Equilibrium Executive Compensation and Shareholder Heterogeneity Milo Bianchiy

Rose-Anne Danaz

Elyès Jouinix

September 28, 2017

Abstract We consider a …rm owned by shareholders with heterogeneous time preferences and beliefs and run by a manager with private information about the chosen production plan. Shareholders trade in a complete asset market and design a compensation scheme so that the manager chooses the plan they prefer and reveals it truthfully. We show that such scheme should restrict the manager from trading in the asset market. Moreover, relative to a linear compensation, the scheme should induce larger weights to output realizations which are extreme and which occur in the distant future. JEL codes: G32, G34, D24, D51, D53, D70 Keywords: heterogeneous shareholders, unanimity, asymmetric information, optimal equilibrium compensation, manager-shareholders equilibrium.

We thank seminar participants at Columbia, Lausanne, D-TEA Paris, Toulouse for very useful discussions. The …nancial support of the GIP-ANR ("Risk" project) and of the Risk Foundation (Groupama chair) is gratefully acknowledged. y Toulouse School of Economics, University of Toulouse Capitole, [email protected] z Université Paris-Dauphine and IPAG Business School, [email protected] x Université Paris-Dauphine, PSL Research University, [email protected]

1

1

Introduction

Corporations are often owned by many investors with possibly di¤erent tastes and beliefs. This begs the question of what the manager should maximize. Jensen and Meckling (1976) famously proposed to de…ne a “corporate objective function”as the result of “a complex process in which the con‡icting objectives of individuals are brought into equilibrium.” It is well known that heterogenous shareholders may agree on the objective of the …rm if they have access to a complete …nancial market. Even under this ideal condition, however, much less is known on how to design a compensation scheme which would induce a manager to act in shareholders’ best interest. We consider a …rm owned by shareholders with di¤erent discount rates and beliefs and run by a manager. The manager chooses a production plan, which determines a ‡ow of uncertain output over time. Shareholders cannot observe the plan chosen by the manager, they only observe the realized production. A contract for the manager speci…es her compensation at each date as a function of the history of production and possibly some constraints to trade in the asset market. Once the manager announces her plan, shareholders and possibly the manager can trade assets in a complete …nancial market. Our …rst result is that the manager would misreport the chosen plan unless the marginal value of her compensation is proportional to her marginal utility of consumption. It follows that her consumption pro…le should coincide with her compensation; that is, the manager should have no incentive to trade in the …nancial market. Building on this result, we show that a production plan that is unanimously preferred by all shareholders, which we call a consensus plan, corresponds to a plan associated to a production equilibrium.1 This is a plan which shareholders would choose if they were to run the …rm by themselves according to pro…t-maximization. We show that, in our economy, a production equilibrium exists and it is unique. Our main question is 1

Unanimity among shareholders is de…ned in a classical way: at equilibrium, shareholders compare what they would get under alternative plans, taking the equilibrium prices as given. An equilibrium plan is a consensus plan if no shareholder would obtain a higher utility under a di¤erent plan.

2

how a contract for the manager can be designed so as to reach such an equilibrium. We show that under complete markets for the manager, a unanimous equilibrium requires that the compensation is linear. At the same time, when the manager is given a linear compensation, she chooses the consensus plan only if she has the same characteristics as the representative shareholder at equilibrium (a …ctitious agent who -if endowed with the entire production - would have no incentive to trade at equilibrium prices). Suppose that shareholders and the manager share the same belief but di¤er in their time preference. The representative shareholder at equilibrium has the same belief as the shareholders and a declining discount rate since aggregate consumption in the distant future is mostly in the hands of more patient individuals (Gollier and Zeckhauser (2005)). Hence, a constant discount rate manager with linear compensation cannot reach the consensus. As we show, this result extends in more general settings and we conclude that in order to implement a unanimous equilibrium, the compensation cannot be linear and the manager should face some trading restrictions. We then consider the case in which the manager is prevented from trading in the …nancial market. We show that she implements the consensus plan only if the marginal utility of her compensation is proportional to that of the representative shareholder at such plan. We provide necessary and su¢ cient conditions on agents’beliefs under which the required compensation depends only on the current production of the …rm and not on its history. We show that, under suitable conditions, there is a unique compensation scheme (up to a positive constant) which leads to a unanimous equilibrium. We then fully characterize the optimal compensation when agents have di¤erent discount rates but homogeneous beliefs: at each date, the compensation is a linear function of the current production but its slope is time-dependent. In particular, it induces the manager to attach an increasing weight to future production. In the case of heterogeneous beliefs, we show that, for any manager whose beliefs are between the "most optimistic" and the "most pessimistic" (in a sense made precise below) of the shareholders, the compensation should increase the weight that the manager attaches to very low or very 3

large levels of production. We conclude by providing an example in which, by imposing further structure on the production set and on shareholders’ characteristics, we can explicitly compute the production equilibrium as well as the required compensation scheme. We think our results have important implications for the study of agency problems and in particular of optimal executive compensation. From a methodological perspective, we highlight the importance of modeling explicitly shareholder heterogeneity and the equilibrium process leading to the de…nition of a representative shareholder. The insights one would get starting directly with a representative agent would be di¤erent and possibly misleading when applied to settings -as large corporations- in which heterogeneity is important. We show it is necessary to impose trading restrictions to the manager in a setting in which the action she takes cannot be observed by shareholders. Without those restrictions, it would be impossible to …nd a compensation scheme which induces the manager to choose the consensus plan and to truthfully reveal her choice. This provides a rationale for the commonly observed restrictions both to insider trading and to non-exclusive contracts. We also qualify the view that agency con‡icts are minimized when the manager owns a substantial part of the …rm’s shares, which has motivated the rise in stock compensation. Our analysis points at the opposite e¤ect in that stock compensation can lead the manager to take Pareto dominated decisions in settings in which shareholders are not completely identical. Stock compensation is also advocated as a remedy for managers’shorttermism. Our results however show that a manager may put too little weight on future performance even when she holds a large fraction of shares. In this respect, we do not emphasize the importance of large stockholding but rather that the compensation rate of the manager should vary over time. Similarly, we emphasize that the compensation rate should vary with the level of production, and in particular it should induce the manager to overweight the occurrence of extreme realizations. This result stands in contrast to the argument in favor of compensations -such as call optionswhich encourage risk taking (see e.g. Kadan and Swinkels (2008)).

4

Related Literature We build on the literature on aggregation of preferences and beliefs in asset markets.2 Our focus on agency problems between a manager and shareholders is however novel in this literature. Similarly, managerial compensation has typically been studied under the perspective of a representative shareholder (see e.g. Murphy (1999) and Murphy (2012) for reviews). We provide new insights by embedding the choice of the compensation in a stock market equilibrium with heterogeneous shareholders. In this spirit, Bolton, Scheinkman and Xiong (2006) consider a market with heterogeneous beliefs and short-selling constraints and show that shareholders may prefer short-term speculative strategies.3 In line with the literature on optimal contracting, we emphasize that, in a setting with asymmetric information, it may be bene…cial to prevent the manager from trading in the stock market (Fischer (1992)).4 Our novelty is to show how trading restrictions, and the form of optimal compensation, are determined by the interaction between moral hazard and shareholder heterogeneity. Finally, we relate to the literature on …rms’ objectives when shareholders are heterogeneous. Magill and Quinzii (2002) review fundamental problems posed by market incompleteness, as well as classic contributions addressing these problems. Bisin, Gottardi and Ruta (2014) study competitive equilibria in a production economy with incomplete markets and agency frictions and derive fundamental welfare properties.5 We instead keep shareholders’ objective as simple as possible by assuming complete markets, and focus on the design of the compensation scheme. 2

Recent contributions include Detemple and Murthy (1994); Gollier and Zeckhauser (2005); Jouini and Napp (2007); Jouini, Marin and Napp (2010); Cvitani´c, Jouini, Malamud and Napp (2012); Xiong and Yan (2010); Bhamra and Uppal (2014). 3 Alternative equilibrium models have instead focused on the labor market equilibrium (e.g. Gabaix and Landier (2008)) or on …nancial market equilibrium with a representative agent (e.g. Diamond and Verrecchia (1982)). 4 This literature has also pointed out at bene…cial aspects of insider trading, such as improving the informational e¢ ciency of market prices (e.g. Leland (1992)). We abstract from this issue as in our settings there are no investors apart from shareholders. We refer to Bhattacharya (2014) for a recent review of these issues. 5 Other recent contributions include Demichelis and Ritzberger (2011), Magill, Quinzii and Rochet (2015), Crès and Tvede (2014).

5

2

Model

We consider a …rm owned by a group of shareholders, who are heterogenous in beliefs and time preferences, and run by a manager. The …rm produces a non-storable consumption good, which we use as numeraire, according to a production plan y: This plan is a random process in which yt (!) de…nes the production of the …rm -net of production costs- at date t in state !:6 There is a …nite set of dates T = f0; :::; T g : Uncertainty is modeled by a …ltered probability space ; (Ft )t2T ; P : We denote by X = p L ; (Ft )t2T ; P the space of progressively measurable random processes x = (x0 ; : : : ; xT ) on ; (Ft )t2T for which Ejxt jp < 1 for all t 2 T and equipped with the norm

kxk =

T X t=0

Ejxt jp

!1=p

;

in which E is the expectation taken under P . It is a Banach space for p 1 whose dual is X 0 = Lq ; (Ft )t2T ; P and q is such that p1 + 1q = 1: For P x 2 X and y 2 X 0 ; let x y = Tt=0 E(xt yt ). We denote by X+ and X+0 the set of nonnegative processes, respectively, in X and X 0 .

2.1

Production

We let Y X denote the set of production plans. The e¢ cient frontier of Y is de…ned as the set E (Y ) of feasible production plans that are not dominated by other feasible production plans, that is E (Y ) = fy 2 Y : @y 0 2 Y; y 0

y 2 X+ and y 0 6= yg :

Denote with NY (y) the normal cone of Y at y; NY (y) = fw 2 X 0 : w (z 6

y)

0; 8 z 2 Y g ;

It is simpler to present our results de…ning the compensation scheme as a function of the net production. One can show the equivalence with a compensation de…ned in terms of gross production, which itself includes the compensation of the manager.

6

which corresponds to the set of linear forms that reach their maximum on Y at y. We say that E (Y ) is smooth if NY (y) reduces to a half-line for all y 2 E (Y ). We make the following assumptions: Assumption (P) 1. Y = K

X+ with K closed and strictly convex,

2. There exist & and

in X+ such that 0 < &

K

;

3. E (Y ) is smooth and all e¢ cient plans have full support: supp(y0 ; : : : ; yt ) = (0; 1)t+1 for all t and all y 2 E (Y ); 4. The …ltration G(y) = (Gt (y))t2T generated by y does not depend on the choice of y 2 E (Y ) and it is denoted by G: From 1 and 2, we have E (Y ) K X+ : Note that K is closed, convex and bounded and therefore weakly compact. Point 4 states that by observing the realizations of an e¢ cient production plan, agents do not gain a …ner assessment of the chosen plan y: Points 3 and 4 are ful…lled in the following example. Example 1 The production process is governed by a Brownian motion W and the variable 2 governs the drift and the volatility of the process. We have Y = Y0 X+ with Y0 = f(yt ( ))t2T : yt ( ) = exp( ( ) t + ( ) Wt );

2

g;

where and are given real valued functions on and takes positive values: All y 2 E (Y ) are of the form (exp( ( ) t + ( ) Wt ))t2T for some 2 and they generate the same …ltration G which equals the …ltration generated by W .

2.2

Shareholders

The …rm is owned by a group of N shareholders, i = 1; :::; N who are only endowed with shares of the …rm. We denote with i agent i’s initial endowment of shares, and we assume i > 0 for all i. 7

Shareholders are heterogeneous in their time preference rate i assumed to be constant and in their subjective probabilities Qi : All subjective probabilities are assumed to be equivalent to P and we denote by M i the density i : of Qi with respect to P; M i = dQ dP A key ingredient in our analysis is that shareholders do not observe the plan chosen by the manager nor the state of the world; they only observe the history of realized production. Therefore, for a given production plan y; the information they hold is represented by G(y) and they only trade G(y)-adapted processes. We further assume that shareholders face complete asset market with respect to the …ltration G(y) and so can trade any claim whose payo¤s are contingent on the realizations of y. Formally, they can trade any G(y)-adapted contingent claim; that is, any c 2 X for which there exists a function C : X+ ! X (where C depends on the choice of y) such that c = C(y) with C(y) = (C0 (y); : : : ; CT (y)); and for which, for every t, there exists a function ct : Rt+1 + ! R such that Ct (y) = ct (y0 ; : : : ; yt ): Shareholders are assumed to be expected utility maximizers. The expected utility of agent i for a contingent consumption plan c is de…ned as T X exp( i t)E Mti u(ct ) ; (1) U i (c) = t=0

in which u is a CRRA instantaneous utility function (the same for all shareholders). That is u(x) =

(

1

x

if x

0;

1 if x < 0;

(2)

for some < 1: We further assume the following: Assumption (C) 1. For all i; M i and M i & 2. For all i; M i & and M i

1

belong to X 0 ; belong to L1

; (Ft )t2T ; P :

Assumption (C) assures that shareholders’marginal utility is well de…ned in all directions and that their utility is well de…ned in E (Y ):

8

2.3

Manager

The …rm is run by a manager with characteristics similar to those of the shareholders. She is an expected utility maximizer with instantaneous utility u, she has a constant time preference rate m and a subjective probability Qm equivalent to P with density M m ; which is adapted to the …ltration F. This describes the nature of asymmetric information between the manager and the shareholders. If the manager chooses a production plan y the information held by the shareholders is represented by G(y) while the information held by the manager is represented by F: Her expected utility of a contingent plan c is therefore de…ned by Um (c) =

T X

m

exp(

t)E [Mtm u(ct )] :

t=0

The manager is given a contract, which is a pair ( ; W ) of a compensation scheme : X+ ! X+ and of a space W X of transaction processes she is allowed to make in the contingent claim market. We …rst consider the compensation scheme. As the shareholders can only observe the history of realized productions, we assume that (y) = ( 0 (y); : : : ; T (y)) is G(y)adapted. Hence, we have for all t t (y)

=

t (y0 ; : : : ; yt );

(3)

t+1 for some t : R+ ! R+ . We further make the following assumption. Assumption (F) The compensation scheme satis…es the following conditions:

t (x0 ; : : : ; xt 1 ; 0)

lim

xt !1

t (x0 ; : : : ; xt )

= 0 for all (x0 ; : : : ; xt 1 ) 2 Rt+ ;

= 1 for all (x0 ; : : : ; xt 1 ) 2 Rt+ :

(4) (5)

According to (4), if the production at date t is zero, then the manager receives no compensation at that date. According to (5), the manager receives an in…nite compensation when production goes to in…nity.7 7

As it will be clear, Assumption (5) is needed only when < 0: Moreover, we take the compensation to in…nity only for simplicity of exposition. Our results require more generally that when < 0 there exists a level of compensation 1 such that

9

We next describe W; the space of transactions allowed to the manager in the contingent claim market. When W = f0g, the manager has no access to the market while when W = X, the manager has access to all possible (Ft )t2T -adapted processes. However, as we consider trading either among shareholders or between shareholders and the manager, prices and consumption processes will be G(y)-adapted. Therefore, given a production plan process y, the manager’s set of feasible plans is C(y) = (y)+W \X G(y) where X G(y) denotes the set of G(y)-adapted processes. For every price process q and production plan y, we denote with Vm (y; q) the maximal utility of the consumption processes that the manager can obtain by trading her compensation under her market and budget constraints: Vm (y; q) = maxfUm (c); c 2 C(y); q c

2.4

q

(y)g:

(6)

Equilibrium

We can now de…ne our concept of equilibrium between shareholders and the manager. Shareholders appoint a manager with a contract ( ; W ) and delegate to her the choice of the production plan. Given prices, the manager chooses a plan which maximizes her indirect utility of production plans as well as her consumption plan. The manager announces the chosen plan to shareholders. Shareholders choose their consumption plans by maximizing their utilities subject to their constraints. As the chosen plan is not observed by shareholders, at equilibrium the manager should have no incentive to misreport the chosen plan and markets should clear. De…nition 1 Let ( ; W ) be given. A ( ; W ) manager-shareholders equilibrium is de…ned by an e¢ cient production process y^ 2 E (Y ); a list of G-adapted consumption processes for shareholders (^ ci )i = (C^ i (^ y ))i , a Gadapted consumption process for the manager c^m = C^m (^ y ), and a G-adapted price process q^ such that: 1. c^i maximizes U i (c) s.t. c is G-adapted and q^ c 2. c^m maximizes Um (c) s.t. c 2 C(^ y ) and q^ c limxt !1

t (x0 ; : : : ; xt )

=

for all (x0 ; : : : ; xt

10

1)

q^

2 Rt+ and all t:

i

(^ q y^),

(^ y ),

3. Vm (^ y ; q^) = maxy2Y Vm (y; q^) where Vm is de…ned in (6), 4. Um (C^m (^ y )) = maxy2Y Um (C^m (y)), 5.

P

i

c^i + c^m = y^ + (^ y ).

Points 1, 2 and 5 de…ne an exchange equilibrium given y^ between the manager and the shareholders with endowments (^ y ) and ( i y^)i in which the manager is constrained to consumption plans c 2 C(^ y ). Together with point 3, they de…ne a concept of equilibrium in the spirit of a production equilibrium in which the choice of the plan is determined by the manager. Point 4 implies that in equilibrium the manager has no incentive to misreport and so there is no asymmetric information between the manager and the shareholders. Notice that for simplicity of exposition we have restricted our attention to e¢ cient production plans. Later, we will restrict even further and de…ne equilibria that are unanimously supported by all shareholders.

3

The manager should not trade

In order to show that there exist contracts ( ; W ) for which the set of manager-shareholders equilibria is not empty, we …rst analyze the properties of these equilibria. Our main result is that the manager truthfully reports the chosen plan y^ only if her compensation is such that, under y^, she has no incentive to trade in the …nancial market. Hence, in a ( ; W ) equilibrium, if it exists, the manager’s consumption coincides with her compensation. We deduce from this result two properties that a ( ; W ) equilibrium should have. First, the manager chooses an equilibrium production plan y^ by maximizing the utility of her compensation over Y ; second, shareholders get the exchange equilibrium consumption when the aggregate endowment is y^ and agents own shares ( i )i of it. We show that these two properties together with the no trade result fully characterize the ( ; f0g) equilibria. These equilibria correspond to the case in which the manager has no access to the …nancial market (W = f0g) and they will be of particular interest in the analysis of Section 5. 11

We start by recalling the de…nition of an exchange equilibrium between shareholders associated to a plan that without loss of generality, we shall assume e¢ cient. De…nition 2 An exchange equilibrium associated to an e¢ cient production plan y is characterized by a set of G-adapted individual consumption processes (ci )i 2 X N and by a G-adapted price process q 2 X 0 such that 1. ci = argmax U i (c); c is G(y)-adapted and q c 2.

P

i

(q y) for all i;

ci = y:

From now on, we shall only deal with G-adapted consumption processes for shareholders. Hence, replacing the density M i by its conditional expectation with respect to G, we can assume wlog that M i is G-adapted. We then show (see Appendix 3), that an exchange equilibrium between shareholders exists and it is unique (uniqueness of the equilibrium price process is to be understood up to a multiplicative constant). The next theorem lists the properties of ( ; W ) manager-shareholders equilibria and it characterizes the ( ; f0g) equilibria. Theorem 1 Let ( ; W ) be given. Let ((^ ci )i ; c^m ; q^; y^) be a ( ; W ) managershareholders equilibrium. Then: 1. The manager does not trade, in other words c^m = (^ y ), 2. The production plan y^ maximizes Um ( (y)); 3. The pair ((^ ci )i ; q^) is a y^ exchange-equilibrium, 4. ((^ ci )i ; c^m ; q^; y^) is a ( ; W 0 ) equilibrium for any G-measurable W 0 W . In particular, it is a ( ; f0g) manager-shareholders equilibrium, 5. If W = f0g, then ((^ ci )i ; c^m ; q^; y^) is a ( ; f0g) equilibrium if and only if Assertions 2-3-4 are ful…lled. The main result of this theorem is Assertion 1, the no-trade result. Let us give a sketch of the proof. From the de…nition of an equilibrium,

12

y^ maximizes two di¤erent functions over Y . From Assertions 2 and 3 in De…nition 1, y^ solves max Um ( (y) + w); (7) y2Y

with w = c^m

(^ y ): From Assertion 4 of De…nition 1, y^ solves max Um (C^m (y)): y2Y

(8)

Condition (7) re‡ects the fact that, by choosing a given y; the manager can a¤ect her compensation (y): Condition (8) comes from the fact that by implementing a given y the manager can change her consumption pro…le C^m (y). As in equilibrium the manager should not misreport her choice, both utilities should be maximal at y^: That is, the marginal utilities of these functions at y^ must be proportional. Solving the resulting stochastic system and imposing the manager’s budget constraint, one obtains that at each date the consumption of the manager should be equal to her compensation, which is the no trade-result. Points 2 and 3 of the theorem follow immediately. Point 4 states that if the manager …nds it optimal not to trade when trading restrictions are given by W , then she would a fortiori …nd it optimal not to trade when facing stricter trading restrictions, as given by W 0 W: This is true in particular for W 0 = f0g. The observation is important as it implies that in order to characterize the compensation schemes which could implement a ( ; W ) manager-shareholders equilibrium for some W , it is su¢ cient to focus on the case W = f0g. We will use this property later on, together with the characterization in point 5, when we analyze the properties of by restricting to W = f0g.

4

Unanimous equilibrium

We now specify how, by comparing di¤erent production plans, shareholders can reach an agreement on the preferred plan. We assume that once a ( ; W ) manager-shareholders equilibrium ((^ ci )i ; c^m ; q^; y^) is reached, shareholders consider an alternative production plan y 2 Y: They take the current market price q^ as given and compare their indirect utility for y^ to the

13

indirect utility they would have obtained with y. If no shareholder would obtain a higher utility under any alternative plan y 2 Y; we say that y^ is a consensus plan. We show that the consensus plan coincides with the production equilibrium plan, we show it exists and it is unique. We then show that given a contract ( ; W ), there exists at most one unanimous ( ; W ) equilibrium that we fully describe. We start by recalling the de…nition of a production equilibrium. It di¤ers from that of an exchange equilibrium associated to a production plan introduced in the previous section in that the production plan is endogenous and it maximizes pro…t at equilibrium price. De…nition 3 A production equilibrium is a pair of a production plan y and exchange equilibrium associated to y , ((c i )i ; q ) such that pro…t at price q is maximized on Y at y , i.e. q 2 NY (y ): The following theorem establishes that the production equilibrium exists and it is unique. It also describes the characteristics of the representative agent and the equilibrium price as a function of the characteristics of this agent, a result that will be extensively used in the next section. We …rst introduce the following notations. Let =

(

2 Rn+ :

X

( i) 1

)

1

=1 ;

i

and N ( ) be de…ned by

Nt ( ) =

X

i

Mti exp(

i

1

t)

i

1

!1

:

Theorem 2 Under Assumptions (P) and (C), 1. There exists a unique production equilibrium ((c i )i ; q ; y ) and a unique vector of equilibrium utility weights ( i ) in such that i

Mti exp(

i

t)u0 (ct i ) = qt for all t and all i:

14

2. At equilibrium, there is a representative agent with instantaneous util~ and a nonnegative discount rate process ~ uniquely ity u, a density M determined by: ~ t = Nt ( ): exp( ~t t)M (9) By de…nition, the equilibrium price is proportional to the marginal utility of the representative agent at y ; which together with equation (9) gives ~ t (yt ) qt = k exp( ~t t)M

1

(10)

;

for some k > 0: In order to de…ne the consensus plan, we specify shareholders’preferences over the di¤erent plans. Shareholder i’s indirect utility of y given the price process q is de…ned by V i (y; q) = max U i (c) s.t. q c

i

(q y):

De…nition 4 Let ( ; W ) be given and ((^ ci )i ; c^m ; q^; y^) be a ( ; W ) managershareholders equilibrium. Shareholders are unanimous about y^ if and only if V i (^ y ; q^) V i (y; q^); for all y 2 Y and all i: Price taking is important to be able to de…ne a consensus plan, and it could also be derived by considering a setting with several small …rms.8 If these …rms are identical, our analysis would not be a¤ected. It may easily be shown that (11) V i (y; q) = i (M i ; i ; q) i q y ; where for a given process q, i (M i ; i ; q) is a constant that depends on the characteristics of the shareholder (M i ; i ) and on q. Hence, given q^; shareholder i prefers y^ to y if and only if q^ y^

q^ y; for all y 2 Y:

(12)

Notice that Equation (12) does not depend on individual i’s characteristics 8

See e.g. Grossman and Stiglitz (1980) for a discussion on price taking behaviors and unanimity.

15

and it corresponds to pro…t maximization under the equilibrium prices. From Theorem 1, Assertion 4, and (12), we obtain the following proposition: Proposition 3 Let ( ; W ) be given and ((^ ci )i ; c^m ; q^; y^) be a ( ; W ) managershareholders equilibrium. If shareholders are unanimous about y^, then ((^ ci )i ; q^; y^) is the production equilibrium ((c i )i ; q ; y ) and ((c i )i ; (y ); q ; y ), the ( ; W ) unanimous manager-shareholders equilibrium.

5

Optimal contract

We now consider the possibility to design a contract ( ; W ) such that the associated ( ; W ) equilibrium implements the unanimous production plan. If this is the case, we say that ( ; W ) is an optimal contract and that is an optimal compensation.

5.1

The manager should face trading restrictions

We …rst show that if W = X G (the manager faces no restrictions to trading in the …nancial market), unanimous equilibria require that the compensation is linear, as de…ned as follows. De…nition 5 We say that the compensation is linear if there exists a > 0 such that (13) t (y) = yt for all t and all y: At the same time, under linear compensation, it is required that the manager shares the same belief and discount rate as the representative shareholder at equilibrium. When shareholders are heterogeneous, however, it is in general not possible to …nd a manager with such characteristics. This impossibility result is illustrated in the next theorem in the two following dual situations: when all agents (including the manager) have the same belief and heterogeneous discount rates, and when all agents (including the manager) have the same discount rate and heterogeneous beliefs (and 6= 0). Theorem 4 Assume that the manager has access to a complete market. Then 16

1. If ((c i )i ; (y ); q ; y ) is a ( ; X G ) equilibrium, then ear and ( m ; M m ) should satisfy exp(

m

should be lin-

~ t for all t 2 T; t)Mtm;G = exp( ~t t)M

~ t and which gives Mtm;G = M

m

(14)

= ~t for all t 2 T:

2. When all agents (including the manager) have the same belief M and i 6= j for some (i; j), there exists no constant discount rate m such that equation (14) holds with M m;G = M . 3. If 6= 0; when all agents (including the manager) have the same discount rate and M i 6= M j for some (i; j), there exists no belief M m such that equation (14) holds with m = . To have an intuition of why the compensation scheme should be linear, notice …rst that since the manager maximizes her utility from consumption Um (^ cm ) in complete markets, her marginal utility at the equilibrium consumption (y ) should be proportional to equilibrium prices. 0 ( (y )) = q : At the same time, the manager maximizes That is, Um on Y her utility from compensation Um ( (y)) at y which implies that 0 0 ( (y )) (y ) 2 NY (y ): As q 2 NY (y ) and NY (y ) reduces to a halfUm 0 0 0 0 line, Um ( (y )) (y ) = ~ Um ( (y )). It follows that (y ) is constant and so is linear. In this way, as the manager has the same possibility to trade as shareholders, she is also given an endowment that is proportional to the output of the …rm. We also show that, when the manager is paid with a linear compensation, she chooses the consensual plan only if she has the same characteristics as the representative shareholder, as de…ned in (9). However, in general, such characteristics cannot be found in an agent with proper beliefs and a constant discount rate. To see this most simply, suppose that shareholders and the manager share the same belief. In this case, there is no constant discount rate which allows to satisfy conditions (9). In particular, a manager paid with linear compensation should have a declining time preference rate since aggregate consumption in the distant future is mostly in the hands of more patient individuals (see e.g. Gollier and Zeckhauser (2005) for a similar insight in 17

a setting with no uncertainty). Similarly, when all shareholders and the manager have the same constant discount rate but di¤erent beliefs and 6= 0, there is no probability distribution over the states of the world which gives to the manager the characteristics of the representative agent (see Jouini and Napp (2007)). In the Online Appendix, we extend this impossibility result by showing that when beliefs and discount rates are heterogeneous the manager cannot have the same characteristics as the representative shareholder at equilibrium, as long as T is su¢ ciently large.9

5.2

Optimal compensation

In the previous subsection, we have shown that it is in general not possible to reach a unanimous manager-shareholders equilibrium when markets are complete for the manager, unless shareholders are homogeneous. We now restrict attention to the case where the manager has no access to the market, W = f0g. We notice that, from Theorem 1, such compensation schemes are the only candidate schemes for a unanimous equilibrium when the manager faces milder trading restrictions, W 6= f0g. From Proposition 3, we know that the equilibrium is of the following type ((c i )i ; (y ); q ; y ); where has still to be determined. From Theorem 1, we have that y is implemented as part of a ( ; f0g) equilibrium if and only if it maximizes Um ( (y)) on Y: Combining these results, we have the following: Proposition 5 Given ( ; f0g), ((c i )i ; (y ); q ; y ) is a unanimous managershareholders equilibrium if and only if y maximizes Um ( (y)) on Y . 5.2.1

Markov compensations

In what follows, we restrict our attention to the case of Markov compensations, which are de…ned as follows: 9

Exploring how the horizon impacts the representative agent and the manager characteristics requires developing new de…nitions and new conditions and those are detailed in the Online Appendix.

18

De…nition 6 We say that the manager’s compensation is Markov if for every t, there exists a function t : R+ ! R+ such that t (y)

=

t (yt )

for all y 2 X:

When the manager’s compensation is Markov, the compensation at some date only depends on current production and not on its whole history. We can then write the …rst order conditions for y to maximize Um ( (y)) on Y as follows: Proposition 6 Let be Markov. If ((c i )i ; (y ); q ; y ) is a unanimous ( ; f0g) manager-shareholders equilibrium, then for some > 0 we have 0 t (yt ) exp(

m

t)Mtm ;G u0 ( t (yt )) =

~ t u0 (y ) for all t: exp( ~t t)M t

(15)

Conditions (15) are necessary and su¢ cient if, for every t, the map u( t ) is concave. As the manager maximizes the utility of her compensation at y , from the …rst order condition, the marginal utility of her compensation at y is in NY (y ). As y is the production equilibrium plan, q 2 NY (y ). Hence the marginal utility of her compensation at y is proportional to the equilibrium price q : From Equation (10) in Theorem 2, that is also proportional to the marginal utility of the representative agent at equilibrium, as expressed in Equation (15). When the utility of the compensation is concave, the …rst order conditions are su¢ cient. In order to characterize Markov compensations, we need to introduce some further notations. For a given date t and production yt , let Ht (yt ) be the -…eld generated by yt . We assume that Ht (yt ) is independent of y for any y 2 E (Y ) and denote it Ht . We consider the following: Assumption (M) Nt ( ) is adapted to Ht for all t: Mtm;G

19

Assumption (M) is equivalent to assume that for every t, there exists a function gt : R+ ! R+ such that exp(

m

t)

Nt ( ) = gt (yt ): Mtm;G

The stochastic term gt (yt ) measures the discrepancy between the belief and discount rate of the representative agent and those of the manager at t. For example, when all agents (including the manager) are homogenous, we have gt (yt ) = 1; when agents (including the manager) have the same beliefs but di¤erent discount rates, gt (yt ) is a constant which depends on t. From point 3 in Assumption (P), we may then rewrite (15) for every z 2 R+ and t as 0 1 = gt (z)z 1 ; (16) t (z) t (z) with > 0. From (16), we observe that t is linear if and only if gt = 1 for every t. Hence, gt can be seen as a measure of the required non-linearity of t . Under suitable integrability conditions detailed below, integrating (16) with respect to z and using condition (4) for > 0 and condition (5) for < 0, we can fully characterize the optimal compensation scheme as: (

t (z)

= t (z) =

Rz gt (u)u 1 du when R0 1 gt (u)u 1 du when z

> 0; < 0:

(17)

Let us next detail what we mean by suitable integrability conditions. For > 0, the integral of gt (u)u 1 must be convergent at 0 as t (0) = 0 and divergent at in…nity as t (1) = 1. For < 0, the integral of gt (u)u 1 must be divergent at 0 as t (0) = 1 and convergent at in…nity as t (1) = 0. In order to ensure this is the case, one may for example assume that there exists some " > 0 such that ( limz!0 gt (z)z " = 0 and gt (z)z is bounded away from 0 at 1 when > 0; limz!1 gt (z)z " = 0 and gt (z)z is bounded away from 0 at 0 when < 0: (18)

20

Finally, the …rst order conditions in (15) are su¢ cient when u( t ) is concave, and this is the case if and only if z ! gt (z)z

1

is non increasing.

(19)

Let us summarize our …ndings in the following theorem. Theorem 7 1. Suppose there exists a unanimous manager-shareholders equilibrium for ( ; f0g): (a) Then

is Markov if only if Assumption (M) is ful…lled.

(b) If is Markov, then, for every t, t is increasing and it is determined by (17) up to a multiplicative constant. (c) If is Markov, then, for any W; any other (Markov or nonMarkov) compensation that implements a unanimous managershareholders equilibrium is proportional to : 2. Assume (18) is ful…lled and let = ( t )t be determined by (17) and such that u( t ) is concave for every t. Then is a Markov compensation that implements a unanimous manager-shareholders equilibrium when W = f0g. The …rst point of the theorem characterize Markov compensation schemes when W = f0g; the second point provides a su¢ cient condition for the existence of a family of compensation schemes that implement a unanimous manager-shareholders equilibrium when W = f0g: This condition is weaker than requiring that t is concave. We now specify a set of exogenous conditions on the beliefs of shareholders and of the manager which imply Assumption (M). We assume that, at each date, shareholders’and manager’s beliefs only depend on the current production and not on its history. More precisely, we assume that the following assumption is veri…ed: Assumption (N) Mti and Mtm;G are adapted to Ht for all t and i:

21

Under Assumption (N), Mti ; Nt ( ) and Mtm;G are Ht -measurable and for every t, there exist functions mit : R+ ! R+ ; nt : R+ ! R+ and mGt : R+ ! R+ such that Mti = mit (yt ); Mtm;G = mGt (yt ) and Nt ( ) = nt (yt ). We may therefore rewrite (17) replacing gt by mntG(u) : Moreover, the utility (u) t

of if

t

is concave if and only if z ! z!

mGt (z) 1 z mit (z)

nt (z) z mG t (z)

1

is decreasing. This is veri…ed

is increasing for every t and every i:

(20)

We then have the following: Corollary 8 Assume (N), (18) and (20). There exists a unique compensation (up to a multiplicative constant) which implements a unanimous manager-shareholders equilibrium when W = f0g and it is given by Equations (17). In the next analysis, we wish to highlight some properties that the compensation schemes de…ned by Equations (17) should have. For this purpose, we …rst focus on a setting in which shareholders have the same beliefs but di¤erent time preferences. We then consider the case in which discount rates are homogenous but beliefs are heterogeneous. 5.2.2

Common beliefs

Suppose that shareholders and the manager have identical beliefs, M i = M m for all i. We remark that Assumption (M) is trivially ful…lled in this case, irrespective of whether discount rates are homogeneous or heterogeneous. It follows from Theorem 7 that the optimal compensation scheme must be Markov. Suppose in addition that there is no heterogeneity in time preferences; that is, i = m for all i. In this case gt (z) = 1 and from (17), we obtain for some > 0; t (z) = z for every z 2 R+ and t. We can state the following result: Corollary 9 If i = m and M i = M m for all i, there exists a unanimous manager-shareholders equilibrium for ( ; f0g) if and only if is linear.

22

Consider then the e¤ect of heterogeneous time preferences, while keep~ = M m and ing M i = M m for all i. We then have M X

exp( ~t t) =

i

exp(

i

1

t)

1

i

!1

:

In the notation of the previous subsection, gt (z) = exp (( m ~t ) t) and from (17), t (z) = exp( 1 ( m ~t ) t)z for every z 2 R+ and t. We can state the following result: Corollary 10 Suppose that M i = M m for all i and i 6= j for some i and j. There exists a unanimous manager-shareholders equilibrium for W = f0g if and only if veri…es t (z)

1 = exp( (

with exp( ~t t) = If

m

X

i

m

exp(

~t ) t)z;

i

1

1

t)

:

1

> mini i , t ! gt (z) is increasing and t ! > 0 and decreasing for < 0.

t (z)

is increasing for

The representative shareholder has a discount rate which declines over time, thereby attaching an increasing value to future production. The manager has a constant discount rate and in order to induce her to act as if she had the representative discount rate, the marginal utility of her compensation should increase with time. This can be achieved by letting t (z) increase. 5.2.3

Heterogeneous Beliefs

We now highlight the e¤ect of heterogeneous beliefs on the shape of the compensation at a given date. We assume that i = m for all i and we impose a further assumption which implies some boundary properties of gt . We say that agent i is more optimistic than agent j (or that agent j is more pessimistic than agent i) with respect to y if for every t Mti Mti = 1 and lim = 0: yt !1 M j yt !0 M j t t lim

23

(21)

We assume that there exists some agent i more optimistic than the manager and some agent j more pessimistic than the manager with respect to y . In particular, for every t, there exists a shareholder i and a shareholder j such that Mti Mtm;G lim = 1 and lim (22) j = 0. yt !1 M m;G yt (!)!0 M t t It is then easy to check that, for any given date t; Nt ( ) Nt ( ) = lim = 1: m;G yt !0 M m;G yt !1 M t t lim

(23)

Equivalently, in the notation of the previous subsection, we further have lim gt (z) = lim gt (z) = 1:

z!0

z!1

(24)

Therefore, the compensation should increase the weight that the manager attaches to extreme realizations of the production. This can be expressed by considering the compensation rate at date t; de…ned by the function t : (0; 1) ! R+ such that t (z)

=

t (z)

z

; z 2 (0; 1):

We can show the following result. Corollary 11 Assume i = m for all i and M i 6= M j for some i and j: Suppose Assumption (M), (18) and (22) hold. If there exists a unanimous manager-shareholders equilibrium for ( ; f0g), then the compensation rate at date t should verify limz!0 t (z) = limz!1 t (z) = 0; for < 0; limz!0 t (z) = limz!1 t (z) = 1; for > 0:

(25)

Under Assumption (22), the manager acts as if she had the representative belief only if the ratio of her instantaneous marginal utility and that of the representative agent is very large when production is either very large or very low. As we show, this can be achieved by having that t (z) ! 1 as z ! 0 or z ! 1; which gives the properties of t (z) depending on the sign of as stated in (25). 24

6

Example

We consider a model with two dates 0 and t. Production takes place only at date t and the production set is: Ya;b;

0

n = exp

( )t +

p

o 0 ;

te x :

2 where x e N (0; 1), ( ) = a b ( 0 ) ; and a, b and 0 are given positive constants. The value of is set by the manager and a given choice of p te x :We assume there generates the production plan y = exp ( )t + are two shareholders with the same discount rate = 0 and CRRA utility as in (2) with = 1: Shareholders only consume at time t and they have heterogeneous beliefs indexed by . An agent of type believes that x e N ( ; 1) and we assume 2 f ; g with > 0. We denote with the endowment (that is, the proportion of stocks) of shareholder of type . In order to guarantee the existence of a production equilibrium, we impose that 1 1 (1 2b) + (26) 2 < 0: 2 1 + exp 2

Proposition 12 Under (26), the production equilibrium is given by (y ; p ; x ; x ) p with y = exp ( )t + te x and =

p = x = where 2

exp(

2b 2b + 1

exp

0

)2

(e x

+ exp

4

(e x + )2 4

4

)2

(e x

+ exp

4

(27)

1) ;

2

(y )

2

;

)2

(e x

exp exp

1 +p (2 t 2b + 1

(e x+ ) 2 4

y ;

2f ;

g;

is the nonnegative solution of p

2

t)

+ exp(

2

)(2

1)+exp(

p

t)(

1) = 0: (28)

We notice from (27) that the unanimous plan induces a larger and so a larger exposure to the random variable x e when shareholder of type is 25

very optimistic ( is large) and when he has many shares ( is large). We look at which compensation scheme would lead the manager to take the consensus plan. Let us assume that the manager has a belief which coincides with the objective one (e x N (0; 1)) and that shareholders have the same endowment, 2 = 1: According to Theorem 7, the compensation is given by Equation (17) with gt (y) = exp 4k

2

t exp( 2k ( )t)y 2k +exp(2k ( )t)y

2k

+exp 2k

2

t ;

which leads to t (y)

=

y 1 2k+1

exp(2k( ( )

2 ) t)y 2k

1 2k 1

( ))t) y 2k (29) where is an arbitrary positive scalar. In order to ensure that the resulting compensation is non-negative for all y 2 (0; 1), we need to impose p t: One can see from (29) that u( t ) = ( t ) 1 is concave as it is the sum of three concave functions, which gives the existence and uniqueness of the compensation scheme up to a constant. We can then state the following: 1+

exp(2k (

2

p t and the manager corProposition 13 Suppose that 2 = 1; rectly believes that x e N (0; 1): There exists a unique compensation up to a constant which implements a manager-shareholders unanimous equilibrium and it is given by Equation (29). It is immediate to verify that limz!0 gt (z) = limz!1 gt (z) = 1 and so the compensation in (29) has the properties highlighted in Corollary 11: the compensation rate (y) goes to zero as y ! 0 or y ! 1:

7

Conclusion

We have shown that, if the manager is allowed to trade as shareholders, it is in general not possible to achieve a unanimous manager-shareholders equilibrium. We have then de…ned some properties of the optimal compensation scheme when shareholders can trade in a complete asset market while the manager is prevented from trading.

26

;

At the other extreme, one may assume that nor shareholders nor the manager have access to a …nancial market. In this case, if there are at least two agents with di¤erent beliefs or discount rates, it is easy to show that one cannot …nd a unanimous production plan. One can then de…ne weaker concepts of consensus among shareholders (e.g. Pareto dominance) and analyze which compensation schemes would lead to the desired plans. It would also be interesting to consider intermediate cases in which shareholders have access to an incomplete market or in which the manager may face milder trading restrictions.10 We have shown that the set of compensation schemes identi…ed when the manager has no access to the market are the only candidate schemes for a unanimous equilibrium even under milder trading restrictions. More generally, one could investigate which minimal trading restrictions would allow to achieve a managershareholders equilibrium. Another natural extension would be to introduce multiple …rms. This would allow to consider a richer contracting space, both in terms of trading restrictions and in terms of compensation, and possibly to investigate issues of competition among …rms for managerial talent. One could also consider more general forms of endowment for shareholders. These are in our view very interesting avenues for future research.

References Acharya, V. V. and Bisin, A. (2009), ‘Managerial hedging, equity ownership, and …rm value’, RAND Journal of Economics 40(1), 47–77. Bhamra, H. S. and Uppal, R. (2014), ‘Asset prices with heterogeneity in preferences and beliefs’, Review of Financial Studies 27(2), 519–580. Bhattacharya, U. (2014), ‘Insider trading controversies: A literature review’, Annual Review of Financial Economics 6(1), 385–403. 10

In Acharya and Bisin (2009) for example, the manager can trade indices and so hedge against aggregate shocks but not against his own …rm. In Bisin, Gottardi and Rampini (2008), the manager can hedge against his compensation and shareholders can monitor at a cost the trading of the manager. See also the discussion in Fischer (1992) on various trading restrictions.

27

Bisin, A., Gottardi, P. and Rampini, A. (2008), ‘Managerial hedging and portfolio monitoring’, Journal of the European Economic Association 6(1), 158–209. Bisin, A., Gottardi, P. and Ruta, G. (2014), ‘Equilibrium corporate …nance and intermediation’. Bolton, P., Scheinkman, J. and Xiong, W. (2006), ‘Executive compensation and short-termist behaviour in speculative markets’, Review of Economic Studies 73(3), 577–610. Crès, H. and Tvede, M. (2014), ‘When markets never fail: Reciprocal aggregation and the duality between persons and groups’. Cvitani´c, J., Jouini, E., Malamud, S. and Napp, C. (2012), ‘Financial markets equilibrium with heterogeneous agents’, Review of Finance 16(1), 285–321. Dana, R.-A. (1995), ‘An extension of milleron, mitjushin and polterovich’s result’, Journal of Mathematical Economics 24(3), 259–269. Demichelis, S. and Ritzberger, K. (2011), ‘A general equilibrium analysis of corporate control and the stock market’, Economic theory 46(2), 221– 254. Detemple, J. and Murthy, S. (1994), ‘Intertemporal asset pricing with heterogeneous beliefs’, Journal of Economic Theory 62(2), 294–320. Diamond, D. W. and Verrecchia, R. E. (1982), ‘Optimal managerial contracts and equilibrium security prices’, Journal of Finance pp. 275– 287. Fischer, P. E. (1992), ‘Optimal contracting and insider trading restrictions’, Journal of Finance 47(2), 673–694. Gabaix, X. and Landier, A. (2008), ‘Why has ceo pay increased so much?’, Quarterly Journal of Economics 123(1), 49–100. Gollier, C. and Zeckhauser, R. (2005), ‘Aggregation of heterogeneous time preferences’, Journal of Political Economy 113(4), 878–896. 28

Grossman, S. J. and Stiglitz, J. E. (1980), ‘Stockholder unanimity in making production and …nancial decisions’, Quarterly Journal of Economics 94(3), 543–566. Jensen, M. C. and Meckling, W. H. (1976), ‘Theory of the …rm: Managerial behavior, agency costs and ownership structure’, Journal of Financial Economics 3(4), 305–360. Jouini, E., Marin, J.-M. and Napp, C. (2010), ‘Discounting and divergence of opinion’, Journal of Economic Theory 145(2), 830–859. Jouini, E. and Napp, C. (2007), ‘Consensus consumer and intertemporal asset pricing with heterogeneous beliefs’, Review of Economic Studies 74(4), 1149–1174. Kadan, O. and Swinkels, J. M. (2008), ‘Stocks or options? moral hazard, …rm viability, and the design of compensation contracts’, Review of Financial Studies 21(1), 451–482. Leland, H. E. (1992), ‘Insider trading: Should it be prohibited?’, Journal of Political Economy 100(4). Magill, M. and Quinzii, M. (2002), Theory of incomplete markets, Vol. 1, MIT press. Magill, M., Quinzii, M. and Rochet, J.-C. (2015), ‘A theory of the stakeholder corporation’, Econometrica 83(5), 1685–1725. Murphy, K. (2012), ‘Executive compensation: Where we are, and how we got there’, Handbook of the Economics of Finance. Elsevier Science North Holland . Murphy, K. J. (1999), ‘Executive compensation’, Handbook of labor economics 3, 2485–2563. Xiong, W. and Yan, H. (2010), ‘Heterogeneous expectations and bond markets’, Review of Financial Studies 23(4), 1433–1466.

29

8

Appendix 1. Proofs

Proof of Theorem 1. Let ( ; W ) be given and ((^ ci )i ; c^m ; q^; y^) be a ( ; W ) manager-shareholders equilibrium. From Assertions 2 and 3 in De…nition 1, the manager solves the following problem: max max Um (c) s.t. c 2 C(y) and q^ c c

y2Y

q^

(y):

As c^m solves Assertion 2 and Um is increasing, c^m = (^ y ) + w for some w 2 W and q^ c^m = q^ (^ y ). Hence q^ w = 0. Therefore any consumption process of the form, c = (y) + w for some y 2 Y veri…es q^ c = q^ (y) and has a lower utility than c^m . Hence y^ maximizes Um ( (y) + w) on Y: Since Um is increasing: max y2Y

T X

m

exp(

t)E[Mtm u( t (y0 ; : : : ; yt )+wt )] = Um ( (^ y )+w) = Um (C^m (^ y )):

s=0

From the …rst order conditions, there exists a G-adapted process p^ 2 NY (^ y) such that T X

Et

s=t

@ s (^ y0 ; : : : ; y^s ) exp( @yt

As the processes p^; T X

EtG

s=t

m

s)Msm u0 (^ cm;s ) = p^t for all t:

(y) and c^m are G-adapted, we have

@ s (^ y0 ; : : : ; y^s ) exp( @yt

m

s)Msm;G u0 (^ cm;s ) = p^t for all t;

(30)

in which EtG denotes the conditional expectation with respect to Gt and Msm;G is the conditional expectation of Msm with respect to Gs . Furthermore from Assertion 4 of De…nition 1, max y2Y

T X

exp(

m

t)E[Mtm u(C^m;t (y0 ; : : : ; yt )] = Um (C^m (^ y ))]:

t=0

30

From the …rst order conditions, there must exist a G-adapted process in NY (^ y ), hence proportional to p^ such that T X

EtG

s=t

"

@ C^m;s (^ y0 ; : : : ; y^s ) exp( @yt

for some every t: T X s=t

m

#

cm;s ) = p^t for all t; (31) s)Msm;G u0 (^

> 0: From Equations (30) and (31), we thus obtain that for

EtG

"

@(C^m;s @yt

s)

(^ y0 ; : : : ; y^s ) exp(

m

#

s)Msm;G u0 (^ cm;s ) = 0:

(32)

In particular, at date T , @ C^m;T @ T (^ y0 ; : : : ; y^T ) = (^ y0 ; : : : ; y^T ): @yT @yT As y^ takes all possible values in (0; 1)T , we have @ T @ C^m;T (x0 ; : : : ; xT ) = (x0 ; : : : ; xT ); 8x 2 RT++1 : @xT @xT Integrating with respect to xT , we obtain C^m;T (x0 ; : : : ; xT ) =

T (x0 ; : : : ; xT )

+ H(x0 ; : : : ; xT

1 );

(33)

for some functions H : RT+ ! R. From (4), we obtain that H = 0 and C^m;T (x0 ; : : : ; xT ) =

T (x0 ; : : : ; xT );

8(x0 ; : : : ; xT ) 2 RT+ :

(34)

Similarly, considering next Equation (32) at t = T 1, we obtain C^m;T 1 = ^ T 1 . By backward induction, we obtain that Cm;t = t for all t and using the equality q^ C^m = q^ (^ y ), that C^m = : To prove Assertion 2, from item 2 of De…nition 1 and from Assertion 2 of Theorem 1, we have Vm (^ y ; q^) = Um (C^m (^ y )) = Um ( (^ y )): As

(y) is a feasible consumption process for the manager when she is 31

given y, Vm (y; q^)

Um ( (y)), therefore

Vm (^ y ; q^) = Um ( (^ y ))

Vm (y; q^)

Um ( (y)); 8y;

proving Assertion 2. The proof of Assertion 3 which is obvious is skipped. To prove Assertion 4, it is obvious that if ((^ ci )i ; c^m ; q^; y^) is a ( ; W ) equilibrium, properties 1, 2, 4 and 5 of a ( ; W 0 ) equilibrium are satis…ed for W 0 W . Since we deal with two sets of possible constraints, let us 0 denote respectively by VmW and VmW , the indirect utility functions of the manager respectively associated to the set W and to the set W 0 : We have 0

VmW (y; q^) = max Um (c); c 2 f (y) + W 0 g; q^ c

q^

(y)

From Assertion 1 of Theorem 1 and De…nition 1, we have VmW (^ y ; q^) = Um ( (^ y )) As f (y)g

f (y) + W 0 g VmW (y; q^)

f (y) + W g, 0

VmW (y; q^)

y ; q^) Hence Um ( (^ y )) = VmW (^ 0 y ; q^) and VmW (^ y ; q^) = VmW (^ 0

VmW (^ y ; q^)

VmW (y; q^); 8 y 2 Y:

Um ( (y)); 8 y 2 Y: 0

y ; q^) VmW (^

VmW (y; q^)

Um ( (^ y )) and Um ( (^ y )) =

0

VmW (y; q^); 8 y 2 Y;

proving 4 for W 0 . To prove Assertion 5, let be given and let ((ci )i ; (y); q; y) satisfy Assertions 1-2-3 of Theorem 1. Let us show that it is a ( ; f0g) equilibrium. Assertions 1 and 2 of De…nition 1 are obviously veri…ed. As W = f0g, Vm (y; q) = Um ( (y)) for all y 2 Y . From Assertion 2 of Theorem 1, Vm (y; q) = Um ( (y)) Um ( (y)) = Vm (y; q) for all y 2 Y , proving Assertion 3 of De…nition 1. From Assertions 1 and 2 of Theorem 1, Um (Cm (y)) = Um ( (y)) and Um (Cm (y)) = Um ( (y)) Um ( (y)) = Um (Cm (y)) proving Assertion 4 of De…nition 1. The converse result is obvious. Proof of Theorem 2. The proof of Assertion 1 may be found in the

32

Online Appendix. To prove the second, from Equation (43), the representative agent’s utility at equilibrium is N ( ) u(y ). Let us de…ne (~t ) and ~ t by induction as follows M Nt exp (~t+1 (t + 1)) ~ t+1 = Nt+1 ( ) exp (~t+1 (t + 1)) : = and M exp (~t t) Et [Nt+1 ( )] ~ t is a martingale and we The process (exp ( ~t t)) is predictable and M ~ t = Nt ( ): Furthermore, this decomposition is unique. have exp( ~t t)M ~ 1 = exp( ~2 t)M ~ 2 , taking the expectation for t = 1; Indeed, exp( ~1t t)M t t t 1 2 1 ~ ~ 2 : If we take now the expectation we obtain ~ = ~ and from there M = M 1

1

1

2

for t = 2 conditional to date 1; we have h i h i ~ 1 = E1 exp( 2~1 )M ~ 1 = E1 exp( 2~2 )M ~2 exp( 2~12 )M 1 2 2 2 2 h i ~ 22 = exp( 2~22 )M ~ 12 : = exp( 2~22 )E1 M ~ 2 : By ~ 11 = M ~ 12 ; we obtain ~12 = ~22 and from there M ~1 = M and since M 2 2 1 2 1 2 ~ ~ induction, we get M = M and ~ = ~ : Proof of Proposition 4. 1. As markets are complete, the manager solves T X max E[exp( m t)Mtm u(ct )]; q c q (y ); t=0

and furthermore (y ) is the optimal consumption process. Therefore, for every t, the marginal utility of the manager at (y ) is proportional to the equilibrium price q : exp( with T X s=t

m

t)Mtm;G u0 ( t (y0 ; : : : ; yt )) = qt for all t,

(35)

> 0. Equation (35) can be rewritten as EtG [

@Zs (y ; : : : ; ys ) exp( @yt 0

m

s)Msm;G u0 ( s (y0 ; : : : ; ys ))] = qt for all t;

(36) where Zs (y0 ; :::; ys ) = ys : As q 2 NY (y ); q is proportional to the p^ introduced in the proof of Theorem 1. Together with Equation (30), (36)

33

implies that, for all t, T X

EtG

@(

Zs )

s

@yt

s=t

m

(y0 ; : : : ; ys ) exp(

s)Msm;G u0 ( t (y0 ; : : : ; ys )) = 0;

where > 0. Using the same arguments as in the proof of Theorem 1, we obtain that the compensation of the manager should be linear. Using (35) and (10), as u0 is homogeneous, we get that exp(

m

~ t: t)Mtm;G = exp( ~t t)M

~ t are martingales and ~t is predictable, we have m = ~t Since Mtm;G and M ~ by the uniqueness of such a decomposition as seen for all t and M m;G = M in Theorem 2. 2. If agents have homogeneous beliefs, we should have exp(

m

t) = exp( ~t t) =

X

i

exp(

i

1

t)

1

1

:

1 1 P i It is easy to check that 1t ln ( exp( i t)) 1 is an increasing function as far as there exists (i; j) such that i 6= j : There is then no solution to the equation above. 3. If agents have homogeneous discount rates, we should have

M m;G =

X

i

i

Mi

1 1

!1

:

1 1 P i i 1 but, as shown in (Jouini and Napp (2007)), ( M ) is not i m a martingale unless = 0. There is then no belief M solution to the equation above for 6= 0: Note that a direct proof can be constructed by adapting inequality (42) below in the case < 0 and by using inequalities (38) and (40) in the case > 0: Proof of Theorem 7. Suppose that there exists a unanimous managershareholders equilibrium for ( ; f0g). Let us …rst prove 1a. If is Markov, then it ful…lls (15), hence, for every t, exp( m t) NMt (m;G) is a function of yt t and Assumption (M) is satis…ed. Conversely assume that Assumption (M)

34

is ful…lled. As T X

EtG

s=t

for some

is an optimal compensation, it veri…es

@ s (y ; :; ys ) exp( @yt 0

m

s)Msm;G u0 ( s (y0 ; :; ys )) = Nt ( )u0 (yt );

> 0 and for all t. At date T , we must have: u0 (

T (y0 ; ; :; yT ))

From Assumption (M),

@ T NT ( ) 0 (y0 ; ; :; yT )) = u (yT ): @xT MTm;G

NT ( ) 0 u (yT ) MTm;G

a function f : R+ ! R+ , such that

is HT measurable, hence there exists NT ( ) 0 u (yT ) MTm;G

= f (yT ). As y takes all

possible values in (0; 1)T , we thus have u0 (

T (x0 ; ; :; xT ))

@ T (x0 ; : : : ; xT ) = f (xT ) for all x 2 RT++1 : @xT

Hence, u(

T (x0 ; ; :; xT )))

= h(xT ) + g(x0 ; :::; xT

1 );

for some functions h : R+ ! R and g : RT+ ! R. For > 0; from (4), u( T (x0 ; :::xT 1 ; 0)) = 0, hence h(0) + g(x0 ; :::xT 1 ) = 0 which implies that g(x0 ; :::; xT 1 ) is constant. Therefore u( T (x0 ; :::; xT )) and T (x0 ; :::; xT ) are functions of xT as was to be proven. For < 0; a similar argument may be given by using xT = 1 and (5). Let us now consider time T 1. As T is only a function of xT , we have MTm;G1

@ T @yT

1 1

(y0 ; : : : ; yT

1 ) exp(

m

(T 1))u0 (

T 1 (y0 ; : : : ; yT 1 ))

= NT

1(

Using the same argument as for time T , we obtain that T 1 only depends on xT 1 . By backward induction, one obtains that for every t, t only depends on xt which shows that is Markov. This ends the proof of 1a. To prove 1b, if is Markov, as shown in the text, t veri…es (16) for every t, hence 0t > 0. Integrating (16), we obtain that is determined by (17) up to a constant. To prove 1c, from 1b, there is a unique Markov compensation up to a constant that implements the unanimous manager-shareholders’equilibrium determined by (17). Furthermore from 1a, Assumption (M) is ful…lled. 35

)u0 (yT

1 ):

Therefore if there exists another possibly non-Markov compensation that implements the unanimous manager-shareholders’equilibrium, then from 1a, it should be Markov, hence proportional to . To prove Assertion 2, let = ( t )t be determined by (17) and assume that u( t ) is concave for every t. Then t veri…es (16) for every t and therefore it veri…es (15) which is a necessary and su¢ cient condition for to implement the unanimous manager-shareholders’equilibrium as u( t ) is concave for every t. Proof of Corollary 11. Assume …rst that < 0. From (24) and (17), for any A > 0 and (z; z) with z < z large enough (or small enough), we have Z z Z z 1 gt (u)u du A u 1 du = A(z z ); t (z) = t (z) z

z

hence, A(z

z )

t (z)

t (z)

t (z)

:

Taking z = 2z, we obtain for z large enough or small enough: t (z)

A(1

2 )z :

Hence limz!0 t (z) = limz!1 t (z) = 1 and limz!0 t (z) = limz!1 t (z) = 0. Similarly, for > 0, from (24) and (17), for any A > 0 and (z; z) with z < z large enough (or small enough), we have t (z)

t (z) =

Z

z

gt (u)u

1

du

A

z

Z

z

u

1

du = A(z

z

hence t (z)

A(z

z ):

Taking z = 2z, we obtain for z large enough or small enough t (z)

and therefore limz!0

t (z)

A 1

= limz!1

36

1 2 t (z)

= 1.

z ;

z );

Proof of Proposition 12. We have p =

2

M u0 (x ; ) = M u0 (x

;

) and y = x

;

+x

;

and (p ; x ; ; x ; ) satis…es the …rst-order conditions for utility maximization as well as the market clearing condition. Note that 2 corresponds to the utility weight of the agent while the equilibrium utility weight of the agent has been normalized to one. We need to check that the budget constraint is also satis…ed, i.e. E [p y ; ] = E [p y ]. After simple calculations, this constraint appears to be equivalent to (28). It is also straightforward to show that (28) admits only one positive solution. Now, let us …nd such that (y ; p ; x ; ; x ; ) is a production equilibrium. Since (p ; x ; ; x ; ) is already a y -exchange equilibrium, we only have to take care of the pro…t maximization constraint. For this purpose let us de…ne g( ; ) = E [p y ] and let us …nd such that g ( ; ) = 0 that can be rewritten as =

1 2b s0 + p (1 2b + 1 t 2b + 1

2

):

For such a ; y = y satis…es the …rst order condition for pro…t maximization. Under condition (26), we can show that g ( ; ) is positive for < and negative for > which means that g( ; ) reaches its maxis a production equilibrium, where imum for = and y ; p ; x ; x p = p ;x = x ; ;x = x ; .

37

9

Online Appendix

Appendix 2. Trading Restrictions for Large T Proposition 14 Assume that the manager has access to a complete market. When T is su¢ ciently large, there is no belief M m and constant discount rate m such that equation (14) holds unless i = j and M i = M j for all (i; j). Proof. Let us …rst give a more precise meaning to this claim. For this purpose, we should be able to consider di¤erent possible horizons. Instead of de…ning …rst the set of dates, let us de…ne …rst the …ltration as well as the beliefs. The …ltration (Fk )k=0;:::;K is given and agent i beliefs are given by (Mki )k=0;:::;K : We only consider the case with heterogeneous beliefs since the case with homogeneous beliefs has already been treated in 2. There exists then k0 such that Mki0 6= Mkj0 for some (i; j): For a given set of dates T = ft0 ; t1 ; :::; tK g ; Fk corresponds to the information structure at date tk and Mki describes the belief of agent i at that date. We have the following result. Claim 15 Let the …ltration (Fk )k=0;:::;K and agent’s beliefs (Mki )i=1;:::;N ;k=0;:::;K be given. For all t > 0; there exists T such that, for all set of dates T = ft0 ; t1 ; :::; tK g with t0 = 0, tk0 t and tK T ; there exists no m m constant discount rate and belief M such that exp( m t)Mtm;G = ~ t ; for all t 2 T: exp( ~t t)M Let us prove our result by contradiction and let us assume that there exists t > 0 such that, for all integer h; there exists Th = (t0;h ; :::; tK;h ) with t0;h = 0; tk0 t and tK;h = Th h as well as a constant discount ~ h; rate m;h and a belief M m;h such that exp( m;h t)Mtm;h;G = exp( ~ht t)M t h ~h for all t 2 Th where (~ ; M ) are the equilibrium representative agent characteristics in the model Mh associated to the set of dates Th . In the next, the utility weights vector associated to Mh is denoted by ( h ) : It is easy to show that we may assume that ti;h ! ti;1 ; for i = 0; :::; K; with t0;1 ::: tk0 ;1 ::: t ::: tK;1 1: Let K 0 be de…ned by K 0 = sup fi : ti;1 < 1g : We have k0 K 0 < K: Let T1 be de…ned by T1 = ft0;1 ; t1;1 ; :::; tK 0 ;1 g, M1 be the model associated to the set of dates 38

T1 and ( 1 ) be the utility weights vector associated to M1 : By Lemma 27, we have ( h ) ! ( 1 ) and 1i > 0 for all i: We have Mki0 6= Mkj0 for some (i; j): Without loss of generality, we assume that k0 = 1 for the rest of the proof: We have then t1 t : Let us assume that there exists (i; j) such 1 1 that i 6= j : For < 0, by concavity of b ! b 1 and since ( h ) 1 2 ; we have exp(

m;h

X

tk;h )=E

X

i

exp(

i i h Mk

tk;h )

1 1

i

1

i h

i

exp(

1

!1

(37)

tk;h ):

i

i Thus, we can’t have m;h 0 for all i: We have then m;h > I = arg min i ; for every h: On the other hand, for all i; we have

2

E4

X

exp(

i

tk;h )

!1 3

1

i i h Mk

5

1

i

i h

sup E i i h

sup i

1

exp(

1

i

exp(

I

i

with

tk;h )Mki

tk;h );

and, in particular, exp(

m;h

Th )

sup i

i h

exp(

i

Th ):

Since hi ! 1i > 0 for all i; for h large enough, supi hi exp( i Th ) = I I I Th ): Therefore for h large enough, m;h which contradicts h exp( m;h > I: 1 For > 0, by convexity of b ! b 1 and since ( h ) 2 ;we have exp(

m;h

2

tk;h ) = E 4 X

X

exp(

i

tk;h )

i i h Mk

i

i h

1

exp(

1

i

tk;h );

1 1

!1 3 5

(38) (39)

i

and, as above, the inequality is strict at least for k = 1: As for (37) we obtain P P 1 m;h < I . Furthermore, for any family (ai ), we have ( ai )1 ai ; 39

1 1

1

therefore 2 X E4 exp(

i

tk;h )

i i h Mk

1 1

i

!1 3 5

X

E

i

exp(

tk;h )

i

X

i h

i i h Mk

!

(40) i

exp(

tk;h );

(41)

i

P i m;h i and therefore 1 tk;h ): For h large enough, we must i h exp( I have m;h which leads to a contradiction as in the case < 0: i Now, if = I for all i and < 0; from (37), we now have m;h I I and as above, m;h for h su¢ ciently large, hence m;h = I for h su¢ ciently large. Furthermore, using the same argument as in (37), we have X X 1 1 1 1 i 1 i 1 i 1 E M = 1; 1 1 1 and the inequality is strict since the M1i are not proportional. We have then exp(

with t1

I

t1 ) = lim exp(

m;h

h!1

= exp(

I

t1 )E

< exp(

I

t1 );

t1;h ) X

i 1

t < 1 which is impossible. The case

1 1

M1i

1

1

1

(42) > 0 is similar.

Appendix 3. Existence of a Production Equilibrium Existence of a production equilibrium is proven by a standard Negishi utility weights method. Although very similar to the pure exchange existence proof, the proof is harder, the continuity of the transfer map being more di¢ cult to prove. The proof of uniqueness extends Dana (1995) by introducing time, processes and a production set and is based on the assumption that shareholders have proportional endowments and homogeneous utility indices. We recall that all consumption and prices processes are assumed to be G-adapted. 40

Concepts, notations and …rst results De…nition 7 An allocation (c1 ; : : : ; cN ; y) is the speci…cation of a G-adapted consumption plan ci 2 X+ for each agent i = 1; : : : ; N and of a production plan y 2 K for the …rm. The allocation is feasible if X

ci = y:

i

As the set of feasible allocations is a (X; X 0 ) closed convex subset of (Y \ X+ )N +1 ; it is then bounded and hence (X; X 0 ) compact. We …rst recall the following classical characterization of Pareto optima. Lemma 16 An allocation (c1 ; : : : ; cN ; y) is a Pareto optimum if and only 1 N if there exists 2 RN + , such that (c ; : : : ; c ) solves max

nX

i

U i (ci ); ci

0; ci G

adapted for all i;

X

o ci = y; y 2 Y \ X+ :

In order to compute explicitly Pareto optima, let us next introduce i some notations. For t 2 T; c 2 R+ and 2 RN + , let u(c; ) and (C (c; ))i be de…ned by u(t; c; ) = max

( X

i

exp(

i

t)Mti u(ci ); ci

i

C i (t; c; ) = arg max

( X

i

exp(

i

0; 8 i;

t)Mti u(ci ); ci

i

X

0; 8 i;

ci X

)

c ; ci

)

c :

One easily veri…es that u(c; ) = N ( )u(c); C i (c; ) = S i ( )c

where

Sti (

)=

iM ie t

P

jMje t

it

1

jt

(43) (44)

1

1

1

represents shareholder’s i stochastic share

) j( of c: For each !; the function u(!; t; c; ) is di¤erentiable with respect to c on ]0; 1[ and we have uc (c; ) = N ( )u0 (c); (45)

41

where the equality is date by date and ! by !: As u(c; ) is homogeneous of degree one in while Cj (c; ) is homogeneous of degree zero in , as in the remainder of the paper, we restrict attention to : def For further use, let us introduce some notations. Let XMi = v : Mi v 2 L1 def

0 for all i; and XM = \i XMi . Let XM = M z; z 2 L1 ; (Ft )t2T ; P P with M = i Mi : We denote by * the convergence with respect to 0 (X; X 0 ) and by *M the convergence with respect to the topology (XM ; XM ). Let V = fv = u(y) : y 2 Kg and co(V ) be its closed convex hull. By Assumptions (C) and (P2), for any y 2 K;

ju(y)j

def

ju( )j + ju(&)j = A 2 XM :

(46)

P P Therefore, coV XM . For 2 ; i i M i M and therefore i i M i 2 0 : XM We need …rst to establish some intermediary results. We start by proving a lemma which will be useful in most of the proofs. Lemma 17

1. There exists a > 0 such that jN ( )j

2. For ( n ) S such that 0 (XM ; XM ):

n

!

0

; we have N (

0 : aM 2 XM n)

! N ( 0 ) for

3. If (yn ) K and yn !a:e: y 0 , then u(yn ) *M u(y 0 ): If (vn ) vn !a:e: v 0 and n ! 0 , then N ( n ) vn !L1 N ( 0 ) v 0 : P

1

1

co(V ),

is bounded over the simplex of Rn , 1 1 P P 1 hence there exists a > 0 such that a i jbi j. We then i jbi j P 0 have Nt ( ) a ( i ( i Mti exp( i t))) aMt and N ( ) 2 XM : 0 2. For n ! and v 2 XM we have vN ( n ) !a:e: vN ( 0 ) and jvN ( n )j aM jvj 2 L1 ; (Ft )t2T ; P : By Lebesgue Theorem, we then have vN ( n ) !L1 vN ( 0 ) for all v 2 XM . 0 3. If yn !a:e: y 0 ; for x 2 XM ; xu(yn ) !a:e: xu(y 0 ) and jxu(yn )j jxAj 2 L1 : By Lebesgue Theorem, we have u(yn ) *M u(y 0 ). If n ! 0 and vn !a:e: v 0 ; N ( n ) vn !a:e: N ( 0 ) v 0 and jN ( n ) vn j aM A 2 L1 and we conclude similarly. We now characterize the Pareto optima. From Lemma 16 and Assumption (P), the search for Pareto optima may be reduced to the following Proof. 1. b !

1 i jbi j

42

; (Ft )t2T ; P

;

representative agent problems P : max y2K

T X t=0

E [u(t; yt ; )] = max F (v; ) with F (v; ) = N ( ) v: v2V

Let us show that P admits a unique solution y (or v = u(y ) depending on the chosen formulation). Lemma 18 The problem P has a unique solution denoted y . Therefore F (:; ) reaches its maximum on V at a unique point v : Proof. Let (yn ) K and " > 0 be such that kyn y 0 kp ! 0 and N ( ) u(yn ) > N ( ) u(y 0 ) + " for all n: There exists a subsequence y'(n) !p:p: y 0 and, by Lemma 17, we have N ( ) u(y'(n) ) !L1 N ( ) u(y 0 ) which contradicts our assumption. Hence y ! N ( ) u(y) is strongly u.s.c. and therefore weakly u.s.c. on K which is weakly compact. Therefore it admits a maximum y . Uniqueness of y follows from the strict concavity of u, that of v from the monotonicity of u. From Lemmas 16 and 18 and Equation (44), as utilities are additively separable, Pareto optima may be described as follows: Lemma 19 An allocation (c1 ; : : : ; cN ; y) is a Pareto optimum if and only i i if there exists 2 RN + , such that y = y and c = C (y ; ) for every i. From (44) and Lemma 19, as the (M i )i are G-adapted, one easily veri…es that Pareto-optimal consumption processes are G-adapted. Let us next recall the de…nition of an equilibrium with transfers. De…nition 8 A G-adapted allocation (c1 ; : : : ; cN ; y) with a strictly positive G-adapted price process p is an equilibrium with transfers if it veri…es: 1. y maximizes pro…t that is : p y

p y for all y 2 Y ,

2. For every i, ci maximizes U i (c) subject to p c

p ci .

We now prove a second welfare theorem: i Lemma 20 For any 2 RN + , the allocation ((C (y ; ))i ; y ) with the price uc (y ; )) is an equilibrium with transfer.

43

Proof of Lemma 20. Let us …rst verify that the price process is in X 0 . From (45), Lemma 17 and Assumption (C), uc (y ; ) is G-adapted and we have uc (y ; ) aM u0 (y ) aM & 1 2 X 0 : To prove Assertion 1, as y solves (P ), we have uc (y ; ) 2 NY (y ): Therefore y maximizes pro…t at price uc (y ; ). The remainder of the proof which is totally standard is skipped. De…nition and Properties of the transfer map We may now de…ne the transfer map T by T i ( ) = uc (y ; ) (C i (y ; )

i

y ); for i = 1; : : : ; N

(47)

= Gi (v ; ) = Gi (g( ); )

i with Gi (v; ) = N ( ) (S i ( ) ) vt and where g : ! V is de…ned by i i g( ) = v . As juc (y ; )(C (y ; ) y )j j jaM A, T is well-de…ned. The subsection is essentially devoted to the proof of the continuity of T . To this end, let us introduce or recall some notations. We recall that F (v; ) = N ( ) v. when Our …rst result shows that F is jointly continuous on co(V ) 0 XM is endowed with the topology (XM ; XM ).

Lemma 21 If (vn ; n ) then F (vn ; n ) ! F (v 0 ;

co(V ) 0 ).

S is such that vn *M v 0 and

n

!

Proof. First, it is easy to check that F is well de…ned on co(V ) Furthermore, we have jF (v 0 ; )

F (vn ;

n )j

jF (v 0 ; 0 )

F (vn ; 0 )j + jF (vn ;

n)

0

;

S.

F (vn ; 0 )j :

It is immediate that jF (v 0 ; 0 ) F (vn ; 0 )j ! 0: Furthermore, jF (vn ; n ) F (vn ; 0 )j jN ( n ) N ( 0 )j A: By Lemma 17, we have jF (vn ; n ) F (vn ; 0 )j ! 0: 44

Using the same type of proof as in the previous lemma, we may also prove that G is jointly continuous on co(V ) when XM is endowed with 0 the topology (XM ; XM ) . S is such that vn *M v 0 and

Lemma 22 If (vn ; n ) co(V ) then G(vn ; n ) ! G(v 0 ; 0 ).

n

!

0

;

From Lemma 22, as T i ( ) = Gi (v ; ), if v was weakly continuous in , we could easily prove the continuity of T i . However, we could not prove such a property. As G( ; ) is linear, we use a weaker convergence property. Lemma 23 Let n ! 0 . Then there exists a subsequence P that n1 nk=1 g( (k) ) *M g( 0 ).

(n)

such

Proof. Let (yn ) K and (vn ) V be de…ned by vn = g ( n ) = u (yn ) : def By Komlos Theorem, there exist y (n) and v (n) such that y 0 (n) = P P def 1 y (k) !a:e: y 0 2 K and v 0 (n) = n1 v (k) !a:e: v 0 . Since y 0 (n) n

A; y 0 (n) * y 0 and v 0 (n) *M v 0 : By concavity of u; we have

and v 0 (n)

def

v 0 (n) u(y 0 (n) ) a.e. and therefore v 0 u(y 0 ) = v 00 2 V: We shall show that v 0 2 V and that F (v 0 ; 0 ) = maxv2V F (v; 0 ) or equivalently that v 0 = g( 0 ). From Lemma 21, F v 0 (n) ; (n) ! F (v 0 ; 0 ). Let us assume that maxv2V F (v; 0 ) > F (v 0 ; 0 ) : There exists " > 0 and v 2 V such that F (v; 0 ) > F (v 0 ; 0 ) + ". From Lemma 21, F (v; (n) ) > F v 0 (n) ; (n) for n large enough. We also have 1X F v n k=1 n

F v

0

(n) ;

(n)

F v 0 (n) ;

(k) ;

F v 0 (n) ;

(n)

1X F v n k=1

(k)

0

n

+ F v 0 (n) ; 1X v n k=1

0

n

n 1 X + v n k=1

N

(k) ;

(k)

(k)

N

(n)

N ( 0)

(k)

N

(k)

N ( 0) 1X N A+ n k=1 n

(n)

N ( 0)

45

(k)

N ( 0)

A

!

!0

Hence 1X F v n k=1 n

(k) ;

F v 0 (n) ;

(k)

(n)

2 1 + " < F (v 0 ; 0 )+ " < F (v; 0 ) 3 3

1 " 3

for n su¢ ciently large. Furthermore, as F (v; ) = lim F (v;

1X F v; (n) ) = lim n k=1

1X F v n k=1

1X < F v; n k=1

n

0

we have

n

n

(k)

;

n

(k) ;

(k)

(k)

for n su¢ ciently large. Hence, there exists some k such that F v (k) ; (k) < F v; (k) which contradicts the de…nition of v (k) : We thus have maxv2V F (v; 0 ) F (v ; 0 ). As v 0 v 00 and since v 00 2 V; we have max F (v; 0 ) = F (v 0 ; 0 ) = F (v 00 ; 0 ): v2V

As F is linear in v and strictly increasing, v 0 = v 00 = g ( 0 ) :

Proposition 24 T is continuous on

:

Proof. Let n ! 0 : Let us show that the sequence T i ( n ) = Gi (g( n ); n ) which is bounded has as unique limit point T i ( ). P First, by Lemma 23, n1 nk=1 g( (k) ) *M g( 0 ) for some subsequence P ( (n) ) and, by Lemma 22, Gi n1 nk=1 g( (k) ); (n) ! Gi (g( 0 ); 0 ) = T i ( 0 ): Next, as for F in Lemma 23, we have 1X g( n k=1 n

Gi

1 n

(k) );

N

(n)

k=1

E N

Pn

(n)

!

Si (k)

1X g( n k=1 n

Gi (n) i

S

(k) );

(k)

!

N ( 0 ) S i ( 0 ) A+ N ( 0) S i ( 0) A (k)

46

!

!0

and therefore 1X g( n k=1 n

G

(k) );

(k)

!

1X = G g( n k=1 n

1X i = T ( n k=1 n

(k) );

(k)

(k) )

! T i( 0)

Hence, the sequence (T i ( n )) has a unique limit point T i ( 0 ) to which it converges. The properties of the transfer map T = (T i ) are summarized in the following lemma. Lemma 25 2 .

1. The transfer map T satis…es

2. The function T is continuous on some i: 3. There exists

i

with

P

T i ( ) = 0 for every

and T i ( ) < 0 when

i

= 0 for

> 0 for all i such that T i ( ) = 0 for all i.

Proof of Lemma 25. The existence of a zero for T is standard. Proof of Proposition 2, Assertion 1. T has a zero to which corresponds a Pareto optimum with zero transfer payments. It is therefore a production equilibrium. Let us show that it is unique. Assume that there are two equilibria (^ c1 ; : : : ; c^N ); y^; p^ and (e c1 ; : : : ; e cN ); ye; pe where without loss of generality, prices are such that p^ y^ = pe ye = 1. Then c^i (resp. c~i ) is the optimal solution to the problem max U i (c) s.t. p^ c

i

i (resp pe c ). From Dana (1995) Proposition 2.1, as utilities are homogeneous, we have for every i,

(^ p Summing over i, we obtain (^ p

pe) (c^i

cei ) < 0:

pe) (^ y

ye) < 0:

From the de…nition of a production equilibrium, we have p^ y^ 47

p^ ye and

pe ye pe y^ which leads to a contradiction. The uniqueness of the vector of equilibrium utility weights follows immediately. Remark 26 Existence and uniqueness of an exchange equilibrium for a given y^ 2 E (Y ) \ X+ can be deduced from the previous theorem. Indeed, let y^ be given and let Y^ = f^ y g X+ : Consider the economy where all the agents have the same characteristics as in the initial economy but where the production set Y is replaced by Y^ . Since Y^ Y , Assumptions (P) and (C) are satis…ed by Y^ and therefore there exists a unique production equilibrium with production plan y^, (^ c1 ; : : : ; c^N ); y^; p^ . By de…nition, (^ c1 ; : : : ; c^N ); p^ is an exchange equilibrium associated to y^, proving the existence of an exchange equilibrium associated to y^. Suppose that there exists another exchange equilibrium associated to y^, c~1 ; : : : ; c~N ); p~ . As y ) and (~ c1 ; : : : ; c~N ); y^; p~ is a production equiliby ) = X+0 ; p~ 2 NY^ (^ NY^ (^ rium of the economy with production set Y^ and is therefore unique. An additional regularity property As in the proof of Proposition 14, let us consider a …ltration (Fk )k=0;:::;K and beliefs (Mki )k=0;:::;K : For a given set of dates T = ft0 ; t1 ; :::; tK 0 g with K 0 K; we de…ne a model MT where, for k K 0 ; Fk corresponds to the information structure at date tk , Mki describes the belief of agent i at that date and where agents’discount rates are given by ( i ) : For h 2 N, let Th = ft0;h ; t1;h ; :::; tK;h g be a set of dates such that ti;h ! ti;1 1 and let K 0 be de…ned by K 0 = sup fi : ti;1 < 1g : Let T1 = ft0 ; t1 ; :::; tK 0 g : To simplify the notations, we will write Mh instead of MTh and M instead of MT1 : The following Lemma provides a regularity result with respect to the set of dates. Lemma 27 For h 2 N[f1g ; there exists a unique production equilibrium (ch ; yh ; ph ) in the model Mh with associated positive equilibrium weights and we have h ! 1 : h 2 Proof. First, remark that, for a given set of dates T; the model MT can be identi…ed with a model MD where the set of dates is f0; 1; :::; Kg P i G i and where agent i maximizes U i;D (c) = K k=0 dk E [Mk u(ctk )] instead of P i maximizing U i (c) = K tk )E G [Mki u(ctk )] where D = (d0 ; :::; dK ) k=0 exp( 48

and dk = exp( tk ); k = 0; :::; K: A simple adaptation of the proof of Theorem 2 permits to show the existence and uniqueness of an equilibrium in MD and, doing so, to construct T D along the same lines as T (Equation 47). We denote by D 2 the vector of equilibrium utility weights that, by construction, satisfy, T D ( D ) = 0: Along the lines of Lemma 25, it is easy to show that T : is a continuous function of (D; ): For h 2 N[f1g, let Dh = (d0;h ; :::; dK;h ) be de…ned by dk;h = exp( tk;h ) (by convention, dk;1 = 0 when tk;1 = 1). We have (d0;h ; d1;h ; :::; dK;h ) ! (d0;1 ; d1;1 ; :::; dK;1 ) and the model Mh , identi…ed with MDh ; admits a unique production equilibrium. Since is compact, we have Dh ! 2 : D1 ( )=0 at least along a subsequence. By continuity of T , we have T and we have = D1 . Therefore, the sequence Dh admits a unique limit point D1 and converges to D1 :

49