Voting in Assemblies of shareholders and Incomplete Markets

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Voting in Assemblies of shareholders and Incomplete Markets

Mich Tvede, Hervé Crès

To cite this version:

Mich Tvede, Hervé Crès. Voting in Assemblies of shareholders and Incomplete Markets. 2001. �hal-

01064884�

Vot ing in A ssemblies of Shar eholder s and I ncom plet e M ar ket s ^¤

M ich T vede ^y University of Copenhagen

H er v¶ e Cr µ es

HEC School of Management

A bst r act

An economy with two dates is considered, one state at the ¯ rst date and a

¯ nite number of states at the last date. Shareholders determine production plans by voting { one share, one vote { and at ½ -major ity stable equilibria, alternative production plans are supported by at most ½£ 100 percent of the shareholders. I t is shown that a ½ -majority stable equilibrium exists provided that

½ ¸ min

½ S ¡ J

S ¡ J + 1 ; B B + 1

¾

where S is the number of states at the last date, J is the number of ¯ rms and B is the dimensions of the sets of e± cient production plans for ¯ rms.

Moreover, an example shows that ½ -majority stable equilibria need not exist for smaller ½ ' s.

K eywor ds: General Equilibrium, Incomplet e Markets, Firms, Vot ing.

JEL -classi¯ cat ion: D21, D52, D71, G39.

¤

Financial support from t he Danish Research Councils and hospit ality of HEC are grat efully acknowledged by Mich T vede.

y

Corresponding author: Inst it ut e of Economics, University of Copenhagen, St udies-

t raede 6, DK -1455 Copenhagen K , Denmark.

1 I nt r oduct ion

If market s are complet e t hen consumers have common shadow prices { name- ly t he vect or of market prices. So shareholders agree t hat ¯ rms should max- imize pro¯ t s wit h respect t o t hese common prices. However, if market s are incomplet e, shadow prices need not be common. T hus, typically shareholders disagree on t he product ion plans t o be chosen. Therefore several suggest ions have been put forward as reasonable object ives for ¯ rms.

It seems nat ural t hat product ion plans should sat isfy t he Paret o crit erion:

t here are no alt ernat ive product ion plans t hat make some shareholders bet t er o® and none worse o®. Unfort unat ely, t he Paret o crit erion is weak: produc- t ion plans sat isfy t he Paret o crit erion if and only if t hey maximize pro¯ t s wit h respect t o some price vect or in t he convex hull of t he shareholders' shadow prices.

Drµ eze (1974) and Grossman & Hart (1979) agree t hat product ion plans should sat isfy t he Paret o crit erion and propose t hat sidepayments between shareholders be allowed. Drµ eze (1974) (resp. Grossman & Hart (1979)) sug- gests t hat product ion plans should re° ect preferences of ¯ nal (resp. init ial) shareholders: t his may be int erpret ed as product ion plans are det ermined after market s close (resp. before markets open). However, sidepayment s depend on informat ion t hat shareholders have incent ives t o manipulat e, a weakness t hat vot ing rules overcome.

Drµ eze (1985) suggest s t hat product ion plans should be st able for simple majority vot ing between shareholders and unanimity between board members (wit hout sidepayments): t here is no alt ernat ive product ion plan t hat makes all board members as well as a majority of shareholders bet t er o®. As in Drµ eze (1974) product ion plans re° ect preferences of ¯ nal shareholders. It appears t o be a drawback t hat unanimity between board members is essent ial for exist ence of equilibria.

DeMarzo (1993) invest igat es some propert ies of equilibria where produc-

t ion plans are st able for simple majority vot ing between shareholders. Typi-

cally t he largest shareholder det ermines t he product ion plan at t hese equilib- ria. Also, DeMarzo shows t hat st ability for simple majority vot ing between shareholders and unanimity between board members imply that board mem- bers determine t he product ion plan. However, as he argues, such equilibria need not exist unless eit her t he degree of market incomplet eness or t he di- mension of t he set of e± cient product ion plans is 1.

In t he present paper, st ability wit h respect t o ½-majority vot ing between shareholders is st udied: t here is no alt ernat ive product ion plan t hat makes more t han ½£ 100 percent of t he shareholders bet t er o®. Indeed, at a ½- majority st able equilibrium (or ½-MSE), consumers do not want t o change t heir port folios, ¯ rms are not able t o make more than ½£ 100 percent of t heir shareholders bet t er o® by changing product ion plans and ¯ nally, market s clear. It is shown t hat if port folios are unbounded t hen a ½-MSE exist s provided t hat

½ ¸ min

½ S ¡ J

S ¡ J + 1 ; B B + 1

¾

where S is t he number of st ates at t he last dat e, J is t he number of ¯ rms and B is t he dimension of t he set of e± cient product ion plans for ¯ rms. If portfolios are bounded t o be non-negat ive t hen a ½-MSE exist s provided t hat

½¸ B =(B + 1).

Di®erent t imings of trade and vot e are considered. Vot ing may t ake place

while market s are open or aft er market s close, in which case ¯ nal sharehold-

ers vot e (as in Drµ eze (1985) and DeMarzo (1993)). And it may t ake place

before market s open, in which case init ial shareholders vot e. In case of vot -

ing before market s open or while t hey are open, shareholders need t o form

expect at ions about price variat ions. Two types of price percept ions are con-

sidered: competitive price percept ions (as int roduced by Grossman & Hart

(1979)) and ¯ xed price percept ions. According t o compet it ive price percep-

t ions consumers perceive t hat income vect ors are valued by t heir shadow

prices; whereas according t o ¯ xed price percept ions t hey perceive t hat prices

are not in° uenced by changes in product ion plans.

In general, changes of product ion plans in° uence t rading opportunit ies t hrough two channels: t hey change t he value of port folios as well as the span of asset s. From t his perspect ive, compet it ive price percept ions and ¯ xed price percept ions represent two ext remes: consumers concent rat e on how changes of product ion plans change t he value of t heir port folios wit h t he former and t he span of asset s wit h t he lat t er.

In case market s are complete, a ½-MSE exist s even for unanimity, i.e.

wit h ½= 0. Ekern & Wilson (1974) have shown t hat t his result ext ends t o t he case of partial spanning , i.e. t he set s of e± cient product ion plans are subset s of t he span of asset s ¹ . In case market s are incomplet e such t hat eit her t he degree of incomplet eness is 1 or the set s of e± cient product ion plans are 1-dimensional, a ½-MSE exist s for simple majority voting, i.e. wit h

½ = 1=2, as argued by DeMarzo (1993). It is shown here t hat in case of a more severe degree of incomplet eness and higher dimensions of t he set s of e± cient product ion plans, super majority rules (½> 1=2) are needed t o ensure exist ence of ½-MSE.

T he social choice lit erat ure o®ers some general result s on exist ence of st able equilibria under super majority vot ing { see, e.g., Ferejohn & Gret her (1974), Greenberg (1979), Caplin & Nalebu® (1988, 1991) and Balasko &

Crµ es (1997). Crµ es (2000) exploit s t he result s of Caplin & Nalebu® (1988, 1991) t o obt ain some condit ions on t he dist ribut ion of consumers' charac- t erist ics under which a ½-MSE exist s for ½between 0.5 and 0.64 in a model wit h a cont inuum of consumers, rest rict ive assumpt ions on production set s and preferences of consumers. Here t he result of Greenberg (1979) is ex- ploit ed t o obt ain a lower bound on t he rat e ½for which a ½-MSE exist s, in a model wit h a ¯ nit e number of consumers, weak assumpt ions on product ions set s and preferences, and no assumpt ions on t he dist ribut ion of consumers' charact erist ics. A di± culty in applying t he result s from t he social choice

1

See Magill & Quinzii (1996), chapt er 6. Act ually, exist ence of ½-MSE for ½= 0 holds

in any model wit h incomplet e market s where equilibrium allocat ions are Paret o opt imal,

e.g., under st rong condit ions for t he CAPM (Borch (1968) and Wilson (1968)).

lit erat ure is t hat preferences of shareholders, as well as shares (i.e. vot ing weight s), are endogeneously det ermined t hrough general equilibrium e®ect s.

Even t hough t he proposed bounds on ½ are quit e high and cannot be improved, as shown by an example, t he results of t he present paper show t hat : (1) t he degree of market incompleteness plays a fundament al role in re- st rict ing t he dimension of t he set of alt ernat ives and thereby in aggregat ing preferences of shareholders and (2) t he lower t he degree of market incom- plet eness t he lower super majority rat e is necessary t o ensure exist ence of a

½-MSE.

T he paper is organized as follows: in Sect ion 2 t he model is int roduced;

in Sect ion 3 assumpt ions are st at ed, exist ence of a ½-MSE, for ½¸ minf (S ¡ J )=(S ¡ J + 1); B =(B + 1)g, is est ablished in case vot ing t akes place aft er market s close, and an example is given showing t hat t he lat t er bound cannot be improved; in Sect ion 4 price percept ions are int roduced and exist ence of a

½-MSE is est ablished in case vot ing t akes place eit her before market s open or while t hey are open, and; ¯ nally Sect ion 5 cont ains some concluding remarks.

All proofs are gat hered in an appendix.

2 T he m odel

Consider an economy wit h 2 dates, t 2 f 0; 1g, 1 stat e at t he ¯ rst dat e, s = 0, and S st ates at t he second dat e, s 2 f 1; : : : ; Sg. There are: 1 commodity at every st at e, I consumers, i 2 f 1; : : : ; I g, and J ¯ rms, j 2 f 1; : : : ; J g. Consumers are charact erized by t heir consumpt ion set s, X i ½ R ^{S+ 1} , endowment s, ! i 2 R ^{S+ 1} , preferences described by correspondences, P i : X i ! X i , and init ial portfolio of shares in ¯ rms, ± i 2 R ^J where ^{P I} _{i = 1} ± i j = 1 for all j . Firms are charact erized by t heir product ion set s, Y j ½ R ^{S+ 1} .

Consumers choose consumpt ion plans, x i 2 X i , and port folios, µ i 2 R ^J .

Firms choose product ion plans, y j 2 Y j . For convenience, let x = (x i ) ^I _{i = 1} ,

µ = (µ i ) ^I _{i = 1} , y = (y j ) ^J _{j = 1} , q = (q j ) ^J _{j = 1} 2 R ^J where q j is the price of shares in

¯ rm j , Y = (y 1 ¢¢¢ y J ) and

Q =

0 B B B B B B

@

q 1 ¢¢¢ q J

0 ¢¢¢ 0 .. . .. . 0 ¢¢¢ 0

1 C C C C C C A

:

Wit h some abuse of not at ion, q j denot es t he price of shares in ¯ rm j as well as t he j 't h column of Q. The budget set of consumer i is

B i (Q; Y ) = f x i 2 X i jx i · ! i + Q± i + (Y ¡ Q)µ i for some µ i 2 R ^J g and x i is a solut ion t o t he problem of consumer i provided that x i 2 B i (Q; Y ) and P i (x i ) \ B i (Q; Y ) = ; . Hence, t here are no st rat egic considerat ions involved in t he choices of port folios. Let U i j (x i ; µ i j ; y j ) denot e the set of product ion plans for ¯ rm j t hat make, at t he considered allocat ion (x; µ; y), consumer i bet t er o®, i.e.

U i j (x i ; µ i j ; y j ) = f y _j ⁰ 2 Y j jx i + (y ⁰ _j ¡ y j )µ i j 2 P i (x i )g:

Next let u j (x; µ j ; y j ; y _j ⁰ ) denot e t he set of consumers who are, at t he considered allocation (x; µ; y), bet t er o® wit h product ion plan y ⁰ _j for ¯ rm j rat her t han y j , i.e.

u j (x; µ j ; y j ; y _j ⁰ ) = f i 2 f 1; : : : ; I gjy _j ⁰ 2 U i j (x i ; µ i j ; y j )g:

Then preferences of ¯ rms are described, for a ¯ xed rat e ½of t he super ma- jority rule, by correspondences, P _j ^½ : ^Q _i X i £ R ^I £ Y j ! Y j , de¯ ned by

P _j ^½ (x; µ j ; y j ) =

8

> >

> >

> >

<

> >

> >

> >

:

; for ^X

i

µ _{i j} ⁺ = 0

f y ⁰ _j 2 Y j j

P

i 2 u

(x ;µ

;y

;y

) µ _{i j} ⁺

P

i µ ⁺ _{i j} > ½g for ^X

i

µ _{i j} ⁺ > 0:

And y j is a solut ion to t he problem of ¯ rm j provided t hat P _j ^½ (x; µ j ; y j ) \ Y j =

; . Thus, if the product ion plan is changed from y j t o y ⁰ _j t hen t his change is

dist ribut ed t o shareholders propot ionally t o t heir shares.

D e¯ nit ion 1 (q ^¤ ; x ^¤ ; µ ^¤ ; y ^¤ ) is a ½-m aj or it y st able equili br ium provided that

² x ^¤ _i · ! i + Q ^¤ ± i + (Y ^¤ ¡ Q ^¤ )µ _i ^¤ and x ^¤ _i is a solution to the problem of consumer i for (q ^¤ ; y ^¤ ) , i.e.

x ^¤ _i 2 B i (Q ^¤ ; Y ^¤ ) and P i (x ^¤ _i ) \ B i (Q ^¤ ; Y ^¤ ) = ; for all i 2 I ,

² y _j ^¤ is a solution to the problem of ¯ rm j for (x ^¤ ; µ ^¤ _j ) , i.e.

y _j ^¤ 2 Y _j ^¤ and P _j ^½ (x ^¤ ; µ ^¤ _j ; y ^¤ _j ) \ Y _j ^¤ = ; for all j 2 J , and,

² markets clear, i.e.

X

i

x ^¤ _i = ^X

i

! _i ^¤ + ^X

j

y _j ^¤ and ^X

i

µ _{i j} ^¤ = 1 for all j 2 J .

3 A ssumpt ions and exist ence of equilibr ium

Assumpt ions on consumers, ¯ rms and t he product ion sect or are imposed in order t o ensure t he exist ence of a ½-majority st able equilibrium.

Consumers are supposed t o sat isfy t he following assumpt ions (a.1) X i = R ^{S+ 1} ,

(a.2) ! i 2 R ^{S+ 1} , (a.3) gr P i is open,

(a.4) f x i g + R ^{S+ 1} ₊ n f 0g 2 P i (x i ),

(a.5) for all x i , t here exist s a unique ¹ i 2 ¢ + such t hat ¹ i ¢(x ⁰ _i ¡ x i ) > 0 for all x ⁰ _i 2 P i (x i ) where ¢ ^S ₊ = f ¸ 2 R ^{S+ 1} ₊ j ^P _s ¸ ^s = 1g, and,

(a.6) if A ½ R ^{S+ 1} is compact t hen t here exist s x i (A) 2 R ^{S+ 1} such t hat if x i 2 R ^{S+ 1} n (f x i (A)g + R ^{S+ 1} ₊ ) t hen A ½ P i (x i ).

Assumpt ions (a.1) and (a.2) imply t hat consumpt ion set s are unbounded as considered by Balasko (1988) while assumpt ions (a.3), (a.4), (a.5) and (a.6) are generalizat ions of equivalent assumpt ions considered by Balasko (1988) t o non-t ransit ive, non-complet e and non-di®erent iable preferences. Assump- t ions (a.3) and (a.4) are st andard cont inuity and monot onicity assumpt ions;

assumpt ion (a.5) st at es exist ence of a unique shadow price vect or ¹ i (x i ) at each consumpt ion bundle x i , and; assumpt ion (a.6) generalizes t he st andard

\ boundedness from below" property of indi®erence sets t o t he present frame- work where preferences are not necessarily t ransit ive nor complet e.

Let Z j ½ R ^{S+ 1} be t he set of e± cient product ion plans, i.e.

Z j = f y j 2 R ^{S+ 1} j(f y j g + R ^{S+ 1} ₊ ) \ Y j = f y j gg t hen ¯ rms are supposed t o sat isfy t he following assumpt ions (a.7) t he product ion set, Y j , is convex and closed, and,

(a.8) t here exist s a compact and B -dimensional a± ne set , B j ½ R ^{S+ 1} , such t hat Z j ½ B j .

Assumpt ion (a.7) is st andard while assumpt ion (a.8) includes \ t runcated"

product ion set s such as

f y 2 R ^{S+ 1} jy ⁰ 2 [y; 0] and y ^s · (y ⁰ ) ^b for all s 2 f 1; : : : ; Sgg where y · 0 and b 2 ]0; 1].

Moreover, t he product ion sect or of t he economy is supposed t o sat isfy t he

following assumpt ion

(a.9) product ion plans for dat e 1, ((y _j ^s ) ^S _{s= 1} ) ^J _{j = 1} , are linearly independent for all product ion plans in t he convex hull of t he closure of t he set of e± cient production plans, y j 2 co cl Z j for all j .

Assumpt ion (a.9) excludes t hat ¯ rms are able t o replicat e product ion plans of each ot her.

T heor em 1 T here exists a ½ -majority stable equilibrium for all economies which satisfy assumptions (a.1) to (a.9) if and and only if

½ ¸ min

½ S ¡ J

S ¡ J + 1 ; B B + 1

¾

:

Remark: The argument t o est ablish t he \ if" of t he assert ion is based on t he proofs of Theorem 2 in Greenberg (1979) and t he t heorem in Shafer &

Sonnenschein (1975). A generalized game is const ruct ed where, among ot her const ruct ions, ¯ rms det ermine product ion plans t hat maximize pro¯ t s wit h respect t o prices which re° ect int erest s of t heir shareholders and groups of shareholders (one per ¯ rm) det ermine prices for which ¯ rms maximize pro¯ t s.

Hence, t he original problem of t he ¯ rm { which is t o ¯ nd a product ion plan for which no alt ernat ive product ion plan can be support ed by a ½-majority of it s shareholders { is decomposed int o pro¯ t maximizat ion wit h respect t o

¯ rm speci¯ c prices and det erminat ion of ¯ rm speci¯ c prices wit h respect t o some art i¯ cial preferences for it s shareholders.

T he argument t o est ablish t he \ only if" of t he assert ion is based on t he const ruct ion of an economy for which no ½-majority st able equilibrium wit h

½< minf (S ¡ J )=(S ¡ J + 1); B =(B + 1)g exist s.

End of remark

In case S¡ J = 1, T heorem 1 ensures exist ence of a simple majority st able

equilibrium. It is easily seen t hat t he prices for which ¯ rms maximize pro¯ t s

are, in t his case, typically not the ones Drµ eze (1974) suggest s. Indeed, in

t heorem 1 the shadow price vect or of t he median shareholder is used whereas

Drµ eze (1974) suggests t hat t he average shadow price vect or should be used.

Trading on t he ¯ nancial market s, when consumers are not const rained in t heir portfolio choices, leads t o suit able normalized shadow prices being contained in some (S¡ J )-dimensional a± ne set (hY ^¤ ¡ Q ^¤ i ^? \ ¢ ^S ₊ ). However, if t here are rest rict ions on port folios, like short sales const raint s, t hen t he degree of market incomplet eness need not rest rict shadow prices.

Cor ollar y 1 Suppose that portfolios are bounded such that µ i 2 [0; 1] ^J for all i and that co cl Z j ½ R ^{S+ 1} ₊ for all j . T hen there exists a ½ -majority stable equilibr ium provided that

½ ¸ B

B + 1 :

It is hard t o love t he assumpt ion t hat co cl Z j ½ R ^{S+ 1} ₊ for all j . However, t he \ Cass-t rick" { one consumer t rades on complet e market s { cannot be applied in corollary 1 because port folios are bounded t o be between 0 and 1. Therefore exist ence of equilibrium is only ensured provided t hat prices of shares are posit ive as explained by Radner (1972) and the assumpt ion ensures t his.

4 Pr ice per cept ions

In t he present sect ion di®erent t imings between t rade and vot e are considered.

At a ½-majority st able equilibrium, (q ^¤ ; x ^¤ ; µ ^¤ ; y ^¤ ), if consumer i considers how t o vot e wit h regard t o a change from (q _j ^¤ ; y _j ^¤ ) t o (q j ; y j ) of price and product ion plan for ¯ rm j (where ± i j > 0 or µ ^¤ _{i j} > 0 because ot herwise consumer i have no vot ing weight ) t hen

² in case vot ing t akes place aft er market s close, she vot es for t he change if and only if

x ^¤ _i + (Y ^¤ jy j ¡ Y ^¤ )µ ^¤ _i 2 P i (x ^¤ _i )

where Y ^¤ jy j is Y ^¤ wit h y j replacing y _j ^¤ ,

² in case vot ing t akes place while market s are open, she vot es for t he change if and only if

x ^¤ _i ¡ (Y ^¤ ¡ Q ^¤ jq j )µ _i ^¤ + (Y ^¤ jy j ¡ Q ^¤ jq j )µ i 2 P i (x ^¤ _i ) for some µ i , and,

² in case vot ing takes place before market s open, she vot es for t he change if and only if

x ^¤ _i ¡ (Q ^¤ ¡ Q ^¤ jq j )± i ¡ (Y ^¤ ¡ Q ^¤ )µ ^¤ _i + (Y ^¤ jy j ¡ Q ^¤ jq j )µ i 2 P i (x ^¤ _i ) for some µ i (here t he vot ing weight s are ± _j ⁺ ).

If port folios are unbounded, i.e. µ i 2 R ^J , t hen ¹ i (x ^¤ _i ) 2 hY ^¤ ¡ Q ^¤ i ^? at a

½-majority st able equilibrium, (q ^¤ ; x ^¤ ; µ ^¤ ; y ^¤ ). T herefore in case vot ing t akes place after market s close (resp. while market s are open or before t hey open), if consumer i vot es for t he change t hen ¹ i (x ^¤ _i ) ¢(y j ¡ y _j ^¤ ) > 0 (resp. ¹ i (x ^¤ _i ) ¢ (q j ¡ q ^¤ _j ) > 0 or ¹ i (x ^¤ _i ) ¢(y j ¡ q j ) 6 = 0). Thus, equivalent ly, in case vot ing t akes place after market s close (resp. while market s are open or before t hey open), if ¹ i (x ^¤ _i ) ¢(y j ¡ y _j ^¤ ) · 0 (resp. ¹ i (x ^¤ _i ) ¢(Q ¡ Q ^¤ ) · 0 and ¹ i (x ^¤ _i ) ¢(y j ¡ q j ) = 0) t hen consumer i vot es against t he change. However, consumers do not know how prices depend on production plans so if vot ing t akes place before or while market s are open t hen t hey need t o form percept ions about t his.

4.1 Com p et it ive pr ice p er cept i ons

Grossman & Hart (1979) int roduced t he not ion of compet it ive price percep- t ions in a model where product ion plans are det ermined by shareholders be- fore market s open. Consider a ½-majority st able equilibrium, (q ^¤ ; x ^¤ ; µ ^¤ ; y ^¤ ), t hen a change of product ion plan from y _j ^¤ t o y j for ¯ rm j is perceived by consumer i t o change the price from q _j ^¤ t o

q i j (x ^¤ _i ; y j ) = 1

¹ ⁰ _i (x ^¤ _i ) ¹ i (x ^¤ _i ) ¢y j :

Consequent ly, if consumer i vot es for t he change and has compet it ive price percept ion t hen ¹ i (x ^¤ _i ) ¢(y j ¡ y _j ^¤ ) > 0 and, equivalent ly, if ¹ i (x ^¤ _i ) ¢(y j ¡ y ^¤ _j ) · 0 t hen consumer i vot es against t he change. This does not depend on whet her vot ing t akes place before market s open, while t hey are open or aft er t hey close. Informally, if consumers have compet it ive price percept ions t hen t hey concent rat e on how changes of product ion plans change values of t heir portfolios rat her t han t he span of asset s.

T hus, t here is a di®erent int erpretat ion of t he model of Sect ion 2 as well as t he result s of Sect ion 3: vot ing t akes place while market s are open and consumers have compet it ive price percept ions. Moreover, Theorem 1 and Corollary 1 ext end t o t he model with vot ing before market s open provided t hat consumers have compet it ive price percept ions. However in t his lat t er case t he set of equilibria is typically not ident ical t o the set of equilibria of t he former case because vot ing weight s typically are not ident ical, i.e. µ _i ^¤ 6 = ± i . Hence, t he next corollary follows from t he proof of Theorem 1 wit h only minor modi¯ cat ions.

Cor ollar y 2 Suppose that voting takes place before markets open, while they are open or after they close and that consumers have competitive price per- ceptions. T hen there exists a ½ -majority stable equilibrium provided that

½ ¸ min

½ S ¡ J

S ¡ J + 1 ; B B + 1

¾

:

4.2 F ixed pr ice p er cept ions

Consider a ½-majority st able equilibrium, (q ^¤ ; x ^¤ ; µ ^¤ ; y ^¤ ), t hen a change of product ion plan from y _j ^¤ t o y j for ¯ rm j is perceived by consumer i not t o change t he price, q _j ^¤ . Informally, if consumers have ¯ xed price percept ions t hen t hey concent rat e on how changes of product ion plans change t he span of asset s rat her t han values of portfolios.

Port folios are bounded between 0 and 1, i.e. µ i 2 [0; 1] ^J , so t he modi¯ ed

budget set of consumer i is

B ~ i (Q; Y ) = f x i 2 X i jx i · ! i + Q± i + (Y ¡ Q)µ i for some µ i 2 [0; 1] ^J g:

Let U ~ i j (q; x i ; y) = f y ⁰ _j 2 Y j jx i + (Y jy _j ⁰ ¡ Q)µ i 2 P i (x i )

for some µ i 2 [0; 1] ^J g

~

u j (q; x; y; y _j ⁰ ) = f i 2 f 1; : : : ; I gjy _j ⁰ 2 ~ U i j (q; x i ; y j )g

t hen preferences of ¯ rms are described by correspondences, ~ P _j ^½ : R ^J £ ^Q _i X i £ [0; 1] ^I £ ^Q _j Y j ! Y j , de¯ ned by

P ~ _j ^½ (q; x; µ j ; y) =

8

> >

> >

> >

<

> >

> >

> >

:

; for ^X

i

µ _{i j} ⁺ = 0

f y _j ⁰ 2 Y j j

P

i 2 ~ u

(x;µ

;y;y

) µ _{i j} ⁺

P

i µ _{i j} ⁺ > ½g for ^X

i

µ _{i j} ⁺ > 0 in case vot ing t akes place while market s are open and ± j replaces µ j in case vot ing t akes place before market s open.

Cor ollar y 3 Suppose that portfolios are bounded such that µ i 2 [0; 1] ^J for all i , that co cl Z j ½ R ^{S+ 1} ₊ for all j and that consumers have ¯ xed price perceptions. Then a ½ -majority stable equilibrium exists provided that

½ ¸ B

B + 1 :

5 Final r emar k s

In t he present paper, bounds on ½are provided such t hat ½-majority st able

equilibria exist . To complement t hese result s on exist ence of equilibrium it

would be nice st udy: (1) t he e± ciency propert ies of equilibrium allocat ions,

and; (2) t he \ size" of t he set of equilibria.

On t he one hand, in many count ries, simple majority vot ing is used in assemblies of shareholders. On t he ot her hand, t he provided bounds on ½ implies t hat simple majority stable equilibria need not exist unless eit her t he degree of incomplet eness is 1 or the set s of e± cient product ion plans are 1-dimensional. T herefore it would be int erest ing t o ¯ nd \ reasonable"

assumpt ions on product ion set s and preferences of consumers t hat ensure exist ence of ½-majority st able equilibria for lower values of ½.

R efer ences

Balasko, Y., Foundations of T he T heory of General Equilibrium , Academic Press (1988).

Balasko, Y. & H. Crµ es, T he Probability of Condorcet Cycles and Super Majority Rules, Journal of Economic T heory 75 (1997), 237-70.

Borch, K ., General Equilibrium in t he Economics of Uncertainty, in K . Borch & J. Mossin eds., Risk and Uncertainty , Macmillan (1968).

Caplin, A. & B. Nalebu®, On 64%-Majority Rule, Economet rica 56 (1988), 787-814.

Caplin, A. & B. Nalebu®, Aggregat ion and Social Choice: A Mean Vot er Theorem, Economet rica 59 (1991), 1-23.

Crµ es, H., Majority St able Product ion Equilibria: A Mult ivariat e Mean Shareholder T heorem, Cahier de Recherche du Groupe HEC (2000).

DeMarzo, P., Majority Vot ing and Corporat e Cont rol: T he Rule of t he Dominant Shareholder, Review of Economic St udies 60 (1993), 713- 734.

Drµ eze, J., Invest ment under Privat e Ownership: Opt imality, Equilibrium

and St ability, in J. Drµ eze, ed., Allocation under Uncertainty: Equilib-

rium and Optimality , Macmillan (1974), 129-166.

Drµ eze, J., (Uncert ainty and) The Firm in General Equilibrium T heory, Eco- nomic Journal 95 (1985), 1-20.

Ekern, S. & R. Wilson, On the Theory of t he Firm in an Economy wit h Incomplet e Market s, Bell Journal of Economics 5 (1974), 171-80.

Ferejohn, J. & D. Gret her, On a Class of Rat ional Social Decision Proce- dures, Journal of Economic Theory 8 (1974), 471-482.

Greenberg, J., Consist ent Majority Rules over Compact Set s of Alt ernat ives, Economet rica 47 (1979), 627-636.

Grossman, S. & O. Hart , A Theory of Compet it ive Equilibrium in St ock Market Economies, Economet rica 47 (1979), 293-330.

Magill, M. & M. Quinzii, T he Theory of I ncomplete Markets , The MIT Press (1996).

Radner, R., Exist ence of Equilibrium of Plans, Prices and Price Expect a- t ions in a Sequence of Market s, Economet rica 40 (1972), 289-303.

Shafer, W. & H. Sonnenschein, Equilibrium in Abst ract Economies wit hout Ordered Preferences, Journal of Mat hemat ical Economics 2 (1975), 345-348.

Wilson, R., The T heory of Syndicat es, Economet rica 36 (1968), 119-132.

A pp endix

P r oof of T heor em 1

In part 1, resp. part 2, it is shown t hat if ½¸ (S ¡ J )=(S ¡ J + 1), resp.

½¸ B =(B + 1), t hen a ½-majority st able equilibrium exist s. In part 3, an

example is provided of an economy for which no ½-majority st able equilibrium

wit h ½< minf (S ¡ J )=(S ¡ J + 1); B =(B + 1)g exist s.

Par t 1: ½¸ (S ¡ J )=(S ¡ J + 1)

The variables t o be det ermined are st at e prices, ¸ 2 ¢ ^S ₊ , consumpt ion bun- dles for consumers, x = (x i ) ^I _{i = 1} ½ ^{Q I} _{i = 1} X i , product ion plans for ¯ rms, y = (y j ) ^J _{j = 1} ½ ^{Q J} _{j = 1} Y j , and prices wit h respect t o which ¯ rms maximize pro¯ t s, º = (º j ) ^J _{j = 1} 2 ^{Q J} _{j = 1} ¢ ^S ₊ .

T he auct ioneer (agent 0) det ermines st at e prices in order t o maximize t he value of excess demand. Consumers (agent k 2 f 1; : : : ; I g) det ermine maximal consumpt ion bundles for their preferences. Firms (agent k 2 f I + 1; : : : ; I + J g) det ermine product ion plans t hat maximize pro¯ t s wit h respect t o prices which re° ect int erest s of t heir shareholders. Groups of shareholders (agent k 2 f I + J + 1; : : : ; I + 2J g, one group per ¯ rm) det ermine prices for which ¯ rms maximize pro¯ t s. Hence, t he original problem of t he ¯ rm { which is t o ¯ nd a product ion plan for which no alt ernat ive product ion can be support ed by a ½-majority of it s shareholders { is decomposed int o pro¯ t maximizat ion wit h respect t o ¯ rm speci¯ c prices and det erminat ion of ¯ rm speci¯ c prices wit h respect t o some art i¯ cial preferences for it s shareholders.

In a ¯ rst st ep, t hese four cat egories of agent s (auct ioneer, consumers,

¯ rms and groups of shareholders) are described. In a second st ep, a suit able correspondence is const ruct ed and K akutani's ¯ xed point t heorem is applied.

Finally, in a t hird st ep, t he ¯ xed point is shown to be a ½-majority st able equilibrium.

St ep 1: descr iption of agents

\ Auctioneer" For agent k = 0, t he st rat egy set , V k ½ R ^{S+ 1} , is de¯ ned by V k = ¢ ^S ₊ \ [ 1

n ; 1] ^{S+ 1}

where n 2 N , t he const raint correspondence, C k : V ! V k (where V is t he product of t he agent s' st rat egy set s, t o be de¯ ned in t he sequel), is de¯ ned by

C k (¸ ; x; y; º ) = V k

and t he preference correspondence, Q k : V ! V k , is de¯ ned by Q k (¸ ; x; y; º ) = f ¸ ⁰ 2 V k j(¸ ⁰ ¡ ¸ ) ¢( ^X

i

x i ¡ ^X

i

! i ¡ ^X

j

y j ) > 0g:

Clearly, V k is compact and convex, C k is cont inuous and gr Q k is open wit h

¸ = 2 co Q k (¸ ; x; y; º ).

\ Consumers" For agent k 2 f 1; : : : ; I g, the st rat egy set , V k ½ X i where i = k, is de¯ ned by

V k = X i \ (f ! i g + ^X

j

± i j co cl Z j + [¡ n; n] ^{S+ 1} )

where n 2 N , t he const raint correspondence, C k : V ! V k , is de¯ ned by C k (¸ ; x; y; º ) = f x ⁰ _i 2 V k j¸ ¢(x ⁰ _i ¡ ! i ¡ Q± i ) · 0g

for k = 1 where q j = (1=¸ ⁰ )¸ ¢y j for all j and

C k (¸ ; x; y; º ) = B i (Y; Q) \ V k

for k 2 f 2; : : : ; I g and t he preference correspondence, Q k : V ! V k , is de¯ ned by

Q k (¸ ; x; y; º ) = co P i (x i ) \ V k :

Clearly, V k is compact and convex, C k is cont inuous and gr Q k is open wit h x i 2 co Q = k (¸ ; x; y; º ) for i = k.

\ Firms" For agent k 2 f I + 1; : : : ; I + J g, t he st rat egy set , V k ½ Y j , is de¯ ned by

V k = co cl Z j

where j = k ¡ I , t he const raint correspondence, C k : V ! V k , de¯ ned by C k (¸ ; x; y; º ) = V k

and t he preference correspondence, Q k : V ! V k , de¯ ned by

Q k (¸ ; x; y; º ) = f y ⁰ _j 2 Z j jº j ¢(y _j ⁰ ¡ y j ) > 0g \ V k :

Clearly, V k is compact and convex, C k is cont inuous and gr Q k is open wit h y j 2 co Q = k (¸ ; x; y; º ) for j = k ¡ I .

\ Shareholders" For agent k 2 f I + J + 1; : : : ; I + 2J g, t he st rat egy set , V k ½ R ^{S+ 1} , is de¯ ned by

V k = ¢ ^S ₊ ;

t he const raint correspondence, C k : V ! V k , is de¯ ned by C k (¸ ; x; y; º ) = f º _j ⁰ 2 V k jº _j ⁰ 2 hY ¡ Qi ^? g

where j = k ¡ I ¡ J . Let t he correspondence F i : V i £ V k ! V k where i 2 f 1; : : : ; I g be de¯ ned by

F i (x i ; º j ) = f º _j ⁰ 2 V k j kº _j ⁰ ¡ ¹ i (x i )k < kº j ¡ ¹ i (x i )kg

where j = k ¡ I ¡ J and let t he correspondence G j : ^{Q I} _{i = 1} V i £ V k £ V k ! f 1; : : : ; I g be de¯ ned by

G j (x; º j ; º _j ⁰ ) = f i 2 f 1; : : : ; I gjº _j ⁰ 2 F i (x i ; º j )g:

Then t he preference correspondence, Q k : V ! V k , is de¯ ned by

Q k (¸ ; x; y; º ) =

8 >

> >

> >

> >

> >

> >

> >

<

> >

> >

> >

> >

> >

> >

> :

; for ^X

i

´ i j (¸ ; x; y) ⁺ = 0

f º _k ⁰ 2 V k j

P

i 2 G

(º

;¹ ;º

) ´ i j (¸ ; x; y) ⁺

P

i ´ i j (¸ ; x; y) ⁺ > ½g for ^X

i

´ i j (¸ ; x; y) ⁺ > 0

where j = k ¡ I ¡ J and ´ i : V 0 £ ^{Q I} _{i = 1} V i £ ^{Q J} _{j = 1} Z j ! R ^J is de¯ ned by

´ i (¸ ; x; y) = arg min kx i ¡ ! i ¡ Q± i ¡ (Y ¡ Q)´ i k; s.t . ´ i 2 R ^J :

Clearly, V k is compact and convex and C k is cont inuous. Lemma 1 below

shows t hat gr Q k is open and t hat º j 2 co (Q = k (¸ ; x; y; º ) \ C k (¸ ; x; y; º )) for

j = k ¡ I ¡ J .

L em m a 1 T he preference correspondence for shareholders, Q k : V ! V k

where k 2 f I + J + 1; : : : ; I + 2J g , has the following properties

² gr Q k is open, and,

² º j 2 = co (Q k (¸ ; x; y; º ) \ C k (¸ ; x; y; º )) for k = I + J + j .

Proof: \ gr Q k is open" Suppose that (x i (n)) t 2 N 2 V i converges t o x i 2 V i

and t hat (¹ i (x i (n))) t 2 N 2 ¢ ^S ₊ converges to ¹ i 6 = ¹ i (x i ) 2 ¢ ^S ₊ . Then t here exist s x ⁰ _i 2 P i (x i ) such t hat ¹ i ¢(x ⁰ _i ¡ x i ) · 0 so t here exist s x ⁰⁰ _i 2 P i (x i ) such t hat ¹ i ¢(x ⁰⁰ _i ¡ x i ) < 0. Therefore t here exist s N 2 N such t hat if n ¸ N t hen º i (x i (n)) ¢(x ⁰⁰ _i ¡ x i (n)) < 0 and x ⁰⁰ _i 2 P i (x i (n)) according t o (a.3). This is a contradict ion t hus ¹ i : V i ! ¢ ^S ₊ is cont inuous. Clearly, if ¹ i : V i ! ¢ ^S ₊ is continuous for all i then gr Q k is open due t o t he de¯ nit ion of Q k : V ! V k .

\ º j 2 = co (Q k (¸ ; x; y; º ) \ C k (¸ ; x; y; º )) for k = I + J + j " Let [r ] 2 Z be de¯ ned by [r ] · r < [r ] + 1 for all r 2 R and let L m = ^P _i [´ i j (¸ ; x; y) ⁺ m] · m ^P _i ´ i j (¸ ; x; y) ⁺ art i¯ cial consumers be de¯ ned by

T l = U i (º j ; x i )

for all l 2 f ^P _i

_{< i} [´ i

j (¸ ; x; y)m] + 1; : : : ; ^P _i

_{· i} [´ i

j (¸ ; x; y)m]g and m 2 N provided t hat m´ i j (¸ ; x; y) ¸ 1. Let t j : ¢ ^S ₊ ! f 1; : : : ; L m g be de¯ ned by

t j (º _j ⁰ ) = f l 2 f 1; : : : ; L m gj´ i j (¸ ; x; y) ⁺ > 0 and º _j ⁰ 2 T l g and let R j ½ ¢ ^S ₊ be de¯ ned by

R j = f º _j ⁰ 2 ¢ ^S ₊ jjt j (º _j ⁰ )j > ½ ^X

i

´ i j (¸ ; x; y) ⁺ mg:

If º _j ⁰ 2 Q k (¸ ; x; y; º ) t hen there exist s " > 0 such t hat

" =

P

i 2 G

(º

;x ;º

) ´ i j (¸ ; x; y) ⁺

P

i ´ i j (¸ ; x; y) ⁺ ¡ ½

due t o t he de¯ nit ion of Q k : V ! V k . Therefore, º _j ⁰ 2 R j provided t hat

m > I =(" ^P _i ´ (¸ ; x; y) ⁺ ) so Q k (¸ ; x; y; º ) ½ [ m R j .

Clearly, º j 2 co (F = i (º j ; x i ) \ C k (¸ ; x; y; º )) due t o t he const ruction of F i : ¢ ^S ₊ £ V i ! ¢ ^S ₊ t herefore º j 2 co (R = j \ C k (¸ ; x; y; º )) for all m be- cause dim C k (¸ ; x; y; º ) = S ¡ J and ½ ¸ (S ¡ J )=(S ¡ J + 1) hence º j 2 co (Q = k (¸ ; x; y; º ) \ C k (¸ ; x; y; º )) according t o Greenberg (1979).

Q.E.D.

St ep 2: construction of correspondence and existence of ¯ xed point

Let K = f 0; : : : ; I + 2J g t hen V k is compact and convex for all k 2 K and V = ^Q _{k2 K} V k . Let t he map f k : V £ V k ! R + be de¯ ned by

f k (z; z _k ⁰ ) = min

(v;v

)2 ( gr Q

)

k(z; z _k ⁰ ) ¡ (v; v _k ⁰ )k;

t he correspondence g k : V ! V k by

g k (z) = arg max

z

2 C

(z) f k (z; z _k ⁰ ):

Then t he correspondence, h : V ! V de¯ ned by h k (z) = co g k (z) is upper hemi-cont inuous and compact and convex valued. Therefore t here exist s (¸ ^¤ ; x ^¤ ₁ ; : : : ; x ^¤ _I ; y ₁ ^¤ ; : : : ; y ^¤ _J ; º ^¤ ) = z ^¤ 2 Z such t hat z ^¤ 2 h(z ^¤ ) according t o t he K akut ani ¯ xed point t heorem. Hence, z _k ^¤ 2 C k (z ^¤ ) and Q k (z ^¤ ) \ C k (z ^¤ ) = ; because z k 2 co Q = k (z) \ C k (z).

St ep 3: existence of ½ -majority stable equilibrium

For consumers, t here exist s x i (f ! i g + ^P _j ± i j co cl Z j ) 2 R ^{S+ 1} such t hat if x i 2 B i (Q; Y ) and P i (x i ) \ B i (Q; Y ) = ; t hen x i ¸ x i (f ! i g + ^P _j ± i j co cl Z j ) according t o (a.6) because f ! i g+ ^P _j ± i j co cl Z j is compact according t o (a.7).

Therefore t here exist z ^¤ 2 h(z ^¤ ) and N C 2 N such t hat if n ¸ N C - recall t hat

V k = X i \ (f ! i g+ ^P _j ± i j co cl Z j + [¡ n; n] ^{S+ 1} ) for k = i - t hen x ^¤ _i 2 B i (Q ^¤ ; Y ^¤ )

and co P i (x ^¤ _i ) \ B i (Q ^¤ ; Y ^¤ ) = ; t hus x ^¤ _i = ! i + Q ^¤ ± i + (Y ^¤ ¡ Q ^¤ )µ _i ^¤ for some

µ _i ^¤ 2 [0; 1] ^J .

For consumers, if ¸ (n) ! ¸ where ¸ ^s (n) ¸ 1=n and ¸ ^s = 0 for s 2 S ⁰ ½ S and x 1 (n) 2 C 1 (¸ (n); x(n); y(n); º (n)) and Q 1 (¸ (n); x(n); y(n); º (n)) \ C 1 (¸ (n); x(n); y(n); º (n)) = ; t hen ^P _{s2 S}

x ^s ₁ (n) ! 1 according t o (a.3) and (a.4) while consumpt ion is bounded from below for all consumers according t o (a.6). Therefore, for t he auct ioneer, t here exist s N A 2 N such t hat if n ¸ N A - recall t hat V k = ¢ ^S ₊ \ [1=n; 1] ^{S+ 1} for k = 0 - and z ^¤ 2 h(z ^¤ ) t hen

P

i x ^¤ _i = ^P _i ! i + ^P _j y _j ^¤ .

For t he ¯ rms, if z ^¤ 2 h(z ^¤ ) t hen y ^¤ _j 2 arg max º _j ^¤ y j ; s.t . y j 2 co cl Z j

t herefore y _j ^¤ 2 arg max º _j ^¤ y j ; s.t . y j 2 Y j and Y j ½ f y _j ^¤ g + hº _j ^¤ i ¡ R ^{S+ 1} ₊ . L em m a 2 I f z ^¤ 2 h(z ^¤ ) and n ¸ N C then P _j ^½ (x ^¤ ; µ _j ^¤ ; y _j ^¤ ) = ; .

Proof: Suppose t hat n ¸ N C , if x ^¤ _i + (y j ¡ y ^¤ _j )µ _{i j} ^¤ 2 P i (x ^¤ _i ) and µ _{i j} ^¤ > 0 t hen ¹ i (x ^¤ _i ) ¢(y j ¡ y _j ^¤ ) > 0 and if ¹ i (x ^¤ _i ) ¢(y j ¡ y _j ^¤ ) 6 = 0 and µ i j > 0 t hen x ^¤ _i + (y j ¡ y _j ^¤ ) = 2 P i (x ^¤ _i ). Hence, if

P

i 2 H

(v

) µ ^¤+ _{i j}

P

i µ _{i j} ^¤+ · ½

for all v j 2 hº _j ^¤ i ^? where

H j (v j ) = f i 2 f 1; : : : ; I gj¹ i (x ^¤ _i ) ¢v j > 0g

t hen P _j ^½ (x ^¤ ; µ _j ^¤ ; y _j ^¤ ) = ; because Y j ½ f y _j ^¤ g + hº _j ^¤ i ^? ¡ R ^{S+ 1} ₊ . Thus, hv j i ^? separat es H j (v j ) from t he rest of t he i 's in t he sense t hat H j (v j ) is above hv j i ^? while t he rest of t he i 's are below or on hv j i ^? , i.e. i 2 H j (v j ) if and only if ¹ i (x ^¤ _i ) ¢v j > 0.

For v j 2 hº _j ^¤ i ^? suppose t hat ^P _s v ^s _j < 1 wit hout loss of generality and let (p(n)) n2 N 2 ¢ where ¢ = f ¸ 2 R ^{S+ 1} j ^P _s ¸ ^s = 1g be de¯ ned by

p(n) = 1

n + ^P _s v ^s _j (nº _j ^¤ + v j ):

for all n t hen (p(n)) n2 N converges t o º _j ^¤ . Let (q(n)) n2 N 2 hp(n) + º _j ^¤ i ^? be de¯ ned by

q(n) = (p(n) ¡ º _j ^¤ ) ¡ (p(n) ¡ º _j ^¤ ) ¢(p(n) + º _j ^¤ )

(p(n) + º _j ^¤ ) ¢(p(n) + º _j ^¤ ) (p(n) + º _j ^¤ ):

for all n. Then some t edious calculat ions show t hat (nq(n)) n2 N converges t o v j and hq(n)i ^? separat es G j (º _j ^¤ ; x ^¤ ; p(n)) from t he rest of t he i 's in t he sense t hat G j (º _j ^¤ ; x ^¤ ; p(n)) is above hq(n)i ^? while t he rest of t he i 's are below or on hq(n)i ^? . Moreover t here exist s N 2 N such t hat if n ¸ N t hen hv j i ^? separat es G j (º _j ^¤ ; x ^¤ ; p(n)) from t he rest of t he i 's in t he sense t hat G j (º _j ^¤ ; x ^¤ ; p(n)) is above hv j i ^? while the rest of t he i 's are below or on hv j i ^? . Thus if n ¸ N t hen G j (º _j ^¤ ; x ^¤ ; p(n)) = H j (v j ). Therefore, P _j ^½ (x ^¤ ; µ ^¤ _j ; y _j ^¤ ) = ; because Q I + J + j (z ^¤ ) = ; .

Q.E.D.

For shareholders, if z ^¤ 2 h I + J + j (z ^¤ ) t hen P _j ^½ (x ^¤ ; µ ^¤ _j ; y _j ^¤ ) = ; provided t hat n ¸ N C according t o lemma 2. T hus, if ½ ¸ (S ¡ J )=(S ¡ J + 1) and n ¸ maxf N A ; N C g t hen a ½-majority st able equilibrium exist s.

Par t 2: ½¸ B =(B + 1)

The variables t o be det ermined are st at e prices, ¸ 2 ¢ ^S ₊ , consumpt ion bun- dles for consumers, x = (x i ) ^I _{i = 1} ½ ^{Q I} _{i = 1} X i , and product ion plans for ¯ rms, y = (y j ) ^J _{j = 1} ½ ^{Q J} _{j = 1} Y j .

Let st rat egy sets, const raint correspondences and preference correspon- dences be de¯ ned as in part 1 of t he proof for k 2 f 0; 1; : : : ; I g.

\ Firms" For agent k 2 f I + 1; : : : ; I + J g, t he st rat egy set , V k ½ R ^{S+ 1} , is de¯ ned by

V k = co cl Z j

where j = k ¡ I , t he const raint correspondence, C k : V ! V k , is de¯ ned by C k (¸ ; x; y; ) = V k

and t he preference correspondence, Q k : V ! V k , is de¯ ned by

Q k (¸ ; x; y) = P _j ^½ (x; ´ j (¸ ; x; y); y j ) \ V k :

Clearly, V k is compact and convex, C k is cont inuous and gr Q k is open wit h y j 2 co Q = k (¸ ; x; y) for j = k ¡ I according t o t he proof of t heorem 2 in Greenberg (1979).

T he rest of t he proof follows from part 1. Thus, if ½¸ B =(B + 1) t hen a

½-majority st able equilibrium exist s.

Par t 3: an example showing that the bound is binding

Consider an economy wit h S consumers wit h ut ility functions linear in period zero consumpt ion and log-linear in period 1 consumption. Consumer i is indexed by weight s, ¼ i = (¼ _i ^s ) ^S _{s= 1} wit h ^{P S} _{s= 1} ¼ _i ^s = 1, on consumpt ion in di®erent st at es. T he ut ility funct ion of consumer i is:

u i (x i ) = x ⁰ _i +

X S

s= 1

¼ _i ^s log x ^s _i wit h

8

<

:

¼ _i ^s = " if s 6 = i

¼ _i ⁱ = 1 ¡ (S ¡ 1)" (1) where " 2 ]0; 1 ¡ 1=(S ¡ 1)[ is small. Alt hough these ut ility funct ions do not sat isfy assumpt ion (a.6), since t he argument is local they can be easily ex- t ended out side t he relevant domain t o ful¯ ll t his assumpt ion. All consumers are endowed wit h ident ical init ial shares of t he J ¯ rms: ± i j = 1=S, for all i ; j , and t he same vector of init ial resources: ! i = (! ; 0; : : : ; 0), all i .

All J ¯ rms have t heir set s of e± cient product ion plans included in t he same (S ¡ 1)-dimensional linear subspace:

Y =

(

y = (y ⁰ ; y ¹ ; y ² ; : : : ; y ^S ) 2 R ^S j

X S

s= 0

y ^s = 0 and y ⁰ = ¡ 1

)

: De¯ ne t he product ion plans y = (y j ) ^J _{j = 1} by:

y ^s _j =

8 >

> <

> >

:

¡ 1 for s = 0

1 for s = j

0 ot herwise

for j · J ¡ 1 and for ¯ rm J y _J ^s =

8 >

> <

> >

:

¡ 1 for s = 0

1=(S ¡ J + 1) for s 2 f J; : : : ; Sg

0 ot herwise

Next , de¯ ne ~ y = ( ~ y j ) ^J _{j = 1} such t hat ~ y J = y J and ~ y j = ®y j + (1 ¡ ®)y J for j · J ¡ 1, wit h ® = J=S. Let , for all j , Z j = Y \ B( ~ y j ; º ) where B( ~ y j ; º ) st ands for t he ball wit h cent er ~ y j and radius º . T his way, an (" ; º )-economy is de¯ ned.

Obser vat ion 1 For all ´ , there exists (" ; º ) such that the (" ; º ) -economy does not have a ½ -majority stable equilibrium for ½< (S ¡ J )=(S ¡ J + 1) ¡ ´ . Consider t he (" ; 0)-economy. It is now shown t hat t here is a unique ½- majority st able equilibrium (for all ½ since º = 0 implies t here is no al- t ernat ive product ion plan), (q ^¤ ; µ ^¤ ; y ^¤ ), wit h y ^¤ = ~ y and q ^¤ = ¯ 1 J where

¯ = S=J ¡ 1.

For t he announced product ion plans y ^¤ , t he expression of t he ut ility level of agent i buying, at price q ^¤ , t he port folio (µ i j ) ^J _{j = 1} is:

! + ¯ J

S ¡ (¯ + 1)

X J

j = 1

µ i j

| { z }

x

+

J ¡ 1 X

s= 1

¼ _i ^s log [®µ i s ]

| { z }

x

;s· J ¡ 1

+ ¼ _i ^V log

"

µ i J + (1 ¡ ®)µ i U

S ¡ J + 1

#

| { z }

x

;s¸ J

;

where U = f 1; : : : ; J ¡ 1g, V = f J; : : : ; Sg, ¼ _i ^V = ^X

s2 V

¼ ^s _i and µ i U = ^X

j 2 U

µ i j . First -order condit ions of t his maximizat ion problem (opt imal port folio choice) gives:

8s · J ¡ 1 : ¼ ^s _i µ i s

+ (1 ¡ ®)¼ _i ^V µ i J + (1 ¡ ®)µ i U

= ¯ + 1; and ¼ _i ^V

µ i J + (1 ¡ ®)µ i U

= ¯ + 1;

which in t urn yields:

8s · J ¡ 1 : µ i s = ¼ _i ^s

®(¯ + 1) ; and µ i J = ¼ _i ^V

¯ + 1 ¡ 1 ¡ ®

®

¼ ^U _i

¯ + 1 :

It is easily checked t hat stock market s clear, as well as market s for good, for t he chosen values of ® = J=S and under t he equilibrium price ¯ = S=J ¡ 1.

Then t he equilibrium port folio is:

8s · J ¡ 1 : µ _{i s} ^¤ = ¼ ^s _i ; and µ _{i J} ^¤ = J

S ¼ _i ^V ¡ (1 ¡ J S )¼ ^U _i ; which is such t hat

X J

j = 1

µ ^¤ _{i j} = J

S for all i .

Suppose now t hat ¯ rm J is given t he opport unity t o propose a small change of it s product ion plan. For " small enough, one has µ _{i J} ^¤ > 0 for J · i · S and µ ^¤ _{i J} < 0 for 0 · i · J ¡ 1. Hence, only t he S ¡ J + 1 last consumers have a posit ive quant ity of shares in ¯ rm J and consequent ly t hey are t he only ones t o vot e, wit h t he same vot ing weight s. T he ut ility funct ion of consumer i , J · i · S, has been const ruct ed such t hat , at t his symmet ric equilibrium, consumer i support s a (t echnically possible) change from y J t o y ⁰ _J in Z J if and only if y _J ⁱ · y _J ⁰ⁱ , i.e. any change that yields more in st ate i . For example, y _J ⁰ = y J + (0; : : : ; 0; ¡ " ; " =(S ¡ J ); : : : ; " =(S ¡ J )) get s t he support of t he last S ¡ J shareholders/ shares. Hence, (q ^¤ ; µ ^¤ ; y ^¤ ) is not st able for any super majority rule of size smaller t han (S ¡ J )=(S ¡ J + 1). Subject t o t he obvious upper hemi-cont inuity of t he equilibrium correspondence in t he present set up, any ½-majority st able equilibrium of t he (" ; º )-economy, for " and º small enough, is such that ½> (S ¡ J )=(S ¡ J + 1) ¡ ´ . Finally, not e t hat S ¡ J · B = S ¡ 1 so t here is no need t o consider t he ot her case which is more obvious.

P r oof of Cor ollar y 1

The variables t o be det ermined are prices, p 2 ¢ ^J ₊ , consumpt ion bundles for consumers, x = (x i ) ^I _{i = 1} ½ ^{Q I} _{i = 1} X i , and product ion plans for ¯ rms, y = (y j ) ^J _{j = 1} ½ ^{Q J} _{j = 1} Y j .

\ Auctioneer" For agent k = 0, t he st rat egy set, V k ½ R ^{J + 1} , is de¯ ned by V k = ¢ ^J ₊ \ [ 1

n ; 1] ^{J + 1}

where n 2 N , t he const raint correspondence, C k : V ! V k , is de¯ ned by C k (p; x; y) = V k

and t he preference correspondence, Q k : V ! V k , is de¯ ned by Q k (p; x; y) = f p ⁰ 2 V k j(p ⁰ ₀ ¡ p 0 )( ^X

i

x ⁰ _i ¡ ^X

i

! _i ⁰ ¡ ^X

j

y _j ⁰ )

+ ^X

j

(p ⁰ _j ¡ p j )( ^X

i

´ i j (p; x; y) ¡ 1) > 0g:

where q j = p j =p 0 for all j and ´ i : V 0 £ ^{Q I} _{i = 1} V i £ ^{Q J} _{j = 1} Z j ! [0; 1] ^J is de¯ ned by

´ i (p; x; y) = arg min kx i ¡ ! i ¡ Q± i ¡ (Y ¡ Q)´ i k; s.t . ´ i 2 [0; 1] ^J for all i . Clearly, V k is compact and convex, C k is cont inuous and gr Q k is open wit h p = 2 co Q k (p; x; y).

\ Consumers" As in part 1 in t he proof of t heorem 1 - rest rict ing port folios t o [0; 1] ^J and disregarding º .

\ Firms" As in part 2 in t he proof of t heorem 1 - rest rict ing port folios t o [0; 1] ^J in t he de¯ nit ion of P _j ^½ (x; µ j ; y j ), replacing µ j wit h ´ j (p; x; y) and disregarding º .

T he rest of t he proof follows from t he last part of part 1 in t he proof t heorem 1.

P r oof of Cor ollar y 3

Proof: Follows from t he proof of T heorem 1 wit h minor changes provided t hat x i 2 ~ B i (Q; Y ) and P i (x i ) \ ~ B i (Q; Y ) imply t hat y j 6 = co ~ U i j (q; x i ; y j ).

Hence, suppose t hat (y j (n)) ^N _{n= 1} 2 ~ U i j (q; x i ; y) where N 2 N t hen t here

exist s (µ i (n)) ^N _{n= 1} such t hat x i + (Y jy j (n) ¡ Q)µ i (n) 2 P i (x i ) for all n 2

f 1; : : : ; ng. Suppose t hat y _j ⁰ = ^P _n ®(n)y j (n) where ®(n) ¸ 0 for all n and

P

n ®(n) = 1 and let µ _j ⁰ and (¯ (n)) ^N _{n= 1} where ¯ (n) ¸ 0 for all n and ^P _n ¯ (n) = 1 be de¯ ned by ®(n)µ ⁰ _{i j} = ¯ (n)µ i j (n) for all n and µ ⁰ _{i j}

= ^P _n ¯ (n)µ i j

(n) for all j ⁰ 6 = j . T hen

x i + (Y jy ⁰ _j ¡ Q)µ _i ⁰ =

X N

n= 1

¯ (n)(x i + (Y jy j (n) ¡ Q)µ i (n)):

Therefore, if x i 2 ~ B i (Q; Y ) and P i (x i ) \ ~ B i (Q; Y ) t hen y j 2 co ~ = U i j (q; x i ; y j ).

It is necessary t o bound port folios t o be non-negative in order t o ensure

t hat t here exist s µ ⁰ _i and (¯ (n)) n such t hat (Y jy _j ⁰ ¡ Q)µ ⁰ _i = ^P _n ¯ (n)(Y jy j (n) ¡

Q)µ i (n).