Monotone Comparative Statics: Topkis’ and Milgrom & Shannon’s Theorems

Topkis’ Theorem is one of the most useful tools to study complementarity with non-differentiable functions. Milgrom and Shannon found the minimal condition for such analysis.

Monotone Comparative Statics (MCS) deals with how the solutions to an optimization problem change with its parameters. It can deliver interesting results under relatively weak assumptions. One of its main tools is Topkis’ Theorem, which allows us to derive comparative statics results without differentiability assumptions and without explicitly solving the model. In order to understand the theorem, we need to know some concepts first. In every concept I introduce, I will adjunct an image with an example for better understanding. The theorem will also have an image with an example and the intuition behind it. Let a,b ∈ R^n. We define:

Meet and Join

We define the meet and join of an and b as the vectors of minimum and maximum coordinate-wise comparison of an and b, respectively.

Strong Set Order
A set A ⊆ R^n is greater than a set B ⊆ R^n in the strong set order if, for every a ∈ A and b ∈ B, (a ∨ b)∈ A and (a ∧ b) ∈ B.

Lattice
A lattice is a set X ⊆ R^n such that (x ∨ y) ∈ X and (x ∧ y) ∈ X for every x,y ∈ X.

Increasing Differences
A function f has increasing differences in (x,θ) if whenever xᴴ ≥ xᴸ and θᴴ ≥ θᴸ,

f(xᴴ,θᴴ) − f(xᴸ,θᴴ) ≥ f(xᴴ,θᴸ) − f(xᴸ,θᴸ).

Supermodularity
A function f:X×Θ→R is supermodular in x if

f(x∨y,θ) − f(x,θ) ≥ f(y,θ) − f(x∧y,θ).

Topkis’ Theorem (1978)
Let x ∈ X and θ ∈ Θ. If X ⊆ R^n is a lattice, Θ ⊆ R^m and f:X×Θ→R has increasing differences in (x,θ) and is supermodular in x, then the solution set X*(θ) is increasing in the strong set order.

Single Crossing

What’s the minimal (necessary and sufficient) condition for Monotone Comparative Statics? Milgrom and Shannon (1994) answered this question. What we need is the Single-Crossing property:

A function f:X×Θ→ℝ is single-crossing in (x,θ) if whenever

xᴴ ≥ xᴸ and θᴴ ≥ θᴸ, we have

f(xᴴ,θᴸ) ≥ f(xᴸ,θᴸ)  ⇒  f(xᴴ,θᴴ) ≥ f(xᴸ,θᴴ),

and

f(xᴴ,θᴸ) > f(xᴸ,θᴸ)  ⇒  f(xᴴ,θᴴ) > f(xᴸ,θᴴ).

Increasing differences implies single-crossing, but the converse is not true because single-crossing means that the ranking between xᴴ and xᴸ can’t reverse, whereas increasing differences means that the increase in the function from going from xᴸ to xᴴ must weakly increase, which is a stronger condition.

Milgrom and Shannon’s Theorem (1994)

Topkis showed that increasing differences and supermodularity are sufficient conditions for Monotone Comparative Statics. Milgrom and Shannon showed that single-crossing is the minimal condition for Monotone Comparative Statics (it’s necessary and sufficient, if and only if).

If f is single-crossing in (x,θ), then the solution correspondence

X*(θ) = argmax f(x,θ)

ₓ∈X

is increasing in the strong set order.

Conversely, if X*(θ) is increasing in the strong set order for every choice set X⊆ℝ, then f is single-crossing in (x,θ).

Application: Consumer Theory

In the next image, we have an application of Topkis’ Theorem to Consumer Theory in both the nondifferentiable and differentiable function cases.

The differentiable case example comes from Amir, R. (2005).

References

Amir, R. (2005). Supermodularity and Complementarity in Economics: An Elementary Survey. Southern Economic Journal, 71(3), 636–660. https://doi.org/10.2307/20062066

Milgrom, P., & Shannon, C. (1994). Monotone Comparative Statics. Econometrica, 62(1), 157–180. https://doi.org/10.2307/2951479

Topkis, D. M. (1978). Minimizing a Submodular Function on a Lattice. Operations Research, 26(2), 305–321. http://www.jstor.org/stable/169636