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 topological assumptions on utility functions and choice sets. We allow for imperfect 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 stochastic 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, Comparative Statics
JEL Classification Codes C61, D11, D81
1
Introduction
Some of the most fundamental insights of economic science derive from comparative statics. The techniques of comparative statics under certainty are important 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 comparative 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,
[email protected], Leonard J Mirman: University of Virginia, [email protected],
Richard Ruble: EM Lyon, [email protected].
† 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 appropriate 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 comparative 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 differentiability. These assumptions are used to employ the implicit function theorem as well
as other topological methods. This approach yields a local and, therefore, a narrow 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 comparative 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 problems 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 problems 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 superextremal functions that yield monotone comparative statics under suitable monotonicity of the constraint sets.1 In particular, various types of superextremal functions are employed, which, when combined with strong monotonicity between
sets, yield variants of monotonic comparative statics. These methods are entirely 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. Moreover, 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 applications 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 system 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
1 See
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 straightforward 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 stochastic 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 expected 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 lotteries. 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 incomparable 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 incompatibility. 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 incompatibility 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 appropriate 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 particular 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 general nonlinear pricing function, and thereby allows for constraint sets that are
nonlinear. Finally Subsection 2.5 takes a first look at stochastic lattices, by considering 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 methods. 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 related
to 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 comparable.
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 ordinal. 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 arg max  ≤
(≤ ) arg max  for all   ⊂ X with  ≤  and arg max  , arg max  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.
3 Milgrom 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 () s.t.  ·  ≤ .
∈<2+
(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 characterize
effects with ªrespect to good . If the constraint sets () =
© the income
( ) ∈ <2+ ,   +   ≤  are 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:
³
³
´
´
Example 4 Let  = 0  and  0 =   0 , so  ∈ () and  0 ∈ ( 0 ).
³
´
Then,  ∧  0 = (0 0) ∈ (), but  ∨  0 =    ∈
 ( 0 ). Therefore,
() £ ( 0 ).
¡
¢
The application of Theorem 3 in <2+  ≤ is thus not informative for shifts
4
in budget
However, this limitation results from the choice of the lat¡ 2 sets.
¢
tice, <+  ≤ , 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 studied. Comparative statics thus requires the determination of a lattice in which
strong constraint set comparability is enabled, and thus the superextremal variant 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
½
 ≤ 0
( ) ≤() (0   0 ) ⇔
.
(5)
 · ( ) ≤  · (0   0 )
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
¡ 2 direct value
¢ order defines a non-Euclidean lattice on the consumption set,
<+  ≤() , to which standard LP techniques can be applied to characterize
normality.
0
For two incomparable points  = ( ) and
= (0  ¢0 ) with    0 and
¡ 
0
0
2
  +      +   , the join and meet in <+  ≤() are:
¶
µ
¶
µ


 ∨  0 = 0 −
( −  0 )   and  ∧  0 =  +
( −  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
5 Antoniadou
[1] extends this order
to many goodsusing the
 lexicographic order.

the up sets differ in <2+  ≤() and <2+  ≤ , utility functions are not generally increasing with respect to ≤() and the LSE property is not a trivial restriction.
7 Antoniadou [1] defines the corresponding set relation, pathwise-lower-than.
6 Because
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, Antoniadou’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 independent 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 expenditure
value order is:9
½
() ≤ ( 0 )
 ≤(())  0 ⇔
.
(6)
  +  () ≤  0 +  ( 0 )
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 by
the 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:
µ
¶
µ
¶
() − ( 0 )
() − ( 0 ) 0
 ∨  0 = 0 −
  and  ∧  0 =  +
  . (7)


8 For example, Mirman and Ruble [9] introduce the radial value order: ( ) ≤
0 0
() (   )
if and only if 0 ≥ 0  and   +   ≤  0 +   0 .
9 We 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
0
)
As 0 − ()−(
  ≥ 0 for such incomparable points,  ∨  0   ∧  0 ∈ <2+

¡ 2
¢
is a lattice. Moreover, budget sets are strong set ranked in
¡so 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.
ª
©
 ∈  (), where  () =  = ( ) ∈ <2+ ,   +  () ≤ ¡  If the utility¢
function is LSE (SSE and the budget constraint is binding) on <2+  ≤(())
then arg max()  ( ) ≤ (≤ ) arg max( 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 relevant
constraint 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 F
is 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 of
choice under uncertainty, our formulation offers two important generalizations:
1 0 The 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).
1 1 The 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
y
X
X X'
-q
X X'
X'
A
B
x
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 contingent 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 technically 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
Z
 ( e) = ( ) () .
(8)
The state utility function  is defined over <2+ and represents the individual’s
preferences over the realizations of consumption. The perfect substitutes specification ( + ) 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 be
specified. 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 uncertainty with budgetary tradeoffs. However, the product lattice provides a convenient 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 relationships 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 incomparable
lotteries (with distribution functions   0 ∈ F respectively), their join is
e∨ = 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 only
0
0
if  ≤
¡  and e ≤  e¢ . With this order, the consumption set is a lattice, <+ × F  ≤¡(× )
¢ , the FOSD product lattice. The analogy with the
Euclidean 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 have
the same quantity of the sure good . However, in a lattice over a restricted
consumption set <+ × F where F ⊆ F is a chain of distribution functions
with respect to ≤  , the analogy is better.
¡
¢
For the same fundamental reasons that the Euclidean lattice <2+  ≤ in
the certainty case is not¡ well suited to problems
¢ with budgetary trade-offs, the
FOSD product lattice <+ × F  ≤(× ) is not well suited to such problems 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 corresponding comparability in the FOSD product lattice. The same observation establishes that lattice theoretic properties in the Euclidean lattice are necessary for
the corresponding property in the FOSD¡ product lattice. That
¢ is,  ( e) has
a lattice property (SM, LSE, SSE)¡ in <+¢ × F  ≤(× ) only if  ( )
has the corresponding property in <2+  ≤ . In the case of the SM property,
1 2 The 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
R
Proposition 8 Suppose
¡  ( e) = (¢) (). 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
0
e0 R¤  e.  is SM if
Rincomparable points
R in0 <+ × F with
R  ≥  and
∧
( ) ()+ (  )0 () ≤ ( ) () + (0  ) ∨ (), which
can be rewritten as:
Z
Z
( ) [ () + 0 ()] + [(0  ) − ( )] 0 ()
(9)
Z
Z
≤ ( ) [ ∧ () +  ∨ ()] + [(0  ) − ( )]  ∨ ().
With the ≤  order,  () + 0 () =  ∧ () +  ∨ () for all , so the
first terms on each side of the inequality are identical. Since  is supermodular,
(0  ) − ( ) is either 0 (if 0 = ), or nondecreasing in . Therefore, since
e∨   e0 ,
Z
Z
[(0  ) − ( )]  ∨ () ≥ [(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 property 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. Let
¡e = {05 05; 1 3} and¢ e0 = {05 05; 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) = 05 × 2 + 05 × 2 = 2  05 × 3 + 05 × 0 =
 ((0 e) ∧ (1 e0 )), and  (1 e0 ) = 05 × 2 + 05 × 2 = 2  05 × 3 + 05 × 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
1 3 Athey [4] studies these issues using convex cones. Our proof of Proposition 8 is less general
but direct.
1 4 The 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 similar. These sufficiency results rely on the desirability of First Order Stochastic
Dominant distributions. They are stated in Proposition 10:
R
Then: (i) If  ( )
Proposition 10 Suppose  ( e) = (¡) ().
¢
2
is
nondecreasing
(in

and
)
and
LSE
in
<

≤
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  ( e) s.t.   +  (e
(
)∈X
(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 state
utility 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 analysis
and 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 not
reflect 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 the
cost 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 <+ have
the 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
 ) ≤ ()  (e
0) .
(A1)
e ≤ ( ) e0 ⇒  (e
 ) ≤ ()  (e
0) .
(A10 )
 (e
 ) =  (e
 0 ) ⇔ e = e0 .
(A2)
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 ≤ :
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:
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
 ) ≤  (e
 0 ).
Assumption (A1) can be derived from a more primitive pricing structure,
fair pricing of lotteries, based on the prices of the underlying outcomes. If
b () : <+ → <+ is an (increasing) price function over the possible outcomes,
the fair price of a lottery F is given by:
Z
 (e
 ) = b ()  ().
(12)
From (12), the prices of the join and meet of two incomparable lotteries, e e0 
with respect to ≤  satisfy:
 (e
 ∧ ) +  (e
 ∨ ) =  (e
 ) +  (e
0) .
1 5 If
 = {; 1} is the degenerate lottery at ,  (
 ) = ().
16
(13)
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 in
the 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 are
comparable 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 a
lattice 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 order
is a generalization of the direct value order.
 )) where  ∈ <++ and  (e
 ) : F → <+ .
Definition 11 Let  = (   (e
Consider ( e)  (0  e0 ) ∈ <+ × F . The First Order Stochastic Dominance
Value (FOSV) order is:
½
e ≤  e0
0 0
( e) ≤  () (  e ) ⇔
.
(14)
  +  (e
 ) ≤  0 +  (e
0)
The FOSV order is a partial order and it generates the First Order Stochastic
Value (FOSV) lattice on the consumption set.
 )) where
 ) : F → <+ satProposition 12 Let
¡  = (   (e
¢  ∈ <++ and  (e
isfies (A1). Then <+ × F  ≤  () is a lattice. For incomparable ( e) 
(0  e0 ) with, without loss of generality,   +  (e
 ) ≤  0 +  (e
 0 ), their join
and meet are given by:
µ
½
¾
¶
∨)
 (e
 0 ) −  (e
 0  e∨
( e) ∨ (0  e0 ) = max 0 +
(15)

©
ª
 e∨ = e ∨  e0 with  ∨ () = min  ()  0 () 
and
µ
¶
 (e
 ) −  (e
∧ ) ∧
 e
(16)

©
ª
 e∧ = e ∧  e0 with  ∧ () = max  ()  0 () 
( e) ∧ (0  e0 ) =
+
0 0
¾
 ) ≤  0 +
Proof. Consider an incomparableµpair (
  + ¶(e
½ e)  ( 0e ) with
∨
(
 )−(
 )
 (e
 0 ). Both  ∨ = (∨  e∨ ) = max 0 +
 0  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 +  (e
 0 )   (e
 ∨ )} ≥  0 +  (e
 0 )  Consider
00 00
0 0
any other upper bound, (  e ) ∈ <+ × F , i.e. ( e)  (  e ) ≤  ()
∨
00
(00  e00 ). Hence
 )   (e
 0 ) ≤  (e
 ∨ ) ≤  (e
 00 ).
½ e ≤0 ∨e , and
¾ by (A1),  (e
(
 )−(
 )
If ∨ = max 0 +
 0 = 0,  (e
 ∨ ) ≤  00 +  (e
 00 ). Otherwise,

½
¾
(
0 )−(
∨ )
(
0 )−(
∨ )
max 0 +
 0 = 0 +
, so  ∨ +  (e
 ∨ ) =  0 +  (e
0) ≤


 00 +  (e
 00 ). In both cases, (∨  e∨ ) ≤  () (00  e00 ). The argument for
the meet is similar and is omitted.
¡ 2The FOSV
¢ lattice is a natural generalization of the direct value lattice
<+  ≤() to the uncertainty case. However, compared with
¾
½ the certainty
(
0 )−(
∨ )

0
,
case, the expression for the deterministic good in the join, max 0 +

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, that
is e∨ 6= e e0 . By (A1), the join lottery e∨ satisfies  (e
 ∨ )   (e
 )   (e
 0 ),
0
∨



−


(
)
(
)
and 0 +
may be negative. Hence the value of the join bundle is

max { 0 +  (e
 0 )   (e
 ∨ )}, i.e. the price of the join lottery can be so high as to
render 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 budget 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
0
 (e
 0 )   (e
 ), and take  =  (e
 ) and  0 =  (e
µ
). Then, ( 0 )¶¤ (). We
(
 )−(
∧ )
have  ∈ () and  0 ∈ ( 0 ), and  ∧  0 =
 e∧ ∈ () (note

 )   (e
 ∧ )). However,  ∨ 0 = (0 e∨ ) ∈
 ( 0 ):
that  ∧ 0 ∈ <+ ×F since  (e
as e∨   e e0 , by (A1)  (e
 ∨ )   (e
0 ) =  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 other
solutions 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
 ).
 )) where  ∈ <++ and  (e
 ) : F → <+ satisDefinition 14 Let  = (   (e
fies (A2). Consider ( e)  (0  e0 ) ∈ <+ × F . The Lottery Expenditure Value
order is:16
½
 (e
 ) ≤  (e
0)
0 0
( e) ≤ (()) (  e ) ⇔
.
(17)
  +  (e
 ) ≤  0 +  (e
0 )
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.
 )) where ¢ ∈ <++ and  (e
 ) : F → <+
Proposition 15 Let ¡ = (   (e
satisfies (A2). Then <+ × F  ≤ (()) is a lattice. For incomparable
 )   (e
 0 ) and   +  (e
) 
( e)  (0  e0 ) with, without loss of generality,  (e
0
0
  +  (e
 ), the join and meet are:
µ
¶
 (e
 ) −  (e
0)
0 −
 e and
(18)
( e) ∨ (0  e0 ) =

µ
¶
 (e
 ) −  (e
0 ) 0
( e) ∧ (0  e0 ) =
+
 e .
(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 FOSV
lattice,
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
 ) satisfies (A2). Then if
<+ × F  Suppose that the lottery price function  (e

(

e
)
is
LSE
(SSE
and
the
budget
constraint
is
binding
at the optimum17 ) in
¡
¢
0
<+ × F  ≤ (()) and  ≤  , arg max()  ( e) ≤ (≤ ) arg max( 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.
1 6 The 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.
1 7 Binding 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 Proposition 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 the
set 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 partial
orders, ≤  () and ≤ (()) coincide on F . Under (A1) and (A2) there
is a more general relation between ≤  () and ≤ (()) , and therefore
between the FOSV and LEV lattices, as stated in Proposition 17:
 )) where  ∈ <++ and  (e
 ) : F → <+
Proposition 17 Let  = (   (e
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 that
 ) ≤  (e
 0 ). This corresponds to (A1). When lotteries are
e ≤  e0 ⇒  (e
restricted to a chain, if  (e
 )   (e
 0 ), e   e0 since F is an ≤  -chain,
20
and if  (e
 ) =  (e
 0 ), e = e0 by (A2) so ≤  -comparability holds, establishing
the equivalence of the two partial order on <+ × F .
¡
¢
Suppose   0 are incomparable in <+ × F  ≤ (()) with  (e
) 
 0 ) and   +  (e
 )¢   0 +  (e
 0 )  Therefore they are incomparable in
¡ (e
<+ × F  ≤  () . Consider the joins given in Propositions 12 and 15. We
∨
have e ≤ 
value
join is½max { 0 +  (e
 0 ) ¾
 (e
 ∨ )}.
µ e , and the
¶ of the first µ
¶
0
0
∨
(

)−





−


( )
( ) ( )
0
0
∨
Therefore,  −
 e ≤  () max  +
 0  e .


The argument for the meet is similar.
¡
¢
Suppose next that   0 are comparable in <+ × F  ≤ (()) with
  + ¡ (e
 ) ≤  0 +  (e
 0 ) and
 ) ≤  (e
 0 )  but incomparable in the FOSV
¢  (e
lattice <+ × F  ≤  ()  Therefore e e0 are incomparable w.r.t. ≤ 
 Let (∨  e∨ ) =  ∨  ()  0  Again  0 +  (e
 0 ) ≤  ∨ +  (e
 ∨ ) and
0
∨
0
e   e . Hence  ≤ (())  ≤  ()  ∨  ()  0  The
argument 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 that
the 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 ≤
R
R
be defined over F by e ≤ e0 if and only if
 () ≥
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
½
R
 ≤
0
e = e ∨ e   () ≡ [ ∨ 0 ] () =

otherwise
½
R
R  (20)
£
¤
()
if

≥
0



e∧ = e ∧ e0   ∧ () ≡  ∧ 0 () =

0 () otherwise
(21)
∨
1 8 For
0
∨
 () if
0 ()
R
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 Dominance
value order is:
½
e ≤ e0
0 0
( e) ≤ () (  e ) ⇔
.
(22)
  +  (e
 ) ≤  0 +  (e
0)
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 +  (e
 0 ), the join and meet are:
µ
½
¾
¶
 (e
 0 ) −  (e
∨)
0 0
0
∨
( e) ∨ (  e ) = max  +
 0  e
 e∨ = e ∨ e0 ,

(23)
µ
¶
 (e
 ) −  (e
∧ ) ∧
0 0
∨
0
( e) ∧ (  e ) =  +
 e
 e = e ∧ e  (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 restrictions 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 closest
to 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 show
that 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 arg max()  ( e) ≤ (≤ ) arg max( 0 )  ( e).
Since all the degenerate distributions are included in F  a possible FOSDchain 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
properties in the¢
¡ easy to give sufficient
¢ conditions for superextremal
¡
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.
1 9 Under LEV, (A2) is used to ensure antisymmetry and thus a partial order which is neccessary 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.
2 0 However, when the chain is arbitrary, F   it is not necessarily the case that all degenerate

distributions are included, and therefore,
necessary condition would be with
 2 the corresponding

respect to the relevant sublattice of <+  ≤() 
23
Proposition 21 Let  =
 )) where  ∈ <++ and ¡(e
 ) : F → <+ satis- ¢
R (   (e
fies (A1). Suppose  = ( ) ()  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:
µ
¶¸
Z ∙
 (e
 ) −  (e
0 )
  0
(0  ) −   +

¶
¸
Z ∙ µ
 (e
 ) −  (e
0 )
0
≤
  −
  − ( )  .

µ
¶
¡ 2
¢
(
)−(
0 )
0
 is supermodular in <+  ≤ , implies (  ) −   +
  is non
decreasing in . Since e   e0 ,
µ
¶¸
Z ∙
 (e
 ) −  (e
0 )
0
  0
(  ) −   +

µ
¶¸
Z ∙
 (e
 ) −  (e
0)
0
≤
(  ) −   +
   .

It is then sufficient that:
µ
¶
µ
¶
 (e
 ) −  (e
0)
 (e
 ) −  (e
0 )
0
0
(  ) −   +
 ≤   −
  − ( ), all 


(
 )−(
0 )
This follows from  concave in  as  +
≤ 0 by construction.

is a rather striking
result. It says that the SM property holds in
¡ This 
¢
<+ × 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 corresponding 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
¡ 2
in <+  ≤() for all  = (   ) ∈ <++ × <+ if and only is  is SM in
24
¡
¢
<2+  ≤ and concave in .21
0
Proof. (⇒) Let ( ) and (0   0 ) be two points in <2+ with   0 and
h 0  ´ .
¡ 2
¢

−

Let b =   These point are incomparable in <+  ≤() for all b ∈ 0 −
0
¡ 2
¢
Then, by definition,  is SM in <+  ≤() if
(25)
 ( ) +  (0   0 ) ≤  ( + b ( −  0 )   0 ) +  (0 − b ( −  0 )  )
¡ 2
¢
The case b = 0 ( = 0) corresponds to  SM in <+  ≤ . In order that  is
concave in  it suffices to show that for all 0    0 −  (and any )
 ( ) +  (0  ) ≤  ( +  ) +  (0 −  )
h
´
0

−
0
b ∈ 0 −
Let b = −
and from (25)
0 . Since 0     −  
0
 ( ) +  (0   0 ) ≤  ( +   0 ) +  (0 −  )
(26)
(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 )
¢
¡ 2
be two incomparable points in <+  ≤() ¡ 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 ,
(29)
 (0  ) −  ( + b ( −  0 )  ) ≤  (0 − b ( −  0 )  ) −  ( ) .
¡
¢
Together, (28) and (29) establish that  is SM in <2+  ≤() .
Example 23 illustrates the importance of the restriction to a ≤  -chain
F in Proposition 21.
¡
¢
√
Example 23 Let ( ) ¡= . This is SM ¢in every <2+  ≤() and
therefore  is SM in any <+ × F  ≤  ()  However, it is not SM on
¢
¡
<+ × F  ≤  ()  Take e = {05 05; 1 4} and e0 = {05 05; 2 3}, so
e∨ = {05 05; 2 4} and e∧ = {05 05; 1 3}  Moreover, assume that prices are
(
∨ )−(
0 )
(
)−(
∧ )
=
= 4, and take  = (5 e),  0 = (10 e0 )
such that


so  ∨  0 = (6 e∨ ) and  ∧  0 = (9 e∧ ). Then,  () +  ( 0 ) = 833,
 ( ∨  0 ) +  ( ∧  0 ) = 828.
¡
¢
Thus, comparing the restricted lattice <+ × F  ≤  () with the gen¢
¡
eral <+ × F  ≤  () , the complexity of the latter resides not only in the
2 1 These 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, since
e∨  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
the LSE (SSE) conditions in the FOSV
¡ cannot be used to determine
¢
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 set
of lotteries Fb , whose supports are bounded above by some  ∈ <+ . Recall
that 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
³ denotes the lottery ´with sure payoff .¡ With lottery supports
¢
so restricted, <+ × Fb  ≤  () is a sublattice of <+ × F  ≤  () 
This ensures that, for large enough incomes, all
³ join lotteries are affordable.
´
Thus, for  0 ≥  ≥  (), () ≤ ( 0 ) in <+ × Fb  ≤  () . 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- o
n
tions on may be further restricted. Let X = ( e) ∈ <+ × Fb |   +  (e
 ) ≥  () 
¡
¢
¡
¢
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
b
set <n
 with price  ()  Consider
+ × F has a FOSD-top degenerate lottery
o
X = ( e) ∈ <+ × Fb |   +  (e
 ) ≥  ()  Suppose that budget constraints
are binding¡ and that the lottery
price function satisfies (A1). If  ( e) is LSE
¢
(SSE) on X  ≤  () and  0 ≥  ≥  (), then arg max()  ( e) ≤
(≤ ) arg max( 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
) ≤
µ
¶
∨
0
¡
¢



−


(
)
(
)
 0 +  (e
 0 ), is 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,
  + ¾(e
 )   0 +  (e
 0 ), and 0  0. If   0 are such
½
∨
0



−


(
)
(
)
that max 0 −
 0 = 0, then  ( ∨  0 ) = 0 ≤  ()  ( 0 ). But

µ
¶
R
R
(
 )−(
∧ )
 ( 0 ) =  (0  ) 0    +


 ∧ =  ( ∧  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 perfect
substitutes, 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 ) be
0
0
two incomparable points with   +  (e
 0 ) ≥   +  (e
 ) and
µ e ¤  e∧. Then
¶
(
)−(
 )
∧
 () =  + e
 and  ( 0 ) = 0 + e
 0 . Also,  ∧  0 =  +


e

(
)−(
∧ )
so  ( ∧  0 ) =  +
+ e
 ∧ =  + e
 =  (), and  ∨  0 =


¾
¶
µ
½
(
∨ )−(
0 )
(
∨ )−(
0 )
 0  e∨ so  ( ∨  0 ) ≥ 0 −
+ e
∨ =
max 0 −


¢
¡
0 + e
 0 =  ( 0 ), so  ( ) is LSE (but not SSE) in <+ × F  ≤  () .
R
Example 27 Let  =  ( e) =³ + () (). Suppose
that  (e
 ) satis´
fies fair pricing. Then  is LSE in <+ × Fb  ≤  () . Letting  = ( e)
and  = (0  e0 ) be two incomparable points with e e0 ≤  -incomparable and
  +  (e
 ) ≤  0 +  (e
 0 ), the LSE conditions are:
Z
 (e
 ) −  (e
∧ )
+ () ∧ ()
 + () () ≥ ()  +

¾ Z
½
Z
 (e
 ∨ ) −  (e
0 )
0
∨
0
+ () () ≥ ()  + ()0 ()
⇒ max 0  −

Z
2 2 Notice
that this does not satisfy assumption (A2), which is however not required here.
27
and
Z
 (e
 ) −  (e
∧ )
()0 () ≥ ()  +
+ () ∧ ()

¾ Z
½
Z
∨
0

(e

)
−

(e

)
+ () ∨ () ≥ ()  + () ().
⇒ max 0 0 −

0 +
Z
Since with ≤  ,  + 0 =  ∧ +  ∨ , these conditions are satisfied if:
½
¾
 (e
 ∨ ) −  (e
 (e
 ) −  (e
∧ )
0 )
(30)
≤
0 − max 0 0 −


and
½
¾
 (e
 ∨ ) −  (e
 (e
 ) −  (e
∧)
0)
0
 − max 0  −
.
≤  − 0 +


(31)
Under fair pricing,  (e
 ) +  (e
 0 ) =  (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 ,
arg max()  ( e) ≤ (≤ ) arg max( 0 )  ( e).
Corollary 29 Consider the consumer problem (11) where the consumption set
<+ × Fb has a SOSD-top degenerate lottery  with price  ()  Suppose the
lottery price function satisfies (A10 ). Then if  (
³ e) is LSE (SSE and
´ the
budget constraint is binding at the optimum) in <+ × Fb  ≤ () and
 0 ≥  ≥  (), then arg max()  ( e) ≤ (≤ ) arg max( 0 )  ( e).
As in the case of the FOSV lattice, a¡possible SOSD chain is¢ that of the degenerate lotteries. Again the sublattice <+ × F  ≤ () coincides with
¢
¡
the expenditure value lattice <2+  ≤()  where the price function is the
projection 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 of
allowable 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 derivative of the state utility function. Restrictions on the third partial derivative are
regularly used in inivariate analysis.23
Proposition 30 Let  =R (   (e
 )) where  ∈ <++ and  ¡(e
 ) : F → <+ satis- ¢
fies (A10 ). Suppose  = ( ) ()  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 +  (e
 0 ) (hence 0  ).  is SM if:
µ
¶¸
Z ∙
 (e
 ) −  (e
0 )
0
(  ) −   +
  0

¶
¸
Z ∙ µ
 (e
 ) −  (e
0 )
0
≤
  −
  − ( )  .

µ
¶
¡
¢
(
 )−(
0 )
Since  is supermodular in <2+  ≤ , (0  ) −   +


is non
0
decreasing
in  From
µ
¶ ∆  =  ( +  ) − ( ) concave in , (  ) −
0
(
)−(
)
 +
  is concave in  Therefore, since e  e0 ,

µ
¶¸
Z ∙
 (e
 ) −  (e
0 )
(0  ) −   +
  0

µ
¶¸
Z ∙
 (e
 ) −  (e
0)
≤
(0  ) −   +
  

It is then sufficient that:
µ
¶
µ
¶
 (e
 ) −  (e
0 )
 (e
 ) −  (e
0 )
  ≤  0 −
  − ( ), all 
(0  ) −   +


This follows from  concave in  as  +
4.4
(
 )−(
0 )

≤ 0 by construction.
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.
2 3 The coefficient of absolute prudence involves the third derivative. It measures the coefficient 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
 ) ∈  let
e ( (e
 )) be a "top" and e ( (e
 )) a "bottom" lottery, e ( (e
 )) 6= e ( (e
 )) 
Consider ( e)  (0  e0 ) ∈ <+ × F . The Augmented Lottery Expenditure Value
order is:25
( e) ≤
(0  e0 ) iff   +  (e
 ) ≤  0 +  (e
 0 ) and
⎧
 )   (e
 0 ) or
⎨  (e
 (e
 ) =  (e
 0 ) and e = e0 or£
¤(32)
⎩
 )) or e0 = e ( (e
 )) 
 (e
 ) =  (e
 0 )  e 6= e0 and e = e ( (e
 ((
))
Note that ≤ (()) is a partial order. For antisymmetry, ( e) ≤ (())
(0  e0 ) and (0  e0 ) ≤ (()) ( e) imply that  = 0 and  (e
 ) =  (e
0 ) 
0

Suppose e 6= e  Then by the augmenting conditions, e = e ( (e
 )) or e0 =
e ( (e
 )) and, e0 = e ( (e
 )) or e = e ( (e
 ))  These conditions form a contradiction 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
 ) =  (e
 0 ) =  (e
 00 ) 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 simultaneously
only 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 antisymmetry 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
2 4 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.
2 5 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.
2 6 Antoniadou [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 µ
( e)  (0  e0 ) with¶ (e
 )   (e
 0 ) and
0
(
)−(
)
  +  (e
 )   0 +  (e
 0 ), the join is 0 −
 e and the meet is

µ
¶
(
)−(
0 )
+
 e0 . For incompararable ( e)  (0  e0 ) with  + (e
 ) ≤  0 +

0

 0 ),  (e
 ) =¢  (e
 0 ), e 6= e0 and
e ( (e
 )) the join is
¡ (e
¡ e 6=
¢ )) and e 6= e ( (e
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.
0
Consider ( e)  (0  e0 ) with   +  (e
 ) ≤  
 (e
 0 ),  (e
¢ ) =  (e
 0 ), e 6= e0
¡ +

0

∨
0 
and e 6= e ( (e
 )) and e 6= e ( (e
 ))  Let  =   e ( (e
 ))  The value of  ∨
0
0
is   + (e
 )  Therefore, by the augmenting condition, ( e)  (0  e0 ) ≤ (())
∨
  Consider any other upper bound,  = (00  e00 )  By definition  0 +  (e
0) ≤
00
00
00
∨
  +  (e
 ) and  (e
 ) ≥  (e
 )  Strict inequality implies  ≤ (()) 
Therefore suppose  (e
 00 ) =  (e
 )  Hence by definition it must be e00 = e ( (e
 ))
and again  ∨ ≤ (())  thus establishing that  ∨ is the join. The argument
for 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 applied:
Proposition 33 Consider the consumer problem (11) where the consumption
set is <+ × F is a lattice with the AEV order under the conditions of Definition 31. If ¡ ( e) is LSE (SSE and
¢ the budget constraint is binding at the
optimum) in <+ × F  ≤ (()) and  0 ≥ , then arg max()  ( e) ≤
(≤ ) arg max( 0 )  ( e).
In example 34 such an AEV lattice, in the case where there are two underlying lotteries that are available at different quantities and different mixtures.
This case is of independent interest, with some extensions of the basic (univariate) model of choice under uncertainty based on the existence of two underlying
distributions.
Example 34 Let S = {1 e1 + 2 e2 , 1  2 ∈ <+ } ⊆ F be the set of lotteries,
where e1 and e2 are two distinct base lotteries satisfying  (e
1 ) =  (e
2 ) = 1.


Suppose lotteries
are
linearly
priced.
Defining

e
()
=
e

,

e
()
=
e

1
2 deter¡
¢
mines a lattice <+ × S ≤ (()) , with e1 a worse lottery than e2 (this may
be 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
0
0 0
Suppose
that  = (
¡
¢ e) and 0  = (  e ) are two incomparable points in
<+ × F  ≤(× ) , with  ≥  and 0 ¤   . The LSE conditions
are:
Z
Z
Z
Z
0
∧
0
∨
(  )0 ≥ () ( ) ⇒ (  ) ≥ () ( ) (33)
and
Z
( ) ≥ ()
Z
∧
( ) ⇒
Z
0
∨
(  ) ≥ ()
Z
(0  )0 ,
and the SSE conditions are:
Z
Z
Z
Z
(0  )0 ≥ ( ) ∧ ⇒ (0  ) ∨  ( )
(34)
(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
Z
Z
(0  ) ∨ ≥ ( ) ∨ ≥ ( ) ,
(37)
∨
which is satisfied if  is nondecreasing and since
 ≤  
R
R . If the first
0
then either (  )0  ( )0 or
R holds strictly,
Rinequality in (33)
∧
0 
(
)
.
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 with
Pr {e
 0 ≥ b}  0 such that (0  b)  ( b). If  is LSE in <2+  ≤ , then for all
 ≥ b, (0  )  ( ). Therefore,
Z
Z
Z
(0  ) ∨  ( ) ∨ ≥ ( ) ,
(38)






and the implication in strict inequalities follows. If  is increasing, the first SSE
condition (35) holds directly since:
Z
Z
Z
(0  ) ∨  ( ) ∨ ≥ ( ) .
(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
R
establish
the implication in strict inequalities, suppose that ( ) 
R
( ) ∧ . Then, there exist  ≤ 0 with
 ≥ } + Pr {e
 ∧ ≥ 0 }  0
¡ 2 Pr¢{e
0
0 0
such that (
<+  ≤ , then (   )  (0  ). It
R  ) 0 ( ). RIf  is0 LSE in
∧
follows
that (
(  ) , and since  + 0 = ∨ + ∧ ,
 
R
R  )
0
∨
0
(  )  (  )0 . The second SSE condition (36) holds if  is increasing 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
R the FOSD
0
product
lattice.
Let
≤
be
defined
by

e
≤

e
if
and
only
if
 () ≥


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.
R
Proposition
¡ 2
¢ 35 Suppose  ( e) = ( ) (). If  (
¡ ) is increasing ¢
on <+  ≤ and strictly concave in , then  ( e) is SSE in <+ × F  ≤(×) .
Proof.
Suppose that ¢ = ( e) and  0 = (0  e0 ) are incomparable points
¡
in <+ × 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 and
e0 are incomparable by SOSD so e∨  e, in which case,
Z
Z
Z
(0  ) ∨ ≥ ( ) ∨  ( ) ,
(40)
since  is strictly concave. Or, (ii) e∨ = e  e0 , and it follows that
   since  and  0 are incomparable, and therefore (0  )  ( ) for all
. Hence:
Z
Z
Z
0
(0  ) ∨ 
( ) ∨ =
( ) .
(41)
Regarding the second LSE condition (36), from e∨  e0 the second
inequality 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 assumptions 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 = {05 05; 0 2}, and e = {1; 1} (degenerate
distribution 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 ) = 05 × 0 + 05 × 6
= 3  25 = 05 × 1 + 05 × 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 → < is
said 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, arg max  ≤ arg max 0  whenever  ≤  0 .
This framework applies to the ≤ (()) order in the absence of (A2),
 ) ≥  (e
 0 ) and
and thus antisymmetry. Take ( e)  (0  e0 ) ∈ <+ × F with  (e
0
0
  +  (e
 ) ≤   +  (e
 ). Then,
½µ
¾
¶
 (e
 ) −  (e
0 )
0 −
 e ,  (e
) ,
(43)
 0 =
 ) =  (e

½µ
¾
¶
 (e
 ) −  (e
0)
 0 =
+
 e ,  (e
0) .
 ) =  (e

 ) =  0 +  (e
 0 ),  0 =  0 =
Note that if  (e
 ) =  (e
 0 ) and   +  (e
{( e ) ,  (e
 ) =  (e
 )}.
¡
¢
Proposition 38 <+ × F  ¹ (()) 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 applications, as with the LEV lattice, the usual monotone comparative statics is not,
in general, implied unless a strong assumption is made about distinguishing between 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 ) ∈ <+ × F
with  (e
 ) ≥  (e
 0 ) and   +  (e
 ) ≤  0 +  (e
 0 ). The sufficient conditions for
lottery choice to be normal given by (42), i.e. -quasipermodularity, are:
⎧
µ
¶
(
)−(
0 )
⎪
⎪
∃e
 ,  [2 (e
 ) − 2 (e
 )] ≥ () 1  +
− 1 ()
⎪

⎪
⎪
⎪
µ
¶
⎪
⎪
(
)−(
0 )
⎪
0
0
0
⎪
 ,  [2 (e
 ) − 2 (e
 )] ≥ () 1 ( ) − 1  −
⎨ ⇒ ∀e

µ
¶
. (44)
0
(

)−


(
)
⎪
0
0
⎪
∃e

,

[
(e

)
−

(e

)]
≥
()

(
)

+
−

⎪

2
2

1
1

⎪
⎪
⎪
µ
¶
⎪
⎪
(
 )−(
0 )
⎪
0
⎪
 ,  [2 (e
 ) − 2 (e
 )] ≥ () 1 () − 1  −
⎩ ⇒ ∀e

Here, e denotes the low-priced lotteries ( (e
 ) =  (e
 0 )), and e denotes the
high-priced lotteries ( (e
 ) =  (e
 )). Suppose that  is concave. Then, the
first condition is satisfied if () 2 agrees with the lottery price ranking  (e
 ),27
and () 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.
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2 7 In the sense that, whenever one high-priced lottery is preferred by  to a lower-priced
2
lottery, the same preference ranking holds for all other lotteries with these prices.
35
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36