Tropical Hypersurfaces in Economics, and the Unimodularity...

Tropical Hypersurfaces in Economics, and theUnimodularity Theorem

Elizabeth Baldwin Paul Klemperer

Nuffield College, Oxford University

May 2016

Material from ‘Tropical geometry to analyse demand’ (2012-14) and‘Understanding preferences: “Demand types” and the existence of

equilibrium with indivisibilities’ (2015)

This work was supported by ESRC grant ES/L003058/1.

E. Baldwin and P. Klemperer Tropical Hypersurfaces in Economics May 2016 1 / 26

Geometric Analysis of Demand: Model

n indivisible goods.

Finite set A ⊂ Zn of bundles available.

Valuation u : A→ R; quasilinear utility u(x)− p.x

Agent demands bundles in set Du(p) = argmaxx∈A

{u(x)− p.x}

Investigate what is demanded where: study where demand changes.

Definition: “Tropical Hypersurface (TH)”

Tu={ prices p ∈ Rn where #Du(p) > 1}.

n indivisible goods. Finite set A ⊂ Zn of bundles available.

{u(x)− p.x}

Exampleof u(x)

{u(x)− p.x}

Exampleof u(x)

{u(x)− p.x}

Exampleof u(x)

(1,0) (0,0)

{u(x)− p.x}

Exampleof u(x)

(1,0) (0,0)

{u(x)− p.x}

Exampleof u(x)

Cells and facets

Definition

Tropical hypersurface Tu: prices p ∈ Rn where #Du(p) > 1.

Definition

A cell of Tu is a non-empty set {p ∈ Tu : X ⊂ Du(p)}, where |X| > 1

A facet is an (n− 1)-dimensional cell of Tu.

The tropical hypersurface is the union of its facets.

In two dimensions, made up ofline segments.

Cells and facets

Definition

(1,1,1)

In 3 dimensions, made up ofpieces of planes (facets / 2-cells),meeting along lines (1-cells).

Cells and facets

Definition

A unique demand region is a connected component of the complement ofTu.

Equivalently, it is the non-empty (absolute) interior of a set of the form{p ∈ Tu : x ∈ Du(p)}.

The Polyhedral Complex

A tropical hypersurface in Rn is the support of a rational polyhedralcomplex of pure dimension (n− 1).

Made of polyhedral pieces(intersections of finite collections ofhalf-spaces): cells.

Every face of a cell is a cell.

The intersection of two cells iseither ∅ or is a face of both the cells.

Tu is contained in the union of thefacets (n− 1-cells).

Cells have rational slope.

(1,1,1)

Geometric versus Economic characterisations of Cells

Recall a unique demand region is a connected component of thecomplement of Tu

C ⊆ Tu is a cell iff it is the intersection of the closures of a collection ofunique demand regions around Tu.

Let C be a cell of Tu, and let C◦ be the relative interior of C. ThenDu(p

◦) is constant for p◦ ∈ C◦. Moreover, the cell is the locus where thisbundle is demanded: C = {p ∈ Rn : Du(p

◦) ⊆ Du(p)}.

How does demand change as you cross a facet?

A tropical hypersurfaceis composed of facets:linear pieces in dimen-sion (n− 1).

(0,2)(1,1)

(1,0)(2,0)p2

If p is in a facet then the agent is indifferent between two bundles:

u(x)− p.x = u(y)− p.y

⇐⇒ p.(y − x) = u(y)− u(x)

Endow all facets with weights: weighted rational polyhedral complex.

(0,2)(1,1)

(1,0)(2,0)p2

u(x)− p.x = u(y)− p.y ⇐⇒ p.(y − x) = u(y)− u(x)

(0,2)(1,1)

(1,0)(2,0)p

(If p is in a facet then the agent is indifferent between two bundles:

The change in bundle is in the direction normal to the facet.

Change in bundle is minus ‘weight w’ times minimal facet normal.

Endow all facets with weights: weighted rational polyhedral complex.E. Baldwin and P. Klemperer Tropical Hypersurfaces in Economics May 2016 6 / 26

Economics from Geometry

4Every tropical hypersurface is balanced:around each (n− 2)-cell,

∑iwivi = 0.

Theorem (Mikhalkin 2004)

A weighted rational polyhedral complex of pure dimension (n− 1) hassupport equal to the tropical hypersurface of a valuation iff it is balanced.

We need not write down valuations of discrete bundles.

We can simply draw tropical hypersurfaces.

Project Aim understand economics via geometry.

Concavity

Definition

A set A ⊆ Zn is discrete-convex if Conv(A) ∩ Zn = A.

Write Conv(u) : Conv(A)→ R for the minimal weakly-concavefunction everywhere weakly greater than u.

u : A→ R is concave if A is discrete-convex and u(x) = Conv(u)(x)for all x ∈ A.

Concavity

Definition

Concavity

Definition

u : A→ R is concave ⇔ Du(p) is discrete-convex for all p⇔ ∀x ∈ Conv(A) ∩ Zn there exists p ∈ Rn with x ∈ Du(p).

x18 00

x1 = 0x1 = 1x1 = 2

This U is not concave.

DU (4) is not discrete-convex.

Theorem (Mikhalkin, 2004)

An (n− 1)-dimensional balanced weighted rational polyhedral complex inRn corresponds to an ‘essentially unique’ concave valuation u.

Concavity

Definition

x18 00

x1 = 0x1 = 1x1 = 2

This U is not concave.DU (4) is not discrete-convex.

Concavity

Definition

Classifying valuations

Economists classify valuations by how agents see trade-offs between goods.

For divisible goods, ask how changes in each price affect each demand.Let x∗(p) be optimal demands of each good at a given price.

∂x∗i∂pj

> 0 means goods are ‘substitutes’ (tea, coffee).

∂x∗i∂pj

< 0 means goods are ‘complements’ (coffee, milk).

With indivisible goods, start by crossing one facet.

(0,0)p2

Economists classify valuations by how agents see trade-offs between goods.For divisible goods, ask how changes in each price affect each demand.Let x∗(p) be optimal demands of each good at a given price.

∂x∗i∂pj

(0,0)p2

Economists classify valuations by how agents see trade-offs between goods.For divisible goods, ask how changes in each price affect each demand.Let x∗(p) be optimal demands of each good at a given price.

∂x∗i∂pj

(0,0)p2

Economic properties from facets

Suppose every facet normal v to Tu...has at most one +ve, one -ve coordinate entry.

(x ,x )1 2

(x +1,x -2)1 2

Decrease price i to cross a facet.

Demand changes from x to x+ v, where v is a facet normal.

By the law of demand, vi > 0.

(x ,x )1 2

(x +1,x -2)1 2

⇒ vj ≤ 0 for all j 6= i.

(x ,x )1 2

(x +1,x -2)1 2

⇒ vj ≤ 0 for all j 6= i.SU

Suppose every facet normal v to Tu...has at most one +1, one -1 coordinate entry.

⇒ vj ≤ 0 for all j 6= i and goods trade-off 1-1.

0( ( 1

By the law of demand, vi > 0. STRO

⇒ vj ≤ 0 for all j 6= i and goods trade-off 1-1.SU

(1,1,1)

By the law of demand, vi > 0. STRO

⇒ vj ≤ 0 for all j 6= i and goods trade-off 1-1.SU

Suppose every facet normal v to Tu...has all positive (or all negative) coordinate entries.

(x ,x )1 2

(x +2,x +3)1 2 p

(Decrease price i to cross a facet.

⇒ vj ≥ 0 for all j 6= i.

Suppose every facet normal v to Tu...has all positive (or all negative) coordinate entries.

(x ,x )1 2

(x +2,x +3)1 2 p

(Decrease price i to cross a facet.

⇒ vj ≥ 0 for all j 6= i.CO

MPLEME

Suppose every facet normal v to Tu...is (1, 4)

(1,4)(2,8)

(3,12)

e.g. (car bodies, car wheels)

By the law of demand, vi > 0. PERFEC

Demand more (1, 4) bundlesCO

MPLEME

Suppose every facet normal v to Tu...is in set D ⊂ Zn.

(1,4)(2,8)

(3,12)

e.g. (car bodies, car wheels)

These facts define structure of trade-offs.

Suppose every facet normal v to Tu...is in set D ⊂ Zn.

Definition: “Demand Type”

u is of demand type D if every facet of Tu has normal in D.

The demand type is the set of all such valuations.

These facts define structure of trade-offs.

Duality: The Demand Complex

Recall that Tu lives in price space. The dual space is quantity space.

Sets ConvDu(p) form a rational polyhedral complex, dimensiondimA.

This is our ‘demand complex’ (= subdivided Newton polytope).

(1,0) (0,0)

value026

(1,0) (0,0)

σ̂ ∈ Rn+1 is a face of the graph of Conv(u) iff the projection of σ̂ to itsfirst n coordinates is a cell of the demand complex.

(1,0) (0,0)

Demand complex cells are dual to the cells of Tu.

k-dimensional pieces ↔ (n− k)-dimensional pieces.

σ ⊂6= σ′ ⇔ Cσ′ ( Cσ

Linear spaces parallel to demand complex and corresp. tropicalhypersurface cells are dual.

σ ⊂6= σ′ ⇔ Cσ′ ( Cσ

✲✲

σ ⊂6= σ′ ⇔ Cσ′ ( Cσ

✲✲

Bundles which are not demand complex vertices are either never demandedor only demanded at prices corresp. to the demand complex cell they’re in.

✲ (1,1)

Aggregate Demand

Many agents: j = 1, . . . ,m, valuations uj : Aj → R.

Definition (Standard)

Aggregate demand at p is the Minkowski sum of individual demands:

Du1(p) + · · ·+Dum(p)

Write A := A1 + · · ·+Aj and U : A→ R so aggregate demand is DU (p).Not hard to see we can use:

U(x) = max{∑

j uj(xj) | xj ∈ Aj ,

∑j x

j = x}

If supply is x, a competitive equilibrium among agents i consists of

allocations xi such that∑

i xi = x.

price p such that xi ∈ Dui(p) for all i.

}x ∈ DU (p) for some p

Refer to supply x ∈ Conv(A) = Conv(A1+ · · ·+Am) as relevant supply.

@ Eqm for some relevant supply iff DU (p) not discrete-convex for some p

iff U is not concave.

Aggregate Demand

Du1(p) + · · ·+Dum(p)

Write A := A1 + · · ·+Aj and U : A→ R so aggregate demand is DU (p).

i xi = x.

Aggregate Demand

Du1(p) + · · ·+Dum(p)

i xi = x.

Aggregate Demand

Du1(p) + · · ·+Dum(p)

i xi = x.

Tropical hypersurface of aggregate demandDU (p) = Du1(p) + · · ·+Dum(p)

Easy to draw TU ,

just superimpose individual tropical hypersurfaces.

Equilibrium fails for concave u1, u2 and some supply x ∈ A1 +A2 iffDu(p) is not discrete-convex for some p ∈ Tu1 ∩ Tu2 .

Easy to draw TU , just superimpose individual tropical hypersurfaces.

Corollary

If u1, . . . , um are of demand type D then so is U .

(0, 0)

(2, 1) (0, 1)

(1, 2)

(1, 0)

Then what is DU (p)?

If p /∈ TU , easy: use “facet normal × weight = change in demand”.

If p ∈ Tui , only one i, and individual valuations concave, also easy.

Interesting case: p ∈ Tui , Tuj for i 6= j.

(0, 0)

F (0, 1)

(1, 2)

(2, 1)

(1, 0)

Then what is DU (p)?

If p /∈ TU , easy: use “facet normal × weight = change in demand”.

If p ∈ Tui , only one i, and individual valuations concave, also easy.

Interesting case: p ∈ Tui , Tuj for i 6= j.

Classic theorems of competitive equilibrium

Theorem (Kelso and Crawford 1982)

Suppose

domain Ai = {0, 1}n for all agents i.

ui : Ai → R is a concave substitute valuation for all agents.

Supply x ∈ {0, 1}n.

Then competitive equilibrium exists.

Seek generalised result of this form:

Suppose we fix a demand type D.

Agents all have concave valuations of demand type D.

‘Relevant’ supply (in the convex hull of domain of aggregate demand).

Ask: does competitive equilibrium always exist?

Yes, iff D has a certain property. . .

Theorem (Milgrom and Strulovici 2009)

Suppose

domain Ai = A, a fixed product of intervals, for all agents i.

ui : Ai → R is a concave strong substitute valuation for all agents.

Supply x ∈ A.

Theorem (Hatfield, Kominers, Nichifor, Ostrovsky, and Westkamp 2013)

Suppose

domain Ai ⊂ {−1, 0, 1}n for all agents i.

ui : Ai → R is a concave ‘full’ substitute valuation for all agents.

Supply x = 0.

Suppose

Supply x = 0.

Suppose we fix a ‘description’.

Agents all have concave valuations of this description.

Suppose

Supply x = 0.

Suppose

Supply x = 0.

Ask: does competitive equilibrium always exist?Yes, iff D has a certain property. . .

Demand types and equilibrium

Recall:

Could only demand (1, 1) at price corresp. to the square demandcomplex cell.

But at this price,

Red demands (1, 0) or (0, 1)Blue demands (0, 0) or (1, 1)

Switching choices takes us between the vertices of the square.

There is no way to get to the bundle in the middle.

Recall:

But at this price,

Recall:

But at this price,

? Area=2.

The problem is that the bundle is in the middle of the square.

There exists a bundle there because the area of the square is > 1.

The area is (abs. value of) the determinant of vectors along its edges.

(1 −11 1

Avoid problems iff all sets of n demand type vectors have det ±1 or 0.⇒ “unimodularity”.∗

∗When vectors in D span Rn, unimodularity ⇔ all sets of n vectors have det ±1 or 0.E. Baldwin and P. Klemperer Tropical Hypersurfaces in Economics May 2016 15 / 26

Theorem (cf. Danilov, Koshevoy and Murota, 2001)

Fix a set D ( Zn. A competitive equilibrium exists for

every finite set of agents with concave valuations of type Dany relevant supply

iff D is unimodular.

The problem is that the bundle is in the middle of the square.

There exists a bundle there because the area of the square is > 1.

The area is (abs. value of) the determinant of vectors along its edges.

(1 −11 1

Avoid problems iff all sets of n demand type vectors have det ±1 or 0.⇒ “unimodularity”.∗

every finite set of agents with concave valuations of type Dany relevant supply

From this, follows existence of equilibrium in:

Gross substitutes (Kelso and Crawford, 1982, Ecta).

Strong substitutes / M \-concave valuations (Danilov et al., 2003,Discrete Applied Math., Milgrom and Strulovici, 2009, JET).

Gross substitutes and complements (Sun and Yang, 2006, Ecta).

Full substitutability on a trading network (Hatfield et al. 2013, JPE).

every pair of agents with concave valuations of type Dany relevant supply

From this, follows existence of equilibrium in:

Gross substitutes (Kelso and Crawford, 1982, Ecta).

Strong substitutes / M \-concave valuations (Danilov et al., 2003,Discrete Applied Math., Milgrom and Strulovici, 2009, JET).

Gross substitutes and complements (Sun and Yang, 2006, Ecta).

Full substitutability on a trading network (Hatfield et al. 2013, JPE).

Transverse Intersections

Definition

Tu1 , Tu2 intersect transversally at p if forminimal C1, C2 of Tu1 , Tu2 containing p,have dim(C1 + C2) = n.

Lemma (See e.g. Maclagan and Sturmfels 2015)

Tu1 , Tu2 intersect transversally after a small generic translation.

Translations in valuations: If uε(x) = u(x) + εv.x then Tuε = Tu + {εv}.

Definition

Translations and Equilibrium

Proposition

If equilibrium does not exist for valuations u1 and u2, and for somerelevant supply y, then for any v ∈ Rn, equilibrium also fails for valuationsu1 and u2ε , in which u2ε (x) = u2(x) + εv.x, for supply y and all sufficientlysmall ε > 0.

Proposition

Proof. Let P j(x) be prices p where x ∈ Duj (p).Failure of equilibrium ⇔