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.

x2

x1

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}


x2

x1








{u(x)− p.x}


Exampleof u(x)

x2

x1026

35 24








{u(x)− p.x}


Exampleof u(x)

x2

x1026

35 24

p1

p2

(0,1)

(1,0) (0,0)

(1,1)








{u(x)− p.x}


Exampleof u(x)

x2

x1013

32 12

p1

p2

(0,1)

(1,0)

(0,0)

(1,1)




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.

pA

pB

In two dimensions, made up ofline segments.


Cells and facets

Definition


Definition




(1,1,1)

p1

p2

p3

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


Cells and facets

Definition


Definition




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.




pA

pB









(1,1,1)

p1

p2

p3







Geometric versus Economic characterisations of Cells

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

Lemma

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

Lemma

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).

(0,0)

(0,1)

(0,2)(1,1)

(1,0)(2,0)p2

p1

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.




(0,0)

(0,1)

(0,2)(1,1)

(1,0)(2,0)p2

p1


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





-1 1

(

(0,0)

(0,1)

(0,2)(1,1)

(1,0)(2,0)p

2

p1

(

11(

(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.





p1

p2



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

v1

v2

v3

v4

w1

w2

w3w

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




Lemma

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




Lemma


x18 00

x1 = 0x1 = 1x1 = 2

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




Concavity

Definition




Lemma





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

p1


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

p1


Economic properties from facets

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

-1 2(

(x ,x )1 2

(x +1,x -2)1 2

p1

p2

(

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.




-1 2(

(x ,x )1 2

(x +1,x -2)1 2

p1

p2

(




⇒ vj ≤ 0 for all j 6= i.




-1 2(

(x ,x )1 2

(x +1,x -2)1 2

p1

p2

(




⇒ vj ≤ 0 for all j 6= i.SU

BSTIT

UTES



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( (1

0( ( 1

-1( (



By the law of demand, vi > 0. STRO

NG

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

BSTIT

UTES




(1,1,1)

p1

p2

p3



By the law of demand, vi > 0. STRO

NG

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

BSTIT

UTES



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

1

p2 2

3(

(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

1

p2 2

3(

(Decrease price i to cross a facet.



⇒ vj ≥ 0 for all j 6= i.CO

MPLEME

NTS



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

(0,0)

p1

p2 1

4(

(

(1,4)(2,8)

(3,12)

e.g. (car bodies, car wheels)



By the law of demand, vi > 0. PERFEC

T

Demand more (1, 4) bundlesCO

MPLEME

NTS



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

(0,0)

p1

p2 1

4(

(

(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).

x1

x2

value

026

35 24

x2

x1

p1

p2

(0,1)

(1,0) (0,0)

(1,1)






x1

x2

value026

35 24

x2

x1

p1

p2

(0,1)

(1,0) (0,0)

(1,1)






x1

x2

value

x2

x1

p1

p2

(0,1)

(1,0) (0,0)

(1,1)

✲

✲

Lemma

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.






x1

x2

value

x2

x1

p1

p2

(0,1)

(1,0) (0,0)

(1,1)

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.






x1

x2

value

x2

x1

p1

p2



σ ⊂6= σ′ ⇔ Cσ′ ( Cσ







x1

x2

value

x2

x1



σ ⊂6= σ′ ⇔ Cσ′ ( Cσ







x1

x2

value



σ ⊂6= σ′ ⇔ Cσ′ ( Cσ







x1

x2

value

??



σ ⊂6= σ′ ⇔ Cσ′ ( Cσ







x1

x2

value

??

✲

✲

✲

✲✲



σ ⊂6= σ′ ⇔ Cσ′ ( Cσ







x1

x2

value

??

✲

✲

✲

✲✲


Lemma

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






3

0

2

2

22

4

44

✲ (1,1)


Lemma







2.5

0

2

2

22

4

44

✲ (1,1)


Lemma



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

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


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).


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.


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

Easy to draw TU ,

just superimpose individual tropical hypersurfaces.

Lemma

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 .

Lemma





(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.

Lemma





(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.

Lemma



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:

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.



? 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.

det

(1 −11 1

)= 2

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.

det

(1 −11 1

)= 2

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.

2

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}.


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 ⇔

P 1(x1) ∩ P 2(x2) = ∅ ∀(x1,x2) ∈ A1 ×A2 s.t. x1 + x2 = y.

Clearly the prices p where x2 ∈ Du2ε(p) are {εv}+ P 2(x2).

P j(x) is a cell, the closure of a unique demand region, or ∅: a polyhedron.If two polyhedra are disjoint, after a sufficiently small translation inany direction, they are still disjoint.

Pick ε so this holds for translation εv and all (x1,x2) ∈ A1 ×A2 s.t.x1 + x2 = y.



Proposition


Corollary

Equilibrium exists for all finite sets of concave valuations of a demand typeand all relevant supplies iff it exists for all finite sets of concave valuationsof that type which intersect transversally.


Unimodularity

For a linearly independent set {v1, . . . ,vs} in Zn, the following areequivalent:

1. {v1, . . . ,v2} are unimodular;

2. There exist vs+1, . . . ,vn ∈ Zn such that the determinant ofv1, . . . ,vn is ±1.

3. the parallelepiped whose edges are these vectors, that is,{∑nj=1 λ

jvj : λj ∈ [0, 1]}, contains no non-vertex integer point;

4. if x ∈ Zn and x =∑s

j=1 αjvj with αj ∈ R, then αj ∈ Z for all j.


Necessity in the Unimodularity Theorem

Finite collections of agents:

Let v1, . . . ,vk ∈ D. Fix p ∈ Rn.

For j = 1, . . . , k choose a concave valuation with facet normal vj ,with weight 1, and with p in its interior.

So Duj (p) = {xj ,xj + vj} for j = 1, . . . , k and some xj .

Thus DU (p) = {x+∑

j δjvj : δj ∈ {0, 1}}, where x =

∑j x

j .

This is the vertices of a parallelepiped⇒ DU (p) is discrete-convex iff v1, . . . ,vk are unimodular.

?

{(1, 1), (−1, 1)} not aunimodular set.

DU (p) notdiscrete-convex.


Necessity in the Unimodularity Theorem

Finite collections of agents:

Let v1, . . . ,vk ∈ D. Fix p ∈ Rn.

For j = 1, . . . , k choose a concave valuation with facet normal vj ,with weight 1, and with p in its interior.

So Duj (p) = {xj ,xj + vj} for j = 1, . . . , k and some xj .

Thus DU (p) = {x+∑

j δjvj : δj ∈ {0, 1}}, where x =

∑j x

j .

This is the vertices of a parallelepiped⇒ DU (p) is discrete-convex iff v1, . . . ,vk are unimodular.

Only two agents

Let k be minimal such that v1, . . . ,vk not unimodular.

Choose valuations u1, . . . , uk as above and let u0 be aggregatevaluation for u1, . . . , uk−1.

Du0(p) is discrete-convex, by minimality, so is concave on Du0(p).

but Du0(p) +Duk(p) = DU (p) which is non discrete-convex.


Transverse intersections and changes in demand

Consider the linear span of changes in demand.

Definition

Lσ := 〈{y − x : x,y ∈ σ}〉R where σ is a demand complex cell.

Lemma (e.g. Gruber, 2007)

∃ a basis for Lσ consisting of edges of σ (i.e. facet normals in T.H.).

For p ∈ Tu1 ∩ Tu2 , write σj , σ for ConvDuj (p),ConvDU (p).

Lemma

Lσ = Lσ1 ⊕ Lσ2 iff intersection is transverse at p

Corollary

If intersection is transverse and demand type is unimodular, and ify,x ∈ σ ∩ Zn, can write y − x = z1 + z2 where zj ∈ Lσj ∩ Zn, j = 1, 2.


Proof of Sufficiency in Unimodularity Theorem

Definition

Lσ := 〈{y − x : x,y ∈ σ}〉R where σ is a demand complex cell.

Suppose u1, u2 concave and p ∈ Tu1 ∩ Tu2 . Write σj , σ forConvDuj (p),ConvDU (p). Wts DU (p) discrete-convex.

Let y ∈ ConvDU (p) ∩ Zn. Wts y ∈ DU (p) = Du1(p) +Du2(p).

Let x ∈ DU (p). Prev slide: ∃ zj ∈ Lσj ∩ Zn with y − x = z1 + z2.

But x ∈ DU (p) so x = x1 + x2, with xj ∈ Duj (p).And y ∈ σ so y = y1 + y2 with yj ∈ σj (not necessarily integer).

So z1 + z2 = (y1 − x1) + (y2 − x2).

Moreover, yj − xj ∈ Lσj , j = 1, 2. So zj = xj − yj . Since xj , zj ∈ Znsee yj ∈ Zn.

As uj concave and yj ∈ σj ∩ Zn, conclude yj ∈ Duj (p).


Examples: Strong substitutes / M \-concave valuations

The strong substitute vectors have at most one +1, at most one -1,otherwise 0s. Substitutes where trade-offs are 1-1.

(1 0 10 1 −1

)0

1( (1

0( ( 1

-1( (

Unimodular set (classic result).

Equilibrium always exists

Model of Kelso and Crawford (1982), Danilov et al. (2003), Milgromand Strulovici (2009), Hatfield et al. (2013).

The model of Sun and Yang (2006) is a basis change.

All ‘product-mix auction’ bids express strong substitute preferences.


Examples: Strong substitutes / M \-concave valuations

The strong substitute vectors have at most one +1, at most one -1,otherwise 0s. Substitutes where trade-offs are 1-1.

1 0 0 1 1 00 1 0 −1 0 10 0 1 0 −1 −1

(1,1,1)

p1

p2

p3

Unimodular set (classic result).

Equilibrium always exists

Model of Kelso and Crawford (1982), Danilov et al. (2003), Milgromand Strulovici (2009), Hatfield et al. (2013).

The model of Sun and Yang (2006) is a basis change.

All ‘product-mix auction’ bids express strong substitute preferences.


Beyond strong substitutes

∃ unimodular demand types, not a basis change from strong substitutes?If n ≤ 3, then no. If n ≥ 4, yes.

Let D be the columns of:1 0 0 1 0 0 1 1 00 1 0 0 1 0 1 0 10 0 1 0 0 1 0 1 10 0 0 1 1 1 1 1 1

front-line workers}

managerInterpretation:

The first three goods (rows) represent front-line workers.

The final good (row) is a manager.

‘Bundles’, i.e. teams, worth bidding for, are:

a worker on their own (not a manager on their own);a worker and a manager;two workers and a manager.

Interpret as coalitions: model matching with transferable utility.Find unimodularity ⇔ core 6= ∅.



∃ unimodular demand types, not a basis change from strong substitutes?If n ≤ 3, then no. If n ≥ 4, yes. Let D be the columns of:

1 0 0 1 0 0 1 1 00 1 0 0 1 0 1 0 10 0 1 0 0 1 0 1 10 0 0 1 1 1 1 1 1

front-line workers}










1 0 0 1 0 0 1 1 00 1 0 0 1 0 1 0 10 0 1 0 0 1 0 1 10 0 0 1 1 1 1 1 1

front-line workers}






Interpret as coalitions: model matching with transferable utility.

Find unimodularity ⇔ core 6= ∅.




1 0 0 1 0 0 1 1 00 1 0 0 1 0 1 0 10 0 1 0 0 1 0 1 10 0 0 1 1 1 1 1 1

front-line workers}








Interval package valuations

D is the basis change of the strong substitute vectors via the uppertriangular matrix of 1s. Vectors with one block of consecutive 1s:

People are ordered from 1 to n.

Subsets of consecutive people can form coalitions.

Where there are ‘gaps’, no complementarity.

This can represent:

small shops along a street considering a merger;

seabed rights for oil / offshore wind.


Summary

Geometric analysis helps us understand

Individual Demand

See an agent’s valuation and their trade-offs.Classify valuations via ‘demand types’.Relate one structure of trade-offs to another.

Aggregate Demand

Always have competitive equilibrium iff ‘demand type’ is unimodular.Count intersections to check for equilibrium in other cases.

Matching with transferable utility

Stability = equilibrium = unimodularity of set of putative coalitions.New models of multiparty stable matching: see ?.


Advert for Research Assistant

We are revising this paper.

We would love to pay a graduate student (or anyone else!) to proof-read,especially the appendices, if anyone knows anyone who might beinterested.

Please talk to me during the week or email [email protected]


References

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N. Sun and Z. Yang. Equilibria and indivisibilities: Gross substitutes andcomplements. Econometrica, 74(5):1385–1402, 2006.


tropical hypersurfaces in economics, and the unimodularity...

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