Majority Stable Production Equilibria: A Multivariate Mean Shareholders Theorem

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Majority Stable Production Equilibria: A Multivariate Mean Shareholders Theorem

Hervé Crès

To cite this version:

Hervé Crès. Majority Stable Production Equilibria: A Multivariate Mean Shareholders Theorem.

2000. �hal-01064883�

M aj or it y St able P r oduct ion Equilibr ia:

A M ult ivar iat e M ean Shar eholder s T heor em ^¤

Herv¶ e Crµ es

^y

HEC School of Management

First version: March 2000 { This version: July 2000

A bst r act

In a simple paramet ric general equilibrium model wit h S st at es of nat ure and K · S

¯ rms | and t hus pot ent ially incomplet e market s| , rates of super majority rule ½2 [0; 1]

are comput ed which guarant ee t he exist ence of ½{ majority st able product ion equilibria:

wit hin each ¯ rm, no alt ernat ive product ion plan can rally a proport ion bigger t han ½of t he shareholders, or shares (depending on t he governance), against t he equilibrium. Under some assumpt ions of concavity on t he dist ribut ions of agent s' types, t he smallest ½are shown to obt ain for announced product ion plans whose span cont ains t he ideal securit ies of all K mean shareholders. These rat es of super majority are always smaller t han Caplin and Nalebu® (1988, 1991) bound of 1¡ 1=e ¼ 0:64. Moreover, simple majority product ion equilibria are shown t o exist for any init ial dist ribut ion of types when K = S ¡ 1, and for symmet ric dist ribut ions of types as soon as K ¸ S=2.

K eywor ds: Shareholders' vot e, general equilibrium, incomplet e market s, super majority.

JEL Classi¯ cat ion N umb er : D21, D52, D71, G39.

¤I am grat eful t o Alessandro Cit anna and Mich T vede for st imulat ing conversat ions and very helpful and const ruct ive comment s.

yHEC School of Management , 78351 Jouy-en-Josas, France; Tel: 33 1 39 67 94 12; cres@hec.fr; Fax:

33 1 39 67 70 85.

1 I nt r oduct ion

In t his paper, a simple paramet ric general equilibrium model wit h S st at es of nat ure and K · S ¯ rms | and t hus pot ent ially incomplet e market s| is st udied. T here is only one good, and t he agent s (consumers/ shareholders) are charact erized by ut ility funct ions exhibiting some quadrat ic feat ure and indexed by a probability vector ¼ in t he (S ¡ 1){

dimensional simplex, ¢

S

, t hat we call t he

type

of t he agent . Agent s' types are supposed t o be dist ribut ed, according a cont inuous measure wit h density f over ¢

S

, and are only endowed wit h init ial shares of t he K ¯ rms. Since t here is no consumpt ion in period zero,

¯ rms are t aken t o be asset s which allocat e a cert ain mass of t he good across st at es in period one.

Rat es of super majority rule ½ are comput ed which guarantee t he exist ence of ½{

majority stable product ion equilibria. The int erpret at ion follows. Given init ially an- nounced product ion plans, a general equilibrium is comput ed: agent s choose their opt imal port folio given t he market prices, and equilibrium prices for shares occur t hat clear t he market s. This product ion equilibrium is shown t o be ½{ majority st able in t he nat ural following sense: within each ¯ rm, t he product ion plans of ot her ¯ rms remaining ¯ xed, no alt ernat ive product ion plan can rally a proport ion bigger t han ½of t he shareholders, or shares, against the equilibrium.

T hese rat es of super majority rule are comput ed (1) under various governances, both of t he `one person-one vot e' and `one share-one vot e' types, and (2) when t he considered shares are t he init ial (pre-t rade) shares or t he equilibrium (post -t rade) shares. Condit ions are given under which t hese rates are smaller t han Caplin and Nalebu®(1988, 1991) bound of 64%. Moreover, it is shown that simple majority product ion equilibria exist for any initial dist ribut ion of types when K = S ¡ 1, and for symmet ric dist ribut ions of types as soon as K ¸ S=2. T hus, even wit h a high degree of market incomplet eness, a product ion equilibrium exist s against which, wit hin each ¯ rm, no alt ernat ive product ion plan can rally more t han half of t he shareholders, or shares.

T he early mot ivat ion of t his paper is t o st udy whet her collect ive choice mechanisms among t he society of shareholders | and in part icular t he simplest one: majority vot ing|

can help de¯ ning or qualifying t he object ive of t he ¯ rm in a cont ext of incomplet e market s.

T he lat t er concept has received a lot of int erest in t he recent years [see, e.g., Cit anna and Villanacci (1997), Dierker, Dierker and Grodal (1999) and Bet t zÄ uge and Hens (2000)].

In t he present set up, t he object ive of a ¯ rm is not invest igat ed from the perspect ive

of e± ciency or maximizat ion of some shareholder's value or pro¯ t funct ion [as in Drµ eze

(1974), Grossman and Hart (1979)], but from t he point of view of st ability wit h respect t o collect ive decision making among shareholders [as in Drµ eze (1987, 1989), DeMarzo (1993)], under di®erent types of governance.

T he result s proposed t end t o show t hat market equilibria exist which are stable with respect t o simple and quit e operat ional collect ive decision mechanisms (here: vot ing rules wit h reasonable rat es of super majority), even when t he degree of market incomplet eness can be considered `high'. Moreover t he less incomplet e t he market s t he smaller t he rat e of super majority necessary t o guarant ee t he exist ence of stable general equilibria. Al- t hough t hese int uit ive ¯ ndings are obtained in a simple set up, it is cert ainly valuable t o have posit ive result s of robust exist ence of majority majority st able product ion equilibria.

Especially given t he fact t hat t he Social Choice lit erat ure is perceived as being dominat ed by impossibility results and considered useless for a general t heory of decision in ¯ rms.

In st andard general equilibrium models of product ion in a cont ext of incomplet e mar- ket s [see, e.g., Magill and Quinzii (1996), Du± e and Shafer (1988) and Geanakoplos, Magill, Quinzii and Drµ eze (1990)], t he ¯ nancial st ruct ure is usually more complex than t he one present ed here. And t he di± culty in de¯ ning an object ive funct ion for a ¯ rm st ems from t he fact t hat, at equilibrium, shareholders can disagree on t he present value of t he product ion plans t hat are not in t he span of t he ¯ nancial st ruct ure: t o discount fut ure income st reams, t hey use shadow prices t hat can be di®erent . T hese shadow prices are endogeneous whereas in t he present paper, t hey are basically always point ing t oward t he ideal security which is exogeneously ¯ xed, by assumpt ion on t he ut ility funct ions.

T here is nevert heless a way t he present paper can shed some light on the debat e on which object ive funct ion t he ¯ rm should opt imize in the cont ext of incomplet e market s.

Firm should make choices t hat are support ed by shareholders, and t he most commonly suggest ed behavior for t he ¯ rm is t hat it should use t he average of t he shareholders' normalized present value vect or, where t he weights for averaging are t he shares of share- holders: a `mean' shareholder is t hus de¯ ned for each ¯ rm. If t he lat t er shares are t he

initial

shares, it is t he Grossman-Hart crit erion, if t hey are t he

equilibrium

shares, it is t he Drµ eze crit erion. The present paper gives some insight s that these two crit eria are likely t o give rise t o majority st able product ion equilibria (see Sect ion 4). The main result of t his paper is t hat

there exist production equilibria such that the

K

mean shareholders¹ can exactly span their type and generate their ideal security (the one they would demand if markets were complete); moreover these are the most stable equilibria

. It is wort h not icing

1Of course, in Drµeze's case, as opposed t o Grossman-Hart 's, t he mean shareholder is endogeneously det ermined at equilibrium.

t hat t he assumpt ions under which t his result holds are weaker in t he case of a governance µ

a la Drµ eze.

T his result has no direct link wit h t he above-ment ioned crit eria since t he announced product ion plan of a ¯ rm does not have t o be t he opt imal product ion plan of it s mean shareholder. But t he collect ion of K product ion plans (called a mult iplan) should be such that t heir span cont ains t he ideal security of

all

mean shareholders; in some way t he mult iplan is opt imal for t he K mean shareholders. T hen t he product ion equilibria are st able for t he lowest possible rat es of super majority. Last ly, t he present paper does not st udy t he quest ion of optimality or const rained opt imality of t he st able equilibria it describes, a subject lying at the core of t he lit erat ure on product ion in a cont ext of incomplet e market s. Especially, it does not pursue t he study of Dierker, Dierker and Grodal (1999) on t he relat ion between majority vot ing and welfare considerat ions

²

.

Technically, t he main result s of t he present paper are based on t hose in Caplin and Nalebu® (1988, 1991). Indeed, the case where agent s are dist ribut ed over ¢

S

and t here is only one ¯ rm (K = 1, and t hen no exchange of shares), is a sub-case of Caplin and Nalebu® (1988, 1991). And of course we get here: ½= 1 ¡ 1=e ¼ 0:632. But alt hough some assumpt ions are less general t han t hose in Caplin and Nalebu® (1988, 1991), t he set up is di®erent , and more general in at least one dimension

³

. It is more general t o t he ext ent that t he number of asset s can be bigger t han one. It is di®erent t o t he ext ent t hat t here is an upst ream market mechanism, wit h equilibrium prices clearing market s for shares. Consequent ly t here is an endogeneous allocat ion of shares and t herefore an

endogeneous distribution over types

for governances µ a la Drµ eze. In t he present set up, t he collect ive choice mechanism is intertwined wit h a general equilibrium market mechanism.

T he paper is organized as follows. Sect ion 2 int roduces t he model and provides some preliminary result s founding t he analysis. Sect ion 3 focuses on t he canonical case where agents are described t hrough charact erist ics t hat are uniformly dist ribut ed over ¢

S

; exact comput at ions are provided illust rat ing how t he less incomplet e t he markets the smaller t he required rat es of super majority. Sect ion 4 discusses the generalizat ion of t he result s obt ained in t he previous sect ion: Caplin and Nalebu®(1988, 1991) general upper bound of 64% for t he rat e of super majority is shown t o hold in case t he dist ribut ions of charact er- ist ics ful¯ ll some condit ions of concavity (Proposit ion 3 and Theorem 3); simple majority

2Dierker, Dierker and Grodal (1999) show t hrough an example t hat majority vot ing and welfare considerat ions can be complet ely unrelat ed.

3Act ually, Caplin and Nalebu® (1991) gives, as an illust rat ion for a possible applicat ion of t heir t heory, t he example of vot ing among shareholders in a cont ext of incomplet e market s.

st able product ion equilibria are shown t o exist under some assumpt ions of symmet ry of t he dist ribut ions of charact erist ics (Proposition 2) or when t he degree of market incom- plet eness is just one (Theorem 2). Appendix A proposes some comment s; in part icular, t hrough paramet ric examples, t hese rat es are shown to decrease wit h t he homogeneity of t he shareholders' types, and t o increase wit h t he shareholders' pessimism. All t echnical proofs are gat hered in Appendix B and Appendix C.

2 T he model

Consider an economy wit h two periods, t = 0; 1 and S st at es of nat ure in period 1, indexed by s, s = 1; : : : ; S. T here is one good, and a cont inuum of agent s, each agent is indexed by probability vect or ¼= (¼

^s

)

^S_{s= 1}

which will be int erpret ed as his

ideal security

once t he ut ility functions ar int roduced. T he agent 's type ¼is t hus t aken in t he (S¡ 1)-dimensional simplex:

¢

S

=

(

¼= (¼

¹

; ¼

²

; : : : ; ¼

^S

) 2 R

^S₊

j

XS s= 1

¼

^s

= 1

)

:

Agent s' types are assumed t o be dist ributed over ¢

S

according t o a cont inuous, at omless density function f : ¢

S

¡ ! R

+

. Consumpt ion t akes place in period one but must be decided in period zero. Agent ¼ is charact erized by a ut ility funct ion: U

¼

[x(¼)], where x(¼) = [x

¹

(¼); : : : ; x

^S

(¼)] is agent ¼'s consumpt ion in period 1. Since t here is only one good, it will be somet imes bet t er t o give a ¯ nancial int erpret at ion to x(¼) as an income vect or. Utility funct ions are of a quadrat ic/ `euclidean' type, described at t he end of t his sect ion.

T here are K ¯ rms indexed by k, k = 1; : : : ; K . All ¯ rms have t he same product ion t echnology, represent ed for t he simplicity of t he analysis by t he span of ¢

S

:

h¢

S

i =

(

y = (y

¹

; y

²

; : : : ; y

^S

) 2 R

^S

j

XS s= 1

y

^s

= 1

)

:

Agent ¼ is endowed wit h init ial shares of t he K ¯ rms: µ

⁰

(¼) = [µ

⁰_k

(¼)]

^K_{k= 1}

. He is t hen t ot ally charact erized by t he vect or [¼; µ

⁰

(¼)]. T he funct ion µ

⁰

: ¢

S

¡ ! R

^K₊

is t aken cont inuous and posit ive over ¢

S

.

A ¯ rm is basically an asset which allocat es an init ial mass ¡

⁰_k

=

Z

¢S

f (¼) µ

_k⁰

(¼) d¼ of the good across st at es in period 1. We do not normalize it t o one t o allow di®erent

¯ rms t o be of di®erent `sizes': t he yield, in t erms of consumpt ion/ income, of ¯ rm k in

period 1 in case st at e s occurs is: ¡

⁰_k

y

_k^s

. To avoid some minor t echnical di± cult ies, it

is preferable not t o impose sign const raint s on product ion plans; t his is ¯ ne wit hin t he

¯ nancial int erpretat ion of t he model. Alt hough it is abusive t o t alk about ¯ rms in such a simple framework, and bet t er t o t alk about securit ies, we st ick to this t erminology and rely on t he forgiveness of t he reader.

M axim izat ion pr ogr am of t he agent s

Given an announced product ion plan y

k

by each ¯ rm (hence an announced mult i-plan Y = (y

k

)

^K_{k= 1}

, where all y

k

's are t aken di®erent ) and a vect or of prices q = (q

k

)

^K_{k= 1}

for t he shares, each agent maximizes his ut ility by choosing t he opt imal vect or of shares

⁴

µ(¼) = [µ

k

(¼)]

^K_{k= 1}

and t he opt imal consumpt ion plan x(¼) according t o the maximizat ion program M (¼):

[µ(¼);x (¼)]

max U

¼

[x(¼)]

s. t .

XK k= 1

q

k

h

µ

k

(¼) ¡ µ

⁰_k

(¼)

i

= 0 (1)

and x(¼) =

XK k= 1

µ

k

(¼) y

k

(2)

T his is of course equivalent t o ~ M (¼):

max

µ(¼)

U ~

¼;Y

[µ(¼)]

s. t .

XK k= 1

q

k

h

µ

k

(¼) ¡ µ

⁰_k

(¼)

i

= 0 where ~ U

¼;Y

[µ(¼)] = U

¼

à _K X

k= 1

µ

k

(¼) y

k

!

.

M aj or it y St able P r oduct ion Equilibr ium

Given t he individual demand funct ions for shares, an equilibrium price will clear t he market for shares.

D e¯ nit i on 1

A Production Equilibrium (PE) is a vector

E = (Y; q; µ(¼))

such that indi- vidual optimization

(C

1

)

, and market clearing

(C

2

)

, are satis¯ ed:

(C

1

)

Given

(Y; q)

, for all

¼

,

[µ(¼)]

solves the maximization program

M (¼) ~

;

(C

2

)

For all

k

,

Z

¢S

f (¼) µ

k

(¼) d¼=

Z

¢S

f (¼) µ

⁰_k

(¼) d¼(= ¡

⁰_k

)

.

4T he choice has been made here not t o impose short -sell const raint s on t he µ's. T he aim is t o prove exist ence of majority st able product ion equilibria, and t he paper is most ly going t o focus on equilibria such t hat µ(¼) > 0 for all ¼.

For a ¯ rm k, given a PE E, a dist ribut ion of vot ing weight s ´ : ¢

S

¡ ! R

^K₊

(´ ´ µ

⁰

or µ), and two product ion plans (y

k

; z

k

), denot e I

E;´

(y

k

) t he subset of agent s ¼ endowed wit h a positive vot ing weight

⁵

in ¯ rm k (i.e., agent s such t hat ´

k

(¼) ¸ 0), and denot e I

E;´

(z

k

; y

k

) [½ I

E;´

(y

k

)] t he subset of agent s ¼ endowed wit h a posit ive vot ing weight in

¯ rm k who prefer z

k

t o y

k

, i.e., such t hat

´

k

(¼) ¸ 0 and U

¼

[x(¼) + µ

k

(¼)(z

k

¡ y

k

)] ¸ U

¼

[x(¼)] ; where x(¼) is de¯ ned t hrough equat ions (2). De¯ ne:

P

E;´

(z

k

; y

k

) =

Z

I_{E ; ´}(z_k;y_k)

f (¼) d¼

Z

I_{E ; ´}(y_k)

f (¼) d¼

and A

E;´

(z

k

; y

k

) =

Z

I_{E ;´}(z_k;y_k)

f (¼) ´

k

(¼) d¼

Z

I_{E ; ´}(y_k)

f (¼) ´

k

(¼) d¼

;

respectively t he fract ion of shareholders (wit h vot ing right s) and t he fract ion of vot e shares who prefer z

k

t o y

k

. De¯ ne moreover

P

E;´

(y

k

) = sup

z_k2 ¢_S

P

E;´

(z

k

; y

k

) and A

E;´

(y

k

) = sup

z_k2 ¢_S

A

E;´

(z

k

; y

k

)

t he maximal fract ions (resp. of t he shareholders/ shares, wit h vot ing right s) against y

k

. D e¯ nit i on 2

For any real

½2 [0; 1]

, a

½

{ Majority Stable Production Equilibrium under

²

the `one person-one vote, pre-trade' governance (in short, a

½

{ MSPEp0) is a PE

E

such that for all

k

,

P

E;µ⁰

(y

k

) · ½;

²

the `one person-one vote, post-trade' governance (

½

{ MSPEp1) is a PE

E

such that for all

k

,

P

E;µ

(y

k

) · ½;

²

the `one share-one vote, pre-trade' governance (

½

{ MSPEa0), is a PE

E

such that for all

k

,

A

E;µ⁰

(y

k

) · ½;

²

the `one share-one vote, post-trade' governance (

½

{ MSPEa1), is a PE

E

such that for all

k

,

A

E;µ

(y

k

) · ½:

For

½= 1=2

, such an equilibrium is a simple{ Majority Stable Production Equilibrium (or s-MSPE).

5Only such agent s have t he right t o vot e in t he present set up.

R em ar k: T he p0 and p1-governance are not dist inct as soon as everybody is posit ively endowed wit h shares of all ¯ rms, bot h init ially and at equilibrium. This will be most ly t he case in t he present paper. It is clear t hat t he most interest ing governance is t he a1-governance. Nevert heless, t here is some di± culty in de¯ ning a ½{ Majority St able Pro- duct ion Equilibrium for t he a1-governance since t he number of post -t rade shares with vot ing rights,

Z

I_{Y ; µ}(y_k)

f (¼) µ

k

(¼) d¼, is endogeneous and can be bigger t han t he init ial

allocat ion of shares, ¡

⁰_k

, in case part of the agent s choose t o be short on k's st ock market

⁶

. But we will concent rat e in t his paper on product ion equilibria where all agent s are allo- cat ed posit ive post -trade shares. For ot her product ion equilibria, one can consider t hat t he excess number of shares is allocat ed in a cont inuous way (i.e., according t o f and µ

⁰

) t o all ot her shareholders, which does not int roduce much dist orsion in t he model.

T he concept of ½{ majority st able equilibrium (for K = 1) is linked t o t he Simpson- K ramer min-max majority [see Simpson (1969), K ramer (1977)]. In t he present paper t he concept is built t o hold for K ¸ 1: min-max majorit ies for product ion equilibria are (resp., for each governance):

½

^¤_p0

= inf

P E (Y;q;µ)

max

k

P

Y;µ⁰

(y

k

) , ½

^¤_p1

= inf

P E (Y;q;µ)

max

k

P

Y;µ

(y

k

)

½

^¤_a0

= inf

P E (Y;q;µ)

max

k

P

Y;µ⁰

(y

k

) and ½

^¤_a1

= inf

P E (Y;q;µ)

max

k

P

Y;µ

(y

k

) :

A ssum pt ions on t he ut ilit y funct ions U

¼

T he ut ility funct ions U

¼

are de¯ ned on R

^S

and assumed t o sat isfy t he two following set s of assumpt ions:

² Assumpt ion (A) : U

¼

is increasing, st rict ly quasi concave, continuously di®erent iable and homot het ic;

² Assumpt ion (E) : The indi®erences surfaces of U

¼

cut h¢

S

i t hrough hyperspheres cent ered on ¼.

Taking homot het ic ut ility funct ions will allow t o focus on consumpt ions in h¢

S

i (since we'll only consider PE wit h

⁷

q = 1

K

, see next subsection). Assumpt ion (E) (said t o be t he `euclidean' assumpt ion) is more problemat ic: it is st andard in Social Choice t heory,

6In fact , t he st ock repurchase plans t hat some ¯ rms implement might be considered as int roducing some type of endogeneity in t he t ot al numbers of shares.

7Not at ion: q = 1_K st ands for q_k = 1; all k.

and t aken for purely t echnical reasons. T he mot ivat ion behind t his assumpt ion is t he following: when asked whet her t hey agree wit h an in¯ nit esimal change

⁸

u 2 R

^S

in t he product ion plan of ¯ rm k, indi®erent shareholders should be on a

hyperplane

in ¢

S

. It is nevert heless clear that such ut ility funct ions exhibit some form of quadrat ic feat ure, an such feat ures are regularly assumed in t he ¯ nance lit erat ure, e.g., in t he CAPM.

When t here is only one ¯ rm (K = 1) as in Caplin and Nalebu® (1988, 1991), it is enough t o t ake ut ility funct ions of t he separable form:

U

¼

[x(¼)] =

XS s= 1

¼

^s

v

^s

[x

^s

(¼)] : (3)

In t hat case, the type ¼ is t he subject ive probability of t he agent over st at es of nat ure.

T he fact that t he element ary ut ility funct ions are common across t he populat ion secures t he needed condit ion [see Grandmont (1978)]. The reason is simple t o see: when K = 1, x(¼) = y

1

is independent of ¼; and for any in¯ nit esimal change u 2 R

^S

in t he produc- t ion plan, shareholders indi®erent t o t he proposed change are described by t he equat ion

P

s

¼

^s

u

^s

D v

^s

[y

1

] = 0 which de¯ nes a hyperplane. If K > 1, shareholders indi®erent t o an in¯ nit esimal change u in the product ion plan of ¯ rm k are described by t he equat ion

P

s

¼

^s

u

^s

D v

^s

[¹x

^s

(¼)] = 0, where ¹x

^s

(¼) st ands for t he opt imal consumpt ion of agent ¼, and di®ers across ¼. For inst ance, in t he log-linear case where v

^s

´ ln, t he lat t er equat ion almost never de¯ nes a hyperplane in ¢

S

. But some of t he result s proposed in t he paper are valid wit h ut ility funct ions of t he form (3); t his discussion is posponed t o Sect ion 4.4.

A last di± culty is t o avoid negat ive consumpt ions/ incomes. We basically discard t his problem: (

i

) in case t he ut ility funct ions are of the separable form (3), by assuming t hat v

^s

sat is¯ es t he Inada condit ions: lim

x¡ ! 0

D v

^s

(x) = + 1 ; (

ii

) in case t he ut ility funct ions sat isfy assumpt ion (E), by endowing t he agents wit h an appropriat e quant ity, x

⁰

(¼), of t he consumpt ion good, what ever t he occuring st at e of nat ure

⁹

.

T he Par et o cr it er ion

Among all product ion equilibria, we will rest rict our at t ent ion t o t hose t hat respect t he Paret o criterion: an eligible product ion plan for majority st ability should be such t hat

8As already writ t en in t he int roduct ion, t he assumpt ion of concavity of t he individual ut ility funct ions ent ails t hat t he most challenging product ion plans are in¯ nit esimally close t o t he st aus quo; see Lemma 2 in Appendix B. T herefore, a challenger is basically an in¯ nit esimal change u in t he product ion plan, wit h, given t he t echnological const raint s,P

su^s= 0.

9Since we will only consider mult iplans Y which spans a hyperplane having a non-empty int ersect ion wit h ¢_S, a uniform upper bound can be found on x⁰(¼), for all ¼.

t here does not exist an alt ernat ive product ion plan preferred by

all

shareholder endowed wit h a vot ing right (i.e., endowed with a posit ive quantity of shares). T he following observat ion shows t hat , in t he present framework, a necessary and su± cient condit ion is t hat st ock prices be all equal

¹⁰

.

Obser vat ion 1

A PE

(Y; q; µ)

satis¯ es the Pareto criterion if and only if

q = 1

K. Proof

: Consider a PE (Y; q; µ) such t hat q 6 = 1

K

. Consider two ¯ rms, k and j , such t hat q

k

> q

j

; t hen t here exist s an alternat ive anounced product ion plan z

k

unanimously prefered t o y

k

by agent s posit ively endowed wit h shares of ¯ rm k. Suppose, wit hout loss of generality, t hat q

1

> q

2

. At t he PE (Y; q; µ), t he gradient of U

¼

[x(¼)] wit h respect t o µ(¼) is colinear t o q. Given q

1

> q

2

, this ent ails that for all ¼: D U

¼

[x(¼)] ¢(y

1

¡ y

2

) > 0.

Consider z

1

= y

1

+ ²(y

1

¡ y

2

), we t hen have, for ² small enough and for all ¼, U

¼

[x(¼) + µ

1

(¼)(z

1

¡ y

1

)] > U

¼

[x(¼)] if µ

1

(¼) > 0. Hence for t he `if' part of t he assert ion. The `only if' part is obviously t rue. 2

In t he sequel of t he paper, we'll de¯ ne a Paret o product ion equilibrium as a PE wit h unit prices: (Y; 1

K

; µ).

Denot e hY i t he vect orial subspace, in h¢

S

i , spanned by Y . At a PE wit h unit prices, t he opt imal choice of an agent is | up t o mult iplicat ion by a scalar, given assumpt ion (A)| the

point of tangency

between hY i and t he sect ions by h¢

S

i of t he agent 's indi®er- ence curves. This optimal point is t he

orthogonal projection

of ¼on hY i when assumpt ion (E) is ful¯ lled.

T his last property ent ails t he following geomet ric int erpret at ion, µ a la Caplin and Nale- bu®, of t he main argument of t he paper (proven in Lemma 2 in Appendix B): trying t o

¯ nd a best challenger t o y

k

, wit hin t he product ion plans of ¯ rm k (t he product ion plans of ot her ¯ rms remaining ¯ xed), reduces t o t ry and cut t he support , ¢

S

, of t he agent s' types by an hyperplane

containing

hY i in such a way as t o maximize t he di®erence in volume of t he two result ing pieces | a volume comput ed using t he dist ribut ion of vot ing weight s, as t he governance speci¯ es it.

10DeMarzo (1993) proves t hat a product ion plan which is st able wit h respect t o a `unanimity responsive' collect ive decision rule should be chosen by using a normalized present value vect or in t he convex hull of t hose of all shareholders. A `unanimity responsive' collect ive decision rule is such t hat it should be able t o implement an alt ernat ive product ion plan t hat Paret o dominat es t he incumbent . See also Proposit ion 31.3 in Magill and Quinzii (1996).

A fundam ent al pr elim inar y r esult

It st at es t hat any vect orial subspace in h¢

S

i can be spanned by a mult iplan Y that can be associat ed wit h a PE wit h equal unit prices.

L em m a 1

Under assumption (A), any multiplan

Y = (y

k

)

^K_{k= 1} generates a vector ial sub- space that can be supported by a production multiplan associated to a PE with unit prices:

there exists a production multiplan

Y = ( ~ ~ y

k

)

^K_{k= 1}, with

y ~

1

= y

1, such that

hY i ´ h~ Y i

, and

( ~ Y ; 1

K

; ~ µ)

is a PE. Moreover,

y

1 can be chosen such that

µ(¼) > 0 ~

for all

¼

.

Proof

: See Appendix A. 2

T his fundament al Lemma allows t o focus only on t he span hY i of a mult iplan Y , and not on t he mult iplan it self. Moreover, t he fact that ~ µ(¼) can be t aken st rict ly posit ive for all ¼ secures t hat all shareholders have t he right t o vot e and t hat t he considered dist ribut ions of vot ing weight s are positive over t he whole support ¢

S

.

3 T he canonical case

We consider t he canonical case of uniform dist ributions of init ial characterist ics in t he set of types ¢

S

. Assumpt ions:

² for t he p0-governance: t he distribut ion f is uniform and µ

⁰_k

(¼) > 0 for all k, all ¼;

² for t he p1-governance: t he distribut ion f is uniform and

^P_k

µ

_k⁰

(¼) > 0 for all ¼;

² for t he a0-governance: t he dist ribut ion f ¢µ

_k⁰

is uniform for all k;

² for t he a1-governance: t he dist ribut ion f ¢

^P _k

µ

⁰_k

is uniform.

It is wort h not icing t hat t he result s of t he present sect ion remain valid under t he assump-

t ion of separable ut ility funct ions of t he type (3) (see Claim 2 in Sect ion 4.4) and t hat

t he preceding set of assumpt ions are weaker for governances µ a la Drµ eze, i.e., based on

post -t rade shares, t han for governances µ a la Grossman-Hart .

3.1 Exist ence of M SP E

For any ¯ xed posit ive int egers S and K , K · S, de¯ ne

¹¹

½

S;K

= 1 ¡

0

@ j S¡ 1

K

k j S¡ 1

K

k

+ 1

1 A

b

^{S ¡ 1}K

c

: (4)

T heor em 1

Fix

K

and

S

. T here always exist, in the canonical case,

½

S;K{ MSPE¹² for all governances of De¯ nition 2. Hence

½

^¤

· ½

S;K for all four governances.

When K = 1, t here are no t ransact ion between agents and everybody keeps it s init ial share of t he ¯ rm; since t he shares are uniformly dist ribut ed accross agent s, t hen t he four governances coincide, and t he above result is a part icular case of Caplin and Nalebu®

(1988), which gives as a uniform upper bound: 1 ¡ 1=e ¼ 0:632. This upper bound is approached for t he present concept of majority vot ing equilibrium in t he case where t he number of asset s (or ¯ rms) is negligible wit h respect t o t he number of st ates of t he world.

In ot her cases t he rat e of super-majority rule t hat guarant ees t he exist ence of a MSPE is lower t han t his previous bound. For example, what ever t he number of st at es of nat ure, if S=3 · K < S=2 [resp. S=4 · K < S=3] t hen a rat e of 56% [resp. 60%] su± ces. Anot her example is t he following immediat e corollary.

Cor ollar y 1

S{ MSPE exist as soon as

K ¸ S=2

for all four governances.

T hus, even wit h a high degree of market incomplet eness, a product ion equilibrium exists against which, wit hin each ¯ rm, no alt ernat ive product ion plan can rally more than half of t he shareholders, or shares. T he sequel of this sect ion is a proof of Theorem 1 which goes t hrough t he design of t he `right ' securit ies.

3.2 B asic const r uct i on of a M SP E

T he aim is t o const ruct a ½{ MSPE for t he lowest possible ½. For ¯ xed S and K , de¯ ne n =

¹

S ¡ 1 K

º

, so t hat S = nK + m, wit h 1 < m · K . We t hen const ruct t he following part it ion of t he set of st at es of nat ure int o K subset s (according t o t he nat ural order, t he m ¯ rst subset s cont ain n + 1 element s, t he K ¡ m ot hers cont ain only n element s):

S

k

= f (k ¡ 1)(n + 1) + 1; : : : ; k(n + 1)g for 1 · k · m T

k

= f m + (k ¡ 1)n + 1; : : : ; m + kng for m + 1 · k · K

11For any real x, we denot e by bxc t he largest int eger smaller or equal t o x, and by dxe t he smallest int eger larger or equal t o x.

12In fact t here is a cont inuum of such MSPE (see t he proof ).

De¯ ne t he K product ion plans ¹ Y = ( ¹y

k

)

^K_{k= 1}

, such t hat : for k · m; ¹y

_k^s

=

8<

:

1

n+ 1

if s 2 S

k

0 ot herwise ; for k ¸ m + 1; ¹y

^s_k

=

8<

:

1

n

if s 2 T

k

0 ot herwise (5) T he main argument revolves around t he following proposit ion which is a more developed rest at ement of Theorem 1.

P r op osit ion 1

Fix

S

and

K

. T hanks to Lemma 1, there exist PE

( ~ Y ; 1

K

; ~ µ)

that are

½

S;K{ MSPE for the four governances. They are such that

h~ Y i ´ h¹ Y i

and for all

¼

, the optimal consumption is

~

x(¼) =

^X

k

µ ~

k

(¼) ~ y

k

=

^X

k

¹µ

_k

(¼) ¹y

k

where

¹µ

is de¯ ned by¹³:

¹µ

_k

(¼) =

8>

>>

<

>>

>:

¼

^Sk if

k · m

¼

^Tk if

k ¸ m + 1

: (6)

Proof

: See Appendix B. 2

Exam ple: When bot h f and µ

⁰

are t aken uniform (and normalized), and all ¯ rms have t he same size (¡

⁰_k

is independent of K ), an example of such a PE ( ~ Y ; 1

K

; ~ µ) is:

for k · m; ~ y

k

= (n + 1)K

S ¹y

k

¡ 1 S

XK j = m+ 1

¹y

j

; and for k ¸ m + 1; ~ y

k

= ¹y

k

; (7)

and µ ~

k

(¼) =

8>

>>

<

>>

>:

S

(n+ 1)K

¼

^Sk

if k · m

¼

^Tk

+

_{(n+ 1)K}¹

¼

^S

if k ¸ m + 1 :

A geomet ric int erpret at ion of Proposit ion 1 will be helpful t o underst and t he proof and t he basic int uit ion of t he const ruct ion. As writ t en before, given t hat market equilibrium prices are 1

K

, t he opt imal choice of an agent is t he point of t angency between hY i and t he sect ions by h¢

S

i of t he agent 's indi®erence curves. Denot e ~ § (¼) t he section by h¢

S

i of t he indi®erence curve going t hrough t he opt imal choice ~ x(¼) (cf. Figure 1.a below).

A change in t he product ion plan ~ y

k

(or equivalent ly ¹y

k

) of ¯ rm k will t hen move h¹ Y i in

13Denot e, for a subset V of t he set of st at es of nat ure, ¼^V = X

s2 V

¼^s.

such a way t hat it st ill goes t hrough all ot her ~ y

j

's. This change, ¯ xing the shares at t heir post -t rade values, project s t he equilibrium consumpt ion ~ x(¼) inward or outward ~ § (¼), hence result ing in an improving or impairing change of t he ut ility level of agent ¼ (cf.

Figure 1.b below).

Lemma 2 in Appendix B shows t hat ¯ nding a best challenger t o ~ y

k

, wit hin t he pro- duct ion plans of ¯ rm k (t he product ion plans of ot her ¯ rms remaining ¯ xed), amount s t o ¯ nding t he

in¯ nitesimal move

of ~ y

k

which improves t he welfare of the biggest propor- t ion of shareholders or shares. Given assumpt ion (E), t his reduces t o t ry and cut ¢

S

by a hyperplane (orthogonal t o t his in¯ nit esimal change) cont aining hY i in such a way as t o maximize t he di®erence in volume of t he two result ing pieces. The best in¯ nit esimal change (of ~ y

k

) is point ing t oward t he largest piece. As in Caplin and Nalebu® (1988) it is shown t hat , when t he dist ribut ion of init ial charact erist ics is uniform, t he most chal- lenging in¯ nit esimal change of t he product ion plan ~ y

k

is t o sacri¯ ce one st at e of nat ure t o t he bene¯ t of all ot hers

¹⁴

, and implement a change

³

¡ ²;

_{S¡ 1}^²

; : : : ;

_{S¡ 1}^²

´

.

3.3 G eom et r i c i ll ust r at i on: S = 3 and K = 2

In t he case S = 3 and K = 2, and under t he assumpt ions given in t he example following Proposit ion 1, wit h ¹y1= (1=2; 1=2; 0) and ¹y2= (0; 0; 1), and t herefore ~y2= ¹y2 and ~y1= (2=3; 2=3; ¡ 1=3), one get s:

[~µ1(¼); ~µ2(¼)] = µ3

4[¼¹+ ¼²]; ¼³+ 1

4[¼¹+ ¼²]

¶

and [¹µ1(¼); ¹µ2(¼)] = ¡

[¼¹+ ¼²]; ¼³¢ : T his is drawn on Figure 1.a; t he indi®erence curve ~§ (¼) corresponding t o t he opt imal ut ility level for agent ¼ is drawn: it is a circle around t he ideal security ¼. An illust rat ion of t he previous discussion is now provided in t his simple case and basically holds for t he four governances.

Opt i m al cut t ing of t he si m pl ex : It should be clear on t he drawing why ( ~Y ; 1₂; ~µ) is majority st able for t he simple-majority rule under all four governances. Indeed, consider, inst ead of ~y₂, anot her proposal

~

y⁰₂ (see Figure 1.b). T he shares being ¯ xed, t he new consumpt ion of agent ¼ will become ~x(¼)⁰which dives inward ~§ , hence result ing in a higher ut ility. But for t he symmet ric (wit h respect t o h¹Y i ) agent , charact erized by type ¼^¿ = (¼₂; ¼₁; ¼₃), who at equilibrium consumes t he same ~x(¼^¿) = ~x(¼), t his is an impairing change. Hence at least half of t he agent s (t he left part of t he t riangle) ¯ nds it impairing t hat any rightward change of t he product ion plan of ~y2 be implement ed. Symmet rically, any, even in¯ nit esimal, leftward change of ~y₂ is going t o be blocked by t he agent s on t he right -hand side of h¹Y i . Finally, since bot h agent s ¼ and ¼^¿ have t he same share of ¯ rm 2, it is obviously t he case t hat t he simple-majority st ability property holds for t he four types of governance. T he same type of argument holds t o prove

14T his is act ually very classical in Social Choice t heory and illust rat ed by t he problem of having t o divide a pie among S individuals; what ever t he init ial allocat ion, t here is a majority of ^{S¡ 1}_S t o expropriat e one individual of his share and dist ribut e it evenly t o t he ot hers.

Figure 1.a Figure 1.b

(1; 0; 0) (0; 1; 0)

~

y₂= ¹y₂= (0; 0; 1)

¹y₁

²

²

~ y1

²

²

² ¼

~ x(¼)

§~

²

²

²

²

²

~ y₂

~ y₂⁰

~ x ~x⁰

¼

²

¼^¿

~ y1 ²

©©*

¾

t hat any change in t he product ion plan ~y1 is going t o be blocked by at least half of t he shareholders, in number and volume of shares.

Moreover it is clear t hat t here are many ways t o cut ¢3int o two pieces of equal sizes. T he two pieces do not have t o be symmet ric. Act ually, Lemma 1 shows t hat any cut t ing of ¢3can be spanned by two product ion plans (y₁; y₂) which will generat e a PE wit h unit prices, hence securing t hat t he fundament al geomet rical int erpret at ion of T heorem 1 be valid. T his ensures a cont inuum of s-MSPE is t he present simple case.

Figure 1.c Figure 1.d

^ y₁

²

²

²

²

² ¼

~ x

^ y⁰₁

^ y2

~ x⁰

a b

²

²

²

¼^¿

¹§

¹y₂

²

²

²

²

¹x(¼)

¼

¹y1 ±¹y₁

±¹x(¼)

~ y1 ²

¡ ¡µ

¡¡µ

?

?

M ult i pl ici t y of t he M SP E: T here are a cont imuun of PE t hat end up wit h t he same \ cut t ing" of ¢S

wit h unit prices q = 1₂for bot h asset s, and unchanged ¹µ (see Lemma 1): for all ®, y₁= (¹₃+ ®;¹₃+ ®;¹₃¡ 2®) and y2 = (¹₃¡ ®;¹₃ ¡ ®;¹₃+ 2®) will always found a PE wit h q = 12 and equilibrium shares ¹µ. (Not ice

~

® = ¹₃.) For example, wit h ^® = ¹₆, one get s ^y₁ = ¹y₁ = (1=2; 1=2; 0) and ^y₂ = (1=6; 1=6; 2=3), wit h:

[^µ₁(¼); ^µ₂(¼)] = ¡

[¼¹+ ¼²] ¡ ¹₂¼³;³₂¼³¢

so t hat ^x(¼) = ~x(¼) = ¹x(¼) (see Figure 1.c). T he drawn change from ^y1 t o ^y₁⁰will be ut ility improving for agent ¼, but ut ility impairing for agent ¼^¿. Not ice here t hat all agent s charact erized by a type ¼such t hat ¼₃¸ 2=3 (i.e., above t he dot t ed line [a; b]), do not have the right to vote under governances based on post -t rade shares since t heir post -t rade shares in ^y₁are negat ive.

Hence t he same rule as before is ful¯ lled: any rightward (resp. leftward) change in t he product ion plan

^

y1will be blocked by (at least ) t he left -hand (resp. right -hand) side of t he t riangle, whose t op has been cut -o®. It is t o avoid t he minor and irrelevant t echnical di± culty of having t o comput e relat ive volumes in a cut -o® simplex t hat PE are cont ruct ed for which all shares allocat ed at equilibrium are posit ive (i.e., wit h ® > 1=3).

A sset s w it h di®er ent pr ices, t he Par et o cr it er i on: One can easily see t hat any proposed change of ~y1 along t he line h¹Y i will be unanimously reject ed. T his fact is linked t o t he reason why t here is no majority st able product ion equilibrium wit h announced product ion plans ( ¹y1; ¹y2): in fact t he PE based on t his mult iplan does not sat isfy t he Paret o crit erion (see Observat ion 1). Indeed, t he equilibrium price vect or is t hen such t hat q₁> q₂: t he shareholders will ¯ nd it opt imal t o `load' more t han in t he above case t heir port folio wit h shares of ¹y2 (see Figure 1.d) t o reach t he opt imal consumpt ion ¹x(¼). As drawn on Figure 1.d, t he opt imal ut ility level will t hen generat e an indi®erence surface ¹§ not t angent t o h¹Y i . Given t he quasi-concavity of t he ut ility funct ions, any change ±¹y₁ of ¹y₁ t oward ~y₁ will be unanimously support ed, since t he consecut ive change ±¹x(¼) is always ut ility improving. T his is t rue unt ill ¹y1reaches

~ y1.

4 M or e gener al cases

In t his sect ion, more general density funct ions, f , and init ial dist ribut ions of shares, µ

⁰

, are invest igat ed. To avoid minor t echnical di± cult ies that would make t he reading less confort able wit hout making t he problem richer, we consider only st rict ly posit ive initial dist ribut ions of charact erist ics: µ

⁰

(¼) > 0 and f (¼) > 0 for all ¼. The aim is t o generalize as much as possible t he result s of t he previous sect ion. In a ¯ rst subsect ion, we invest igat e, for unspeci¯ ed f and µ

⁰

, t he case of complete market s, along wit h t he case of incomplete market s wit h only one dimension of incomplet eness. T hen we consider t he case of

symmetric

dist ribut ions of charact erist ics (subsect ion 4.2). For t hese two cases,

simple

majority product ion equilibria are shown t o exist . Finally, t he case of º -concave dist ribut ions of charact erist ics is considered (subsect ion 4.3), an assumpt ion regarded as imposing some measure of consensus in t he society of shareholders. Caplin and Nalebu®

(1991) result s are t hen used t o provide rat ios of ½-majority st able product ion equilibria.

4.1 T he cases K = S, K = S ¡ 1

T he case K = S is t rivial, since for a PE (Y; 1

K

; µ) | whose exist ence is secured by Lemma 1| every agent of type ¼ is able t o generat e it s idiosyncrat ic ideal security:

[

^P _k

µ

⁰_k

(¼)] ¢¼. In t his case, in equilibrium, all y

k

's are

unanimously

support ed against any alt ernat ive product ion plan for any f and any init ial dist ribut ion of shares µ

⁰

; i.e., P

Y;µ⁰

(y

k

) = A

Y;µ⁰

(y

k

) = P

Y;µ

(y

k

) = A

Y;µ

(y

k

) = 0 for all k, as t he t heory of complet e market s predict s. We t hus have t he following observat ion.

Obser vat ion 2

I f

K = S

, for any density

f

and any initial distributions of shares

µ

⁰, there exist PE which are stable for any voting rule (even infra-majority voting rule¹⁵).

T he case K = S ¡ 1 is more di± cult and int erest ing. As far as the p0 and p1- governances are concerned, t he argument is st raighforward since t he same dist ribut ion of vot ing weight s, f , is t aken for all ¯ rms. Therefore a median-vot er-like argument allows t o go t hrough: For a PE (Y; 1

K

; µ), we know t hat ¯ nding a best challenger t o t he announced product ion plan y

k

amount s t o cut t he support of agent s' types by a hyperplane cont aining hY i . But t here is a unique such hyperplane, i.e., hY i it self. Therefore, t o prove exist ence of a s{ MSPE, it is enough to choose hY i such t hat it separat es ¢

S

int o two pieces of equal measure wit h respect t o f . T his is obviously always possible, and t here is an in¯ nit e number of ways t o do so as soon as S > 2. T hanks t o Lemma 1, we know t hat such a hyperplane can be supported by a PE wit h unit prices and posit ive shares.

T he argument is more complicat ed for governances based on shares, e.g., t he a0 and a1-governances. Indeed, hY i should be chosen such t hat it separat es ¢

S

int o two pieces of equal measure

simultaneously

wit h respect t o K (= S ¡ 1) dist ribut ions of vot ing weight s.

Hence a `mult ivariat e-median-vot er' argument is necessary. The following proposition, based on degree t heory and using t he Borsuk-Ulam t heorem, is shown.

¹⁶

T heor em 2

I f

K = S ¡ 1

, there exist s-MSPE for any

f

and any

µ

⁰, for all four gover- nances.

Proof

: To prove exist ence of s{ MSPEa0 one has t o choose hY i t hat separates ¢

S

int o two pieces of equal measure wit h respect t o t he dist ribut ions f ¢µ

⁰_k

, for all k.

15An infra-majority vot ing rule is a majority rule wit h rat e ½< 1=2, i.e. such t hat an alt ernat ive a defeat s an alt ernat ive b if a proport ion bigger t han ½of t he populat ion prefers a t o b; hence it is possible t hat two alt ernat ives defeat each ot her at t he same t ime.

16It is wort h not icing t hat it remains valid under assumpt ion (A) only (cf. Claim 1 in Sect ion 4.4) on t he ut ility funct ions for t he governance based on pre-t rade shares.

Consider t he (S ¡ 2){ unit sphere (of dimension S ¡ 2) S

S¡ 2

. For any point Á on t he sphere, denote hÁi t he hyperplane (of dimension S ¡ 2) in h¢

S

i t hat is ort hogonal to t he vect or

^{¡ !}

0Á and divides ¢

S

int o two pieces of equal measure wit h respect t o t he dist ribut ion f ¢

^P _k

µ

_k⁰

. Denot e hÁi

⁺

t he one of t hese two pieces t oward which

^{¡ !}

0Á points. For any k, 1 · k · K ¡ 1 (= S ¡ 2), denot e ¹

⁰_k

(Á) t he (cont inuous) measure of hÁi

⁺

wit h respect t o t he dist ribut ion f ¢µ

_k⁰

. A generalizat ion of t he Borsuk-Ulam t heorem

¹⁷

st at es t hat t here exists a point Á

0

such t hat for all k, 1 · k · K ¡ 1, one has:

¹

⁰_k

(Á

0

) = ¹

⁰_k

(¡ Á

0

) :

T herefore, given t hat hÁ

0

i = h¡ Á

0

i , hÁ

0

i divides ¢

S

int o two pieces of equal measure wit h respect t o t he dist ribut ions f ¢µ

_k⁰

, for all k, 1 · k · K ¡ 1. Since by const ruct ion it also divides ¢

S

int o two pieces of equal measure wit h respect t o f ¢

^P _k

µ

_k⁰

, it does so with respect t o f ¢µ

_K⁰

. Hence t he proof for t he a0-governance.

To prove exist ence of s{ MSPEa1 one has t o choose a hyperplane t hat separat es ¢

S

int o two pieces of equal measure wit h respect t o the dist ribut ions f ¢µ

k

, for all k. The argument is more complicat ed because t he lat t er dist ribut ions are endogeneously de¯ ned.

Nevert heless, t he argument also relies on t he Borsuk-Ulam t heorem applied t o funct ions de¯ ned t hrough anot her principle. T his is post poned t o Appendix B. 2

4.2 Sym m et r ic densit i es

It is possible t o de¯ ne more general assumpt ions under which simple majority st able product ion equilibria exist for all four governances | i.e., Corollary 1 holds t rue. We de¯ ne symmet ric dist ribut ions of types: for all permut at ions ¾of f 1; : : : ; Sg, if ¼

^¾

denot es t he vect or of probabilit ies: (¼

^¾(1)

; : : : ; ¼

^¾(S)

), t hen for all ¼, f (¼

^¾

) = f (¼).

P r op osit ion 2

Assume that

f [

resp.

f ¢µ

⁰_k for all

k

,

f ¢

^P _k

µ

⁰_k

]

is symmetric over

¢

S, then s{ MSPEp0 and p1

[

resp. s{ MSPEa0, s{ MSPEa1

]

exist as soon as

K ¸ S=2

.

Proof

: Thanks t o Lemma 2, t his goes by proving t hat any hyperplane t hrough ¹ Y (as de¯ ned by equat ions (5)) cut s ¢

S

int o two equal part s, in t erms of shareholders (¯ rst

17See T heorem 3.2.7 in Lloyd (1978): Let D be a bounded, open, symmet ric subset of Rⁿ cont aining 0;

let ¹ : @D ¡ ! R^m be cont inuous, and m < n; t hen t here is Á 2 @D such t hat ¹ (Á) = ¹ (¡ Á). Here D is t he unit ball, n = S¡ 1, @D ´ SS¡ 2, and m = S¡ 2: ¹ (Á) = (¹⁰₁(Á); : : : ; ¹⁰_{K ¡ 1}(Á)). An illust rat ion is t hat t here exist two ant ipodal point s on t he eart h wit h same t emperat ure and pressure. See also Guillemin and Pollack (1974), pages 91-93.

assert ion of t he proposit ion) as well as in t erms of shares (second assert ion). Since K ¸ S=2, one has S = K + m wit h m · K . To any ¼, associat e it s symmet ric t hrough h¹ Y i :

¼

^¿

= (¼

²

; ¼

¹

; : : : ; ¼

^2m

; ¼

^{2m ¡ 1}

; ¼

^{2m+ 1}

; : : : ; ¼

^S

) :

Generically, ¼and ¼

^¿

are strict ly on each side of h¹ Y i , and t hen will always count er-balance each ot her in any collect ive decision making under `one person-one vot e' governances.

Under t he assumpt ions of t he proposit ion t hey have t he same amount of shares of each

¯ rm, and will always count er-balance each ot her in any collect ive decision making under

`one share-one vot e' governances. 2

In fact , as easily seen from t he proof, much lighter assumpt ions of symmet ry can ensure t he result . Indeed, t he argument developed here shows some similarity with t he underlying analysis in Grandmont (1978): in t hat paper, exist ence of majority-st able equilibria (in t he case wit hout exchange: K = 1) was shown for cent rally-symmet ric support s of agent s' types. The present argument relies on t he same principle: t he simplex ¢

S

is symmet ric, not wit h respect t o a point , but wit h respect t o K -dimensional subspaces (wit h K ¸ S=2), and t he only needed assumpt ion is t hat t he dist ribut ions of charact erist ics be symmet ric wit h respect t o one of t hese subspaces

¹⁸

.

4.3 º -concave densi t ies

A density funct ion f is º -concave over ¢

S

if for all ¼; ¼

⁰

2 ¢

S

, 8¸ 2 [0; 1], f [(1 ¡ ¸ )¼+ ¸ ¼

⁰

] ¸ [(1 ¡ ¸ )f (¼)

^º

+ ¸ f (¼

⁰

)

^º

]

^1=º

:

T his assumpt ion is regarded as imposing some measure of consensus in t he society. Not ice t hat for º = 1 , one get s t he uniform dist ribut ion of Sect ion 3. De¯ ne

¹⁹

:

½(S; º ) = 1 ¡

Ã

S ¡ 1 + 1=º S + 1=º

! _{S+ 1=º}

:

Consider a PE (Y; 1

K

; µ). As in t he canonical case, ¯ nding a best challenger t o t he equilibrium product ion plan of a ¯ rm reduces t o try and cut t he support ¢

S

by an

18T here is t he implicit feat ure, in Caplin and Nalebu® (1988), t hat t he simplex is, as a support of vot ers' type, t he geomet rical shape t hat allows t he most uneven cut t ing t hrough t he cent er of gravity (see t he principle of symmet rizat ion of Schwart z on which t hey found t his feat ure): if an upper bound works for t he simplex, it works for any ot her convex support . T his feat ure might not be t rue anymore as far as cut t ing t he support t hrough a well-chosen K -dimensional subspace is concerned.

19T he rat io ½(S; º ) is bounded above by 1 ¡ 1=e when º ¸ 0.

hyperplane cont aining hY i in such a way as t o maximize t he di®erence in volume of t he two result ing pieces. When t he dist ribut ion of shareholders' voting weight s is exogeneously

¯ xed (as for t he p0, p1 and a0 governances), given t hat t he support of all considered dist ribut ions is convex, one can direct ly import Caplin and Nalebu® (1991) main result on º -concave dist ribut ion of charact erist ics t o get t he following proposit ion.

P r op osit ion 3

I f

f

is

º

-concave, then for

º ¸ ¡ 1=S

, any PE

(Y; 1

K

; µ)

such that

hY i

contains the mean shareholder' s type²⁰

¼

g of distribution

f

is a

½(S; º )

{ MSPEp0 and p1.

I f

f ¢µ

₀^k is

º

-concave for all

k

, then for

º ¸ ¡ 1=S

, any PE

(Y; 1

K

; µ)

such that

hY i

contains the

K

mean shareholder types

(¼

_g^k

)

^K_{k= 1} of the

K

distributions

(f ¢µ

^k₀

)

^K_{k= 1} is a

½(S; º )

{ MSPEa0.

I n both cases, there exist a continuum of such

½(S; º )

{ MSPE.

It is clear t hat, for t he for t he `one person-one vot e' governances, t he higher K , t he smaller t he rat e of super-majority ½t hat is necessary t o guarantee t he exist ence of ½{

majority stable product ion equilibria. Indeed, on t op of having t o cut ¢

S

t hrough it s cent er of gravity, one can add as many const raint s as t here are ¯ rms, each added const raint lowering t he di®erence in size of t he two pieces result ing from t he cut t ing. We leave for furt her research act ual comput at ions of t he ext ent t o which t he subsequent rat e ½can be improved, i.e., by comput ing t he t rue

²¹

min-max ½(S; K ; º ). For t he a0-governance, one does not have t hese K ¡ 1 added const raint s on t he way t o cut t he simplex. It is easy t o prove in t hat set up t hat t he rat io ½(S; º ) cannot be improved for t he a0-governance.

When t he dist ribut ion of shareholders' vot ing weights is endogeneously det ermined by t he market mechanism from t he announced mult iplan Y , as for t he a1-governance, a result similar t o Proposit ion 3 is more di± cult t o obtain. One has t o prove t he exist ence of a PE (Y; 1

K

; µ) such t hat , for all k,

1. hY i cont ains, for all k, t he cent er of gravity of t he `equilibrium' dist ribut ion f ¢µ

k

; 2. f ¢µ

k

º -concave for some º .

T he following mult ivariat e mean shareholder t heorem can be proposed.

T heor em 3

I f the distribution

f ¢

^P_k

µ

_k⁰ is

º

-concave, then for

¡ 1=S · º · 1

, there exist

½(S; º )

{ MSPEa1.

20T he mean shareholder's type is t he one t hat lies at t he cent er of gravity of t he dist ribut ion; it is de¯ ned as: ¼_g= (¼¹_g; : : : ; ¼^S_g) wit h for all s: ¼_g^s =

Z

¢S

f (¼) ¼^s d¼.

21For º = 1 , ½(S; K ; 1 ) = ½_S;K as de¯ ned by (4).

Proof

: See Appendix B. In fact t he proof shows t hat t here are, generically wit h respect t o f ¢

^P _k

µ

⁰_k

, up t o

0

@

S ¡ 1 K ¡ 1

1

A

di®erent subspaces hY i for which the theorem holds. 2 T his last result sheds some light on t he debat e on which object ive funct ion t he ¯ rm should opt imize in t he cont ext of incomplet e market s. Firm should make choices t hat are support ed by shareholders. In t he present set up, a shareholder is basically charact erized by it s type ¼, which can be ident i¯ ed as his ideal security. For example T heorem 3 shows existence of product ion equilibria which are st able for `accept able' rat es of super majority;

t hey are such t hat , for ¯ rm k, t he shareholder whose type, ¼

g;k

, is at t he cent er of gravity of t he equilibrium dist ribut ion of shares (i.e., t he above-ment ioned mean shareholder) can exact ly span it s type, and generat e it s ideal security: [

^P_`

µ

⁰_`

(¼

g;k

)] ¢¼

g;k

; he could not do bett er if market s were complet e. But t o span his ideal security he needs, in general, t o buy

all

securit ies. (T he same line of reasoning holds for t he a0-governance t hrough Proposit ion 3.)

T his result has no direct link wit h t he Drµ eze crit erion. Indeed, t he announced se- curity/ production plan, y

k

, of ¯ rm k does not have to be t he ideal security/ product ion plan, ¼

g;k

, of this mean shareholder (in general, t he mult iplan Y , wit h y

k

= ¼

⁰_g;k

cannot be support ed as a Paret o PE). But t he multiplan Y should be such t hat it cont ains t he ideal security of

all

mean shareholders; in some way t he mult iplan Y is opt imal for t he K mean shareholders. Then t he product ion equilibria are st able for t he lowest possible rat es of super majority. Finally, it is also t he case here t hat t he assumpt ions securing t he result are weaker for t he governances µ a la Drµ eze, i.e., based on post -t rade shares.

4.4 Ext ensi ons

I m m ediat e ext ension t o a br oader class of ut ilit y funct ions

As easily seen from the proof, t he exist ence of s-MSPE (for governances based on pre-t rade shares) in the case K = S ¡ 1 (or, t rivially, K = S) are st ill valid under assumpt ion (A) and t he assumpt ion t hat ¼= Argmax f U

¼

(x) j x 2 ¢

S

g. Indeed, given a PE (Y; 1

K

; µ), t he ¯ rst order conditions of agent s' maximizat ion programs give, for all k ¸ 2, all ¼:

D U

¼

[x(¼)] ¢(y

k

¡ y

1

) = 0 : (8)

Suppose K = S ¡ 1, when asked whet her t hey agree wit h an in¯ nit esimal change u 2 R

^S

(which can be taken ort hogonal t o h¹ Y i , given equat ions (8)) in t he product ion plan of