Lattices and Lotteries

Elena Antoniadou, Leonard J Mirman, Richard Ruble∗,†.

September 25, 2009

Abstract

This paper develops techniques for comparative statics in the consumer

problem under uncertainty that involve order-based rather than topolog-

ical assumptions on utility functions and choice sets. We allow for imper-

fect substitutability and for a broad set of lottery choices. Building on the

value lattice approach to the consumer problem developed by Antoniadou

(1996), we introduce a partial order that integrates nonlinear pricing. We

discuss two extensions to take uncertainty into account, one based on sto-

chastic dominance and the other on lottery pricing. We identify specific

difficulties that arise in a general lottery choice setting (ranking budget

sets with respect to the strong set order, ensuring antisymmetry). We

then provide different ways to adapt our approach, depending on whether

choice involves small lotteries or whether partial orders can be refined, or

by appealing to a related technique that does not require antisymmetry.

Keywords: Lattice Programming, Choice under Uncertainty, Compara-

tive Statics

JEL Classification Codes C61, D11, D81

1 Introduction

Some of the most fundamental insights of economic science derive from com-

parative statics. The techniques of comparative statics under certainty are im-

portant tools, which yield insights into the effect of exogenous changes in the

environment on optimal choice. This is especially important in economics since

much of the empirical evidence is generated and can be explained by compar-

ative statics. Indeed, monotone comparative statics is at the heart of much

of the empirical evidence. This is best understood in the context of demand

∗Contacts: Elena Antoniadou: Australian National University,

elena.antoniadou@anu.edu.au, Leonard J Mirman: University of Virginia, lm8h@virginia.edu,

Richard Ruble: EM Lyon, ruble@em-lyon.com.†We would like to thank the Department of Economics, Australian National University,

and EM Lyon for their hospitality during this research project. We would also like to thank

an anonymous referee and participants at seminars at the Australian National University and

HEC Montréal for their comments and suggestions.

1

theory. The insights from demand theory e.g., normality and monotonicity, are,

in general, from the certainty context and only in special cases carry over to

uncertainty. However, economics is replete with examples in which uncertainty

plays a major role. E.g. the study of portfolio choice models and consumption

savings models with uncertain rate of return. In fact, the problem of portfolio

choice under uncertainty has been studied mainly in the context of univariate

preferences for money, where the effect of each good on the utility is exactly

the same. Monotone comparative statics are then derived in the very special

case of pure substitutes using the notion of risk aversion. However, the problem

is much deeper and more encompassing when the utility function is allowed to

depend on the effects of each of the goods separately, i.e. there is some degree

of complementarity between the goods, as in the consumption savings problem.

There are several reasons that the study of general comparative statics in

the economics of uncertainty has been neglected. In fact, the techniques of

comparative statics, based on differentiability and uniqueness, are not appro-

priate for studying uncertainty with general lotteries. These assumptions are as

inappropriate under uncertainty as under certainty, but the fact that lotteries

are, in general, infinite dimensional objects makes the use of the usual tools of

topology inconceivable. It is the purpose of this paper to study a general model

of comparative statics by introducing techniques that are suitable for the study

of comparative statics under uncertainty. We introduce monotone compara-

tive statics techniques in the lattice theoretic context generalizing the structure

for lattice programming already developed in the certainty case. We build a

foundation for the study of more complex comparative static problems using

generalized lattice theoretic techniques as well as a framework that can be used

to derive monotonic comparative statics integrating lotteries into constrained

maximization problems.

In order to study comparative statics using the usual topological methods it

is necessary that there be a locally unique maximum and enough differentiabil-

ity. These assumptions are used to employ the implicit function theorem as well

as other topological methods. This approach yields a local and, therefore, a nar-

row version of monotone comparative statics. The assumptions underlying the

analysis are very restrictive; for example, they do away with the possibility of

analyzing models of discrete choice, non-differentiability, and correspondences.

In the maximization problem under uncertainty conditions for monotone com-

parative statics that are a closer fit to the problems being studied are needed.

The difficulties that arise in the uncertainty case are both similar to the prob-

lems that arise with certainty as well as additional problems due to the more

general nature of distribution functions embedded in the use of lotteries.

Lattice programming was introduced in the certainty case to avoid the prob-

lems inherent in using topological methods and has been applied to constrained

maximization problems, e.g., without differentiability. In the certainty case

monotone comparative statics deals with choices on the real line, i.e., a chain

with the natural order. In our model, the choice variables are not, in general,

defined on a chain, but are lattices; hence, not all points are comparable. The

problem of choice under uncertainty that we study is a natural extension of

2

the certainty problem of maximizing utility by an agent subject to a budget

constraint. To study comparative statics with respect to changes in income

in the general problems of uncertainty, we generalize the lattice programming

techniques used in the certainty case.

The work of Veinott [13], and LiCalzi and Veinott [7] characterize superex-

tremal functions that yield monotone comparative statics under suitable monotonic-

ity of the constraint sets.1 In particular, various types of superextremal func-

tions are employed, which, when combined with strong monotonicity between

sets, yield variants of monotonic comparative statics. These methods are en-

tirely order based; only the notion of "bigger than" applies. Although suggested

by the Euclidean lattice, Veinott’s method encompasses a wide variety of spaces

and lattices and is applicable to a large assortment of monotone comparative

static problems. Thus, the Veinott method supports a more general approach to

the monotone comparative statics than does the use of Euclidean lattices. More-

over, the Euclidean lattice is not applicable to economic choice problems since

budget constraints are not compatible with Euclidean lattices. Hence, lattices

consistent with budget constraints must be used in order to employ Veinott’s

method.

Antoniadou [1] introduced a "value" lattice to employ Veinott’s method.

The value order makes the budget sets consistent with Veinott’s theorem. It

was shown by Antoniadou [1], and [3] and Mirman-Ruble [9] that when using

value lattices monotone comparative statics can be applied to a wide variety of

situations not covered by the traditional topological methods. In these applica-

tions the optimal choice may be a set or a correspondence, thus the comparative

statics are much richer, and the notion of monotonicity much more varied. To

study the comparative statics of a general maximization problem, Antoniadou

showed, the ordering of the constraint sets plays a pivitol role.

Although under certainty, prices play a role in determining comparative

statics, with uncertainty the price system plays a crucial role. The price sys-

tem that is consistent with lattices of general lotteries must be nonlinear and

therefore more complex than under uncertainty. One step in the direction of

understanding the effect of prices on lattices and monotone comparative statics

is to study lattices that are consistent with nonlinear prices in the certainty case.

To that effect we introduce an "expenditure" lattice to deal with the problem

of nonlinear prices and then extend this expenditure lattice to the uncertainty

case.

The application of lattice programming to models of uncertainty is -more

difficult than in the case of certainty. For example, monotonicity may be in

the form of the lattice representing first order stochastic dominance (FOSD),

second order stochastic dominance (SOSD), or the monotone likelihood ratio

property. Even in the case of expected utility the ordering of the constraint set

is different with each criterion and the comparative statics must take account

of these differences. Different notions of comparability and monotonicity give

rise to different notions of comparative statics. Moreover, the set of lotteries is

1See also Milgrom and Shannon [8] and Topkis [12] for other, related, functional properties.

3

not, in general, a chain but has a lattice structure. The underlying lattices that

are consistent with Veinott’s Theorem must be understood. Indeed, a straight-

forward generalization of the value lattices to uncertainty does not work. As

noted, the price system for lotteries also plays an important role. In particular,

the choice of a lottery rather that a single point, as in the certainty case, cannot,

except in very special cases, be described using a linear pricing system because

it is not the good itself that is being purchased but the lottery. There must

exist a price system that is consistent with the choice set and the value lattice

structure in order to yield results using Veinott’s Theorem.

In this paper, we begin by reviewing the structure of value lattices in the

certainty case and then introduce a new lattice, the expenditure lattice, to deal

with nonlinear prices in the certainty case. This is the expenditure lattice. The

value and expenditure lattices combined with the stochastic product lattices

form the basis for dealing with lattices and lotteries. However, as in the certainty

case, Euclidean lattices are not consistent with strong budget set orderings and,

therefore, are not well suited to use with Veinott’s Theorem.

We next introduce two orders in the uncertainty case. The first is the sto-

chastic value lattice for the general stochastic choice problem when the set of

lotteries are ordered by FOSD (or SOSD). We then extend the expenditure value

order, introduced in the certainty case. The lottery expenditure value lattice,

defined over the entire set of lotteries, provides a link between the two classes of

orders (certainty and uncertainty). We show that under a strong condition on

the lottery price function, the lottery expenditure value order enables a lattice

and is consistent with the strong set order, which is necessary to apply Veinott’s

Theorem. The use of these two lattices allows us to study comparative statics

for ordinal utility functions. Thus our conclusions are consistent with both ex-

pected and nonexpected utility. In the former case we give conditions for ordinal

preferences to be supermodular and, therefore, LSE in our stochastic lattices.

To study the stochastic value lattices, we turn to the natural orders on lot-

teries. Firstly we consider the FOSD.. The first order stochastic value (FOSV)

order enables a lattice. However for the FOSV lattice another difficulty, not

encountered in the expenditure value lattice approach, in applying Veinott’s

Theorem, emerges. Since the lottery space is a lattice, the sup of two incom-

parable points may be outside the budget space and, thus, incompatible with

the strong set ordering necessary to invoke Veinott’s Theorem. Even though

the FOSV lattice is a value lattice, Veinott’s Theorem may not apply, since

budget constraint compatibility may fail due to this price and lattice incompat-

ibility. This problem does not arise with the expenditure value lattice because

the strong assumption we employ on prices has the effect of making the set of

lotteries a chain on prices and thus it is compatible with Veinott’s Theorem.

There are several ways to address the problem of the price lattice incom-

patibility introduced by the stochastic value lattice. We adapt the stochastic

value lattice so it is compatible with the price lattice. The first adaptation is

to assume that the set of FOSD lotteries is a chain. In this way the sup of two

lotteries, is one of the lotteries and, therefore by construction, satisfies the ap-

propriate budget constraint. In this case, Veinott’s Theorem is applicable and

4

monotone comparative statics follows. In the case the FOSD lattice is restricted

to be a chain the expenditure lattice and the value lattice are identical and yield

the same result. However, except when restricted to chains, these lattices are

different. A second approach is to assume that the support of the lotteries is

bounded above. This is the case when the lottery choice is small relative to

income. In this case, the price of any lottery is also bounded above and thus the

price of the sup satisfies the budget constraint and therefore, Veinott’s Theorem

may be invoked. The same approaches can be used in the case of the stochastic

value lattice with SOSD.

Finally, the expenditure order may be refined without resorting to the strong

assumption on prices. Instead of insuring that different lotteries have different

prices, suppose that the set of distinct lotteries with the same price has an

inf and a sup, then the partial order defines a lattice and Veinott’s Theorem

can be applied. Each of these approaches represents a different avenue towards

application of Veinott’s Theorem and their suitability depends on the partic-

ular problem investigated and the particular monotone comparative statics of

interest.

2 Background and Preliminary Results

We provide a summary of lattice methods (Subsection 2.1) and review existing

results on the application of Lattice Programming (LP) to comparative statics

with budgetary tradeoffs under certainty (subsection 2.2). The reader may refer

to Antoniadou [1], [3] and Mirman and Ruble [9] for further material, and also

to Li Calzi and Veinott [7], Veinott [13], Milgrom and Shannon [8] and Davey

and Priestly [6].

In the second half of Section 2 we introduce the consumer problem under

uncertainty which is studied in the remainder of the paper (Subsection 2.4) and

give some preliminary results. In Subsection 2.3 we introduce a new order (and

hence, a new lattice) that complements our earlier work in the certainty case,

the expenditure value order. The resulting lattice framework is built on a gen-

eral nonlinear pricing function, and thereby allows for constraint sets that are

nonlinear. Finally Subsection 2.5 takes a first look at stochastic lattices, by con-

sidering a product lattice using First Order Stochastic Dominance (we address

the product lattice with Second Order Stochastic Dominance in Appendix 5.2).

2.1 Lattice programming tools for comparative statics

In economics, comparative statics is typically studied using topological meth-

ods. These methods require important restrictions. In order to apply the main

tool, the implicit function theorem, to an optimal solution, the problem must

be smooth enough. It therefore does not apply in the absence of continuity

(as with discrete choices), differentiability (as with Leontief functions), or in

the presence of nonconvexities (as with multiple non-locally unique solutions).

These assumptions are made on technical grounds, and are not rooted in empir-

5

ical observation of behavior. In addition, the application of this methodology is

not straightforward on function spaces (the subject of this paper).

LP is an alternative approach for comparative statics that does not impose

smoothness restrictions. Using order-based properties, LP exploits the natural

relationships inherent in the optimization problem, and the ensuing monotone

comparative statics results are global.

Let (X ≤X ) be a lattice. The following two sets of concepts, one relatedto sets and the other to functions, are needed to state the main comparative

statics theorem:

Definition 1 (Set orders - Veinott [13]) Let and be two sets in (X ≤X ).Strong set order: ≤ iff for all ∈ , 0 ∈ , ∧ 0 ∈ and

∨ 0 ∈ .

Chain-lower-than: ≤ iff ≤ and all ∈ , ∈ are compara-

ble.

Strongly-lower-than: ≤ iff for all ∈ , ∈ , ≤X .

Definition 2 (Supermodular - Superextremal properties) Let : X → < be a

real-valued function on (X ≤X ).2 is supermodular (SM) iff, for all 0 ∈ X ,

( ∨ 0) + ( ∧ 0) ≥ () + ( 0) , (1)

is lattice superextremal (LSE) iff, for all 0 ∈ X ,() ≥ () ( ∧ 0)⇒ ( ∨ 0) ≥ () ( 0). (2)

is strictly superextremal (SSE) iff, for all incomparable 0 ∈ X ,() ≥ ( ∧ 0)⇒ ( ∨ 0) ( 0). (3)

The SM property is cardinal, whereas the LSE and SSE properties are ordi-

nal. An SM function is LSE, and an increasing transformation of an SM function

is LSE (while an affine transformation of an SM function is SM).

A real-valued function on a poset (X ≤X ), : X → <, is increasing if ≤X (X ) 0 ⇒ () ≤ () ( 0), and nondecreasing if ≤X 0 ⇒ () ≤ ( 0). An increasing function on a lattice is SSE (thus LSE, but

not necessarily SM). The power of the ordinal lattice theoretic properties for

comparative statics resides in Theorem 3 (Veinott’s theorem), which establishes

equivalence between the LSE (SSE) property and the ordering of optimum sets:

Theorem 3 (Veinott [13], Li Calzi and Veinott [7])3 Given a lattice (X ≤X ),let : X → < and ⊆ X . Then is LSE (SSE) if and only if argmax ≤(≤) argmax for all ⊂ X with ≤ and argmax , argmax 6=∅.

2 SM is used in Topkis [12], while Li Calzi and Veinott [7] and Veinott [13] define LSE

and SSE functions, and other variants. Milgrom and Shannon [8] define quasi-supermodular,

which is the same as LSE.3Milgrom and Shannon [8] also have the LSE part of Theorem 3.

6

2.2 Value lattices and comparative statics

We focus on choice problems with two goods. Under certainty and linear prices

∈ <2++, the consumer choice problem is:

max∈<2+

() s.t. · ≤ . (4)

The consumption set, <2+, is usually ordered with the Euclidean order ≤.It is therefore natural to begin by embedding this problem in the Euclidean

lattice¡<2+≤¢. One might try to reason as follows in order to character-

ize the income effects with respect to good . If the constraint sets () =©( ) ∈ <2+, + ≤

ªare increasing in (with respect to the set order-

ing ≤), then by Veinott’s Theorem (Theorem 3), if is LSE (SSE) in¡<2+≤¢,

the set of optimal choices is increasing (with respect to the set ordering≤ (≤)),which implies normality of both goods and thus normality of good .

Such reasoning yields an economically vacuous result: any increasing utility

function is SSE on¡<2+≤¢, so optimum sets increase over any pair of strong

set ranked constraint sets. Since clearly not all increasing utility functions yield

all normal goods, this suggests that the LSE restriction is not informative with

regard to normality (or for that matter, price effects). The reason for this is

that this lattice is not suited to the problem of consumer choice. The strong set

order induced by the Euclidean order does not rank budget sets, i.e. for ≤ 0,() £ (). Relevant joins may fail to belong to the bigger set when this

involves budgetary trade-offs:

Example 4 Let =³0

´and 0 =

³ 0´, so ∈ () and 0 ∈ ( 0).

Then, ∧ 0 = (0 0) ∈ (), but ∨ 0 =³

´∈ ( 0). Therefore,

() £ (0).

The application of Theorem 3 in¡<2+≤¢ is thus not informative for shifts

in budget sets.4 However, this limitation results from the choice of the lat-

tice,¡<2+≤¢, and not from Veinott’s Theorem. The use of the Euclidean

lattice is not fruitful because the lattice method requires that the ordering of

the underlying consumption space be consistent with the problem being stud-

ied. Comparative statics thus requires the determination of a lattice in which

strong constraint set comparability is enabled, and thus the superextremal vari-

ant conditions can be used.

It is this insight that led Antoniadou [1] to introduce a partial order adapted

to the context of income effects, the direct value order.

4 In fact, the problem is deeper than just pertaining to income effects. A subset of the

Euclidean lattice is not a Euclidean sublattice unless it is a box. Therefore, a constraint

set with budgetary trade-offs cannot be a Euclidean sublattice, and by extension relevant

constraint sets cannot be ordered by the strong set order.

7

Definition 5 (Antoniadou [1], [3]) Let ∈ <++×<+ Consider ( ) (0 0) ∈<2+. The direct value order (for two goods) is:5

( ) ≤() (0 0)⇔½

≤ 0

· ( ) ≤ · (0 0) . (5)

The direct value order ranks bundles by both their value and the amount of

one of the two goods (here, ). It thus reflects the fact that what matters for

the income effect for one good is not whether one bundle is larger than another

in the Euclidean sense, but whether the bundle reflects a higher consumption

of one of the goods, and whether this bundle is more valuable at market prices.

The direct value order defines a non-Euclidean lattice on the consumption set,¡<2+≤()¢, to which standard LP techniques can be applied to characterizenormality.

For two incomparable points = ( ) and 0 = (0 0) with 0 and+

0 + 0, the join and meet in

¡<2+≤()¢ are: ∨ 0 =

µ0 −

( − 0)

¶and ∧ 0 =

µ+

( − 0) 0

¶.

By construction, this lattice solves the problem of budget set comparability,

as · ( ∨ 0) = ( ·) ∨ ( · 0) (also, · ( ∧ 0) = ( ·) ∧ ( · 0)):hence, () ≤ ( 0) for ≤ 0. Theorem 3 can therefore be invoked to

conclude that, when is LSE (SSE), the optimal choice set is nondecreasing

with respect to ≤ (≤) in income, which implies that the optimal choice set of is nondecreasing in income in a set theoretic sense, i.e. the good is normal.6

For this, and most other economic problems, the application of Theorem 3

generally relates to sufficiency: when satisfies a superextremal property, the

behavior of the set of optimizers is determined. The necessity part of Veinott’s

Theorem places restrictions on optimal behaviour over constraint sets that do

not naturally arise in economic optimization problems.

Multiple optimum solutions can arise, since (strict) quasiconcavity is not

imposed on the objective, and are consistent with Theorem 3. This means that

the notion of a normal good must be refined. We say that good is pathwise

normal if every optimal choice at high income is greater than or equal to some

optimal choice at low income, and every optimal choice at low income is smaller

than or equal to some optimal choice at high income.7 We say that good is

strongly normal if every optimal choice at high income is weakly greater than

every optimal choice at low income.

Several standard preferences, such as the Cobb-Douglas ( ) = , are

LSE (SSE) on some or every direct value lattice, thus establishing normality.

Moreover, preferences remain LSE in the sublattice¡<× ℵ≤()¢ in which

5Antoniadou [1] extends this order to many goods using the lexicographic order.6Because the up sets differ in

<2+≤() and <2+≤, utility functions are not gen-erally increasing with respect to ≤() and the LSE property is not a trivial restriction.

7Antoniadou [1] defines the corresponding set relation, pathwise-lower-than.

8

the choice of good is discrete, which is beyond the scope of the implicit function

approach to comparative statics.

As there are many partial orders similar to the direct value order, Antoni-

adou’s work is the source of a whole class of value lattices which are particularly

suited to budgetary tradeoffs.8 We next introduce a new lattice that is of in-

dependent interest in the certainty case and has applications in the uncertainty

case.

2.3 The expenditure value order

Antoniadou [1], [3], and Mirman and Ruble [9] focus on the consumer problem

under linear pricing, but the method is readily extended to address nonlinear

pricing. This is particularly useful in the uncertainty case since a general price

function is more suitable for lotteries.

Suppose that the price of is given by an increasing function () : <+ →<+. The linear price () = is a special case. Then, the following is a

partial order on <2+:

Definition 6 Let = ( ()) where ∈ <++ and () : <+ → <+ an

increasing function. Consider = ( ) 0 = (0 0) ∈ <2+. The expenditurevalue order is:9

≤(()) 0 ⇔½

() ≤ (0)+ () ≤

0 + (0). (6)

Reflexivity and transitivity are straightforward, and antisymmetry follows

from the fact that () is increasing and thus uniquely identifies the level of

. The expenditure value order requires that the expenditure on a good is

non-decreasing, which in the certainty setting is equivalent to the level of the

good being non-decreasing, since () ≤ () (0) if and only if ≤ () 0 bythe invertibility of the price function. This monotonicity allows us to define

normality in the same way as under linear pricing, using the expenditure value

order in a more general non-linear pricing context. However, under uncertainty,

and more generally in the context of a function space, this equivalence does not

hold so readily. The monotonicity of expenditure and the monotonicity of the

choice set are not in general identical.

The expenditure value order enables a lattice on the consumption set, the

expenditure value lattice,¡<2+≤(())¢. For two incomparable points and

0 with + () 0 + (0) and () (0), the join and meet are:

∨ 0 =µ0 − ()− (0)

¶and ∧ 0 =

µ+

()− (0)

0¶. (7)

8For example, Mirman and Ruble [9] introduce the radial value order: ( ) ≤() (0 0)if and only if 0 ≥ 0 and + ≤

0 + 0.

9We define the expenditure value order over two goods. Multi-dimensional extensions can

be constructed, for example, following Antoniadou [1] in the case of the direct value order, or

as in Mirman and Ruble [9].

9

As 0 − ()−(0)

≥ 0 for such incomparable points, ∨ 0 ∧ 0 ∈ <2+so¡<2+≤(())¢ is a lattice. Moreover, budget sets are strong set ranked in¡<2+≤(())¢ and thus Theorem 3 can be applied to give sufficient conditions

for good to be normal:10

Proposition 7 Let = ( ()) where ∈ <++ and () : <+ → <+an increasing function. Consider the consumer problem max∈<2+ () s.t.

∈ (), where () =© = ( ) ∈ <2+, + () ≤

ª If the utility

function is LSE (SSE and the budget constraint is binding) on¡<2+≤(())¢

then argmax() ( ) ≤ (≤) argmax(0) ( ) for all ≤ 0 and good

is pathwise (strongly) normal.

The expenditure value order and lattice allow us to address comparative

statics under arbitrary pricing rules, which could involve quantity discounts or

premia, or a combination.11 In all cases the ensuing budget frontier is downward

sloping and involves tradeoffs between the two goods. Budget sets differing by

the level of income cannot be strong set ordered in the Euclidean lattice, and

may not be strong set ordered in any direct value lattice, but they can always be

strong set ordered in the relevant expenditure value lattice. Figure 1 illustrates

this in the case of a quantity discount on good . The shaded area is the up set

of 0. The join and meet of two incomparable points, which lie on the relevantconstraint sets, are shown.

The two lattices¡<2+≤(())¢ and ¡<2+≤()¢ are identical when the

price function is linear. In the certainty setting, the expenditure value lattice

generalizes the direct value lattice under non-linear pricing, while preserving the

relation between expenditure and underlying quantity. In the uncertainty case,

however, the relationship between the price function and the underlying lottery

is not necessarily one-to-one or increasing, so stochastic generalizations of these

lattices yield different results.

2.4 Consumer problem in the presence of a risky good

In the uncertainty problem we study, the consumer faces a choice between a

deterministic good, and a second good that is subject to risk. The consumer

has preferences ( e) defined over the consumption set <+ × F, where Fis the set of distribution functions on <+. We denote a typical element of

the consumption set as ( e) where e is the random variable, a lottery, with

distribution function ∈ F Compared with the classic Arrow-Pratt model ofchoice under uncertainty, our formulation offers two important generalizations:

10The expenditure value order addresses a criticism of our earlier work by Quah [10], namely

that it only deals with the consumer problem under linear pricing. Proposition 7, like Quah’s

“C-flexible” set order deals with constraint sets that are horizontal translations (as income

changes).11The expenditure value order as defined here does not allow for bundling of the two goods,

only quantity bundling of one good. In order to allow for product bundling a general price

function ( ) must be used.

10

x

X X'

X X'X

X'

y

-q

A B

Figure 1: The Expenditure Value Order

(1) we do not assume that the two goods are perfect substitutes, and (2) there is

not a single underlying lottery or distribution function which can be purchased

in different quantities. Notice that in our formulation the quantity of a lottery

purchased need not be explicitly specified, since it is embedded in the availability

of different distribution functions on <+. Degenerate distribution functions

which remove all uncertainty are also available.

The consumer chooses a single distribution function or lottery and therefore

there is no arbitrage available. The prices of the lotteries facing the consumer

need not be aggregated from the prices of (a continuum of) underlying contin-

gent commodities. This problem cannot be mapped to a (finite dimensional)

Arrow-Debreu model with contingent commodities and it is therefore different

from the standard consumer problem under certainty. Even if it is mapped to

a contingent commodity space, this would have to be an infinite dimensional

function space, with all the difficulties that that presents. Our model is techni-

cally different, and makes available more possibilities about what is available to

the consumer.

The sufficient conditions we derive relate to properties of ordinal utility.

This differs from the standard analysis which seeks to derive sufficient (and

sometimes necessary) conditions based on cardinal utility in an expected utility

framework, and in particular based on risk aversion and related concepts. This

has the important implication of not being restricted to the expected utility

framework, thus enabling a framework which compares the comparative statics

conclusions of expected and non-expected utility.

11

Under the expected utility hypothesis the consumer objective is

( e) = Z ( ) () . (8)

The state utility function is defined over <2+ and represents the individual’spreferences over the realizations of consumption. The perfect substitutes speci-

fication (+ ) is a special case of this formulation.

In the presence of uncertainty and in the rich consumption set we assume,

the concept of change in the consumption of the risky good is not as obvious as

in the certainty case, or when there is only one underlying lottery. The notion

of increase in a bundle ( e), and in particular of an increase in e must bespecified. This is straightforward in the certainty case and when there is only

one underlying distribution function in the consumption set, i.e. all that can

change is the quantity of the specified distribution function. However, in the

general case, it is necessary to specify how different distribution functions are

to be compared. A natural starting point for this is to use the standard notions

of stochastic dominance, First and Second Order Stochastic Dominance (FOSD

and SOSD).

In the case of income effects under certainty, even when a good is desirable,

it is not necessarily the case that more of it is purchased as income increases.

Sufficient conditions in the consumer problem under certainty, based on the class

of value orders, ensure that a more desirable consumption of the good is chosen

as income increases. When this is translated to the uncertainty framework, at

least under expected utility, a more desirable lottery for everybody, irrespective

of (increasing state) preferences, is one that first order stochastically dominates

another. We study the comparative statics question of when such a desirable

change, a first order stochastically dominant change, in the consumption of

the risky good occurs when income increases. A more restricted notion of more

desirable is represented by SOSD. A more desirable lottery from the perspective

of a risk averse expected utility maximizer is one that second order stochastically

dominates another. Therefore,we also study SOSD changes in the consumption

of the risky good occurs as income increases.

In Section 3 we construct stochastic value orders and lattices to address

these comparative statics questions. However, before that in Subsection 2.5

we study the product lattice that arises from crossing the usual order and the

FOSD order. This is a counterpart of the Euclidean lattice in the deterministic

problem. Even though it is not suitable to comparative statics with budgetary

trade-offs, it offers an introduction to the stochastic value orders of Section 3.

Furthermore, it allows us to adapt the result linking desirability with FOSD,

to a lattice theoretic framework with bivariate preferences, and to link lattice

theoretic properties of cardinal and ordinal preferences in the expected utility

framework (in Appendix 5.2 we discuss the corresponding product lattice with

the SOSD order).

12

2.5 Preferences and the FOSD product lattice

Like the Euclidean product lattice under certainty, the FOSD product lattice

does not enable ranking of budget sets with respect to the strong set order, and

therefore does not characterize the comparative statics of choice under uncer-

tainty with budgetary tradeoffs. However, the product lattice provides a con-

venient setting in which to study the relationship between the superextremal

properties of the state utility function, and expected utility, . We show

that this relationship is not straightforward, even in the simple product lattice

setting, in the sense that superextremal properties of the state utility function

are not necessarily inherited by the expected utility function.12 The relation-

ships that we establish do not carry over to the more complex stochastic value

lattices.

FOSD defines a partial order on the space of distribution functions. Let

≤ be defined on F by e ≤ e0 if and only if 0 () ≤ () for

all ∈ <+. In fact (F≤) is a lattice. If e e0 are incomparablelotteries (with distribution functions

0 ∈ F respectively), their join ise∨ = e ∨ e0 with ∨() = [ ∨ 0 ] () = min {() 0()} and their

meet is e∧ = e ∧ e0 with ∧() = [ ∧ 0 ] () = max {() 0()} (i.e. the component-wise min and max respectively).

The FOSD product order is defined as ( e) ≤(×) (0 e0) if and onlyif ≤ 0 and e ≤ e0. With this order, the consumption set is a lat-tice,

¡<+ ×F≤(×)¢, the FOSD product lattice. The analogy with theEuclidean lattice

¡<2+≤¢ is not perfect, because (F≤) is itself a lattice,and not simply a chain. As a result, if the two associated distributions are not

ranked by ≤, two bundles can be incomparable even though they havethe same quantity of the sure good . However, in a lattice over a restricted

consumption set <+ × F where F

⊆ F is a chain of distribution functionswith respect to ≤, the analogy is better.For the same fundamental reasons that the Euclidean lattice

¡<2+≤¢ inthe certainty case is not well suited to problems with budgetary trade-offs, the

FOSD product lattice¡<+ ×F≤(×)¢ is not well suited to such prob-

lems in the uncertainty case. The manifestation of the problem is again that

budget sets are not strong set comparable, with joins of incomparable pairs

in¡<+ ×F≤(×)¢ not necessarily belonging to the larger budget set.

In fact, since the set of distribution functions includes all degenerate lotteries,

strong set comparability in the Euclidean lattice is necessary for the correspond-

ing comparability in the FOSD product lattice. The same observation estab-

lishes that lattice theoretic properties in the Euclidean lattice are necessary for

the corresponding property in the FOSD product lattice. That is, ( e) hasa lattice property (SM, LSE, SSE) in

¡<+ ×F≤(×)¢ only if ( )has the corresponding property in

¡<2+≤¢. In the case of the SM property,

12The product lattice is used by Athey [5] to derive results on comparative statics under

uncertainty in a setting (distributions are ordered with respect to maximum likelihood ratio

rather than stochastic dominance), in which the consumption set of lotteries is a parameterized

family of distributions.

13

sufficiency also holds:13

Proposition 8 Suppose ( e) = R( ) (). Then is SM in the

FOSD product lattice¡<+ ×F≤(×)¢ if and only if is SM in

¡<2+≤¢.Proof. (⇒) Follows from the fact that F contains the degenerate distributions.(⇐) Suppose that is SM, and let = ( e) and 0 = (0 e0) be two

incomparable points in <+ × F with 0 ≥ and e0 ¤ e. is SM ifR( )()+

R(0 )0() ≤

R( )∧() +

R(0 )∨(), which

can be rewritten as:Z( ) [() + 0()] +

Z[(0 )− ( )] 0() (9)

≤Z

( ) [∧() + ∨()] +Z[(0 )− ( )] ∨().

With the ≤ order, () + 0() = ∧() + ∨() for all , so thefirst terms on each side of the inequality are identical. Since is supermodular,

(0 ) − ( ) is either 0 (if 0 = ), or nondecreasing in . Therefore, sincee∨ e0,Z[(0 )− ( )] ∨() ≥

Z[(0 )− ( )] 0(), (10)

so (9) holds.

This result is special to the SM property and does not extend to the ordinal

lattice theoretic properties. We show this by example in the case of the LSE

property (the rest follow). Since the SM property is sufficient for the LSE prop-

erty Proposition 8 gives a sufficient condition for the expected utility function

to be LSE in¡<+ ×F≤(×)¢.

Example 9 Consider the points (sublattice) in <2+ : {0 1}×{0 1 2 3} Suppose takes the values 3 2 0 2 at the point (0 0) (0 3) respectively, and the

values 2 0 2 3 at the points (1 0) (1 3), respectively. Then is LSE at

these points (there are six incomparable pairs to verify) but not increasing. Lete = {05 05; 1 3} and e0 = {05 05; 0 2}, so that e0 e; (0 e) (1 e0) ∈¡<+ ×F≤(×)¢ are incomparable, with (0 e)∨(1 e0) = (1 e) and (0 e)∧(1 e0) = (0 e0). Then (0 e) = 05 × 2 + 05 × 2 = 2 05 × 3 + 05 × 0 = ((0 e) ∧ (1 e0)), and (1 e0) = 05 × 2 + 05 × 2 = 2 05 × 3 + 05 × 0 = ((0 e) ∨ (1 e0)). Thus is not LSE.14

Thus, the LSE property of the state utility is necessary for the LSE property

of the expected utility in the FOSD product lattice, but not sufficient. The SM

13Athey [4] studies these issues using convex cones. Our proof of Proposition 8 is less general

but direct.14The expected utility function is not LSE and therefore it is not SM either, which from

Proposition 8 can only be true if the state utility function is not SM in the Euclidean

(sub)lattice. This can be verified in this example.

14

property provides a sufficient condition for LSE, as does nondecreasing state

utility (provided the state utility is LSE). The case of the SSE property is sim-

ilar. These sufficiency results rely on the desirability of First Order Stochastic

Dominant distributions. They are stated in Proposition 10:

Proposition 10 Suppose ( e) = R( ) (). Then: (i) If ( )

is nondecreasing (in and ) and LSE in¡<2+≤¢ then ( e) is LSE in¡<+ ×F≤(×)¢ and (ii) if ( ) is increasing in and then ( e)

is SSE in¡<2+ ×F≤(×)¢

Proof. See Appendix 5.1.

The characterization of lattice properties is more delicate in the case of the

related SOSD product lattice. These questions are taken up in Appendix 5.2.

3 Stochastic Value orders and lattices

We now construct stochastic value lattices to carry out comparative statics in

the consumer problem under uncertainty. Recall that the consumption set is

<+ ×F or a subset therein, X ⊆ <+ ×F. The consumer problem is:

max()∈X ( e) s.t. + (e) ≤ . (11)

We need not restrict ourselves to ordinal preferences that are derived under the

expected utility hypothesis, and unless explicitly stated, we do not assume that

ordinal preferences satisfy the expected utility hypothesis. Under the expected

utility hypothesis, ( e) = ( e) In this case we assume that the stateutility function, , is increasing in both elements. Therefore, from Proposition

10, the expected utility function is SSE in¡<+ ×F≤(×)¢.

Observe that the price of the risky good is not linear. The lottery price

function (e) in the budget constraint is an important element of the analysisand is addressed separately in Subsection 3.1. Three stochastic value lattices

are constructed. In Subsection 3.2 we construct a value lattice using FOSD, the

First Order Stochastic Value lattice (FOSV). In Subsection 3.3 we adapt the

expenditure value order to the stochastic framework and construct the Lottery

Expenditure Value lattice (LEV). The FOSV lattice deals with choices that are

comparable with respect to FOSD. In the LEV lattice, the notion of increase is

defined with respect to the lottery price function (e), which may or may notreflect stochastic dominance of the underlying lotteries. The two lattices are

compared in Subsection 3.4. Finally in subsection 3.5 we construct the Second

Order Stochastic Value lattice (SOSV) using SOSD. With all of these lattices,

the lottery price function (e) plays a significant role.3.1 Lottery price function

We use a general lottery price function, (e) : F → <+, to represent thecost of a given lottery. This corresponds to the price function () discussed

15

in the certainty case,15 with one important difference. In the certainty case

the domain of the price function is a chain, whereas here it is (a lattice over) a

function space. Thus, while the invertibility of the price function in the certainty

case is an innocuous assumption, this is not the case with uncertainty. Under

uncertainty, invertibility is a much more restrictive assumption. Although it is,

in principle, possible that the price function is one-to-one since F and <+ havethe same cardinality.

Some structure on (e) is needed to construct value lattices. To begin, fix (0) = 0 where 0 = {0; 1} is the degenerate lottery at 0. More substantively,the lottery price function is assumed to be consistent with the partial order on

the lottery space, where one is used. In the case of the FOSV lattice the price

function is required to be consistent with ≤:

e ≤ () e0 ⇒ (e) ≤ () (e0) . (A1)

Assumption (A1) states that a lottery that (strictly) dominates another by

≤ is (strictly) more expensive. In other words, the ordering on prices

reflects the ordering on lotteries. Under the expected utility hypothesis and

increasing state preferences, a lottery that is preferred by everybody would cost

more than a less preferred lottery.

In the case of the SOSV lattice, the price function is required to be consistent

with ≤:

e ≤ () e0 ⇒ (e) ≤ () (e0) . (A10)

For the LEV lattice, when there is not necessarily an underlying order on

the set of lotteries, we assume that the lottery price function is one-to-one:

(e) = (e0)⇔ e = e0. (A2)

This property is strong, insofar as no two distinct lotteries have the same price,

but is instrumental in the construction of the LEV lattice. It also makes a chain

of the space of lotteries F, by means of the lottery expenditure order ≤()defined by e ≤() e0 if and only if (e) ≤ (e0).Assumption (A1) can be derived from a more primitive pricing structure,

fair pricing of lotteries, based on the prices of the underlying outcomes. Ifb () : <+ → <+ is an (increasing) price function over the possible outcomes,the fair price of a lottery F is given by:

(e) = Z b () (). (12)

From (12), the prices of the join and meet of two incomparable lotteries, e e0with respect to ≤ satisfy:

(e∧) + (e∨) = (e) + (e0) . (13)

15 If = {; 1} is the degenerate lottery at , () = ().

16

Assumption (A1) follows directly. However, Assumption (A2) need not hold

under fair pricing.

Under fair pricing (with b () also linear) the lottery price function is linear inthe amount purchased, with (e) = (e) for all ∈ <+. However, (A1) and(A2) do not imply linear pricing. Furthermore, linear pricing, (e) = (e)for all ∈ <+ does not suffice to compare the prices of distributions that arecomparable with respect to FOSD but are not generated by shifts in the same

underlying distribution, as in (A1), and it does not imply (A2).

3.2 FOSV lattice

We seek a lattice framework that can both capture the idea of FOSD increases

in the lottery choice e and increased value of a bigger bundle. We propose alattice that builds on the value lattices in the certainty case, ordering bundles

with respect to both their value and the comparative statics variable — the

lottery choice. In fact, as F contains the degenerate distributions, this orderis a generalization of the direct value order.

Definition 11 Let = ( (e)) where ∈ <++ and (e) : F → <+.Consider ( e) (0 e0) ∈ <+ × F. The First Order Stochastic Dominance

Value (FOSV) order is:

( e) ≤ () (0 e0)⇔ ½ e ≤ e0+ (e) ≤

0 + (e0) . (14)

The FOSV order is a partial order and it generates the First Order Stochastic

Value (FOSV) lattice on the consumption set.

Proposition 12 Let = ( (e)) where ∈ <++ and (e) : F → <+ sat-isfies (A1). Then

¡<+ ×F≤ ()¢ is a lattice. For incomparable ( e) (0 e0) with, without loss of generality, + (e) ≤

0 + (e0), their joinand meet are given by:

( e) ∨ (0 e0) = µmax½0 + (e0)− (e∨)

0

¾ e∨¶ (15)

e∨ = e ∨ e0 with ∨ () = min© ()

0 ()

ª

and

( e) ∧ (0 e0) = µ+ (e)− (e∧)

e∧¶ (16)

e∧ = e ∧ e0 with ∧ () = max© ()

0 ()

ª

Proof. Consider an incomparable pair ( e) (0 e0) with + (e) ≤ 0+

(e0). Both ∨ = (∨ e∨) = µmax

½0 +

(0)−(∨)

0

¾ e∨¶ and ∧ =µ

+()−(∧)

e∧¶ are well-defined in <+ × F. To check that ∨ is the

17

least upper bound, first observe that ∨ is an upper bound of ( e) (0 e0) since

∨ + (e∨) = max {0 + (e0) (e∨)} ≥ 0 + (e0) Consider

any other upper bound, (00 e00) ∈ <+ × F, i.e. ( e) (0 e0) ≤ ()(00 e00). Hence e∨ ≤ e00, and by (A1), (e) (e0) ≤ (e∨) ≤ (e00).If ∨ = max

½0 +

(0)−(∨)

0

¾= 0, (e∨) ≤

00 + (e00). Otherwise,max

½0 +

(0)−(∨)

0

¾= 0+

(0)−(∨)

, so ∨+ (e∨) =

0+ (e0) ≤

00 + (e00). In both cases, (∨ e∨) ≤ () (00 e00). The argument forthe meet is similar and is omitted.

The FOSV lattice is a natural generalization of the direct value lattice¡<2+≤()¢ to the uncertainty case. However, compared with the certaintycase, the expression for the deterministic good in the join,max

½0 +

(0)−(∨)

0

¾,

is more complex. This results from the fact that for lotteries e e0 that are≤-incomparable the join with respect to ≤ is a distinct lottery, thatis e∨ 6= e e0. By (A1), the join lottery e∨ satisfies (e∨) (e) (e0),and 0 +

(0)−(∨)

may be negative. Hence the value of the join bundle is

max {0 + (e0) (e∨)}, i.e. the price of the join lottery can be so high as torender it unaffordable under the same budget constraints as the original pair.

(A1) does not restrict the magnitude of the price of the join. Hence, strong bud-

get set comparability may fail and therefore, Veinott’s theorem does not apply.

This result would not change under fair pricing. In other words, if the price

of the join lottery makes the join more expensive than the constituent bundles,

then budget sets are not ranked with respect to ≤, as the next example shows.

Example 13 Let = (0 e) and = (0 e0) be two incomparable bundles, with (e0) (e), and take = (e) and 0 = (e0). Then, ( 0) ¤ (). We

have ∈ () and 0 ∈ ( 0), and ∧ 0 =µ()−(∧)

e∧¶ ∈ () (note

that ∧ 0 ∈ <+×F since (e) (e∧)). However, ∨ 0 = (0 e∨) ∈ ( 0):as e∨ e e0, by (A1) (e∨) (e0) = 0.

The FOSV lattice thus fails to be an immediate extension of the LP method

to the uncertainty problem when the lottery space is a general lattice. It does

work however, insofar as budget sets are ranked in the lattice and Veinott’s

Theorem can be applied, when the underlying space of lotteries is restricted to

be a chain, in which case e∨ ∈ {e e0}. We return to this point and propose othersolutions in Section 4, but first we examine another extension of the certainty

case, the LEV lattice.

3.3 LEV lattice

An alternative approach is to extend the expenditure value order we introduced

in the case of certainty to deal with nonlinear prices to <+ × F. The lottery

18

expenditure value order imposes no quantitative order on the set of lotteries,

but is consistent with the nonlinear lottery price function (e).Definition 14 Let = ( (e)) where ∈ <++ and (e) : F → <+ satis-fies (A2). Consider ( e) (0 e0) ∈ <+ × F. The Lottery Expenditure Valueorder is:16

( e) ≤ (()) (0 e0)⇔ ½ (e) ≤ (e0)

+ (e) ≤ 0 + (e0) . (17)

Assumption (A2) on the lottery price function ensures that antisymmetry

holds and therefore the lottery expenditure value order is a partial order. (A2)

is much more restrictive than the corresponding assumption in the certainty

case. Nonetheless, under (A2) the consumption set is a lattice with the lottery

expenditure value order, the Lottery Expenditure Value (LEV) lattice. This is

stated in Proposition 15. The proof is straightforward and is omitted.

Proposition 15 Let = ( (e)) where ∈ <++ and (e) : F → <+satisfies (A2). Then

¡<+ ×F≤ (())¢ is a lattice. For incomparable

( e) (0 e0) with, without loss of generality, (e) (e0) and + (e)

0 + (e0), the join and meet are:( e) ∨ (0 e0) =

µ0 − (e)− (e0)

e¶ and (18)

( e) ∧ (0 e0) =

µ+

(e)− (e0)

e0¶ . (19)

The LEV lattice avoids the problem of affordability of the join lottery in the

FOSV lattice, because under Assumption (A2) the set of distribution functions

is a chain (with the price order ≤()). Therefore, in contrast to the FOSVlattice, budget sets are strong set comparable, i.e. () ≤ ( 0) for ≤ 0 in¡<+ ×F≤ (())¢. Hence, Theorem 3 can be applied:

Proposition 16 Consider the consumer problem (11) with the consumption set

<+ × F Suppose that the lottery price function (e) satisfies (A2). Then if ( e) is LSE (SSE and the budget constraint is binding at the optimum17) in¡<+ ×F≤ (())¢ and ≤ 0, argmax() ( e) ≤ (≤) argmax(0) ( e).Thus, the SSE property implies from Proposition 16, that on every expansion

path expenditure on the risky good increases. The LSE property implies that

optimal expansion paths, where expenditure on the risky good increases, exist.

16The lottery expenditure value order is structurally identical to the expenditure value order

in the certainty case. The prefix lottery is used to indicate that the set on which it is applied

is a stochastic function space and not a product of two chains.17Binding budget constraints ensure that chain-lower-than comparability implies strongly

lower than comparability. In the case of strong set comparability of the argmax sets this

assumption is not required.

19

This is a strong result. Still, it implies nothing about the change in optimal

lotteries, absent more information on the lottery price structure. In particular

it does not imply that optimal lottery consumption is increasing with respect to

FOSD, or any other quantitative measure. The content and impact of Propo-

sition 16 depends on the nature of the lottery price function that implies (A2)

holds, and that yields further quantitative information on lotteries. It may be

that (A2) is better justified on a restricted set of lotteries. Next we address

some of these issues by comparing the LEV and FOSV lattices.

3.4 Comparison of FOSV and LEV lattices

Under certainty, the expenditure and direct value lattices are closely linked. In

particular, if pricing is linear, the two lattices are identical. The corresponding

lattices under uncertainty, the LEV and the FOSV lattices, respectively contain

the expenditure and direct value lattices as sublattices. But the introduction of

uncertainty underscores the difference between the two approaches, one of which

is based on the pricing of and the other on the quantity of . Under certainty,

there is a one-to-one relationship between these notions. When uncertainty is

introduced, the notion of an increase in (or rather, e) is less clear, since theset of lotteries is not necessarily a chain (as is the real line) and the lottery price

function need not be one-to-one. (A2) is a very strong assumption, it turns

the set of lotteries to a chain. The fact that the underlying lottery space is

restricted to be a chain does not imply that the price system is linear, i.e., (e)may remain in general nonlinear.

When the set of lotteries is already a chain by some order, then (A2) becomes

less troublesome, if it is consistent with that order. Let F ⊆ F denote a chain

with respect to ≤. Then (A1) implies Assumption (A2) and the partialorders, ≤ () and≤ (()) coincide on F

. Under (A1) and (A2) there

is a more general relation between ≤ () and ≤ (()), and thereforebetween the FOSV and LEV lattices, as stated in Proposition 17:

Proposition 17 Let = ( (e)) where ∈ <++ and (e) : F → <+satisfies (A1) and (A2). Consider = ( e) 0 = (0 e0) ∈ <+ ×F. Then,

( e) ≤ () (0 e0)⇒ ( e) ≤ (()) (0 e0) ,and the join and meet in the corresponding lattices satisfy

∨ (()) 0 ≤ () ∨ () 0

∧ () 0 ≤ () ∧ (()) 0

If 0 ∈ <+×F then ( e) ≤ (()) (0 e0)⇒ ( e) ≤ () (0 e0)

and the two partial orders and lattices are equivalent.

Proof. Since both partial orders are value orders, we need only establish thate ≤ e0 ⇒ (e) ≤ (e0). This corresponds to (A1). When lotteries arerestricted to a chain, if (e) (e0), e e0 since F

is an ≤-chain,

20

and if (e) = (e0), e = e0 by (A2) so ≤-comparability holds, establishingthe equivalence of the two partial order on <+ ×F

.

Suppose 0 are incomparable in¡<+ ×F≤ (())¢ with (e)

(e0) and + (e) 0 + (e0) Therefore they are incomparable in¡<+ ×F≤ ()¢. Consider the joins given in Propositions 12 and 15. We

have e ≤ e∨, and the value of the first join is max {0 + (e0) (e∨)}.Therefore,

µ0 − ()−(0)

e¶ ≤ () µmax½0 + (0)−(∨)

0

¾ e∨¶.

The argument for the meet is similar.

Suppose next that 0 are comparable in¡<+ ×F≤ (())¢ with

+ (e) ≤ 0 + (e0) and (e) ≤ (e0) but incomparable in the FOSV

lattice¡<+ ×F≤ ()¢ Therefore e e0 are incomparable w.r.t. ≤

Let (∨ e∨) = ∨ () 0 Again 0 + (e0) ≤

∨ + (e∨) ande0 e∨. Hence ≤ (()) 0 ≤ () ∨ () 0 Theargument for the meets is similar.

Thus, the two orders are equivalent on subsets of the consumption set <+×F where F

is a chain with respect to ≤. However, the assumption thatthe set of lotteries forms a chain imposes a strong restriction. Even though the

two partial orders can be ordered, and the joins and meets of incomparable pairs

can be compared, it is not the case that lattice theoretic properties of functions

in the two lattices can be compared in the general case (when equivalence does

not hold). They are different lattices and they have different properties.

3.5 SOSV lattice

There are ways to compare lotteries, other than FOSD, that can be used to

construct value lattices.18 We consider Second Order Stochastic Dominance

(SOSD). Under the expected utility hypothesis with univariate preferences SOSD

corresponds to desirability by all concave utility functions. Thus, it yields a

framework by which to address the comparative statics question of when a more

desirable lottery, according to univariate concave expected utility maximization,

is chosen as income increases. This gives a mechanism for addressing the link

beween concavity and lattice theoretic properties of (ordinal) preferences.

SOSD defines a partial order on the set of distribution functions. Let ≤be defined over F by e ≤ e0 if and only if R () ≥ R 0() for all ∈ <+. In fact (F≤) is a lattice. If e e0 are two incomparable lotteries(with distribution functions

0 ∈ F respectively), their join and meet are

e∨ = e ∨ e0 ∨() ≡ [ ∨ 0 ] () =½

() ifR

≤R

0

0 () otherwise

(20)e∧ = e ∧ e0 ∧() ≡ £ ∧ 0¤ () = ½ () ifR

≥R

0

0 () otherwise

(21)

18For example, Monotone Likelihood Ratio (MLR) or Monotone Probability Ratio (MPR).

21

The Second Order Stochastic Dominance value (SOSV) lattice is constructed

in a manner analogous to the FOSV lattice. First, we define the SOSV order:

Definition 18 Let = ( (e)) where ∈ <++ and (e) : F → <+.Consider ( e) (0 e0) ∈ <+ × F. The Second Order Stochastic Dominancevalue order is:

( e) ≤ () (0 e0)⇔ ½ e ≤ e0+ (e) ≤

0 + (e0) . (22)

The analysis then proceeds as in the case of the FOSV lattice, noting though

that the join and meet of two incomparable lotteries are now defined with respect

to ≤.

Proposition 19 Let = ( (e)) where ∈ <++ and (e) : F → <+satisfies (A10). Then

¡<+ ×F≤ ()¢ is a lattice. For incomparable

( e) (0 e0) with + (e) ≤ 0 + (e0), the join and meet are:

( e)∨ (0 e0) = µmax½0 + (e0)− (e∨)

0

¾ e∨¶ e∨ = e∨ e0,

(23)

( e) ∧ (0 e0) = µ+ (e)− (e∧)

e∧¶ e∨ = e ∧ e0 (24)

Despite the similarity in the construction of joins and meets in the FOSV

and SOSV lattices, the two are different, since the join and meet lotteries are

different. In particular, ≤ is a refinement of ≤ This does not mean

that the joins and meets of incomparable lotteries in the two lattices can be

compared. Furthermore, assumption (A10) is stronger than assumption (A1).For example, fair pricing (condition (12)) satisfies (A1) but not necessarily (A10).Budget set strong set comparability encounters the same problems in the

SOSV lattice¡<+ ×F≤ ()¢ as in ¡<+ ×F≤ ()¢. Similar re-

strictions can be imposed in order to apply the LP method in both lattices, as

shown in Section 4.

4 Frameworks for comparative statics

We discussed in Section 3 issues that arise when introducing uncertainty into the

value lattice framework. In particular, the FOSV and SOSV lattices generally

fail to rank budget sets, whereas the LEV order requires a strong structural

assumption on the lottery price function in order to generate a lattice. We now

outline ways in which these issues can be addressed.

We introduce two different sublattices of the FOSV lattice within which

budget sets are ranked. The first, the space of lotteries F is a chain, is closestto the existing literature of comparative statics under uncertainty. At the price

of being restrictive, this framework yields a sufficient condition for the expected

utility to be SM and thus LSE in terms of the state utility . The second

22

sublattice allows for lotteries that do not form a chain, but bounds the support

of the lotteries. Budget sets are then ranked so long as income is high enough,

and the LSE property, though harder to characterize, can be established in

particular cases. This is the case where the lottery choice is small relative to

income.These two approached can be applied in the case of the SOSV lattice as

well.

The next approach builds on the LEV order by relaxing (A2) at the cost of

introducing pairs of extreme lotteries that can be used to construct joins and

meets of lotteries which cannot be distinguished by price alone. The advantage

of this approach is that it does not require the set of lotteries to be a chain, or

the lotteries to be small relative to income. The applicability of each approach

depends on the specific problem.19

4.1 FOSV lattice: lottery chains

When the consumption set is a sublattice of the form¡<+ ×F

≤ ()¢(A1) implies (A2) and by Proposition 17

¡<+ ×F ≤ ()¢ is equivalent

to¡<+ ×F

≤ ()¢. Therefore, Proposition 16 can be applied to showthat not only is the expenditure on the lottery increasing but also the choice

set of the lottery is increasing with respect to FOSD.

Corollary 20 Consider the consumer problem (11) with the consumption set

<+×F Suppose that the lottery price function satisfies (A1). If ( e) is LSE

(SSE and the budget constraint is binding at the optimum) in¡<+ ×F

≤ ()¢and ≤ 0, then argmax() ( e) ≤ (≤) argmax(0) ( e).Since all the degenerate distributions are included in F a possible FOSD-

chain is one that includes only the degenerate distributions, F Then the

sublattice¡<+ ×F

≤ ()¢ coincides with the expenditure value lattice¡<2+≤()¢ where () : <+ → <+ is defined by () = ({; 1}) i.e.the price of the denegerate lottery at (which under (A1) is increasing) This

makes clear that a necessary condition for any lattice theoretic property of a

function in¡<+ ×F≤ ()¢ is the same property in ¡<2+≤()¢ 20

It is not easy to give sufficient conditions for superextremal properties in the

whole¡<+ ×F≤ ()¢ However, in a sublattice ¡<+ ×F

≤ ()¢Proposition 21 gives sufficient conditions, on the state utility function for the

expected utility function to be SM (and hence LSE), under the expected utility

hypothesis.

19Under LEV, (A2) is used to ensure antisymmetry and thus a partial order which is nec-

cessary to construct a lattice. In Appendix 6 we show how a framework that builds on a

quasi order, in the absence of antisymmetry, may yield some comparative statics results (see

Shirai [11]). The cost of the absence of antisymmetry is that joins and meets are replaced

by set-valued functions, thus making the sufficient conditions on the objective based on those

even more difficult to check and justify.20However, when the chain is arbitrary, F

it is not necessarily the case that all degenerate

distributions are included, and therefore, the corresponding necessary condition would be with

respect to the relevant sublattice of<2+≤()

23

Proposition 21 Let = ( (e)) where ∈ <++ and (e) : F → <+ satis-fies (A1). Suppose =

R( ) () Then is SM on

¡<+ ×F ≤ ()¢

if is SM on¡<2+≤¢ and concave in

Proof. Suppose that is SM in¡<2+≤¢ and concave in . Consider ( e)

and (0 e0) two incomparable points in the lattice ¡<+ ×F ≤ ()¢ with

0 and e e0. is SM if:Z ∙(0 )−

µ+

(e)− (e0)

¶¸0

≤Z ∙

µ0 − (e)− (e0)

¶− ( )

¸.

is supermodular in¡<2+≤¢, implies (0 ) −

µ+

()−(0)

¶is non-

decreasing in . Since e e0,Z ∙(0 )−

µ+

(e)− (e0)

¶¸0

≤Z ∙

(0 )−

µ+

(e)− (e0)

¶¸.

It is then sufficient that:

(0 )−

µ+

(e)− (e0)

¶≤

µ0 − (e)− (e0)

¶− ( ), all

This follows from concave in as +()−(0)

≤ 0 by construction.

This is a rather striking result. It says that the SM property holds in¡<+ ×F ≤ ()¢ irrespective of the properties of the pricing function

(other than (A1) which ensures the lattice is well defined). Concavity and the

SM property in¡<2+≤¢ combine to give a strong sufficient condition for the SM

property in any¡<+ ×F

≤ ()¢ We should not expect this to be neces-sary for the ordinal LSE/SSE properties in any particular

¡<+ ×F ≤ ()¢

However, Proposition 22 shows that the SM property in the Euclidean lattice

plus concavity in are not only sufficient but also necessary for the correspond-

ing property in every direct value lattice, when all possible prices are allowed.

Therefore, the SM property on every direct value lattice is itself sufficient for

the SM property on any¡<+ ×F

≤ ()¢ Proposition 22 is also of independent interest in the context of the consumer

problem under certainty, and in terms of establishing a relation between our

approach and that of Quah [10].

Proposition 22 Suppose that : <2+ → < is continuous in Then is SM

in¡<2+≤()¢ for all = ( ) ∈ <++ × <+ if and only is is SM in

24

¡<2+≤¢ and concave in .21

Proof. (⇒) Let ( ) and (0 0) be two points in <2+ with 0 and 0.

Let b = These point are incomparable in

¡<2+≤()¢ for all b ∈ h0 0−−0´

Then, by definition, is SM in¡<2+≤()¢ if

( ) + (0 0) ≤ (+ b ( − 0) 0) + (0 − b ( − 0) ) (25)

The case b = 0 ( = 0) corresponds to SM in¡<2+≤¢. In order that is

concave in it suffices to show that for all 0 0 − (and any )

( ) + (0 ) ≤ (+ ) + (0 − ) (26)

Let b = −0 . Since 0 0 − b ∈ h0 0−

−0´and from (25)

( ) + (0 0) ≤ (+ 0) + (0 − ) (27)

By the continuity of in (27) must hold at the limit as 0 → thus implying

(26).

(⇐) Let any = ( ) ∈ <++ × <+ and b = . Let ( ) and (0 0)

be two incomparable points in¡<2+≤()¢ with +

0 + 0 (so

0 + b ( − 0)) and 0. By SM in¡<2+≤¢,

(0 0)− (+ b ( − 0) 0) ≤ (0 )− (+ b ( − 0) ) , (28)

and by concavity in ,

(0 )− (+ b ( − 0) ) ≤ (0 − b ( − 0) )− ( ) . (29)

Together, (28) and (29) establish that is SM in¡<2+≤()¢.

Example 23 illustrates the importance of the restriction to a ≤-chainF in Proposition 21.

Example 23 Let ( ) =√. This is SM in every

¡<2+≤()¢ andtherefore is SM in any

¡<+ ×F ≤ ()¢ However, it is not SM on¡<+ ×F≤ ()¢ Take e = {05 05; 1 4} and e0 = {05 05; 2 3}, soe∨ = {05 05; 2 4} and e∧ = {05 05; 1 3} Moreover, assume that prices are

such that(∨)−(0)

=

()−(∧)

= 4, and take = (5 e), 0 = (10 e0)so ∨ 0 = (6 e∨) and ∧ 0 = (9 e∧). Then, () + ( 0) = 833,

( ∨ 0) + ( ∧ 0) = 828.

Thus, comparing the restricted lattice¡<+ ×F

≤ ()¢ with the gen-eral

¡<+ ×F≤ ()¢, the complexity of the latter resides not only in the21These conditions correspond to Quah’s [10] “concave-modular” condition which thus, in

the two-good setting, amounts to the SM condition on in the whole class of direct value

lattices. However, the SM condition on any direct value lattice does not require continuity.

25

more elaborate expression for the join and budget set ordering, but also in the

weaker relationship between the lattice properties of the state utility function

and ordinal preferences. The fact that the underlying lattice has the form of

a chain on its lotteries simplifies the expression of the join and meet, sincee∨ e∧ ∈ {e e0}. This makes the LSE (SSE) conditions easier to verify, when

is not SM.

Proposition 21 hinges on the fact that F is a chain with respect to ≤,

and thus cannot be used to determine the LSE (SSE) conditions in the FOSV

lattice¡<+ ×F≤ ()¢ or an arbitrary LEV lattice. Nonetheless, the

standard hypothesis that only the quantity of a unique underlying lottery is

chosen, is captured by the case where the lottery domain is a chain.

4.2 FOSV lattice: small lotteries

In this subsection we suggest another way to apply the LP approach to the

FOSV lattice. Rather than assuming that the lotteries in the choice set lie on

a ≤ −chain, the supports of the lotteries are restricted. Consider the setof lotteries bF, whose supports are bounded above by some ∈ <+. Recallthat the FOSV lattice does not rank budget sets with respect to the strong

set order because the join of two incomparable lotteries can be unaffordable.

However, if supports are bounded above, then the price of any lottery, and in

particular of any join lottery, is bounded by (), where, in slight abuse of

notation, also denotes the lottery with sure payoff . With lottery supports

so restricted,³<+ × bF≤ ()´ is a sublattice of ¡<+ ×F≤ ()¢

This ensures that, for large enough incomes, all join lotteries are affordable.

Thus, for 0 ≥ ≥ (), () ≤ ( 0) in³<+ × bF≤ ()´. This

is a case where the lottery good is “small enough” relative to income. Then,

Veinott’s Theorem can be applied.

In fact, since in order to ensure strong set budget set comparability income is

restricted to be high enough, ≥ () the sublattice we impose sufficient condi-

tions on may be further restricted. Let X =n( e) ∈ <+ × bF | + (e) ≥ ()

o

Then¡X ≤ ()¢ is itself a sublattice of ¡<+ ×F≤ ()¢ Some

functions may satisfy the LSE conditions in the smaller sublattice but not in

the bigger.

Proposition 24 Consider the consumer problem (11) where the consumption

set <+ × bF has a FOSD-top degenerate lottery with price () Consider

X =n( e) ∈ <+ × bF | + (e) ≥ ()

o Suppose that budget constraints

are binding and that the lottery price function satisfies (A1). If ( e) is LSE(SSE) on

¡X ≤ ()¢ and 0 ≥ ≥ (), then argmax() ( e) ≤(≤) argmax(0) ( e).In order to apply Proposition 24, the LSE (SSE) conditions must be checked

directly. These conditions are more demanding in this framework since the

26

lotteries do not form a chain. However, the sublattice¡X ≤ ()¢ offers a

simplification in the construction of joins, thus making the verifcation process

simpler. In particular, the join of incomparable ( e) (0 e0) with + (e) ≤

0 + (e0), is µ0 − (∨)−(0)

e∨¶ in¡X ≤ ()¢. This is not only

critical in enabling strong set budget set comparability, but it also makes it more

likely that specific functions will satisfy the ensuing LSE/SSE properties. In the

general lattice¡<+ ×F≤ ()¢ some common state utility functions do

not result in LSE expected utility:

Example 25 Let = ( e) and suppose that is increasing in for

0, with (0 ) = 0 (notably, Cobb-Douglas preferences satisfy these con-

ditions). Let = ( e), 0 = (0 e0) ∈ ¡<+ ×F ≤ ()¢ with e e0≤-incomparable, + (e)

0+ (e0), and 0 0. If 0 are such

that max

½0 − (∨)−(0)

0

¾= 0, then ( ∨ 0) = 0 ≤ () ( 0). But

( 0) =R (0 ) 0

R

µ+

()−(∧)

¶∧ = ( ∧ 0). There-

fore, is not LSE on¡<+ ×F≤ ()¢.

LSE expected utility functions can still be identified on the general lattice¡<+ ×F≤ ()¢, e.g., a linear utility function with goods that are perfectsubstitutes, or quasi-linear preferences:

Example 26 Let = ( e) and suppose ( ) = + . Suppose that

the price function satisfies (e) = e.22 Let = ( e) and 0 = (0 e0) betwo incomparable points with

0+ (e0) ≥ + (e) and e0 ¤ e. Then() = + e and ( 0) = 0 + e0. Also, ∧ 0 =

µ+

()−(∧)

e∧¶so ( ∧ 0) = +

()−(∧)

+ e∧ = + e = (), and ∨ 0 =µmax

½0 − (∨)−(0)

0

¾ e∨¶ so ( ∨ 0) ≥ 0 − (∨)−(0)

+ e∨ =

0 +e0 = ( 0), so ( ) is LSE (but not SSE) in¡<+ ×F≤ ()¢.

Example 27 Let = ( e) = +R()(). Suppose that (e) satis-

fies fair pricing. Then is LSE in³<+ × bF≤ ()´. Letting = ( e)

and = (0 e0) be two incomparable points with e e0 ≤-incomparable and+ (e) ≤

0 + (e0), the LSE conditions are:+

Z()() ≥ ()+ (e)− (e∧)

+

Z()∧()

⇒ max

½0 0 − (e∨)− (e0)

¾+

Z()∨() ≥ ()0 +

Z()0()

22Notice that this does not satisfy assumption (A2), which is however not required here.

27

and

0 +Z

()0() ≥ ()+ (e)− (e∧)

+

Z()∧()

⇒ max

½0 0 − (e∨)− (e0)

¾+

Z()∨() ≥ ()+

Z()().

Since with ≤, + 0 = ∧ + ∨, these conditions are satisfied if:

0 −max½0 0 − (e∨)− (e0)

¾≤ (e)− (e∧)

(30)

and

−max½0 0 − (e∨)− (e0)

¾≤ − 0 +

(e)− (e∧)

. (31)

Under fair pricing, (e) + (e0) = (e∧) + (e∨), hence both conditions hold.4.3 Comparative statics with SOSV lattice

The same approaches that were used in Subsections 4.1 and 4.2 to enable strong

set budget set comparability in the FOSV lattice, can be used in the case of the

SOSV lattice. We state these rather obvious results in the following corollaries:

Corollary 28 Consider the consumer problem (11) with the consumption set

<+ ×F where F

is a chain with respect to SOSD. Suppose that the lottery

price function satisfies (A10). Then if ( e) is LSE (SSE and the budget

constraint is binding at the optimum) in¡<+ ×F

≤ ()¢ and ≤ 0,argmax() ( e) ≤ (≤) argmax(0) ( e).Corollary 29 Consider the consumer problem (11) where the consumption set

<+ × bF has a SOSD-top degenerate lottery with price () Suppose the

lottery price function satisfies (A10). Then if ( e) is LSE (SSE and the

budget constraint is binding at the optimum) in³<+ × bF≤ ()´ and

0 ≥ ≥ (), then argmax() ( e) ≤ (≤) argmax(0) ( e).As in the case of the FOSV lattice, a possible SOSD chain is that of the de-

generate lotteries. Again the sublattice¡<+ ×F

≤ ()¢ coincides withthe expenditure value lattice

¡<2+≤()¢ where the price function is theprojection of the stochastic price function to the domain of degenerate lotteries.

Hence, a necessary condition for any lattice theoretic property of a function on¡<+ ×F≤ ()¢ is the same property on ¡<2+≤()¢ When the set ofallowable lotteries is restricted to any SOSD-chain, we can also give sufficient

conditions for the ordinal utility function to be SM (and thus LSE), under the

expected utility hypothesis. Proposition 30 is the analogue of Proposition 21.

In comparison to Proposition 21, an extra condition is needed in Proposition

30. This condition is analogous to the first partial derivative of the state utility

28

function being concave, and thus it is a restriction on the third partial deriva-

tive of the state utility function. Restrictions on the third partial derivative are

regularly used in inivariate analysis.23

Proposition 30 Let = ( (e)) where ∈ <++ and (e) : F → <+ satis-fies (A10). Suppose =

R( ) () Then is SM on

¡<+ ×F ≤ ()¢

if is SM on¡<2+≤¢ and concave in and ∆ = (+ ) − ( ) is

concave in , for all ∈ <+.

Proof. Consider incomparable points ( e) and (0 e0) in ¡<+ ×F ≤ ()¢

with e e0 and + (e) ≤ 0 + (e0) (hence 0 ). is SM if:Z ∙

(0 )−

µ+

(e)− (e0)

¶¸0

≤Z ∙

µ0 − (e)− (e0)

¶− ( )

¸.

Since is supermodular in¡<2+≤¢, (0 ) −

µ+

()−(0)

¶is non-

decreasing in From ∆ = (+ ) − ( ) concave in , (0 ) −

µ+

()−(0)

¶is concave in Therefore, since e e0,

Z ∙(0 )−

µ+

(e)− (e0)

¶¸0

≤Z ∙

(0 )−

µ+

(e)− (e0)

¶¸

It is then sufficient that:

(0 )−

µ+

(e)− (e0)

¶≤

µ0 − (e)− (e0)

¶− ( ), all

This follows from concave in as +()−(0)

≤ 0 by construction.

4.4 LEV lattice: refining the order

In the case of the LEV order, (A2) is needed to ensure antisymmetry. The

problem is that there is no way to distinguish between different lotteries that

have the same price, absent some quantitative ordering on these lotteries, which

the LEV order does not provide. We propose a way to distinguish between

lotteries with the same price, without imposing an explicit order on the lotteries.

23The coefficient of absolute prudence involves the third derivative. It measures the coeffi-

cient of absolute risk aversion of the first derivative. Quah [10] derives analogous conditions

in a model of consumption and savings with variable choice of a single lottery, assuming

differentiability and a unique optimum.

29

The advantage of this approach is that it does not turn the set of lotteries into

a chain. However, this is done at the cost of arbitrarily assigning a top and a

bottom amongst the set of lotteries that have the same cost.24 We use these

two "extreme" lotteries to impose an augmenting condition on the LEV order

which makes the resulting order a partial order without Assumption (A2), and

which allows us to construct a lattice.

Definition 31 Consider <+ × F Let = ( (e)) where ∈ <++ and

(e) : F → where ⊆ <+ is the range of (e). For every (e) ∈ lete ( (e)) be a "top" and e ( (e)) a "bottom" lottery, e ( (e)) 6= e ( (e)) Consider ( e) (0 e0) ∈ <+ ×F. The Augmented Lottery Expenditure Valueorder is:25

( e) ≤ (()) (0 e0) iff + (e) ≤ 0 + (e0) and⎧⎨⎩ (e) (e0) or

(e) = (e0) and e = e0 or (e) = (e0) e 6= e0 and £e = e ( (e)) or e0 = e ( (e))¤ (32)

Note that≤ (()) is a partial order. For antisymmetry, ( e) ≤ (())(0 e0) and (0 e0) ≤ (()) ( e) imply that = 0 and (e) = (e0) Suppose e 6= e0 Then by the augmenting conditions, e = e ( (e)) or e0 =e ( (e)) and, e0 = e ( (e)) or e = e ( (e)) These conditions form a contra-diction and therefore e = e0 For transitivity suppose ( e) ≤ (()) (0 e0)and (0 e0) ≤ (()) (00 e00) The only case that needs to be verified is (e) = (e0) = (e00) and e 6= e0 6= e00 In this case e = e ( (e)) or e0 =e ( (e)) and e0 = e ( (e)) or e00 = e ( (e)) These can hold simultaneouslyonly if e = e ( (e)) or e00 = e ( (e)). Therefore ( e) ≤ (()) (00 e00) The role of the augmenting condition in the AEV order is to enable anti-

symmetry without affecting comparability in cases in which the lotteries have

different prices. Pairs with equally priced lotteries are incomparable except in

the special case where these are one or both of the "extreme" lotteries. This is

fairly innocuous when it comes to the partial order, but not in the construction

of the lattice, since in that case the augmenting condition determines the nature

of joins and meets. However, it does yield a lattice to be constructed without

requiring the set of lotteries to be a chain.26

Proposition 32 Consider¡<+ ×F≤ (())¢ subject to the conditions of

Definition 31. Then¡<+ ×F≤ (())¢ is a lattice. There are two types

24 If an underlying order such as FOSD or SOSD is used on the set of lotteries, this may

guide the selection of such "extreme" lotteries. Or the setting of the problem may suggest

what these may be.25 Instead of having one set of "extreme" lotteries for each price, it is possible to have one set

of "extreme" lotteries for the whole set, and require that those are available at all quantities

and linearly priced. The resulting lattice is different.26Antoniadou [2] has used the technique of augmenting the partial order with a non-binding

condition for pairs that satisfy the required comparability, which nonetheless determines the

nature of the lattice.

30

of incomparable pairs. For incomparable ( e) (0 e0) with (e) (e0) and + (e)

0 + (e0), the join is µ0 − ()−(0)

e¶ and the meet isµ+

()−(0)

e0¶. For incompararable ( e) (0 e0) with + (e) ≤ 0+

(e0), (e) = (e0), e 6= e0 and e 6= e ( (e)) and e0 6= e ( (e)) the join is¡0 e ( (e))¢ and the meet is ¡ e ( (e))¢.Proof. The first type of incomparable pairs as the same as in the LEV lattice

and therefore the proof is omitted.

Consider ( e) (0 e0) with + (e) ≤ 0+ (e0), (e) = (e0), e 6= e0

and e 6= e ( (e)) and e0 6= e ( (e)) Let ∨ = ¡0 e ( (e))¢ The value of ∨is

0+ (e0) Therefore, by the augmenting condition, ( e) (0 e0) ≤ (())∨ Consider any other upper bound, = (00 e00) By definition 0+ (e0) ≤

00 + (e00) and (e00) ≥ (e) Strict inequality implies ∨ ≤ (()) Therefore suppose (e00) = (e) Hence by definition it must be e00 = e ( (e))and again ∨ ≤ (()) thus establishing that ∨ is the join. The argumentfor the meet is similar and is omitted.

As with the LEV lattice, budget sets are strongly set comparable, i.e. () ≤( 0) for ≤ 0 in

¡<+ ×F≤ (())¢. Therefore, Theorem 3 can be ap-

plied:

Proposition 33 Consider the consumer problem (11) where the consumption

set is <+ × F is a lattice with the AEV order under the conditions of Defi-

nition 31. If ( e) is LSE (SSE and the budget constraint is binding at theoptimum) in

¡<+ ×F≤ (())¢ and 0 ≥ , then argmax() ( e) ≤(≤) argmax(0) ( e).In example 34 such an AEV lattice, in the case where there are two under-

lying lotteries that are available at different quantities and different mixtures.

This case is of independent interest, with some extensions of the basic (univari-

ate) model of choice under uncertainty based on the existence of two underlying

distributions.

Example 34 Let S = {1e1 + 2e2, 1 2 ∈ <+} ⊆ F be the set of lotteries,where e1 and e2 are two distinct base lotteries satisfying (e1) = (e2) = 1.

Suppose lotteries are linearly priced. Defining e () = e1, e () = e2 deter-mines a lattice

¡<+ × S≤ (())¢, with e1 a worse lottery than e2 (this maybe with respect to ≤ for example, but not necessarily).

Which refinement of the consumption set is appropriate depends on the

specific problem. The value of each of our comparative statics propositions

depends on the nature of the problem studied. Our work shows clearly the

complexity of the problem.

31

5 Appendix - Product lattices

5.1 Proof of Proposition 10

Suppose that = ( e) and 0 = (0 e0) are two incomparable points in¡<+ ×F≤(×)¢, with 0 ≥ and 0 ¤ . The LSE conditions

are:Z(0 )0 ≥ ()

Z( )∧ ⇒

Z(0 )∨ ≥ ()

Z( ) (33)

andZ( ) ≥ ()

Z( )∧ ⇒

Z(0 )∨ ≥ ()

Z(0 )0 ,

(34)

and the SSE conditions are:Z(0 )0 ≥

Z( )∧ ⇒

Z(0 )∨

Z( ) (35)

and Z( ) ≥

Z( )∧ ⇒

Z(0 )∨

Z(0 )0 . (36)

We successively examine (33) and (35), and (34) and (36).

For the first LSE condition (33), the implication in weak inequalities holds

if: Z(0 )∨ ≥

Z( )∨ ≥

Z( ), (37)

which is satisfied if is nondecreasing and since ≤ ∨. If the firstinequality in (33) holds strictly, then either

R(0 )0

R( )0 orR

( )0 R( )∧. In the latter case, since +0 = ∨ +∧ ,R

( )∨ R( ) and the implication in strict inequalities follows

when is nondecreasing. In the former case, there therefore exists b withPr {e0 ≥ b} 0 such that (0 b) ( b). If is LSE in ¡<2+≤¢, then for all ≥ b, (0 ) ( ). Therefore,Z

(0 )∨

Z ( )

∨ ≥Z ( ), (38)

and the implication in strict inequalities follows. If is increasing, the first SSE

condition (35) holds directly since:Z(0 )∨

Z( )∨ ≥

Z( ). (39)

For the second LSE condition (34), the implication in weak inequalities

follows directly from the FOSD order on lotteries if is nondecreasing. To

32

establish the implication in strict inequalities, suppose thatR( ) R

( )∧. Then, there exist ≤ 0 with Pr {e ≥ } + Pr {e∧ ≥ 0} 0

such that ( 0) ( ). If is LSE in¡<2+≤¢, then (0 0) (0 ). It

follows thatR(0 )

R(0 )∧, and since + 0 = ∨ + ∧ ,R

(0 )∨ R(0 )0 . The second SSE condition (36) holds if is in-

creasing since e0 e∨, from which the second inequality follows directly.

5.2 SOSD product lattice

The definition of the SOSD product lattice proceeds by analogy with the FOSD

product lattice. Let ≤ be defined by e ≤ e0 if and only if R () ≥R 0() for all ∈ <+, e e0 ∈ F. SOSD defines a partial order and a lattice

on F The join and meet of incomparable pairs of lotteries are given by (20)and (21) of Subsection 3.5, respectively. They are used to construct the SOSD

product lattice,¡<+ ×F≤(×)¢.

Properties of the state utility function may again be linked to the lattice

properties of the expected utility function . In the case of SOSD, this involves

the fact from the univariate case that for concave, a second order stochastic

dominant lottery is always preferred. Proposition 35 is the analog to Proposition

10.

Proposition 35 Suppose ( e) = R( ) (). If ( ) is increasing

on¡<2+≤¢ and strictly concave in , then ( e) is SSE in ¡<+ ×F≤(×)¢.

Proof. Suppose that = ( e) and 0 = (0 e0) are incomparable pointsin¡<+ ×F≤(×)¢, with 0 ≥ and e0 ¤ e. ( e) is SSE if it

satisfies conditions (35) and (36). Suppose that is increasing. Beginning with

the first SSE condition (35), since and 0 are incomparable, either: (i) e ande0 are incomparable by SOSD so e∨ e, in which case,Z(0 )∨ ≥

Z( )∨

Z( ), (40)

since is strictly concave. Or, (ii) e∨ = e e0, and it follows that0 since and 0 are incomparable, and therefore (0 ) ( ) for all

. Hence: Z(0 )∨

Z( )∨ =

Z( ). (41)

Regarding the second LSE condition (36), from e∨ e0 the secondinequality holds since is strictly concave.

Proposition 35 restricts the state utility function; e.g. strict monotonicity

rules out Leontief preferences, and strict concavity in rules out risk-neutrality

if the goods are perfect substitutes. It is not clear how to weaken these as-

sumptions to obtain sufficient conditions for to be LSE in the FOSD product

lattice. Example 36 illustrates that conditions analogous to those of Proposition

10 fail to ensure that is LSE:

33

Example 36 Suppose that is defined on {0 1}×{0 1 2}. The values of at(0 0) (0 1) (0 2) are 1 3 4 respectively, and at (1 0) (1 1) (1 2), are 0 3 6

respectively. Then, is LSE, non-decreasing, and (consistent with) concave

in . Define two lotteries, e0 = {05 05; 0 2}, and e = {1; 1} (degeneratedistribution at 1). Then, e0 e. Take = (0 e) and 0 = (1 e0)as two incomparable points. Their meet and join are ∨ 0 = (1 e) and ∧ 0 = (0 e0). Then, () = 3 = ( ∨ 0), but ( 0) = 05× 0+ 05× 6= 3 25 = 05× 1 + 05× 4 = ( ∧ 0) and therefore is not LSE.

6 Appendix: Beyond the LP framework

Shirai [11] introduces a variant of the LP framework that applies to ordered

spaces in which antisymmetry does not hold. The analogs to the standard

definitions are the following. Let X be a set ordered bya quasi order ¹X that

satisfies reflexivity and transitivity. Given two points and 0, let 0

and 0 be the sets of least upper bounds and greatest lower bounds (in

X ). If ¹X is antisymmetric, we would have 0 = { ∨ 0} and 0 =

{ ∧ 0}. (X ¹X ) is said to be a pre-ordered lattice structure if for all 0 ∈X , 0 0 6= ∅.Given two sets 0, the -strong set order ≤ is defined by ≤ 0 if and

only if, for all 0 ∈ X , 0 ⊆ and 0 ⊆ 0. A function : X → < issaid to be -quasisupermodular if for all 0 ∈ X ,

∃ ∈ 0 , () ≥ () ( )⇒ ∀ ∈ 0 , () ≥ () ( 0). (42)

Then,

Theorem 37 (Shirai [11]) If (X ¹X ) is a pre-ordered lattice structure and :X → < is -quasisupermodular, argmax ≤ argmax0 whenever ≤ 0.

This framework applies to the ≤ (()) order in the absence of (A2),and thus antisymmetry. Take ( e) (0 e0) ∈ <+ × F with (e) ≥ (e0) and+ (e) ≤

0 + (e0). Then,0 =

½µ0 − (e)− (e0)

e¶ , (e) = (e)¾ , (43)

0 =

½µ+

(e)− (e0)

e¶ , (e ) = (e0)¾ .Note that if (e) = (e0) and + (e) =

0 + (e0), 0 = 0 =

{( e) , (e) = (e)}.Proposition 38

¡<+ ×F¹ (())¢ is a preordered lattice structure.Thus, Theorem 37 applies to the LEV lattice structure, and condition (42)

is sufficient fo the monotonicity of argmax sets with respect to ≤. However,an increase with respect to ¹ (()), reflects an increase in expenditure on

34

the lottery as income increases, but makes no distinction between lotteries of

a given price. The -strong set order thus reflects a strong prediction with

respect to optimizers if choice sets are not singletons. This is a weak version

of monotone comparative statics and although perhaps useful in some applica-

tions, as with the LEV lattice, the usual monotone comparative statics is not,

in general, implied unless a strong assumption is made about distinguishing be-

tween points if the choice set is not a singleton. In the LEV lattice, the price

function has the strong separating property (A2), while in this framework the

-quasipermodularity remains opaque.

We finish with two illustrative examples.

Example 39 Let ( ) = 1() + 2(). Taking ( e) (0 e0) ∈ <+ × Fwith (e) ≥ (e0) and + (e) ≤

0 + (e0). The sufficient conditions forlottery choice to be normal given by (42), i.e. -quasipermodularity, are:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

∃e , [2 (e)− 2 (e )] ≥ () 1µ+ ()−(0)

¶− 1()

⇒ ∀e, [2 (e)− 2 (e0)] ≥ () 1(0)− 1

µ0 − ()−(0)

¶∃e , [2 (e0)− 2 (e )] ≥ () 1µ+ ()−(0)

¶− 1(

0)

⇒ ∀e, [2 (e)− 2 (e)] ≥ () 1()− 1

µ0 − ()−(0)

¶. (44)

Here, e denotes the low-priced lotteries ( (e ) = (e0)), and e denotes thehigh-priced lotteries ( (e) = (e)). Suppose that is concave. Then, thefirst condition is satisfied if () 2 agrees with the lottery price ranking (e),27and () if a given lottery is preferred to other lotteries at a given price, then

there is indifference between all higher-priced lotteries. This characterization of

normality, although restrictive, obtains with no assumptions regarding the space

of lotteries, or the lottery price function.

Example 40 Same as Example 39, but suppose that 1() is concave, 2() =

, lotteries are fairly priced with (e) = e, and = 1. Then, (44) holds.References

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[3] E. Antoniadou, Comparative Statics for the Consumer Problem, Economic

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27 In the sense that, whenever one high-priced lottery is preferred by 2 to a lower-priced

lottery, the same preference ranking holds for all other lotteries with these prices.

35

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