Generalized Monotonicity Analysis∗
Bruno H. Strulovici†

Thomas A. Weber‡

Abstract
Complex economic models often lack the structure for the application of standard techniques in monotone comparative statics. Generalized Monotonicity Analysis (GMA) extends the available methods in several directions. First, it provides
a way of finding parameter moves that yield monotonicity of model solutions. Second, it allows studying the monotonicity of functions or subsets of variables. Third,
GMA naturally provides bounds on the sensitivity of variables to parameter changes.
Fourth, GMA may be used to derive conditions under which monotonicity obtains
with respect to functions of parameters, corresponding to imposed parameter moves.
Fifth, GMA contributes insights into the theory of comparative statics, for example, with respect to dealing with constraints or exploiting additional information
about the model structure. Several applications of GMA are presented, including
constrained optimization, non-supermodular games, aggregation, robust inference,and monotone comparative dynamics.

Keywords: Aggregation, Comparative Statics, Comparative Dynamics, Monotone Comparative Statics, Parameterized Equations, Parameter Transformation, Quantitative Monotonicity Analysis, Robust Inference, Supermodular Games.
JEL-Classification: C61, C43, C72, D11.

∗

We are very grateful to David Greenstreet, Paul Milgrom, John Quah, Kevin Roberts, and an anony-mous reviewer for helpful comments and discussion.
†
Department of Economics, Northwestern University, 2001 Sheridan Road, Evanston, IL 60208-2600.
E-mail: b-strulovici@northwestern.edu.
‡
Department of Management Science and Engineering, 442 Terman Engineering Center, Stanford
University, Stanford, CA 94305-4026. E-mail: webert@stanford.edu.

11Introduction

The comparison of model predictions, typically described in terms of solutions to an
optimization or equilibrium problem, for different parameter values is the subject of comparative statics (Hicks 1939; Samuelson 1941). Of particular interest is the monotonicity
of solutions in model parameters, giving rise to monotone comparative statics (MCS).
Until the appearance of ordinal methods, the standard tool of MCS was, beyond directcomputation of solutions, the implicit function theorem (IFT). The introduction of ordinal
methods, particularly by Topkis (1968), Milgrom and Shannon (1994), and Athey (2002),
has allowed researchers to characterize MCS in terms of sufficient (and in some sensenecessary) conditions on the primitives of the model.
The standard approach to MCS takes the formulation of the problem (which represents
the model) as given. It cannot deliver positive results when monotonicity does not obtainwith respect to the original variables and parameters. However, interesting monotonicity
properties may generally be found once one broadens the horizon to the monotonicity
of functions of variables with respect to functions of parameters. This paper presents a
simple method, called Generalized Monotonicity Analysis (GMA), to systematically uncover and analyze monotonicity properties of solutions to optimization or equilibrium
problems. In GMA, parameters are allowed to move simultaneously in the parameter
space, and monotonicity is investigated for functions of model solutions. The approach,which is fundamentally geometric, also sheds new light on existing comparative-statics
results. GMA introduces several concepts, such as ‘pseudo-gradients’ and ‘monotonicity
directions,’ which may help applied economists to disentangle various issues that arise
when trying to analyze the monotonicity properties of their models, even when model
complexity rules out the use of standard MCS tools. The GMA approach raises questions
beyond standard comparative-statics analysis. For example, one may ask what structure isrequired for the variables of a problem to be monotonic in some parameter aggregate (Section 4.3). In problems where parameters represent heterogeneity across some agents, one
may ask what notion of “increased homogeneity” improves their incentives (Section 4.2).
More generally, Section 3 shows that GMA discloses all monotonicity properties relating
solutions of a problem to its parameters given the available information (Theorem 1).
GMA extends comparative statics in yet another direction, namely the study of the rates
at which solutions vary with parameters. This application of GMA is referred to as quantitative monotonicity analysis. As one of the many potential applications of quantitative2monotonicity analysis, we employ it in Section 4.4 to obtain rate constraints in a wellknown problem of robust inference concerned with the empirical determination of firms’
productivity levels.
The gist of GMA can be described as follows. Consider a model described by the parameterized equation
f (x, t) = 0,
where x ∈ Rn is the variable, t ∈ Rm is the parameter, and f takes values in Rn . For any t,
let x(t) denote the solution to the above system, which, for this introductory exposition,is assumed to be unique. For each t, let R(t) ⊂ Rn denote an information set known tocontain the solution: x(t) ∈ R(t). Assume, still for this introduction, that the Jacobianmatrix Dx f of f with respect to x is everywhere invertible. For each nonzero vector vof Rm , consider the vector
W (x, t, v) = −[Dx f ]−1 (x, t)(∂f /∂v),
where ∂f /∂v is the directional derivative of f in direction v. We call this vector apseudo-gradient along direction v, because it coincides with the actual gradient of x(t)with respect to v if one substitutes x = x(t) in the expression for W (x, t, v). The family W̄ (t, v|R) = {W (x, t, v)}x∈R(t) of all pseudo-gradients on the information set contains
essential information about solution sensitivity to parameters: even though the exact location of x(t) may be unknown, the reaction of x(t) to a parameter change in direction v iscontained in W̄ (t, v|R). The key, then, is to determine properties of W̄ (t, v|R) that permit
statements about which parameter moves will affect solutions monotonically. The simplest
case is when W̄ (t, v|R) is a singleton. Suppose, for example, that W̄ (t, ej |R) = {wj (t)},where ej is the j-th basis vector of Rm . In general, wj (t) may have some negative components: moving from t along the arbitrary direction ej does not necessarily increase
all components of x(t). However, suppose that we can find real numbers λj (t) suchPthat
j λj (t)wj (t) is a positive vector. Then, moving from parameter t in directionPv(t) = j λj (t)ej increases all components of the solution x(t), at least locally. We callsuch directions of parameter changes monotonicity directions at t. It turns out that theset V (t|R) of all monotonicity directions at t is a convex cone,1 which is referred to as the
GMA-cone at t. In general, W̄ (t, v|R) contains multiple elements. One can show that thesmaller the set W̄ (t, v|R) (in the inclusion order) and the larger the GMA-cone V (t|R)
1

The cone may be empty. In that case, given one’s information about the sensitivity of x to theparameters, it is impossible to construct parameter moves that guarantee an increase in the solution(cf. Theorem 1).

3

at t, the easier it becomes to find parameter moves along which solutions are monotonic.
Selecting a vector field of monotonicity directions (contained in V (t|R) at each t) yields
trajectories along which a model solution (or smooth selections thereof, in the general caseof a set-valued solution) is necessarily monotonic. — Instead of considering the monotonicity of a model solution directly, GMA can be applied to a function of the solution,such as an average (discussed in Section 4.3). Lastly, we stress that the method permits the natural use of additional information about solutions through the information
set R(t). For example, if the equation f (x, t) = 0 represents the first-order necessary
optimality condition of a maximization problem, then R(t) may consist of all x that alsosatisfy the second-order condition, Dx f (x, t) ≤ 0. A smaller information set R(t) resultsin smaller pseudo-gradient sets W̄ (t, v|R), and hence in larger GMA-cones V (t|R) (Theorem 2). Note that instead of its straightforward use to merely guarantee the monotonicity
of model solutions along parameter paths, GMA can also be employed to influence the rate
at which solutions vary with parameters, a more quantitative approach to MCS developedin Section 3.5.
When analyzing monotonicity properties of solutions to a given problem, results naturally
depend on the concept of monotonicity that is being used. For example, a comparativestatics analysis based on ordinal methods usually describes the monotonicity of solutions in the strong set order.2 In noncooperative games with strategic complementarities, equilibrium monotonicity is typically limited to the smallest and largest equilibria(Milgrom and Roberts 1990). However, Echenique (2002) shows that, in such games,monotonicity results can be extended to all stable equilibria. Allowing for less structured (e.g., non-supermodular) environments, we allow for any differentiable (or at least
right-differentiable) selection of the solution set X(t) and propose a systematic way toinvestigate properties of such a selection. GMA can generate statements of the following
nature.
Each differentiable solution is nondecreasing as parameters are moved in thisdirection or along that trajectory.
The differential nature of such statements may or may not restrict the strength of theresults, and can sometimes be relaxed to yield more general statements. The following five2

X(s) ≤ X(t) in the strong set order (Veinott 1989) if and only if (x, y) ∈ X(s) × X(t) implies

that (x ∧ y, x ∨ y) ∈ X(s) × X(t), where x ∧ y denotes the componentwise minimum and x ∨ y thecomponentwise maximum of the two vectors x and y.

4

points further clarify how GMA can be applied. First, as mentioned earlier, GMA may
be used as an exploratory step to elicit monotonicity properties of a complex problem.
Directly applying ordinal methods to a complex problem may often require much ingenuity or expertise, or may simply fail if one restricts attention to the original variables andparameters. In contrast, the method proposed here is based on elementary differential
calculus and provides a helpful start to explore monotonicity. The results of this first step
may then be derived under more general conditions using ordinal methods.3 In general,
one should think of differential and ordinal methods as complements, whose strengths canbe combined to yield sharp and general results when analyzing the monotonicity of solutions to an optimization or equilibrium problem. Second, in many problems the solution
is generically unique, so that the question of which selections to consider naturally disappears when X(t) is a singleton that is differentiable in t. If there are multiple solutions
that are locally unique, GMA considers monotonicity of each corresponding selection, andthus necessarily also the most interesting ones (e.g., stable selections).4 Third, in equilibrium analysis, one may be interested precisely in equilibrium evolutions that are smoothwith respect to parameters. Indeed, ruling out solutions that suddenly jump after an
infinitesimal parameter move may be a desirable assumption when analyzing equilibriummonotonicity.5 Fourth, GMA easily accommodates the introduction of additional constraints to discriminate between solutions. For example, nonnegativity of solutions may
be imposed as an additional inequality to the information set R(t). Fifth, monotonicity
can be weak or strict, or even stronger. While ordinal methods yield weak monotonicity,
the differential nature of GMA makes is possible to establish strict monotonicity results.6
As noted earlier, GMA can be used in the form of ‘quantitative monotonicity analysis’ toobtain explicit bounds for the response rate of model solutions to changes in parameters.7
GMA extends the boundaries of MCS conceptually by providing a systematic method forinvestigating the interplay between model structure and available information (including
voluntary restrictions) about model solutions. The approach is constructive, as it endogenously generates the largest cones of directions in the parameter space that are compatible
with monotonicity.8 Parameter directions found this way are likely to have an intuitive3
4

For example, such exploration was used in Quah and Strulovici (2008).
The transversality theorem provides simple conditions under which differentiability of solutions ob-

tains generically, as a consequence of the invertibility of Dx f (Guillemin and Pollack 1974, p. 68).
5
Similar assumptions are often made in optimization problems, cf. Persico (2000, Assumption A1).
6
Edlin and Shannon (1998) focus on strict monotonicity of solutions. Their analysis is based on
differentiability and first-order conditions.
7
See Section 3.5 and the application in Section 4.4.
8
Jensen (2007) also considers general cones for comparative statics. However, in his analysis the5interpretation (for example, they can express a notion of “proximity,” as in Section 4.2),
which can be difficult to determine otherwise.
The paper builds on Strulovici and Weber (2008), providing more general foundations andresults for monotonicity analysis, examining new issues such as quantitative monotonicity
analysis and robustness, and proposing economic applications that illustrate the versatilityof the method. Section 2 introduces the concepts of the paper. Section 3 contains core
theoretical results. Section 4 provides applications of GMA to constrained optimization,
non-supermodular games, aggregation theory, quantitative monotonicity analysis, andmonotone comparative dynamics. Section 5 concludes.

2

Concepts and Notation

The GMA approach seeks to establish the monotonicity of a ‘criterion’ in certain parameter movements. The (monotonicity) criterion is a function of a variable that solves a‘primitive equation.’ The approach can accommodate additional solution requirements
which are not encoded in the primitive equation, but are available in the form of an‘information structure.’ An information structure describes a subset of solutions to theprimitive equation. The method proceeds to compute ‘pseudo-gradients’ to estimate thecriterion change in the different parameter directions. These estimates are then used toobtain ‘monotonicity directions’ in the parameter space, along which a criterion change
is guaranteed to be nonnegative.
Primitive Equation. Let X(t) denote the set of solutions x(t) to the primitive equation
f (x, t) = 0,(1)

where the (decision) variable x lies in a smooth manifold X ⊆ Rn , the parameter t lies
in a convex subset T of Rm , and f : Rn × Rm → Rn (with n ≥ 1) is a continuously
differentiable (smooth) function.9
Monotonicity Criterion. Instead of considering only the monotonicity of solutions x(t)to the primitive equation, we introduce a monotonicity criterion x̂ = φ(x), where φ is adirection cones are treated as exogenous primitives.
9
All of our results can easily be extended to allow for infinite-dimensional parameters, although, forsimplicity, we focus on the case with finite-dimensional parameter vectors. Some of the results also
generalize to infinite-dimensional variables, as in Section 4.5.

6

Figure 1: GMA Primitives (for x̂ = φ(x) ≡ x).
d-dimensional differentiable function of x (the notation x̂ will sometimes be used insteadof φ(x) for simplicity). For example, monotonicity of the first component of x(t) can be
examined by setting φ(x) = x1 . The criterion can also be used to examine the monotonicity of aggregates of the original variables. If, for example, x = (x1 , . . . , xn ) represents
consumption decisions of n individuals, then the comparative statics of aggregate consumption can be examined by choosing a criterion of the form φ(x) = x1 + · · · + xn . To
avoid obscuring the exposition of the GMA method, we often de-emphasize the presenceof the criterion φ by simply writing x̂ instead of φ(x). Analyzing the monotonicity of theoriginal variables amounts to setting φ(x) = x or, equivalently, x̂ = x.
Information Structure. For any t ∈ T , let R(t) ⊂ X(t) denote the subset of solutions
to (1) which satisfy a number of additional constraints, given a parameter value of t. Suchadditional constraints may include nonnegativity, second-order conditions, or stability
requirements, depending on whether (1) derives from an optimization or an equilibrium
problem.10 The collection R = {R(t)}t∈T defines a parameterized information structureof the problem. The crucial point is that, when searching for monotonicity directions, one
may be able to exploit not only the fact that any element x(t) of R(t) solves the primitive
equation (by definition), but also that it satisfies the additional requirements implied by
the inclusion x(t) ∈ R(t). A map x : t 7→ x(t) ∈ R(t) is a selection of R. We say that R
is a refinement of R0 , denoted by R ⊆ R0 , if R(t) ⊆ R0 (t) for all t.
10

GMA (particularly Theorem 3 and Corollary 1) can be extended to conditions that cannot be sum-

marized by a subset of X , for example conditions involving derivatives of x(t). Such conditions can be
useful to narrow down the set of pseudo-gradients, as in the example of Section 4.1.

7

Symbol

Dimension

Interpretationx∈Xn

Variablex̂ = φ(x) ∈ φ(X )d ∈ {1, . . . , n}Monotonicity criteriont∈TmParameter

v∈R

m

m

Parameter directionnnPseudo-gradient of f at (x, t) in direction v, satisfies (2)ddCriterion function

γ : [0, 1] → TmPath in the parameter space

R(t)

n

Information set at t (= subset of X(t))

X(t)

n

Set of solutions to (1) at t

V (t|R)m

GMA-cone at t given R, defined in (4)W (x, t, v)n

Set of pseudo-gradients at (x, t) in direction v

W̄ (t, v|R)w(x, t) ∈ R
φ:X →Rn

Set of pseudo-gradients at t in direction v (= ∪x∈R(t) W (x, t, v))nn

Set of feasible decisions (= manifold)mm

Parameter set (and the underlying vector space)

X ⊂R
T ⊂R

Table 1: Summary of Notation.
Parameter Paths. A continuously differentiable map γ : [0, 1] → T is called a path of theparameter space. A parameter t ∈ T is on γ if t = γ(λ) for some λ ∈ [0, 1]. A path γ starts
at t ∈ T if γ(0) = t. It starts at t in the direction v ∈ Rm if γ̇(0+ ) = dγ/dλ|λ=0+ = v. A
path γ that starts at t in the direction v will be referred to as a (t, v)-path. A selection xof R is γ-differentiable if λ 7→ x(γ(λ)) is differentiable on (0, 1), and is right-differentiable
at 0.
Pseudo-Gradients. A pseudo-gradient of f at (x, t) in direction v is a vector w in Rn suchthat
Dx f (x, t) w + Dv f (x, t) = 0(2)

for some x ∈ R(t). The set of pseudo-gradients is denoted by W (x, t, v). This is the set of
potential gradients assumed by any smooth selection of R, as the parameter moves from t
in direction v. Since any solution to (1) at t which may be of interest is localized to theinformation set R(t), the set of all possible pseudo-gradients for a parameter movementfrom t in direction v isW̄ (t, v|R) = {w ∈ W (x, t, v) : x ∈ R(t)}.

(3)

The set W̄ (t, v|R) contains all vectors of Rn which may be actual gradients of somedifferential solution to (1) as the parameter moves from t in direction v.

8Monotonicity Path. γ is a monotonicity path if x̂ = φ(x) is nondecreasing along γ forany γ-differentiable selection x.
Monotonicity Directions. Given an information structure R and a parameter t ∈ T , let
V (t|R) = {v ∈ Rm : Dφ(x)w ≥ 0, x ∈ R(t), w ∈ W (x, t, v)}(4)

denote the set of parameter directions in which the monotonicity criterion is nondecreasing.11 As is shown in the next section, V (t|R) forms a convex cone, containing the initial
directions of all monotonicity paths starting at t. It will henceforth be referred to as the
GMA-cone at t.
Table 1 summarizes the notation, and Figure 1 depicts the main GMA primitives.

3

Theoretical Results

In this section, we first show that the set of pseudo-gradients contains all actual gradientsof smooth selections of R (Proposition 1). We then establish that the set V (t|R) of monotonicity directions is a convex cone (Proposition 2), that this cone contains all directions
guaranteeing monotonicity of φ evaluated at the solutions (Theorem 1, the “necessity”
part), and that the cone widens as R becomes more informative (Proposition 2). We
then establish that trajectories following directions in the GMA-cone are monotonicity
paths for φ (Theorem 3, the “sufficiency” part), and provide a simple characterizationof monotonicity when Dx f is invertible (Corollary 1). After summarizing the different
steps of GMA, we show how information contained in R may be folded into the primitive
equation (1), which establishes a form of equivalence between various initial descriptionsof any given problem. The analysis is then modified to consider several other importantconcepts of monotonicity: strict monotonicity and quantitative monotonicity (Propositions 4 and 5). The latter concept, which to the best of our knowledge is new, can beused to derive “robust” comparative statics, as illustrated by Section 4.4 in the context
of productivity estimation. Another theoretical aspect of the method, concerning theinclusion of constraints, is treated as an illustrative application in Section 4.1.
11

While the GMA-cone clearly depends on the criterion function φ, we choose to ignore this dependence

in the notation.

9

3.1

Construction and Properties of Monotonicity Paths

Proposition 1 (Pseudo-Gradients) If γ is a (t, v)-path and x a γ-differentiable selection of R, thenDv x(t) =dx(γ(λ)) ∈ W (x(t), t, v).
dλ λ=0+

Proof. Right-differentiating both sides of f (x(γ(λ)), γ(λ)) = 0 at λ = 0+ yields
Dx f (x(t), t)dx(γ(λ))+ Dv f (x(t), t) = 0,dλ
λ=0+which satisfies (2) with x(t) ∈ R(t).

The next result characterizes the shape of the set of all directions v in which the monotonicity criterion increases.
Proposition 2 (Monotonicity Directions) For any t, V (t|R) is a convex cone.
Proof.

Let t ∈ T . We first show that V (t|R) is a cone. It clearly contains theorigin. For any t, v ∈ V (t|R), α > 0, and x ∈ R(t), we have W (x, t, αv) = αW (x, t, v),
since Dαv f = αDv f . Therefore, Dφ(x)w ≥ 0 for w ∈ W (x, t, αv) if and only if Dφ(x)w ≥0 for w ∈ W (x, t, v). To establish the convexity of V (t|R), consider any v, v 0 ∈ V (t|R)
and x ∈ R(t). Since Dv+v0 f (x, t) = Dv f (x, t) + Dv0 f (x, t), it follows that W (x, t, v + v 0 ) =W (x, t, v) + W (x, t, v 0 ), and hence that Dφ(x)w ≥ 0 for all w ∈ W (x, t, v + v 0 ). Since thelast inequality holds for all x ∈ R(t), it follows that v + v 0 ∈ V (t|R).



We now show that V (·|R), when considered on its entire domain T , generates all monotonicity paths along which the criterion is monotonic. Since GMA is concerned withsmooth solutions, we assume (possibly by restricting the information structure) that for
all t, each element of R(t) can be reached by a smooth selection of R. Precisely, R
is smooth if for any path γ, parameter t on γ, and vector x ∈ R(t), there exists a γdifferentiable selection y of R such that y(t) = x. Smooth information structures ruleout isolated points in the graph of X(t), which are irrelevant for the GMA concept ofmonotonicity.
Theorem 1 (Necessity) Suppose that R is smooth and that v ∈/ V (t|R). Then there
exists a (t, v)-path γ and a γ-differentiable selection x such that some component of thecriterion x̂ = φ(x) is decreasing in a right-neighborhood of λ = 0.
10

Proof.

By definition, v ∈/ V (t|R) implies that there exist x ∈ R(t), w ∈ W (x, t, v),

and i ∈ {1, . . . , d} such that the i-th component of Dφ(x)w is negative. Let γ be any pathsuch that γ(0) = t and γ̇(0+ ) = v. By smoothness of R, there exists a γ-differentiableselection x of R such that x(0) = x. By construction, the i-th component of Dφ(x(0))w
is negative for w ∈ W (x, t, γ̇(0+ )), implying that φi (x(γ(λ))) is decreasing in a rightneighborhood of λ = 0.

Thus, the set V (t|R) is in effect the largest cone of directions v (i.e., the “GMA-cone”)
at t, for which the criterion x̂(t) is guaranteed to be monotonic at the beginning ofany (t, v)-path.
The mapping V (·|R) can be viewed as a set-valued vector field: trajectories generated
by V (·|R) are parameter paths with the property that at any t on such a path, the directionof the path belongs to V (t|R). As the notation indicates, the shape of the cone V (t|R) depends on the point t in the parameter space and on the available information structure R.
Standard methods in comparative statics consider only the case where the GMA-cone is
fixed to the positive orthant, i.e., where V (t|R) = Rm
+ . Trajectories generated by sucha cone require that all components of the parameter path be nondecreasing. Here, bycontrast, the GMA-cone V (t|R) is determined endogenously by the primitives and theavailable additional information about solutions to the problem (1). As the information
structure becomes finer, the GMA-cone cannot become smaller.
Theorem 2 (Information-Structure Refinements) If R ⊆ R0 , then V (t|R0 ) ⊆V (t|R) for all t ∈ T .
Proof. For a given t ∈ T fix an arbitrary v ∈ V (t|R0 ). Then for any x ∈ R(t) ⊂ R0 (t)and any w ∈ W (x, t, v) we have Dφ(x)w ≥ 0, which implies by (4) that v ∈ V (t|R).

The finer the available information structure about the solution set, the larger the setof monotonicity directions in the parameter space. This establishes a partial order over
information structures R, which is similar to Blackwell’s (1951) order for the comparisonof information sources.12
12

An information source A is at least as informative as an information source B in the sense of Black-well, if for any decision, subject to some minor technical requirements, the (outcome-contingent) payoffs
attainable using information source B can also be attained using information source A. Analogously, an
information set R is at least as useful for generating MCS as the information set R0 if R ⊆ R0 , sincethen W (t|R0 ) ⊆ W (t|R), i.e., the set of all monotonicity directions generated by R contains the one
generated by R0 . If R0 is obtained from R by the removal of constraints, then the partial order is valid

11

The next result is the backbone of the GMA method: it states that the criterion x̂ isnondecreasing along any direction of the cone V .
Theorem 3 (Sufficiency) If γ is a path such thatγ̇(λ) ∈ V (γ(λ)|R)(5)

for all λ ∈ [0, 1], then it is a monotonicity path.
Proof. Fix any γ-differentiable selection x of R. For any λ ∈ [0, 1), let t = γ(λ) and v =γ̇(λ) (or the right-derivative of γ if λ = 0). By Proposition 1, Dv x(t) ∈ W (x(t), t, v). By
construction of γ, this implies that Dφ(x(t))Dv x(t) ≥ 0. Now, for λ < λ0 , the fundamental
theorem of calculus implies that
0

Z λ0

0

x̂(γ(λ )) − x̂(γ(λ)) = φ(x(γ(λ ))) − φ(x(γ(λ))) =Dφ(x(γ(`)))Dγ̇(`) x(γ(`))d` ≥ 0,
λhence that the criterion x̂(γ(λ)) is nondecreasing in λ.

The differential inclusion (5) describes the dynamics of monotonicity paths (interpreting λas time).13 When Dx f is invertible and T is finite-dimensional, the pseudo-gradient is
unique and can be directly computed, which simplifies the statement of Theorem 3.
Corollary 1 Suppose that v : T → Rm is a smooth vector field such thathi−Dφ(x) (Dx f )−1 (x, t) Dt f (x, t) v(t) ≥ 0(6)

for all t ∈ T and x ∈ R(t). Then, trajectories generated by v are monotonicity paths withrespect to φ.
Proof. With Dx f invertible, the only vector w satisfying (2) is −[Dx f ]−1 Dv f , so thatthe condition (4) for all x ∈ R(t) and w ∈ W (x, t, v) reduces to (6).

3.2

Method Summary

The GMA approach can be summarized as follows.
for any problem of the form (1) in which X and T are fixed.
13
Differential inclusions are a standard tool in the description of dynamic systems (see, e.g.,
Smirnov (2002)).

12

1. Step 1 (Pseudo-Gradients) For all relevant t and v, use (3) to compute the set ofpseudo-gradients. This set describes the possible changes of any smooth selectionof R along any (t, v)-path, or – in other words – the solution sensitivity to any(smooth) change of parameters at t in direction v.
2. Step 2 (GMA-Cone) For each t, solve (4) to determine the convex cone V (t|R) of
monotonicity directions at t.
Alternatively, one may wish to impose a cone of monotonicity directions and deriveconditions on the primitives under which parameter changes of interest are monotonicity paths. For example, this is what standard comparative statics does, when V
is identically equal to the positive orthant. This approach is further explored in Section 3.3.
3. Step 3 (Monotonicity Paths) When the GMA-cone is not reduced to zero, selecting one monotonicity direction for each t generates a vector field. It is typicallypossible to generate a smooth vector field (at least for a large subset of T ). Eachof its trajectories is then a monotonicity path. By Theorem 3 this can be accomplished by solving the differential inclusion (5). If, however, V (t|R) = {0}, one
needs to gather more information about the solutions to (1) to narrow down the setof pseudo-gradients which by Proposition 2 can only widen the GMA-cone. Alternatively, it may be useful to choose a less demanding criterion function φ, for example
by reducing its dimension d (which may be achieved by dropping or aggregating
variables).
4. Step 4 (Optional - Reparametrization) If monotonicity paths cover T , they may
naturally be used to provide a new parametrization of the initial problem, underwhich monotonicity obtains with respect to the first of these new parameters. Moreprecisely, if one can find a (Lipschitz-continuous) function σ : T × [0, 1] → Rm , suchthat
γ(0) = t, γ̇(λ) = σ(γ(λ), λ) ∀ λ ∈ (0, 1)⇒

γt (λ) = γ(λ) ∀ λ ∈ (0, 1)

for all t ∈ T , then, by the rectifiability theorem for direction fields (see, e.g., Arnold
and Il’yashenko (1988)), it is possible to (at least locally) transform coordinates
such that monotonicity is guaranteed. The function σ can be obtained by pasting
together the differential equations that describe γt by selecting a continuous σ suchthat
σ(t, λ) = γ̇t (λ) ∈ V (γt (λ)|R)13

for all t ∈ S. One method is to use a (k − 1)-dimensional plane P that is transverse(i.e., never collinear) to the (Lipschitz-continuous) vector field described by v(t) =σ(γt (0), 0) for all t ∈ S. Then, any t ∈ S corresponds to a unique point on theplane (described by the m − 1 contra-variant coordinates s2 , . . . , sm ) and the time ittakes to get to or from the plane P to t (described by the co-variant coordinate s1 ),depending on the direction of the flow. This defines a new parametrization t = ϕ(s)in s = (s1 , . . . , sm ) on S such that the criterion x̂(ϕ(s1 , s2 , . . . , sn )) is nondecreasingin s1 .14

3.3

Imposed Monotonicity Paths

The method described thus far discloses all paths in the parameter space along which thecriterion x̂ = φ(x) is nondecreasing for all smooth solutions x. In some problems, one may
wish to reverse the question, and find conditions under which some particular directionsof the parameter space are guaranteed to yield monotonicity of φ(x). For example, usualcomparative statics consider the case in which V contains the positive orthant of theparameter space. Such conditions can be tested by checking that extreme rays of thecorresponding GMA-cone, which are unit vectors of the parameter space, belong to V .
Imposing that extreme rays of the positive orthant belong to V generates inequalities
that are equivalent to the differential characterization of supermodularity as applied toobjective functions of optimization problems.
More generally, suppose that one wishes to ensure that a particular cone V0 of parameter
directions guarantees the monotonicity of smooth solutions (or some criterion thereof),
and let ∆ denote a set of vertices (directions in T ) generating the cone V0 . Then thefollowing result obtains, the proof of which is immediate and therefore omitted.
Proposition 3 If ∆ ⊂ V (t|R), then the criterion x̂ = φ(x) is nondecreasing along any
trajectory generated by V0 , i.e., any path γ with γ̇ ∈ V0 is a monotonicity path.
Section 4.3 illustrates this result in the context of parameter aggregation.
14

Further details on global reparametrizations, using additional tools in the theory of ordinary dif-ferential equations, have been developed elsewhere (Strulovici and Weber 2008). As the applications in
Section 4 demonstrate, it is often enough to restrict attention to the first three steps of GMA to obtain afairly complete picture of the monotonicity properties of the solutions to a given problem of the form (1).

14

3.4

Folding Information into the Primitive Equation

The analysis so far has maintained a clear distinction between the primitive equation (1)and the additional conditions imposed by the information structure R. In fact, the additional information encapsulated in R can often be folded into the primitive equation. To
see this, consider an example. Suppose that
R(t) = {x ∈ X : g(x, t) = 0 and h(x, t) ≤ 0},
where the smooth constraint function h(x, t) takes values in Rk (with k ≥ 1). Thissituation arises when (1) represents the first-order necessary optimality condition of a
maximization problem. The corresponding necessary second-order optimality condition(at any interior solution x, not on the boundary of X ) is that fx (x, t) ≤ 0. Define Z =X × Rk , z = (x, y) with y ∈ Rk . The information from R(t) can be folded into the initial
equation if f is extended tof (x, t)fˆ(z, t) = g(x, t),h(x, t) + y 2

where y 2 = ((y1 )2 , . . . , (yk )2 ) ∈ Rk+ represents a vector whose elements are the nonnegative
slacks associated with the components of the inequality constraint.
To further illustrate this technique, suppose that x is one-dimensional, and that R(t) ={x : f (x, t) = 0 and fx (x, t) ≤ 0} ⊂ R. Consider any smooth selection x : t 7→ x(t)of R. If fx (x(t), t) < 0, then x0 (t) = −ft (x(t), t)/fx (x(t), t) by the implicit function
theorem. However, this theorem cannot be applied if fx (x(t), t) = 0. Suppose now thatthe second-order condition is incorporated into f . The resulting equation is
"
#
f
(x,
t)
fˆ(x, y, t) == 0.
fx (x, t) + y 2
Differentiating these equations yields15 relations: fx x0 + ft = 0 and fxx x0 + fxt + 2yy 0 = 0.
For y(t) 6= 0, fx is invertible and the implicit function theorem can be used. If y(t0 ) = 0 forsome parameter t0 , the second relation yields x0 (t0 ) = −fxt (x(t0 ), t0 )/fxx (x(t0 ), t0 ), which
describes the sensitivity of the selection x(t) at t0 , despite the singularity of the equation
at that particular parameter and solution (as long as fxx (x(t0 ), t0 ) 6= 0). Thus, folding
15

Smoothness of y(t) obtains by the relation fx (x(t), t) + y 2 (t) = 0, as well as the smoothness of fx

and x in t.

15

the second-order condition into f implies sharp predictions for the pseudo-gradient, evenat singularity points.16

3.5

Quantitative Monotonicity Analysis

Another innovation of this paper is to examine at some degree of generality the rate at
which solutions are monotonic. We believe that this question, for which ordinal methods
are ill-suited, may arise in many important instances, as illustrated by the application in
Section 4.4. In contrast, this question is a natural extension of our method. Quantitative
monotonicity analysis investigates the magnitude of the change in the solution to theprimitive equation (1) as a consequence of parameter variations. We start by extending
our previous result to the strict monotonicity of solutions to (1), considering directions inV̂ (t|R) = {v ∈ Rm : Dφ(x)w > 0, x ∈ R(t), w ∈ W (x, t, v)} ,which for all t ∈ T is a subset of the GMA-cone V (t|R).
Proposition 4 If γ is a differentiable path such that γ̇(λ) ∈ V̂ (γ(λ)|R) for all λ ∈ [0, 1],
then φ(x) is (strictly) increasing along γ.
Proof. Consider any γ-differentiable selection x of R and λ ∈ [0, 1), and let t = γ(λ)
and v = γ̇(λ) ∈ V̂ (t|R). From Proposition 1, Dv x(t) ∈ W (x(t), t, v). By assumption,Dφ(x(t))Dv x(t) > 0. For 0 ≤ λ < λ0 ≤ 1, it isZ λ00
φ(x(λ )) − φ(x(λ)) =Dφ(x(γ(`)))Dγ̇(`) x(γ(`))d`.
λThe last integral is positive, which shows the result.



If one can find a positive lower bound for the sensitivity of the criterion, the result canbe strengthened as follows.
Proposition 5 (Quantitative Monotonicity Analysis) Let γ be a differentiable
path. If there exists a vector ρ = (ρ1 , . . . , ρd ) with strictly positive components such thatw ∈ W̄ (γ(λ), γ̇(λ)|R) , x ∈ R(γ(λ))
16⇒Dφ (x(γ(λ))) w ≥ ρ

As long as R(t) describes a closed subset of the solution X(t), the above method works. Indeed,a theorem by Whitney (Postnikov 1987, p. 20) states that any closed set Z at t can be written in theform fˆ(z, t) = 0 with an appropriate function fˆ that is smooth in z. The smoothness of fˆ in (z, t) can
also be guaranteed by Whitney’s theorem, as long as the graph of the set-valued mapping, which assigns
the relevant constraint set Z(t) to each parameter value t, is closed.

16

for all λ ∈ (0, 1), then0 ≤ λ < λ0 ≤ 1⇒x̂(γ(λ0 )) − x̂(γ(λ)) ≥ ρ(λ0 − λ).

The proof is essentially identical to the proof of Proposition 4 and is therefore omitted.17
By a similar argument, one can derive upper bounds on the rate of monotonicity. Proposition 5 can be used, for example, to show that x increases faster (or slower) than aparameter (taking ρ = 1). The value of ρ provides valuable information about the rate ofincrease (or decrease), giving rise to quantitative statements. A simpler result than Proposition 5 has been used when φ(x) = xi for some i ∈ {1, . . . , n}, t is finite-dimensional,
and the path γ amounts to increasing a single component tj of the parameter vector(Samuelson 1947). Indeed, suppose that Dx f is invertible that i = j = 1. Then Corollary 1 with v = (1, 0 . . . , 0)T and Dφ = (1, 0, . . . 0) yields the condition m(x, t)T a(x, t) ≥ρ, where m(x, t) is the first row of the matrix −[Dx f ]−1 (x, t) and a(x, t) is the vector (df1 /dt1 , df2 /dt1 , . . . , dfn /dt1 ). An application of quantitative monotonicity analysis
to robust inference is provided in Section 4.4.

4

Applications

In Section 4.1 we show how constraints can be incorporated into the framework of Section 3 and give two illustrations of GMA with constraints. The first recovers Chipman’s(1977) normal-good theorem for supermodular, strongly concave utility functions (Section 4.1). The second derives a condition on the marginal rate of substitution of a utility
function in order for a good to be a Giffen good. Section 4.2 is a key application of thepaper. It considers a non-supermodular game in which each player has two decisions(investment and location) and a one-dimensional type. Using GMA, we obtain conditions for investment decisions to be nondecreasing in parameter directions that increase acertain ‘proximity’ between players. This application also illustrates the use of criterion
functions (to select a subset of the variables) and of the GMA-cone (to describe parameter
directions in which the players’ proximity increases). The notion of proximity is derived
endogenously. The subsequent application, presented in Section 4.3, illustrates anotheraspect of GMA, demonstrating the consequences of imposing a large cone of monotonicity directions. Specifically, it is shown that requiring monotonicity with respect to any
additive increasing function of the parameters implies that the solution depends only
17

Proposition 5 can easily be extended to the case in which ρ is a function of the parameter.

17

on that function. The result is proved for the case where the aggregate is a parametersum. A simple transformation of each parameter component generalizes this proof toany additive, increasing function of the parameters. This result, which goes beyond purecomparative-statics analysis, provides a good illustration of the use of GMA-cones. In
Section 4.4, quantitative monotonicity analysis is used as a tool for robust inference about
unobservable parameters in the context of productivity estimation. The last application,
discussed in Section 4.5, concerns comparative dynamics. More than all of the preceding applications, it shows the importance of pseudo-gradients for monotonicity analysis.
Proof is provided for a general theorem for comparative dynamics, which implies a simple
result about the implications of local properties of the law of motion on the monotonicity
of solutions (related to Huggett (2003)). It is then shown how global properties of the
dynamic equation also can be exploited by the theorem, in a result that can be interpreted
as a stylized model of the influence of positive global cycles on the growth of individual
firms.

4.1

Constrained Optimization

Constrained optimization problems often prevent the use of ordinal methods, by violating
crucial lattice assumptions.18 GMA can accommodate problem constraints in a naturalmanner, as long as they are described in terms of smooth functions.19 The general argument is described first, then applied both to prove Chipman’s normal-good theorem forsupermodular, strongly concave utility functions, and to derive a condition under which
a good is a Giffen good. Consider the problem
maxy∈Rn G(y, t),s.t. g(y, t) = 0,
where G : Rl × Rm → Rl and g : Rl × Rm → Rk are twice differentiable. A necessary condition for optimality (Bertsekas 1995, p. 255) is the existence of a k-dimensional vector νsuch that at the optimum
F (y, ν, t) = Dy G(y, t) + ν · Dy g(y, t) = 0.
18

A notable exception is Quah’s (2007) ordinal method, which provides conditions on the transformationof constraints to guarantee monotonicity for the entire vector of decision variables.
19
As pointed out in Footnote 16 at the end of Section 3.4, provided that the problem constraints confine
the variable to a closed set, the assumption of a constraint representation in terms of smooth functions
is not restrictive, at least from a theoretical point of view.

18

Together with the k equations g(y, t) = 0, this determines a system of n = l + k equationsin n variables x = (y, ν),"f (x, t) =Dy G(y, t) + ν · Dy g(y, t)#g(y, t)= 0,

corresponding to the primitive equation (1) of our analysis. The approach can easily be
extended to allow for both equality and inequality constraints (cf. Section 3.4).

Example: Normal Goods. As an illustration, consider the budget-constrained optimalconsumption problem
U (t) =

max G(y),y: g(y,t)=0

where g(y, t) = p · y − t, p  0 denotes the price vector for a commodity bundle y,and t > 0 is the available budget. Our goal is to show that if G is increasing, strongly
concave, and supermodular, then any smooth optimizer selection y(t) is nondecreasing
in t. The constraint g(y, t) = 0 does not define a lattice if there are three or more goods.20
With G concave and differentiable, the first-order condition and budget constraint can be
written in the form"f (x, t) =Dy G(y) − λpg(y, t)#= 0,

where x = (y, λ) and λ is the nonnegative Lagrange multiplier associated with the budget
constraint.
In this problem, the natural criterion function is φ(x) = φ(y, λ) = y, since only monotonicity in demand matters. The information set R(t) consists of two constraints. The
first constraint is the equation itself, f (x, t) = 0. The second constraint stems from theobservation that the value function U is concave in t as21 the maximum of a concave function subject to the convex constraint p · y ≤ t. Since λ = U 0 (t), concavity of U implies
that λ0 (t) ≤ 0. The constraints f (x, t) = 0 and λ0 (t) ≤ 0 define the information structure of the problem. Finally, normal-good monotonicity means that any wealth increase
should increase demand in all goods. This implies that the GMA-cone must be V = [0, ∞)(since t lies in R, a nontrivial GMA-cone is necessarily one of the rays [0, ∞) or (−∞, 0]).
20With two goods, adopting the new order (reversing the sign of the quantity of the second good) isenough to recover the lattice structure and show the result.
21
See for example Luenberger (1969, p. 216).

19

For any x and t, the pseudo-gradients w(x, t) must solve, by definition,"#
Dy f wy (x, t) − pwλ (x, t)= 0,
p · wy (x, t) − 1
where wy and wλ are the components of w corresponding to y and λ. Strong concavityof G implies that Dy f is invertible. Since G is concave and supermodular, the diagonal
elements of Dy f are nonpositive, while its off-diagonal elements are nonnegative. Thisimplies (see, e.g., Samuelson 1947) that all elements of [Dy f ]−1 are nonpositive. Therefore,wy (x, t) = wλ (x, t)[Dy f ]−1 p has nonnegative components. This proves the following result.
Proposition 6 (Normal Goods) If G is strongly concave, supermodular, and twice
differentiable, then yi (t) is nondecreasing in t for all i ∈ {1, . . . , n}.
The result is identical to Chipman (1977). The proof there is very similar to ours, butdoes not proceed from a general approach to comparative statics: it uses properties of theindirect utility function instead of treating Lagrange multipliers as part of an enlarged
system of equations. This leads to one subtle difference: while we exploit the fact that theLagrange multiplier has a nonpositive derivative to prove good normality, Chipman firstshows that demand either increases for all goods or decreases for all goods, and concludes
by observing that, the budget having expanded, all demand has necessarily increased forsome good.22
Example: Giffen Goods. Consider the budget-constrained optimal-consumption problem
maxu(x, y)(7)(x,y)∈B(p,q)

where u is increasing and concave, p, q are positive prices, and consumption (x, y) is
constrained by the budget set
B(p, q) = {(x, y) ≥ 0 : px + qy ≤ 1} .
We use GMA to derive conditions under which the second good, say, behaves as a Giffen
good, i.e., such that the demand y(p, q) for the second good increases in q. We focus oninterior solutions. From our earlier analysis, the problem can be rewritten in the formux (x, y) − λp uy (x, y) − λq  = 0,(8)px + qy − 1
22

See also Quah (2007) for a treatment of comparative statics of this and other constrained optimization

problems without differentiability assumptions.

20

where λ is the Lagrange multiplier associated with the budget constraint.
Applying Step 1 of the GMA-method (cf. Section 3.2) yields a description of the set ofpseudo-gradients as a solution to


 

"
#
uxx (x, y) uxy (x, y) −p
wx
−λ 0


 
 vp
 uxy (x, y) uyy (x, y) −q   wy  +  0 −λ 
=0


 
 v
q
p
q
0
wλ
x
y

(9)

in terms of w = (wx , wy , wλ )T for any given parameter vector (p, q), candidate solution (x, y), and (nonzero) parameter direction (vp , vq ). Using a symbolic solver (and theequalities ux /p = uy /q = λ), we find that the unique solution for wy is

 

ux
ux
ux
1 (xµx + µ)vp + (yµx − µ2 )vq
1 uy x (xvp + yvq ) + uy vp − uy vq
 
 
,
=
wy =
ux
ux
ux
q
q
µµy − µx
−
uy
uy
uy
y

(10)

x

where µ(x, y) = ux (x, y)/uy (x, y) > 0 is the consumer’s marginal rate of substitution
between the two goods. To characterize situations where the second good is a Giffen
good, we use the alternative Step 2 (cf. Section 3.3) of the method by imposing theparameter change vp = 0 and vq = 1, and asking that wy be nonnegative. Concavity23of u can be used to show that the denominator in the rightmost expression of (10) is
positive. This yields the following result.
Proposition 7 (Giffen Goods) y(p, q) is (locally) increasing in q if and only if µ2 <yµx evaluated at (x(p, q), y(p, q)).
This result illustrates how constraints can be dealt with to obtain conditions on theprimitive of the problem (here, the marginal rate of substitution between goods) thatyield monotonicity. Intuitively the condition states that the second good is Giffen provided
that the marginal rate of substitution of the first good with respect to the second is small
compared to the elasticity of that rate of substitution with respect to x: if the price ofthe second good increases, an increase in x would reduce the marginal value of the firstgood compared to the second by so much that it is better to increase consumption of thesecond good instead. Sørenson (2007) provides simple examples for Giffen goods, which
rely on a Leontief-type kink in the demand curve, whereas the condition in Proposition 7
works with smooth primitives.
23

More precisely, we assume that the Hessian of u is negative definite.

21

4.2

Equilibrium of a Non-Supermodular Game

Supermodular games have been widely studied with comparative-statics techniques, yielding interesting results about the existence and monotonicity of equilibria (Vives 1990;
Milgrom and Roberts 1990; Echenique 2002). Far less is known about monotonicity properties of non-supermodular games. Such games provide a good illustration of how GMA
can be used to explore a complex comparative-statics problem and get some insights
about its structure. We consider a game with two players in which each player chooses
two actions: a nonnegative quantity qi and a location zi on the real line. Players have atype ti ∈ R corresponding to some preference in the location space. When choosing qi ,player i ∈ {1, 2} obtains the payoff
Πi (qi , zi , zj , ti ) = qi Ri (zi , zj , ti ) − C i (qi ),where C i is a strictly convex cost function, Ri is player i’s revenue function, and j denotes
the index of the other player. For example, if two firms engage in a joint venture, qi may
represent firm i’s investment, zi its final product positioning, ti its initial preference or
position, Ri is i’s return and C i its opportunity cost from investing qi in the joint venture
rather than in other projects. We assume that Ri is supermodular in (zi , zj ) and (zi , ti )and strictly concave in zi . Note, however, that Πi is not in general supermodular in (qi , zi ),so that standard results from the theory of supermodular games do not apply. In fact, it
is easy to verify that qi cannot be monotonic in ti or tj over the entire parameter space.
Therefore, monotonicity results are possible only if one looks for other parameter moves.
Our goal is therefore to find parameter moves that jointly increase players’ equilibrium
investments (q1∗ , q2∗ ). The above supermodularity assumptions ensure that an increase inplayer types increases their location decision. Still, such an increase does not necessarily
increase player i’s revenue Ri or his action qi .
Equilibrium conditions, based on the first order conditions for the two actions of each of
the two players, form a system of four equations:Ri − (C i )00


i


R
1

f (x, t) =  Rj − (C j )00  (x, t) = 0,jR1
where x = (q1 , z1 , q2 , z2 ) and t = (t1 , t2 ), and Rki denotes Ri ’s partial derivative with22

respect to its k-th variable. Hessian and cross-partial derivative matrices of f are
Dx f = 









−(C 1 )00

0

0

R21

0

1
R11

0

0

R22

−(C 2 )00

0

2
R12

0




 R1

1 
R12

 13 0 
 , Dt f = 
.
 0
0 
R32 



2
2
R11
0 R13

R31

0

Since we are interested in variations of q1∗ and q2∗ , we use the criterionφ(q1 , z1 , q2 , z2 ) = (q1 , q2 ).
From Corollary 1, monotonicity directions v = (v1 , v2 ) are given by the condition−Dφ(x)[(Dx f )−1 Dt f ](x, t)v(t) ≥ 0.
After simplification, this yields Av ≥ 0, wherejiAii = κi [R3i + δR13
R2i R12],andji
Aij = κi [δR13
R2i (−R11
)],
i
i
2 −1
1
2
1
) > R12) . We assume that (−R11
R12
− R12
R11
with κi = −[(C i )00 ]−1 > 0 and δ = (R11for i ∈ {1, 2}, ensuring that δ is positive. This condition means that the marginal returnof a player’s action is more sensitive to his own action than to the other player’s. We
also assume that if R1i = R1j = 0 and ti > tj , then R2i and R3j are nonnegative while R2j
and R3i are nonpositive. This means that, in equilibrium (when R1i = R1j = 0), player
returns decrease if one widens the gap between their preferences (conditions on R3 ’s),and player i’s return increases if player j’s action moves in the direction of i’s preference(conditions on R2 ’s). Those assumptions will be satisfied, for example, if one can check
independently that ti > tj implies that zi ≥ zj in equilibrium (this is likely to hold if player
returns are symmetric), and that on the domain {(zi , zj , ti , tj ) : zi ≥ zj and ti ≥ tj }, theabove inequalities hold, as can easily be checked for any particular form of the return
functions.
Under those assumptions, one would expect that, as player preferences get closer to eachother, players will benefit more from their interaction and thus invest more. Indeed, theoff-diagonal elements of the matrix A are positive for i and negative for j, which already
implies that if ti > tj , an increase of ti (higher type) causes player j to reduce investment,
while an increase in tj causes player i to increase investment. To obtain global comparative
statics, however, a joint move of player types is required, which brings them closer in a23

particular direction. Thus, suppose, without loss of generality, that t1 > t2 , and that t1
1
2. The effect on (q1∗ , q2∗ ) is proportional to, while t2 is increased by R13
is decreased by R13
2
1
2
1
2
1
R13
(−R31 ) + γ((−R11) − R12), R13
R32 + γ((−R11) − R12),2
1> 0. As a result, investments increase in equilibrium as long as a
R13
where γ = R13

player’s action has more impact on his marginal return than on the other player’s, that isji12
if (−R11) > R12. The slope R13/R13
can be interpreted as the ratio of players’ action-typecomplementarity. If player 1 exhibits more action-type complementarity than player 2,
increasing investments will be guaranteed only if 2’s preference is increased by a larger
amount than that by which 1’s preference is decreased. That ratio is constant if players
incur a quadratic cost by moving from ti to zi : Ri = −µi (zi − ti )2 + ψ i,1 (zi , zj ) + ψ i,2 (zj , ti )for i = 1, 2. With symmetric complementarity (µ1 = µ2 ), investments increase as player
types get closer to each other, i.e., as |t1 − t2 | decreases.24

4.3

Aggregation

GMA can be used to examine the comparative statics of a given criterion in terms of
aggregates of parameters, such as an arithmetic mean or, equivalently, a sum. As an
example, we use the GMA-method to establish, under fairly general conditions, that asolution to (1) is monotonic with respect to the sum t1 + · · · + tm of the parameters only
if it is independent of the individual parameters conditional on that sum.25 The proof ofthis result illustrates the use of imposed monotonicity paths (cf. Section 3.3). Given acriterion φ and an information structure R, by Corollary 1, the condition(x, v) ∈ R(t) × V (t|R)⇒−Dφ(x)[(Dx f )−1 Dt f ](x, t)v ≥ 0,

implies that trajectories generated by V (t|R) are monotonicity paths. By Theorem 1, thelast condition is also necessary to guarantee monotonicity.
Let t = (t1 , . . . , tm ) be a parameter vector, and consider the sum of its components,
sm = t1 + · · · + tm . Solution monotonicity in sm is stronger than solution monotonicity inany given component of t: the former requires not only that the solution is nondecreasing
24

For this symmetric quadratic case, there is an n-player equivalent to the result: types get “closer”

if the vector t = (t1 , . . . , tn ) gets closer (for the usual Euclidean distance) to the first bisector of theparameter space, i.e., the line whose direction is given by the vector (1, . . . , 1).
25
By a simple nonlinear re-scaling of the parameter vector, this result can be extended to any additive
function that is strictly increasing in the parameters.

24

in each component of t, but also that it (weakly) increases when a parameter ti is increasedby more than the amount by which another parameter tj is decreased, keeping otherparameters fixed. For simplicity, it is assumed that X(t) is nonempty for all t, andthat fx < 0 everywhere. These assumptions are naturally satisfied when (1) is obtained
as the first-order necessary optimality condition for the maximization of a strongly concave
objective function on a convex set. The solution set X(t) ≡ R(t) contains only a single
element, denoted by x(t).
First, consider the simple case when m = 2. The first three steps of the GMA-method in
Section 3.2 yield that the GMA-cone with respect to monotonicity in s2 is
V (t|R) = {(τ1 , τ2 ) ∈ R2 : τ1 + τ2 ≥ 0}.
This cone itself is generated by ∆ = {(−1, 1), (1, −1), (1, 1)}. Hence, f must satisfy
hi(x, v) ∈ R(t) × ∆ ⇒ −fx−1 ft1 (x, t) ft2 (x, t) v ≥ 0.
(11)
By substituting the different v ∈ ∆, the last relation can be rewritten in the formx ∈ R(t)⇒min {ft1 + ft2 , ft1 − ft2 , −ft1 + ft2 } (x, t) ≥ 0,

which implies that ft1 (x, t) = ft2 (x, t) ≥ 0 for all x ∈ R(t). Let s = (s1 , s2 ) = (t1 , t2 ) = t
define a smooth change of parameters by t = ϕ(s) = (s1 , s2 − s1 ), and let
F (x, s) = f (x, ϕ(s)).
Then s1 is a contra-variant coordinate, as Fs1 (x, s) = ft1 (x, ϕ(s)) − ft2 (x, ϕ(s)) vanisheson R(ϕ(s)). Fix s and consider a differentiable selection p 7→ x(s, p). Since Fx (x(ϕ(s)), s) =f (x(ϕ(s)), ϕ(s)) < 0 by assumption, the solution in the new coordinates, x(ϕ(s)), must be
independent of s1 . It can therefore depend only on the parameter aggregate s2 . Note thatthe information structure in this example is given by R(t) ≡ {x(t)}: the mere knowledge
that x(t) is a solution to (1), along with the stipulated requirement that x be nondecreasing in s2 , is enough to imply that that x(ϕ(s)) depends only on s2 . This conclusion
generalizes to more than two parameter dimensions.
Theorem 4 (Monotonicity and Dependence on Aggregates) Assume that thesolution to (1) is a singleton, i.e., X(t) ≡ {x(t)} on T . If
s0m = t01 + ... + t0m > t1 + · · · + tm = sm⇒x(t0 ) ≥ x(t)for all t0 = (t01 , . . . , t0m ), t = (t1 , . . . , tm ) ∈ T , then there exists a function y : R → X suchthat x(t) = y(sm ) on T , i.e., the solution depends only on the aggregate sm .
25

Proof.

The result has been already established for m = 2. For m > 2 consider

the following induction step. Assume that the result is true for k ≥ 2 parameters, andexamine the case of k + 1 parameters, so that t = (t1 , . . . , tk+1 ). For a fixed tk+1 , thesolution x(t) is nondecreasing in sk and thus depends only on sk , by induction hypothesis.
Thus, x depends only on sk and tk+1 . Moreover, x is monotonic in the sum s = sk + tk+1 .
Applying the argument for the two-dimensional result to the parameter vector (sk , tk+1 )then establishes that there is a function y : R → X such that x(t) = y(sk +tk+1 ) = y(sk+1 ),which concludes our proof.



Theorem 4 implies that the only way a unique solution to a parameterized equation of theform (1) can be monotonic in a parameter average is if this average is a “sufficient statistic”
for all the parameters involved in its computation. There have been numerous studies
in aggregation and statistics concerning the question of which structures of problem (1)imply that solutions depend solely on a given parameter aggregate (see, e.g., the treatiseby Blackorby and Shorrocks (1996)). The monotonicity analysis performed here considers
the converse question and therefore yields necessary structural conditions: monotonicity
in an additive aggregate implies that the aggregate must be a sufficient statistic for its
components.26
In aggregation theory, a central question is to determine when some macroeconomic variable will be monotonic in a certain aggregate value of household characteristics. Combining Theorem 4 with previous aggregation results, the above analysis suggests that
aggregate monotonicity can occur only if the macroeconomic variable is determined byan equation where the aggregate value is the only determinant parameter. Individual
parameters may influence the function only through a multiplicative factor that does notaffect the root of the equation.
26

Another way to make the connection between monotonicity and dependence on additive aggregatesis as follows. Suppose that x(t) is monotonic in the sum s2 = t1 + t2 . Fix t0 = (t01 , t02 ), and let s02 = t01 + t02 ,a = supt:t1 +t2 <s02 {x(t)}, and b = inf t:t1 +t2 >s02 {x(t)}. By assumption, a ≤ x(t0 ) ≤ b. Continuity of x(t)in t, as a consequence of the maximum theorem (Berge 1963), then implies that a = x(t0 ) = b. This istrue for all x(t) such that t1 + t2 = s02 . Therefore, x(t) must be constant on the line t1 + t2 = s02 . Thisshows that x depends only on the parameters through their sum.

26

4.4

Robust Productivity Estimates

An important problem in applied industrial organization is the comparison of firms’ private productivity forecasts given their observable investment decisions. Olley and Pakes(1996) consider a model in which firms base investment decisions on current capital stock(observed by outsiders) and productivity forecasts (unobserved by outsiders). They provide conditions under which investment decisions are monotonic in productivity forecasts.
Clearly, if a firm’s productivity forecast were the only parameter influencing its investment decision, one could simply rank the firms’ productivity forecasts based the observed
investment decisions. However, since capital stock materially affects investment decisions,such a comparison of productivity forecasts across firms is valid only for a fixed level ofcapital stock.
Quantitative monotonicity analysis can be used to show that it is possible to rank two
firms’ productivity estimates, as long as their capital stocks are close, and their investment
decisions differ sufficiently. In that sense the monotonicity disclosed by GMA is robustwith respect to perturbations, and thus allows for robust inference. To simplify the exposition, we first consider a static setting where investment decisions are based on a simple
point estimate of a firm’s productivity, before discussing the dynamic setting in whichinvestment decisions are always based on the current forecast for the next realization of
the underlying Markovian productivity process.

4.4.1

Static Setting

Suppose that the first-order condition of a firm’s investment problem isf (x, k, p) = 0,
where x is the investment decision, k is the firm’s capital stock, and p is its productivity
estimate. Assuming invertibility of fx , the pseudo-gradient is (with t = (k, p))hi−1w(x, t) = −fx
fk fp (x, t).
Since we are interested only in the monotonicity of any solution x(k, p), the criterion
function here is simply φ(x) ≡ x. Since fx < 0 at any solution (second-order condition,
assuming strict concavity of the objective in the underlying maximization problem), thefirm’s investment decision increases if fk dk + fp dp is positive. Suppose one has observed
27

two firms’ investment decisions, x1 = x(k1 , p2 ) and x2 = x(k2 , p2 ), and it is known that
their capital stocks are close, i.e., such that |k2 − k1 | ≤ κ for some κ > 0. Then, if fp > ρfor some ρ > 0 and |fk | < η for some η < 0 on the entire domain of f , then x2 −x1 > θ > 0
implies p2 > p1 + (θ − κη)/ρ. In particular, an observed investment difference between
firm 2 and firm 1 of more than κη implies that firm 2’s productivity forecast is surely
larger than firm 1’s productivity forecast, no matter what the firms’ precise capital stocks
are.

4.4.2

Dynamic Setting

In the dynamic setting considered by Olley and Pakes, productivity follows a stochastic
Markov process. The only parameters affecting a firm’s investment decision are capitalstock, productivity level, and the productivity forecast.27
Letting p̃ (or p̃|p) denote the (random) next-period productivity level given that thecurrent productivity level is p, a firm’s value function Π(k, p) solves the Bellman equation
Π(k, p) = max {π(k, p) − c(x, k) + βE[Π(x + (1 − δ)k, p̃)|p]} ,xwhere π is the current-period gross payoff, c is the adjustment-cost function, β is the
per-period discount factor, and δ represents the per-period depreciation rate. Convexityof c in x and concavity of π imply that the optimal dynamic investment policy x(k, p)satisfies the first-order necessary optimality condition
f (x, k, p) = βE[Πk (x + (1 − δ)k, p̃)|p] − cx (x, k) = 0.
Applying the above analysis yields the following result.
Proposition 8 If there exists η > 0 such that|β(1 − δ)E[Πkk (x + (1 − δ)k, p̃)|p] − cxk (x, k)| < η
for all (x, k, p), thenx(k2 , p2 ) > x(k1 , p1 ) > η|k2 − k1 |⇒p2 > p1 ,where x(ki , pi ), for i ∈ {1, 2}, denotes the optimal investment given a current capital stockof ki and a current productivity estimate of pi .
27

In accordance with recent literature (cf. Olley and Pakes (1996)), we abstract from the firm’s age inthis example. For simplicity, we also rule out exit, which means that salvage values are sufficiently low.

28

Therefore, any upper bounds on the concavity of the firm’s value function and on the crossderivative of the cost function with respect to capital and productivity allow for a robustinter-firm comparison of private productivity estimates based on observed investments,
as long as capital stocks in the industry are not too different.

4.5

Comparative Dynamics of Equilibrium Paths

Given a parameterized description of a time-varying process in a Euclidean space, monotone comparative dynamics obtain when a parameter increase results in the process taking
on higher values at all points after the initial time. Such an approach to comparative dynamics, with applications to growth, has been studied by Brock and Mirman (1972),
Mendelsohn and Sobel (1980), Becker (1983), and Amir et al. (1991).
In what follows, we take the view that comparative dynamics is equivalent to a comparativestatics problem in which the variable is the entire process, and the equation f = 0 describes
its law of motion.
This section exploits the concept of pseudo-gradients to derive comparative dynamics.
The variable x is a function of time, the evolution of which is described by some dynamic equation. The goal is to determine nontrivial changes of the dynamic equationthat increase the value of x at each instant. Cast in the GMA framework, comparative
dynamics imply that pseudo-gradients must satisfy a differential equation. This equation
is first used to prove a simple result relating local properties of the dynamic equation to
the monotonicity of x, and then to show how global properties of the dynamic equation
can be incorporated into monotonicity analysis, providing a highly stylized interpretationof the impact of global cycles on comparative dynamics (Proposition 10).
For this application, X is infinite-dimensional. In keeping with standard notation in
dynamic systems, the parameter is denoted by α instead of t, and t now denotes time.
Suppose that the evolution of a process can be described in terms of the initial value
problemẋ(t) = g(x, t, α),x(0) = x0 ,(12)

for all t ∈ [0, T ], where x0 ∈ RN , T ∈ R++ , and α ∈ A = [α, ᾱ] ⊂ Rm (with α < ᾱ)¯¯are given constants, and g : RN × [0, T ] × Rm → RN is a smooth function. If x(t, α) is
any given solution to the initial value problem (12), then monotone comparative dynamics

29

(MCD) for this dynamic system obtains ifα̂ ≥ α ⇒x(t, α̂) ≥ x(t, α), ∀ t ∈ [0, T ] .

(13)

To treat this problem within our framework, we first split the time interval [0, T ] into nsubintervals Ik = [tk−1 , tk ], k ∈ {1, . . . , n}, where tk = kT /n and n ≥ 1. Discretizing theinitial value problem (12) accordingly yields
 Txk − xk−1 =gk (xk , α), k ∈ {1, . . . , n},
n

(12’)

where xk = x(tk ) and gk (·, α) = g(·, tk , α). Interpreting (x1 , . . . , xn ) as an (nN )-dimensionalvariable and (g1 , . . . , gn ) as an (nN )-dimensional function, the discretized initial value
problem (12’) is of the form (1) with f = (f1 , . . . , fn ), andfk (x1 , . . . , xn , α) = xk − xk−1 − (T /n)gk (xk , α).
The corresponding pseudo-gradient with respect to any direction (α, v) ∈ A × Rm is
Wn (α, v) =

w∈R

N


 

 

T ∂gk (xk , α)T ∂gk (xk , α)
: wk+1 − 1 +wk =· v, 0 ≤ k ≤ n − 1 ,
n
∂xk
n
∂α

where w0 = 0. By taking the limit as n → ∞, we therefore obtain an N -dimensional
pseudo-gradient with functional componentsW∞ (α, v) =∂g(x, t, α)∂g(x, t, α)w ∈ C ([0, T ], R ) : ẇ(t) −w=
v, w(0) = 0, t ∈ [0, T ]
∂x
∂α
1

m

for any (α, v) ∈ A×Rm . The set of directions w that constitutes the pseudo-gradient W∞ (α, v)
is defined by the linear initial value problemẇ(t) −∂g(x(t, α), t, α)∂g(x(t, α), t, α)
w=
v,
∂x
∂α

w(0) = 0,

for all t ∈ [0, T ], any solution of which can be written in the formZ tg(x(s, α), s, α)−1ds v,
w(t, α, v) = F (t, α)F (s, α)∂α0where F is an N × N fundamental matrix28 satisfying the matrix differential equationḞ (t) =∂g(x(t, α), t, α)
F (t)
∂x

for all t ∈ [0, T ] and F (0) = 1. The positive cone V ⊂ Rm that admits MCD in the senseof (13) corresponds to the set of directions v such that w(t, α, v) ≥ 0.
28

For N = 1, F (t, α) = exphRt ∂g(x(s,α),s,α)0∂xids , t ∈ [0, 1]. When N > 1, the fundamental matrix

cannot be given in closed form.

30

Theorem 5 If γ is a differentiable (α, v)-path such thatZ t
g(x(s, γ(λ)), s, γ(λ))−1F (t, γ(λ))
F (s, γ(λ))
ds γ̇(λ) ≥ 0,
∂α
0

(14)

for all λ ∈ [0, 1] and all t ∈ [0, T ], thenx(t, γ(λ̂)) ≥ x(t, γ(λ))
for all λ̂ ≥ λ ∈ [0, 1] and all t ∈ [0, T ].
As a first application of Theorem 5, suppose that α is one-dimensional, and that one
wishes to find conditions under which x(t, α) is nondecreasing in α. Since F is positive,
inequality (14) reduces to
 Z t

Z u
∂g
∂g
exp −
(x(s, α), s, α)ds(x(t, α), t, α)dt ≥ 0
∂α
0
0 ∂x
for u ∈ [0, T ]. This inequality is always true if ∂g/∂α is nonnegative.
Proposition 9 If m = 1 and ∂g/∂α is nonnegative, then x(t, α) is nondecreasing in α.
This proposition can also be proved by dynamic analysis,29 but is much simpler to proveusing Theorem 5. Huggett (2003) proves a similar result in discrete time, allowing for
Markov uncertainty.
As another application, suppose that x represents a firm’s capital, the growth of which attime t depends on some investment return h(x, t) and some exogenous factor k(t). The
goal is to assess the impact of k(t) on the growth of x: with ẋ(t) = h(x, t) + αk(t), wewish to derive conditions on h and k so that an increase in the weight α of the external
shock k increases x. If k changes sign (e.g., exhibiting cyclical behavior), ∂g/∂α is notalways positive, so that Proposition 9 does not apply. From Theorem 5, monotonicity
obtains if

Z u
0

 Z t

exp −
hx (x(s, α), s)ds k(t)dt ≥ 00

for all u ∈ [0, T ]. In order to obtain monotonicity, there must be some sense in which k has
an overall positive effect on x, despite sometimes having locally negative effects. Thus,
29

Let α < β and y(t) = x(t, β) − x(t, α). Then y(0) = 0 and y 0 (t) = g(x(t, β), t, β) − g(x(t, α), t, α),so y 0 (t) ≥ 0 whenever y(t) = 0. We wish to show that y(t) ≥ 0 for all t. The only problem is if y(t) = 0
and y 0 (t) = 0 for some t. Then consider the smallest s > t such that y 0 (s) 6= 0. By construction, y(t0 ) = 0
on [t, s], so y 0 (s) ≥ 0, implying that y 0 (s) > 0. Thus, y can never become negative.

31

we assume that for all t, K(t) =Rt0k(s)ds ≥ 0. We now use the following lemma, whose30

proof is omitted.

Rt
Lemma 1 If ψ is such that 0 ψ(s)ds ≥ 0 for t ∈ [0, T ] and v is a decreasing function,
Rt
Rtthen 0 v(s)ψ(s)ds ≥ v(t) 0 ψ(s)ds.
 R
t
Applying the lemma to ψ(t) = k(t) and v(t) = exp − 0 hx (x(s, α), s)ds , which isdecreasing in t if hx is nonnegative, we have proved the following result.31
Proposition 10 Suppose that ẋ(t) = h(x, t)+αg(t) and x(0) = x0 , with h nondecreasingRt
in x and 0 g ≥ 0 for all t.32 Then x(t, α) is nondecreasing in α for all t.
This result can be interpreted (within a highly stylized, deterministic context) as follows:
if a firm’s growth is subject to exogenous global cycles, each of which has an overall positiveeffect, an increased impact of these cycles on the firm results in the firm possessing higher
capital at all times.

5

Conclusion

Analyzing the monotonicity of solutions to optimization or equilibrium problems is often a difficult task. Despite the recent advances in MCS, many problems in economic
theory and other fields do not fit the mold of standard techniques, which are based onsupermodularity on lattices or on other ordinal concepts, and one is led to either focus onvery simple models or make strong assumptions in order to obtain intuitive monotonicity
results. GMA extends the previous literature on comparative-statics analysis in several
directions. First, it provides a new way of endogenously generating parameter moves thatyield monotonicity. This may result, for example, in partitions of the parameter space into
regions where simple but region-dependent parameter moves do indeed yield monotonicity. It may also result in the emergence of some particular aggregate of the parameters,such as the notion of parameter closeness derived in Section 4.2. This construction is30

The lemma is nontrivial, since ψ has no sign requirement. Quah and Strulovici (2008) prove a

different version of the lemma, which is equivalent to this one by a change of variables.
31
To our knowledge, this result cannot be proved using existing theorems of comparative dynamics(since g can take negative values). It can be derived along the lines of Footnote 29.
32
As usual, we assume that h and g are smooth, to guarantee existence and uniqueness of x(·, α) for
all α.

32

achieved thanks to the introduction of a new concept, that of a pseudo-gradient, whichsummarizes all available information (provided in the form of an information set) for thedetermination of monotonicity properties. Second, GMA naturally permits considerationof the monotonicity of functions of a solution instead of the monotonicity of the solution
itself as in the standard analysis. This flexibility proves important when analyzing themonotonicity properties of complex multivariate models. Third, GMA may be used toderive conditions under which monotonicity obtains with respect to imposed moves ofthe parameters, for example corresponding to parameter aggregates. Fourth, the method
introduced here is easily extended to derive bounds on the sensitivity of variables (orfunctions thereof) to parameters. Such quantitative results largely escape the reach ofstandard ordinal methods. Fifth, GMA introduces several concepts and insights regarding
the nature of comparative statics, for example in dealing with constraints or exploiting
additional information about the problem structure. Owing to its simplicity and systematic nature, GMA is a natural and potentially crucial first step to analyze monotonicity
in complex problems, and is in many ways complementary to ordinal methods.

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