ORDINAL COMPARATIVE STATICS
Alexei Savvateev
New Economic School and Central Economics and Mathematics Institute, Moscow. The
financial support of the grant В # NSh-929.2008.6, School Support, is gratefully acknowledged.
I am grateful for comments from participants of seminars at CEMI, FIAN, SEI (Irkutsk),
SPb-EMI and EU (St-Petersburg), as well as from students at NES and participants of the
Makarov Readings at CEMI RAS.
Abstract. In this note, I introduce a pre-order on the space of all preference relations defined on
the fixed universal lattice. Using this order, together with the Veinott order on the power set of the
universal lattice, I state the monotonicity theorem which claims that the satiation correspondence
Argmax(, S) is monotone with respect to its both arguments, if one confines himself to qusisupermodular preference relations, on the one hand, and sublattices on the other. This theorem generalizes
and clarifies the results presented in the seminal paper [1].
Key words: Comparative statics, Veinott monotonicity, Intensity of preference relation, Lattices,
Argmaximum
Introduction
In their celebrated paper [1], the authors claim that most of results important in economics
are to be naturally formulated in ordinal terms, as dependencies of the “more – less” type,
rather then having been expressed by means of numerical values. Their approach is powerful,
as becomes clear from their paper.
However, the authors have not been attached to this point of view up to its logical point of
destination, that is to formulate the whole theory of comparative statics in purely ordinal terms.
Indeed, when the goal is to solve a maximization problem, it is rarely when one is interested in
the value of maximum, but rather in the set of points where maximum is being reached, that
is, in the Argmaximum set. And this latter concept does not depend on the specific function to
be maximized, rather on the class of functions defined up to a (strictly increasing) monotonic
transformation. This means that, in essence, usually people consider maximization (satiation)
of preference relations (i.e. complete preorders), under the label of maximization of a given
function defining this relation.
In this note, I am interested in the monotonic behavior of this Argmaximum as a function of
the two variables: a preference relation, and a subset on which it is maximized. Let us call any
such pair of a preference relation and a subset “an experiment”. This is the most general approach
to what is called “monotone comparative statics”. (The counterpart, “continuous comparative
statics”, also presents interest, but is skipped in what follows.)
Argmaximum is a set-valued correspondence from the set of all experiments (which coincides
with the Cartesian product of the power set of the universal lattice and the set of all preference
relations) to the universal lattice itself. In order to be able to state any propositions concerning
monotonic behavior of Argmaximum, one needs to specify monotonic structures on the set of
all experiments.
I show that it is possible to introduce a pre-ordering on the set of all preference relations such
that, combined with the Veinott’s ordering on the power set of the universe lattice, it makes
the set-valued correspondence of Argmaximum monotonic. Two precautions are to enter the
story, however: (1) the result holds only on the set of quasisupermodular preference relations;1
and (2) nontrivial applications run only for the class of sub-lattices, out of the whole power set
of the universal lattice.
Effectively, all the results known from monotone comparative statics as summarized
in [1] follow from this unified ordinal approach, being interpreted as the monotonicity of
Argmaximum correspondence per se.
Necessary definitions and the main result
Consider a problem of finding a satiation set for a given preference relation, restricted
to a given subset S ⊂ X of the universal lattice X (the lattice on which all our preference
relations to be considered are defined):
Argmax(, S) := {x ∈ S | ∀y ∈ S x y}.
This problem is called “a problem of finding Argmaximum”. Traditional approach, however,
deals with a given function f : X → R:
f (x) → max, x ∈ S,
and for any given subset S ∈ X produces two objects: the number M axx∈S f (x), as well as
the set Argmaxx∈S f (x) = {x ∈ S | f (x) ≥ f (y) ∀y ∈ S}.
In a wide class of problems, however, there is not much interest in the precise value
M axx∈S f (x) of the maximization problem (since this value changes with a re-scaling, i.e. under any increasing transformation of the function f ). The set Argmax, on the contrary, is the
same for all increasing transformations g(f (x)) of the initial function f (x) to be maximized.
This suggests that the natural formulation of maximization problems refer to classes of increasing transformations of functions, not to functions themselves. But these classes are exactly
parametrized by alternative preference relations on X. This justifies the undertaken approach.
Next, let us turn to parametric families of maximization problems of the type
f (x, θ) → max, x ∈ S[θ].
If we replace them with families of satiation problems, we encounter a certain difficulty.
Namely, in which sense a parametric family depends on parameters monotonically? For changing
subsets on which preferences are to be satiated, corresponding notion of monotonicity was
introduced in [3] and discussed extencively in [2], as well as in [1] cited above. This is the
notion of strong set order which extends the partial ordering on the lattice to the power set
of that lattice and which, being confined to the set of nonempty sublattices, defines a partial
ordering. Here it is:
Definition (Veinott 1989). Subset S1 lies not below subset S2 (notation: S1 S2 ), if
∀x ∈ S1 , y ∈ S2 x ∧ y ∈ S2 , x ∨ y ∈ S1 .
Things are not that clear for changing preference relations. In [1], where functions are to
be maximized, as well as in all the literature following that paper, any difficulties were hidden
under the assumption that the function has two arguments: one is the variable with respect to
which maximization is contemplated, and the second is a parameter, and monotonicity with
respect to a parameter was expressed via the following single crossing condition: for any two
ordered pairs (x0 , θ0 ) and (x, θ), where ordered means that x0 x and θ0 θ, we have that
if f (x0 , θ) ≥ (>)f (x, θ), then f (x0 , θ0 ) ≥ (>)f (x, θ0 ).
1
I give an alternative definition of quasisupermodularity, in purely ordinal terms — for preference relations
directly, instead of defining it on utility functions, as it is usually being done.
The essence of this single crossing condition, or their earlier more restrictive version of
increasing differences, is still to be clarified. For that matter, I offer alternative approach.
Namely, I define a pre-ordering on the set of all preference relations directly, and then show that
monotonicity with respect to parameters directly corresponds to the single crossing condition
of (Milgrom, Shannon 1994). I call the introduced pre-order intensity of preferences: a given
preference relation is more intensive then another preference relation if, speaking loosely, it is
closer to the partial ordering exogenously defined on the universal lattice under consideration.
Formally:
Definition. A preference relation 1 is called at least as intensive as a preference relation
2 (notation: 1 ≥ 2 ) iff ∀x ≥ y we have that
(i) x 2 (2 )y implies x 1 (1 )y
Now, it is time for a final point. Monotonicity results proved in (Milgrom, Shannon 1994,
Topkis 1998) follow immediately from Veinott monotonicity of Argmaximum with respect to
this intensity pre-ordering. Moreover, monotonicity with respect to a pre-ordering implies that
if the two distinct preference relations are equivalent (“equally intensive”), they must produce
one and the same Argmaximum correspondence, which means that they give identical sets of
satiation on any sublattice! The very fact that distinct preference relations could not be told
from each other in any of the choice experiments involving sublattices, is truly amazing.
To state the main result, I still need to reformulate the notion of quasisupermodularity also
in purely ordinal terms;
Definition. A preference relation is called quasisupermodular iff
(i) x x ∧ y implies x ∨ y y, and
(ii) y x ∨ y implies x ∧ y x.
Now, the monotonicity theorem in its most general form, in purely ordinal terms, holds
when formulated as follows:
Ordinal theorem on monotonw comparative statics. The set-valued correspondence
Argmax(, S) is monotonic with respect to its both arguments, being considered on the
Cartesian product of the space of all quasisupermodular preference relations and the subset of
the power set of the universal lattice formed by all nonempty sublattices.
Список литературы
[1] Milgrom P. and Shannon C. Monotone Comparative Statics. - Econometrica, vol. 62, no. 1,
1994, pp. 157-180.
[2] Topkis, Donald Supermodularity and complementarity. - Princeton University Press, 1998.
[3] Veinott, Arthur F. Lattice programming. - Unpublished notes from lectures delivered at
Johns Hopkins University, 1989.