 5∂S
σ0∗(p, b)∂beα0 (ε−1)∂s∗∂s∗(p, e−α0 b) − eα1 (ε−1)(p, e−α1 b)∂β∂β.
(b/β0∗ )ε−1 − eα1 (ε−1)(A.15)∂s∗(p, e−α1 b) 5 σ0∗ < 0, one needs to have α1 → +∞, together with (4.19),∂β∂s∗to make eα1 (ε−1)(p, e−α1 b) go to 0. This condition is actually sufficient to∂β∂s∗(p, e−α0 b) = 0,
make the right hand side of (A.15) go to 0 when σ1∗ = 0, hence∂βeven when α0 is bounded below. Under the additional more general condition(4.18), where one may have a1 > 0 when σ1∗ < 0, one verifies that the absolutevalue of the numerator of the right hand side of (A.15) is bounded above by

Since

a1 bν1 (ε−1) eα0 (ε−1)(1−ν1 ) + a0 bν0 (ε−1) eα1 (ε−1)(1−ν0 ) ,(A.16)

which tends to 0 when α0 → −∞, α1 → +∞ under the conditions specified in(4.18), (4.19). Q.E.D.
6. Proposition 6. (Locally increasing and concave aggregate budget share)
Let (H.0) hold, as well as (D.O) with σ1∗ > σ0∗ = 0, and consider an increasing
exponential heterogeneity distribution with ε = 1.
∂2S∂S1) One has b 2 (p, b) > (ε − 1) (p, b) if∂b∂b[log σ1∗ − log σ0∗ ]/(α1 − α0 ) > ε − 1 = 0.

(A.17)

The aggregate budget share is then locally strictly increasing,∂S(p, b) > 0, and∂bstrictly convex, b∂2S(p, b) > 0, under condition (3.6).
∂b2

312) Assume that the individual "base" budget share is non-decreasing with in∂s∗
∂2S
∂S
come,
(p, β) = 0 for all β > 0. Then the aggregate income elasticity b 2 (p, b)/ (p, b)
∂β
∂b
∂b
∂s∗
converges uniformly to ε − 1 = 0 when α0 → −∞, α1 → +∞, if(p, b) a) is∂βbounded above or does not tend to +∞ too fast when β1∗ 5 β → +∞, and b) goesto 0 fast enough when β0∗ = β → 0 :a) 0 < σ1∗ 5b) 0 5∂s∗(p, β) 5 a1 β ν1 (ε−1) for β1∗ 5 β, with a1 > 0, 0 5 ν1 < 1, (A.18)∂β∂s∗(p, β) 5 a0 β ν0 (ε−1) for 0 < β 5 β0∗ , with a0 > 0, ν0 > 1.
∂β(A.19)

Proof. 1) As seen in the proof of Proposition 5, (D.0) implies (A.12). Since
γ0 > 0 and 0 5 σ0∗ < σ1∗ here, the right hand side is positive if and only if (A.17)∂S
holds. Since(p, b) > 0 under condition (3.6) (Proposition 2.a), the proof is∂bcomplete.
2) Let [α0 , α1 ] become large, with α0 < log(b/β1∗ ), log(b/β0∗ ) < α1 . Under the∂S∂s∗assumption(p, β) = 0 for all β > 0, from (A.13), a lower bound for(p, b) is∂β∂bgiven by∂S
(p, b) =∂b

Z log(b/β ∗ )1

γ0 σ1∗ eα(ε−1) dα,α0γ0
which is equal to σ1∗[(b/β1∗ )(ε−1) −eα0 (ε−1) ] > 0 when ε > 1, and to σ1∗ γ0 [log(b/β1∗ )−ε−1α0 ] > 0 when ε = 1.
One gets then from (4.15) when ε = 1
∂s∗
∂s∗
−α0
∂2S
(p,
e
b)
−
(p, e−α1 b)b 2 (p, b)
1
∂β
∂β
∂b
5 ∗
.
∂S
σ1
log(b/β1∗ ) − α0(p, b)∂b
Assumption (A.18) implies that∂s∗(p, e−α0 b) is bounded above by a1 > 0 when∂β∂s∗(p, e−α1 b) 5 σ0∗ from (D.0). So the above income elasticity∂βgoes to ε − 1 = 0 when α0 → −∞ (even if α1 is bounded above).
When ε > 1, (4.15) implies
ε = 1, while 0 5

32

∗
∗
∂2S
α0 (ε−1) ∂s (p, e−α0 b) − eα1 (ε−1) ∂s (p, e−α1 b)||eb 2 (p, b)
ε−1
∂β
∂β
∂b
.
− (ε − 1) 5
∗
∗
α
(ε−1)
ε−1
0
∂S
σ1
(b/β1 )
−e(p, b)∂b(A.20)∗∂s−α
Since 0 < σ1∗ 5(p, e 0 b), one needs to make α0 go to −∞, together with∂β∂s∗
(A.18), to make eα0 (ε−1)(p, e−α0 b) go to 0. The condition is in fact sufficient to∂β∂s∗make the right hand side of (A.20) go to 0 when σ0∗ = 0, hence(p, e−α1 b) = 0,∂βeven when α1 is bounded above. Under the additional more general condition(A.19), where one may have σ0∗ > 0 and a0 > 0, the absolute value of the numeratorof the right hand side of (A.20) is bounded above by an expression identical to(A.16). This expression goes to 0 under conditions (A.18), (A.19) when α0 →−∞, α1 → +∞. Q.E.D.

7. Proposition 7. (Locally increasing and concave aggregate budget share)
Let (H.0) hold, as well as (D.1) with σ0∗ > σ1∗ = 0, and consider an increasing
exponential heterogeneity distribution with 0 5 ε 5 1.
1) One has b∂2S∂S(p, b) < (ε − 1) (p, b) if
2∂b∂b[log σ1∗ − log σ0∗ ]/(α1 − α0 ) < ε − 1 5 0.

(A.21)∂S
The aggregate budget share is then locally strictly increasing,(p, b) > 0, and∂b∂2S
strictly concave, b 2 (p, b) < 0, if either ε = 0, or under condition (3.6) when∂b
ε > 0.
2) Assume that the individual "base" budget share is non-decreasing with in∂s∗
∂2S
∂S
come,
(p, β) = 0, for all β > 0. The aggregate income elasticity b(p, b)/ (p, b)
∂β
∂b
∂b
∂s∗
(p, β) a) goes
converges uniformly to ε − 1 5 0 when α0 → −∞, α1 → +∞, if∂β
to 0 fast enough when β1∗ 5 β tends to +∞, and b) is bounded above or does not
go too fast to +∞ when β0∗ = β tends to 0 :a) 0 5∂s∗(p, β) 5 a1 β ν1 (ε−1) for β = β1∗ , with a1 = 0, ν1 > 1, (A.22)∂β33b) 0 < σ0 5∂s∗(p, β) 5 a0 β ν0 (ε−1) for β = β0∗ , with a0 > 0, 0 5 ν0 < 1. (A.23)∂β

Proof. 1) Under (D.1), (4.15) implies
h
i
∂2S
∂S
α0 (ε−1) ∗α1 (ε−1) ∗
−
e
σ
(p,
b)
−
(ε
−
1)
(p,
b)
5
γ
e
σ
0
1
0 .
∂b2
∂b
If σ0∗ > σ1∗ = 0, since γ0 > 0, the right hand side is negative if and only if (A.21)∂S
holds. To complete the proof, one needs to show
(p, b) > 0. Under (H.0), this∂bis true if ε = 0 (Proposition 1), and under condition (3.6) when ε > 0 (Proposition
2.a).
2) Let [α0 , α1 ] become large with α0 < log(b/β1∗ ), log(b/β0∗ ) < α1 . Under the∂S∂s∗(p, β) = 0 for all β > 0, from (A.13), a lower bound for(p, b) is
assumption∂β∂bgiven byb∂S
γ0(p, b) = σ0∗[(b/β0∗ )ε−1 − eα1 (ε−1) ]∂b
1−εwhen 0 5 ε < 1, and by∂S(p, b) = σ0∗ γ0 [α1 − log(b/β0∗ )]∂bwhen ε = 1.
One gets then from (4.15) when ε = 1
b

∂s∗
∂s∗
∂2S
(p, e−α0 b) −(p, e−α1 b)
(p,
b)
1 ∂β
∂β
∂b2
.
5 ∗
∂S
σ0
α1 − log(b/β0∗ )(p, b)∂b

Assumption (A.23) implies that 0 < σ0∗ 5∂s∗(p, e−α1 b) 5 a0 when ε = 1 while∂β∂s∗(p, e−α0 b) ≤ σ1∗ from (D.1). So the above income elasticity goes to 0 = ε−1∂βwhen α1 tends to +∞ (even when α0 is bounded below).

0≤

When ε < 1, (4.15) implies

34

∂s∗
∂s∗
∂2S
|eα0 (ε−1)(p, e−α0 b) − eα1 (ε−1)(p, e−α1 b)|(p,b)|ε − 1|∂β∂β∂b2.
− (ε − 1) 5∂S
σ0∗(b/β0∗ )ε−1 − eα1 (ε−1)(p, b)∂b(A.24)∂s∗Since(p, e−α1 b) = σ0∗ > 0, one needs to make α1 → +∞, together with (A.23),∂β∂s∗to make eα1 (ε−1)(p, e−α1 b) go to 0. This condition is infact sufficient to make∂β∂s∗(p, e−α0 b) = 0,
the right hand side of (A.24) go to 0 when σ1∗ = 0, hence∂βeven when α0 is bounded below. Under the additional more general condition(A.22), where one may have σ1∗ > 0, a1 > 0, the absolute value of the numeratorof the right hand side of (A.24) is bounded above by the same expression statedin (A.16). Under the conditions (A.22), (A.23), this expression goes to 0 when
α0 → −∞, α1 → +∞. Q.E.D.
b35