Capital Ownership Under Market Incompleteness: Does it Matter?

The implications of a context with household heterogeneity and incomplete financial markets have been mostly studied under the assumption that households own the physical capital and undertake the intertemporal investment decision. Further, firms rent capital and labor from the households to maximize period profits. The present paper provides the conditions under which this assumption is still irrelevant when markets are incomplete. It is shown that, if firms choose the optimal investment to maximize their asset value, in the sense that they discount future cash flows with state price processes that are consistent with security prices, the equilibrium allocations are the same as in the standard setting with static firms. On the other hand, the firm valuation of future cash flows only coincides with the valuation of the unconstrained shareholders. Given this, value maximization might still lead to shareholder disagreement in the presence of effectively binding portfolio restrictions.


Introduction
Following Bewley (1977Bewley ( , 1986, an extensive macroeconomic literature has studied the quantitative implications of models with heterogeneous agents and incomplete …nancial markets. Among others, Aiyagari (1994), Huggett (1997), Storesletten, Telmer and Yaron (2001, 2004 and Smith (1997, 1998) have analyzed the e¤ects of such a framework on the aggregate saving rate, the shape of the wealth distribution, the asset returns, and the welfare costs of business cycles.
An important assumption shared by the previous models is that households are the owners of the physical capital stock and undertake the intertemporal investment decisions. Further, …rms simply rent capital and labor from the households to maximize pro…ts on a period by period basis. Whereas this assumption is innocuous in a complete markets setting, things are very di¤erent when markets are incomplete. In this last case, when …rms make the intertemporal investment decisions, shareholder disagreement may result in equilibrium, and the standard objective of value maximization is no longer well de…ned. This is due to the fact that the available markets do not provide su¢ cient information to value future pro…ts unambiguously.
In this paper, we characterize the competitive equilibrium for the class of incomplete markets models typically studied in the macroeconomic literature under the assumption that …rms decide on the level of investment. In particular, we show that there exists a natural generalization of the concept of value maximization to an incomplete markets context that gives rise to the same equilibrium allocation as in the economies cited above, in which …rms solve static optimization problems. Hence, the competitive equilibrium allocations are the same regardless of whether the stock of physical capital is owned and accumulated by households, who rent it to …rms, or by …rms who, as in…nitely-lived entities owned by shareholders, make the intertemporal investment decisions.
The framework for our analysis is an in…nite horizon economy with one good, aggregate uncertainty, idiosyncratic risk, and general portfolio restrictions. Markets are incomplete and …rms are assumed to make the investment decision in order to achieve "value"maximization. Under incomplete markets, value maximization is de…ned as the objective that discounts cash ‡ows with present value processes that are consistent with security prices, in the sense that they satisfy a no arbitrage relation between security prices and their payo¤s. Following the general equilibrium literature (see for example DeMarzo (1988DeMarzo ( , 1993 and Du¢ e and Sha¤er (1986b)), we …rst de…ne a value maximizing competitive equilibrium as an allocation where both households and …rms optimize, and the objective of the latter is value maximization. We then show that a value maximizing competitive equilibrium is also an equilibrium in the standard setting where …rms maximize period by period pro…ts. Conversely, an equilibrium in the standard setting is a value maximizing equilibrium in an economy where in…nitely lived …rms make the intertemporal investment decisions. One of the important implications of these results is that the equivalence of equilibrium allocations holds regardless of the particular present value process used by value maximizing …rms to discount future pro…ts.
Several remarks are worth noting. First, our main characterization result is derived under constant returns to scale in production and expected discounted utility for the households, two key assumptions that are typically satis…ed in the macroeconomic literature mentioned above. The latter guarantees convergence of present value sums and it rules out stock price bubbles in equilibrium for all present value processes that are consistent with security prices. Further, in the absence of price bubbles, the assumption of constant returns to scale guarantees that the capital stock that is chosen by a value maximizing …rm is equal to its stock market value. It turns out that this is also the level of capital that households would choose if they were making the intertemporal investment decision in a setting with one period lived …rms. Thus, if aggregate capital is the same, the consumption opportunity set for the households is the same across the two economies, providing a justi…cation for the equivalence of the equilibrium allocations in the versions of the model with one period lived and value maximizing …rms.
Second, the set of consistent present value processes is a singleton under complete markets, while there may exist a continuum of present value processes if markets are incomplete, potentially leading to a continuum of value maximizing equilibria. However, using our main characterization result, we can establish that a value maximizing equilibrium will be invariant with respect to the particular discount factor of the …rm if the equilibrium with one period lived …rms is unique.
Third, our competitive equilibrium concept does not take into account the ownership structure of the …rm. In the paper, we de…ne a value maximizing competitive equilibrium with shareholder agreement as one where the valuation of future cash ‡ows of the …rm coincides with the valuation of shareholders in the control group. Using this de…nition, we show that only the shareholders who are unconstrained in the stock market every period agree with a value maximizing production plan. Note that this is due to the fact that their marginal rates of substitution (or valuation of future pro…ts) belong to the set of consistent present value processes. Therefore, they agree with the valuation of the …rm. 1 In contrast to this, our main characterization result is independent of whether shareholders are constrained or not in the asset market, an appealing feature of our approach.
Last, whereas value maximization is a natural objective, others are clearly possible. Under di¤erent …rm objectives, however, the equivalence of allocations with the standard setting is not guaranteed. In other words, otherwise similar incomplete markets models might give rise to di¤erent quantitative implications than the ones established in the macroeconomic literature mentioned earlier. As an example, authors like Aiyagari (1994), Huggett (1997) or Smith (1997, 1998) have shown that imperfect risk sharing in the standard setting can lead to an increase in the aggregate capital due to precautionary savings. In contrast, aggregate household wealth, which is equal to the stock market value of …rms, need not be the same as the aggregate capital stock when …rms do not maximize their market value. In other words, precautionary savings are not necessarily be re ‡ected in the aggregate capital stock in the absence of value maximization. This is illustrated by Carceles-Poveda (2007), who studies the quantitative implications of alternative …rm objective functions in a two agent model with incomplete markets. In such a framework, it is shown that the aggregate capital stock is very sensitive to the assumption on …rm's objectives. In addition, it would be interesting to study further implications of …rm objectives on issues such as the wealth distribution. Furthermore, a di¤erent setup where this could be important are the standard overlapping generation models with production, where social security typically crowds out aggregate capital when households make the investment decisions. 2 In such a setting, if …rms decide on the investment level but do not maximize their value, they will not necessarily invest less when social security is introduced, potentially alleviating the crowding out e¤ect. These are important issues that we leave for further research.
Finally, our work is related to the general equilibrium literature with incomplete markets and production in a multiperiod context. 3 In this literature, the concept of value maximization that we have adopted has been used before by DeMarzo (1988DeMarzo ( , 1993 and Du¢ e and Sha¤er (1986b). The former demonstrates the validity of the Modigliani-Miller theorem while the latter study the issue of shareholder disagreement and show the existence of equilibria. Di¤erently from these authors, who focus exclusively on …rms as intertemporal decision-makers, we study the relationship between the allocations obtained in settings in which …rms accumulate physical capital and the allocations obtained in the standard macroeconomic setting in which …rms solve static decision problems. Therefore, our work can be viewed as establishing a link between this general equilibrium literature and the quantitative macro-…nance literature mentioned in the opening paragraph.
The rest of the paper is organized as follows. The following section presents the model and section three discusses the main equivalence results. These are further discussed in section four. Section …ve summarizes and concludes.

The Model Economies
In this section, we …rst introduce a common general environment and then present the two di¤erent model economies. The …rst economy is the one typically considered in the macroeconomic literature, where households own the stock of physical capital and make 1 Carceles-Poveda and Coen-Pirani (2006) discuss the issue of shareholder disagreement in a similar context. However, they focus on the preferences of shareholders with respect to the investment decision of the …rm without postulating a particular …rm objective. In addition, Du¢ e and Sha¤er (1986b) show that shareholders generically disagree with value maximization in a more general setting.
2 For a survey of this literature see Imrohoroglu, Imrohoroglu and Joines (1999). 3 In a multiperiod context with incomplete markets, authors such as Hernandez and Santos (1996), Levine (1989), Quinzii (1994a,1994b), and Levine and Zame (1996) have established the existence of an equilibrium in exchange economies. intertemporal investment decisions, whereas the representative …rm simply rents capital and labor from the households to maximize pro…ts on a period by period basis. In this sense, the …rm can be considered as being static or short lived. Second, we consider the case in which the …rm is the owner of the stock of physical capital. Here, the …rm is dynamic and is assumed to undertake all the intertemporal investment decisions. 4 2.1. The General Environment. We consider an in…nite horizon economy with aggregate uncertainty, idiosyncratic income shocks and sequential trading. Time is discrete and indexed by t = 0; 1; 2::: Further, the resolution of uncertainty is represented by an information structure or event-tree N . Each node or date-state s t 2 N , summarizing the history of the environment through and including date t, has a …nite number S(s t ) of immediate successors. We use the notation s r js t with r t to indicate that node s r belongs to the sub-tree with root s t . Further, with the exception of the unique root node s 0 dated at t = 0, each node has a unique predecessor dated at t 1, which we denote by s t 1 . The probability of date-event s t at period zero is denoted by (s t ), with (s 0 ) = 1, since the initial realization s 0 is given. In addition, (s r js t ) denotes the probability of s r given s t . Throughout the text, Technology. At each node s t 2 N , there exists a spot market for a single consumption good y(s t ), produced with the following aggregate technology: where k(s t 1 ) 2 R + and n(s t ) 2 R + denote the aggregate physical capital and labor, z s t is an aggregate productivity shock, and the initial stock of capital k(s 1 ) 2 R ++ is given. We make the following assumptions.
(A.1) The technology shock follows a stationary (Markov) process with state space S z = fz m : m 2 M z , z m 2 [z; z]g, where M z is a …nite set of integers, 0 < z < z < +1, and the initial realization z(s 0 ) is given.
(A.2) Given z, the production function f (z; ; ) : R 2 + ! R + is continuously di¤erentiable on the interior of its domain, strictly increasing, strictly quasiconcave, and homogeneous of degree one in the two arguments. We also assume that f (z; 0; n) = 0, f k (z; k; n) > 0 and f n (z; k; n) > 0 for all k > 0 and n > 0. Further, lim k!0 f k (z; k; n) = 1 and lim k!1 f k (z; k; n) = 0 for all n > 0.
The previous two assumptions are standard in the macroeconomic literature. Assumption (A.1) models the technology shock as a stationary Markov chain. It is important to note that our results only require that the shock takes a …nite number of positive values, and we can therefore relax the Markov assumption. Further, assumption (A.2) imposes standard conditions on the production process. In particular, the homogeneity assumption implies that f (z; k; n) = f k (z; k; n)k + f n (z; k; n)n via Euler's theorem. As we will see later, this last property is crucial to obtain our results.
The aggregate capital stock depreciates at the rate 2 (0; 1), and we denote the total supply of goods available from production at s t including undepreciated capital by: F (z(s t ); k(s t 1 ); n(s t )) = f (z(s t ); k(s t 1 ); n(s t )) + (1 )k(s t 1 ): Financial Markets. At each date-state s t , there exist spot markets for a …nite number L of securities. The …rst is a claim to productive activity that is indexed by l = 1. The rest are …nancial assets whose returns are denominated in units of the consumption good.
A security l 2 L traded at s t is de…ned by its current price q l (s t ) 2 R + and by the payo¤s it promises to deliver at future nodes. Holding a portfolio of securities a(s t 1 ) 2 R L at the end of period t 1 entitles the owner to a one period payo¤ of R(s t ) 0 = d(s t ) + q(s t ) 0 a(s t 1 ) if date-state s t is realized, where q(s t ) = (q 1 (s t ); :::; q L (s t )) 0 and d(s t ) = (d 1 (s t ); :::; d L (s t )) are the vectors of prices and dividends respectively. 5 A security traded at s t is of …nite maturity if there exists a date T such that R l (s r js t ) = 0 for all s r js t with r T . Otherwise, the security is in…nitely lived. Further, security markets are one period complete at node s t if the rank of the matrix de…ned by [R(s t+1 ) 0 ] s t+1 js t , where one row corresponds to R(s t+1 ) 0 for each node s t+1 js t , is equal to S(s t ). Markets are complete if they are one period complete at every date-state. We make the following assumptions.
Assumption (A.3) requires that dividends are nonnegative. This is consistent with the fact that free disposal of securities implies nonnegative security prices. In addition, it imposes the additional restriction that dividends on the productive claim are bounded away from zero at each node. As we will show later, this rules out stock price bubbles in equilibrium. 6 Clearly, a necessary condition for markets to be complete is that L S(s t ) at all s t 2 N . On the other hand, since we are particularly interested in the case where markets are incomplete, assumption (A.4) limits the number of available assets at each node.
No Arbitrage Pricing. The security price process q is arbitrage free at s t if there does not exist a portfolio a(s t ) 2 R L such that R(s t+1 ) 0 a(s t ) 0 for all s t+1 js t and q(s t ) 0 a(s t ) 0, with at least one strict inequality. In other words, arbitrage free prices have to be such that it is not possible to construct a portfolio with non-positive value and nonnegative payo¤s at every successor node. While this must be the case in equilibrium, the presence of no arbitrage at date-state s t implies the existence of positive present value prices (s t ) > 0 and (s t+1 ) with (s t+1 ) > 0 for each s t+1 js t , such that: Given (q; d), the absence of arbitrage at each date-state s t allows us to de…ne processes = (s t ) s t 2N for the entire information structure such that the previous no arbitrage equation holds. In what follows, we denote the set of such processes for the sub-tree with root s t by Q s t (q; d). Further, we de…ne s r t = (s r )= (s t ) for s r js t . Note that the present value ratios s t+1 t that are consistent with security prices are uniquely determined by (3) if markets are complete. On the other hand, the number of linearly independent equations is not su¢ cient to uniquely determine the ratios when markets are incomplete.
The previous present value prices can be used to evaluate future streams of consumption goods. In particular, for a non-negative stream x that speci…es x(s t ) 2 R + for all s t 2 N , the present value at s t of the subsequent stream with respect to some 2 Q s t (q; d) can be de…ned as: Similarly, we can de…ne the fundamental value v d l (s t ; ) of security l with respect to some 2 Q s t (q; d). In addition, using some algebra, the bubble component of the security with respect to 2 Q s t (q; d) can be expressed as: As shown by Santos and Woodford (1987), if a security price is non-negative, its funda- Further, whereas the fundamental value need not be the same for all state prices satisfying equation (3), the authors show that it must lie between the …nite bounds In this case, the fundamental value is uniquely de…ned for all 2 Q s t (q; d).
Households. The economy is populated by a countable set of in…nitely lived households I. Households'preferences %= (% i ) i2I over consumption plans c i satisfy the following assumption.
(A.5) For every i 2 I, % i can be represented by the following function: where i 2 (0; 1) is the individual discount factor, and the period utility function u i : R + ! R is strictly increasing, strictly concave and continuously di¤erentiable in the interior of its domain, with lim c i !0 u 0 i (c i ) = 1 and lim c i !1 u 0 i (c i ) = 0. The class of preferences in assumption (A.5) is standard in the macroeconomic literature. It is important to note that this class of preferences satis…es the property that there is a "uniform lower bound on the impatience" of each agent. This last property, which is stated formally in the appendix, has been assumed by several authors studying in…nite horizon exchange economies with incomplete markets, such as Santos and Woodford (1997), Quinzii (1994a,1994b), Hernandez and Santos (1996) and Levine and Zame (1996). In essence, it requires that at each node s t an agent is willing to give up a fraction of his future consumption after node s t in exchange for a multiple of the current aggregate endowment. Further, the fraction of future consumption that each agent is willing to give up (or the degree of impatience), is uniform across all nodes and feasible consumption plans. As we will see later, this property is crucial to establish the absence of price bubbles in the present setup.
Each household i 2 I enters the markets at t = 0 with a …nite initial endowment a l i (s 1 ) of each security, whose sum across households determines the net supply of the security at each node, which we denote by A l = P i2I a l i (s 1 ). Without loss of generality, the supply of the productive claim and of the rest of securities is normalized to one and zero respectively, and we let A = (A 1 ; :::; A L ) 0 . At each date-state s t 2 N , households are also endowed with one unit of time that is entirely allocated to labor, and which they can transform into i (s t ) e¢ ciency labor units that will be used to produce output in exchange of wages. Given this, the labor income of the household at s t is given by w i (s t ) = w(s t ) i (s t ), where w(s t ) is the fraction of output allocated to labor payments. We make the following assumptions.
(A.7) The labor income shock i follows a stationary (Markov) process with state space where M is a …nite set of integers, 0 < < < 1, and the initial realization i (s 0 ) is given.
Assumption (A.6) guarantees that the supply of each security is non-negative. Further, assumption (A.7) models the labor income shock as a discrete state Markov chain. As before, our results only require that the shock takes a …nite number of positive values, and we can therefore relax the Markov assumption. The aggregate and idiosyncratic shocks could potentially be correlated, and we denote their joint transition matrix by in what follows.
At each node s t , household i 2 I chooses consumption c i (s t ) 2 R + and a portfolio of securities a i (s t ) 2 R L subject to the following constraints: Equation (5) is the standard budget constraint with sequential markets and equation (6) is the law of motion of the individual wealth ! i (s t ). At t = 0, equation (6) where we have used the fact that d l (s 0 ) = 0 for l 2. Finally, to avoid Ponzi schemes, equation (7) imposes a …nite limit of B i (s t ) on the total amount that households can borrow at every node. 8 A possible trading restriction that one can impose is the present value constraint, which is e¤ectively never binding at any …nite date. In particular, it is the tightest borrowing limit such that the portfolio holdings satisfy the budget constraint with c i (s t ) 2 R + for all s t 2 N , and wealth is always non-negative after a …nite date. As shown by Santos and Woodford (1997), this constraint can be formally speci…ed as: In essence, the restriction implies that households can borrow at most the lowest present value of their individual endowments in order to be solvent. The two production economies are described in what follows.
2.2. The k-economy. In the k-economy, we make the usual assumption in the macro-…nance literature, implying that households are the owners of the physical capital stock and make the inter-temporal investment decision. In this case, the problem of the …rm is particularly simple. At each date-state s t , after observing the realization of the productivity shock z, the …rm chooses capital and labor to maximize period pro…ts. Thus, it solves a sequence of static problems: leading to the following necessary and su¢ cient …rst order conditions: where w(s t ) 2 R + and r(s t ) 2 R + are the competitively determined wage and gross capital rental rates respectively. Further, each household i 2 I maximizes the preferences in (A.5) subject to the following constraints: In the previous equations, k i (s t ) is the amount of physical capital held by the household at the end of period t, illustrating the fact that households make the inter-temporal investment decision. If we denote by k i (s 1 ) and a i (s 1 ) the initial asset holdings of i at t = 0, the period zero budget constraint takes the same form with A k-economy is speci…ed by a set of preferences %, a transition matrix , initial val- and borrowing limits B = (B i ) i2I , where a 1 i (s 1 ) = k i (s 1 )=k(s 1 ) represents the initial endowment of capital shares of household i 2 I. A k-economy is therefore described by E k = f%; (k 0 ; a 0 ; z 0 ; 0 ); ; d a ; Bg.
De…nition 2.1. The vector of processes o is a CE for E k = f%; (k 0 ; a 0 ; z 0 ; 0 ); ; d a ; Bg if (i) for each i 2 I and for each s t 2 N , c i ; k i ; (a l i ) l 2 is optimal under the preferences % given q l l 2 , (w; r), (k 0 ; a 0 ; z 0 ; 0 ), , d a and B (ii) (w; r) satis…es the …rm's optimality conditions (iii) all markets clear, i.e., for all s t 2 N , n(s t ) = P i2I i (s t ), Before discussing the framework with dynamic …rms, it is important to note that the constraints faced by the household sector in the k-economy can be directly mapped into the framework of the general environment if we de…ne the shares of physical capital held by household i at date-state s t as a 1 i (s t ) = k i (s t )=k(s t ). With this normalization, the total supply of shares is positive and equal to A 1 = 1. Further, q 1 (s t ) = k(s t ), R 1 (s t+1 ) = r(s t+1 )k(s t ) and d 1 (s t ) = r(s t )k(s t 1 ) k(s t ) = F (z(s t ); k(s t 1 ); n(s t )) w(s t )n(s t ) k(s t ).
The e-Economy. In the e-economy, we assume that the …rm owns the entire stock of capital and undertakes the inter-temporal investment decision by solving a dynamic optimization problem. Further, households are entitled to the future dividend payments through their ownership of a perfectly divisible equity share in the …rm that is traded at price q 1 s t .
At each node s t , households maximize the preferences in (A.5) subject to constraints (5)-(7). Further, the …rm produces output, pays wages to the total labor employed and decides on the amount of investment. Investment is entirely …nanced with retained earnings, and the residual of gross pro…ts (output net of labor payments) and investment is paid out as dividends to the …rm equity owners, i.e., where N f (s t ) is the net cash ‡ow of the …rm. Unfortunately, the de…nition of an appropriate …rm objective is more complicated than before, since the standard approach, that …rms maximize their share value, is not well speci…ed under market incompleteness. The reason is that the available markets do not provide su¢ cient information to value future dividend streams unambiguously. To see this, consider the case of e¤ective complete markets and let m s t+r t be the t + r-period ahead pricing kernel. Note that m s t+r t represents the period t price of one unit of time t + r consumption, contingent on the economy being at date-state s t+r js t . Since all the shareholders will agree on the pricing kernel under complete markets, the objective of the …rm at date-state s t can then be naturally speci…ed as follows: for all i 2 I As usual when markets are complete, the …rm maximizes the present discounted value of its net cash ‡ows, using as a discount factor the unique present value process of its shareholders, which is also the only element of the set Q s t (q; d). In addition, both the agents and the …rm value future output in each state identically, and all shareholders will therefore agree with the investment choice made by the …rm. On the other hand, since a unique present value process that is consistent with market prices is not necessarily available under market incompleteness, the previous objective is no longer well de…ned, and shareholder disagreement may result in equilibrium. 9 In what follows, we brie ‡y discuss some of the approaches that have been proposed in the literature following the seminal paper of Diamond (1967). Moreover, a discussion of shareholder disagreement in the present setup is provided in a later section. 10 Value Maximization. As noted by DeMarzo (1993), a natural generalization of the previous Arrow Debreu …rm objective to an incomplete markets setup is to require …rms to maximize the value of their output according to some consistent present value prices, in the sense that they satisfy the no arbitrage condition in (3). The two period value maximizing …rm objective postulated by the author can therefore be expressed in our multi-period setup as follows: This approach has also been followed by DeMarzo (1988) and Du¢ e and Sha¤er (1986b), who study the validity of the Modigliani Miller theorem and the existence of equilibrium and shareholder agreement in a general incomplete markets context. As noted by the authors, one could alternatively assume that the …rm maximizes its share price according to some valuation function that assigns a price process to a given stream of cash ‡ows. Further, as long as this valuation does not predict security prices that allow for arbitrage opportunities, there exist some positive present value prices 2 Q s t (q; d) such that the valuation conjectured by the …rm is equal to the objective function above.
The optimization problem under value maximization can be characterized by the follow-9 Note that shareholder disagreement would not be an issue if one assumed privately owned …rms, as in Angeletos (2007) and Angeletos and Calvet (2005Calvet ( , 2006. 1 0 An excellent survey on the existing unanimity results and a discussion of shareholder disagreement under value maximization is provided in Grossman andStiglitz (1977, 1980). Further, see Du¢ e and Shafer (1986) for a discussion of unanimity under value maximization in a general multiperiod setup, and Carceles-Poveda and Coen-Pirani (2006) for a discussion of unanimity in a …nite period version of the present setting. ing necessary and su¢ cient …rst order conditions: w(s t ) = f n (z(s t ); k(s t 1 ); n(s t )) (17) The …rst equation determines the equilibrium aggregate wage rate. Further, the second equation determines the optimal production plan and it illustrates that the inter-temporal investment decision in this economy is made by the …rm.
The e-economy is speci…ed by a set of preferences %, initial values (k 0 ; a 0 ; z 0 ; 0 ) = n k(s 1 ); a i (s 1 ); i (s 0 ) i2I ; z(s 0 ) o , a transition matrix , security processes d a = d l l 2 and limits B = (B i ) i2I . The e-economy is then described by E e = f%; (k 0 ; a 0 ; z 0 ; 0 ); ; d a ; Bg.
De…nition 2.2. The vector of processes f(c i ; a i ) i2I ; q; w; kg is a value maximizing CE (VM CE) for E e = f%; (k 0 ; a 0 ; z 0 ; 0 ); ; d a ; Bg if (i) for each i 2 I and for each s t 2 N , (c i ; a i ) is optimal under the preferences % given (q; w), (k 0 ; a 0 ; z 0 ; 0 ), , d a and B (ii) (w; k) satis…es the …rm's optimality conditions for some 2 Q s t (q; d) (iii) all markets clear, i.e., for all s t 2 N , n(s t ) = P i2I i (s t ), Several remarks are worth noting. First, the previous equilibrium de…nition implies that the set of allowable present value processes Q s t (q; d) that the …rm can use to discount its net cash ‡ows has to satisfy a …xed point problem in the following sense. When the …rm discounts pro…ts with some that belongs to the set of admissible present value prices Q s t (q; d), its production choice k( ) generates a new asset structure (q( ); d( )) and a new set of admissible present value prices Q s t (q( ); d( )) to which the original has to belong. Thus, if we de…ne a mapping from the admissible set of present value prices to the set of present value prices that it generates, the equilibrium set of discount factors can be seen as a …xed point of this mapping. Moreover, if the set satisfying the previous …xed point problem is not single valued, the presence of incomplete …nancial markets might generate indeterminacy of equilibria with respect to the …rm discount factor (see Du¢ e and Sha¤er (1986b)). Second, since the state process can be interpreted as the discount factor used by the …rm to value future net cash ‡ows, value maximization will generate shareholder disagreement if does not agree with the valuation of the controllers of the …rm.
Following the seminal paper of Diamond (1967), several authors have proposed alternative criteria concerning the discount factor of the …rm. In a two period context, Dreze (1974) has proposed as a discount factor a weighted average of the marginal rates of substitution of the di¤erent shareholders, with the weights re ‡ecting the …nal holdings of shares. Grossmann and Hart (1979) have extended this idea to a multi-period setting, arguing that the weights should re ‡ect the initial allocation of shares among shareholders. Dreze (1985) and DeMarzo (1993) have introduced a control mechanism based on majority voting to decide among alternative production plans, implying that the …rm should discount future pro…ts using some weighted average of the marginal rates of substitution of the controllers of the …rm. In addition, other authors, such as Radner (1972a), Sandmo (1972), Sondermann (1974) or Leland (1972), have simply assumed the existence of a utility function for the …rm, de…ned exogenously over pro…ts. In the present paper, we focus on value maximization, since this is the objective that will yield the same allocations as in the standard setting with one period lived …rms.
Competitive Equilibrium with Shareholder Agreement. It is important to note that the equilibrium concept in de…nition 2.2 does not take into account the relationship between the investment decision of the …rm and its ownership structure. In other words, the investment decisions can potentially be made without taking into account the preferences of the shareholders.
To address this issue, let I c s t I be the subset of shareholders that have control over the …rm at s t . We can now extend the previous equilibrium de…nition to an equilibrium concept with shareholder agreement, in the sense that all shareholders in the control group support the production plan chosen by the …rm. To do this, we should replace condition (ii) in De…nition 2.2 with the following condition: (ii)'(w; k) satis…es the …rm's optimality conditions for some 2 Q s t (q; d) that coincides with the valuation of future cash ‡ows of all i 2 I c s t . If this condition is satis…ed, a production plan that is unilaterally chosen by the value maximizing …rm also maximizes the utility of the shareholders in the control group. The issue of shareholder disagreement is brie ‡y analyzed in section four.

Equivalence of the Production Allocations
This section shows the equivalence of the set of equilibria in the two production economies under value maximization. Throughout the section, we assume that the assumptions of Section 2 are satis…ed. We start by stating several results that we will use to prove the main theorems, and relegate most of the proofs to the appendix. To distinguish the allocations, the caret bearing variables always denote k-economy allocations.
Proposition 3.1 Consider a CE for E k or E e . For each node s t 2 N and for each security l 2 L traded at s t that is either (i) of …nite maturity or (ii) in positive supply, we have that q l (s t ) = v d l (s t ; ) for all 2 Q s t (q; d). In addition, lim r!1 P s t+r js t s t+r t k(s t+r ) = 0 for all 2 Q s t (q; d).
Proposition 3.1 establishes the absence of price bubbles for securities that are of …nite maturity or in positive supply. Further, it implies that the discounted value of the aggregate capital converges to zero as time goes to in…nity. These results can be established by showing that our preference assumptions imply that bubbles cannot exist for any security that is …nitely lived or in positive supply and for any present value process 2 Q s t (q; d) such that the present value of the aggregate labor endowment v wn (s t ; ) is …nite when this state price process is used. This part directly follows from Santos and Woodford (1997), and we only include the proof for completeness. Second, we show that the presence of trade in a claim to productive activity implies that the present value of the aggregate labor endowment is …nite for any consistent present value process. To do this, we rely on the de…nition of the productive dividend payments. Clearly, the previous two results imply that bubbles cannot exist for any asset that is …nitely lived or in positive supply. 11 The next lemma shows that the aggregate capital stock chosen by a value maximizing …rm in the absence of price bubbles is equal to the ex-dividend …rm value q 1 .
Lemma 3.1 If the e-economy …rm discounts its net cash ‡ows with some 2 Q s t (q; d), the equilibrium investment plan satis…es k(s t ) = q 1 (s t ) = v N f (s t ; ) for all s t 2 N .
To prove the lemma, recall that the …rst order conditions of the …rm's problem under value maximization imply that: where 2 Q s t (q; d). Multiplying the previous expression with k(s t ), adding and subtracting k(s t+1 ) on the right hand side, and using the homogeneity condition of the production function, we obtain: Further, substituting iteratively for k(s t+r ) for 1 r T , we have that: The …rst term on the right-hand side of the previous equation has a well de…ned limit, and the second term converges to zero as T goes to in…nity by proposition 3.1. Thus, taking limits of the previous equation as T goes to in…nity, the aggregate capital stock can be expressed as: On the other hand, the ex-dividend …rm value is equal to the value of equity q 1 (s t ), whose dividends are given by d 1 (s t ) = N f (s t ) at all s t 2 N . Further, since equity is in positive supply, proposition 3.1 implies that q 1 (s t ) = v N f (s t ; ) for all 2 Q s t (q; d), establishing the result.
The result of the previous lemma is crucial to establish the equivalence of the equilibrium allocations in the two production economies. In particular, as shown by lemma 3.2 below, it implies that the set of budget feasible allocations is the same across the two production economies as long as they are characterized by the same preferences, initial values, transition matrices for the shocks, …nancial asset structure, portfolio constraints and production plans.
Lemma 3.2 Consider optimal allocations in the k-and e-economies. Further, assume that (k 0 ; a 0 ; z 0 ; 0 ), , d l ; q l l 2 , B and k are the same. If the …rm in the e-economy has a value maximizing objective, the set of budget feasible allocations is the same in the two production economies.
Lemma 3.2 shows that households in the k-economy can achieve the same consumption allocation as in the e-economy (and vice-versa) by choosing the same portfolio of assets a l i l 2 and a physical capital investment k i that is equal to the total equity investment q 1 a 1 i they would choose in the e-economy. If the production plan k is the same across the two production economies, this asset choice generates the same …nancial wealth, implying that a consumption plan that is feasible in one economy is also feasible in the other. As before, the result relies on the homogeneity of the production function, since it requires that k(s t ) = q 1 (s t ). We are now ready to state our main results.
Then, there exist processes b k i and b r such that Theorem 3.1 asserts that a value maximizing equilibrium in the e-economy is also an equilibrium in a k-economy with the same characteristics. The argument of the proof is very simple. We …rst note that the k-economy aggregate capital stock b k is equal to the value of the …rm q 1 in the e-economy, which is in turn equal to the e-economy capital stock k by lemma 3.1. Given this, the returns on labor and capital de…ned by (w; b r) satisfy the …rm's optimality conditions in the k-economy, and lemma 3.2 implies that an optimal household allocation in the e-economy is also optimal in the k-economy. Finally, market clearing in the latter economy follows from market clearing in the …rst. Theorem 3.2 below states that the reverse is also true.
Theorem 3.2 Let ; b q l l 2 ; b w; r o be a CE for the economy speci…ed Then, there exist processes for a 1 i , and q 1 such that n b c i ; a 1 i ; b a l i l 2 i2I Theorem 3.2 can be proved using similar arguments. In particular, since the aggregate capital is the same in the two economies, the fact that q 1 = b q 1 implies that both economies have the same asset structure and the same set of consistent present value prices. Given this, the k-economy aggregate wage rate b w and aggregate capital stock b k, which satis…es the no arbitrage pricing condition in (3), satisfy the …rm's optimality conditions in the e-economy for some 2 Q s t (q; d). Finally, lemma 3.2 implies that an optimal household allocation in the k-economy is also optimal in the e-economy, and market clearing in the latter economy directly follows from market clearing in the …rst.
The previous two theorems imply that value maximization leads to the same dimension of the set of equilibria in the two production economies. On the other hand, the equilibrium in the e-economy might depend on the particular …rm discount factor 2 Q s t (q; d) (see Du¢ e and Sha¤er (1986b)). As stated by the following theorem, however, if the equilibrium with one period lived …rms exists and is unique, the equilibrium under dynamic …rms is independent from the discount factor of the …rm.
Theorem 3.3 If a CE in the k-economy exists and is unique, the e-economy VM CE is invariant with respect to the …rm discount factor 2 Q s t (q; d).
The proof of this theorem follows from the previous two. In particular, theorems 3.1 and 3.2 establish that the set of equilibria has the same dimension in the two production economies. Thus, if we consider two equilibria in the e-economy that just di¤er in the discount factor chosen by the …rm, the theorems show that they are also equilibria in the k-economy. On the other hand, if the k-economy equilibrium is unique, the two e-economy equilibria will clearly result in the same allocations.

Discussion
Theorems 3.1 and 3.2 extend to an incomplete markets setting with general portfolio restrictions the result, well known under complete markets, that capital ownership is irrelevant under value maximization. In other words, equilibria are the same whether the agents own the capital and rent it to the …rm, or whether the …rm, considered as an in…nitely lived corporation that is owned by the agents as shareholders, holds the capital stock. Several remarks are worth noting.
First, in addition to the assumption of constant returns to scale in production, the results require the standard assumption that agents are impatient, which implies convergence of the present value calculations in equilibrium. To get some intuition for the previous …ndings and for why these assumptions are needed, consider the investment decision in the two production economies. In a k-economy with portfolio restrictions, the aggregate capital stock is determined by the unconstrained households, whose marginal rate of substitution between periods t and t + 1 is denoted by m s t+1 t . If we substitute for r s t+1 and multiply by k s t the …rst order condition of an unconstrained household, it follows that the capital stock in this economy has to satisfy the following Euler condition: Similarly, consider an analogous e-economy where the investment decision is made by a value maximizing …rm. In the absence of price bubbles, the homogeneity of the production function implies that the stock of a value maximizing …rm is equal to the aggregate capital stock, k s t = q 1 s t , as shown by Lemma 3.1. Given this, the aggregate investment plan in the e-economy has to satisfy the following Euler condition for some 2 Q s t (q; d): In essence, the results of section three can be seen as the equivalence of equations (19) and (20). In this case, a value maximizing …rm in the e-economy will choose the same capital stock as if households were making the inter-temporal investment decision in the keconomy, and viceversa. To see that our assumptions guarantee that this is the case, consider the process m 2 Q s t (q; d), where m s t+r t is de…ned by the marginal rate of substitution of the particular household that is unconstrained at each node between s t and s t+r js t , and which may or may not be the same household every period. First, the absence of price bubbles implies that we can substitute with m in equation (20). This is due to the fact that q 1 s t is equal to its fundamental value v d 1 s t ; , which is uniquely de…ned for all 2 Q s t (q; d) and in particular for m 2 Q s t (q; d). Second, under constant returns to scale, Euler's Theorem implies that d 1 s t + q 1 s t = F k (z(s t+1 ); k(s t ); n s t+1 )k(s t ). It therefore follows that equations equations (19) and (20) are equivalent. In turn, this also implies that both economies will have the same budget sets (Lemma 3.2) and the same equilibrium allocations (Theorems 3.1 and 3.2). On the other hand, since the assumptions needed are relatively standard in the macro-…nance literature, our results can be applied to a wide class of economies. Two important extensions are brie ‡y discussed in what follows.
Second, theorems 3.1-3.3 can be easily extended to economies with (i) external …nance and (ii) heterogeneous …rms. In the …rst case, we assume that …rms in the two production settings can rise capital by issuing di¤erent assets. In the second case, we assume that …rms di¤er in their productivity process. In addition, the e-economy …rms can also di¤er in their discount factors as long as they belong to the set of consistent present value prices, which is common across …rms. Using similar arguments to the ones in Proposition 3.1, price bubbles can be ruled out in these two cases. In the presence of external …nancing, we can show that the ex-dividend …rm value in the e-economy, which is equal to the market value of the assets in its capital structure, is equal to the economy wide capital stock. In the economy with heterogeneous …rms, we can show that each …rm j will set its investment level to its market value, k j s t = q j s t = v d j s t ; j where j 2 Q s t (q; d). Given this, the results of Lemma 3.2 and Theorems 3.1-3.3 follow through in both cases. 12 Third, in spite of the fact that a value maximizing …rm in the e-economy chooses the same production plan as if households were making the intertemporal investment decision, this does not necessarily imply that the plan is unanimously approved by the shareholders that belong to the control group I c s t I of the …rm. In particular, a plan that is chosen by a value maximizing …rm will only be approved by the shareholders if the discount factor coincides with the future cash ‡ow valuation of the controllers of the …rm. On the other hand, equation (20) implies that the valuation of the …rm only coincides with the valuation of future pro…ts of shareholders that are unconstrained every period. To see this, note that the equation can be rewritten as: The …rst equality represents that fundamental value of equity with respect to . Further, since this is uniquely de…ned for all consistent present value processes, the fact that m 2 Q s t (q; d) implies that v d 1 s t ; = v d 1 s t ; m . Here, it is important to note that only the marginal rate of substitution of a shareholder that is unconstrained every period belongs to the set of consistent present value prices Q s t (q; d). Given this, only if the shareholders that belong to the control group of the …rm are unconstrained, unanimity will obtain with respect to value maximization. As an example, this would happen if the controllers of the …rm were subject to the present value constraint in (8), since it is e¤ectively never binding.
This result relates to the unanimity …ndings in Carceles-Poveda and Coen-Pirani (2006), where the preferences of the shareholders with respect to investment alternatives are derived explicitly without postulating any objective for the …rm. In particular, it is shown that, with a constant returns to scale technology, all the shareholders of a …rm agree on setting k s t = q s t as long as the borrowing constraints are not binding. Clearly, this implies that a value maximizing investment plan maximizes the utility of the unconstrained shareholders. In contrast, a value maximizing equilibrium with shareholder agreement might not exist if portfolio restrictions are e¤ectively binding.
Last, whereas value maximization is a natural objective, others are clearly possible. In the absence of value maximization, however, the equivalence of allocations result might not hold. To see this, consider a …rm with the general objective function: where u f is the period utility function and f is the …rm's discount factor. It is easy to show that the optimality condition determining aggregate capital implies that: This equation re ‡ects that the aggregate capital stock k(s t ) under a general objective will not be equal to the value of the …rm q 1 (s t ) unless u f is linear and f 2 Q s t (q; d), corresponding to value maximization. Further, if k s t 6 = q 1 s t , Lemma 3.1 implies that the set of feasible allocations in the economy with in…nitely lived …rms is not the same as in the standard setting, potentially leading to di¤erent equilibrium allocations. In sum, in the absence of value maximization, otherwise similar incomplete market models might lead to di¤erent quantitative implications with respect to the ones established in the macroeconomic literature. 13 5. Summary This paper characterizes the competitive equilibrium in a class of incomplete market economies where …rms make the intertemporal investment decision and households are subject to general portfolio restrictions. In particular, it shows that there is a speci…c objective function for …rms that yields the same equilibrium as the economies recently considered in the macroeconomic literature. Under such objective, capital ownership is irrelevant, in the sense that the equilibrium allocations are the same regardless of whether consumers or …rms own the stock of physical capital in the economy. Thus, the paper extends to an incomplete markets setting the standard result about the irrelevance of capital ownership obtained under complete markets.
Speci…cally, it is shown that, if …rms undertake the intertemporal investment decision to maximize their market value, de…ned as the present value of future pro…ts using as a discount factor present value processes that do not allow for arbitrage opportunities, the equilibrium allocations are the same as in the standard setup where households make the investment decision and …rms rent capital and labor to maximize pro…ts on a period by period basis. The result is derived under constant returns to scale in production and expected discounted utility for the households. Since these two assumptions are relatively standard in the macro-…nance literature, the result can be applied to a relatively large class of models that have been used in the literature for quantitative analysis.
We conclude by noting that the implications of models with incomplete markets and dynamic …rms in the absence of value maximization might di¤er from the standard setup with static …rms. A quantitative assessment of some of these implications is an important issue which we leave to future research.
Since v wn (s t ; ) < +1 by assumption, it follows that v w i (s t ; ) < +1 for all i, and the right hand side of the previous equation has a …nite limit equal to (s t )v w i (s t ; ) < +1. Since w n (s t ) + d(s t )A = w c (s t ) and v dA (s t ; ) q(s t ) 0 A < +1, v wn (s t ; ) < +1 also implies that v wc (s t ; ) < +1, and it follows that v c i (s t ; ) < +1 for all i 2 I. Given this, the …rst term on the left hand side of the previous equation also has a well de…ned and …nite limit equal to (s t )v c i (s t ; ) < +1. Finally, since v wc (s t ; ) < +1 and w c (s t+T ) 0 for all date-states s t+T js t , we have that lim T !1 (1 i ) 1 P s t+T js t (s t+T )w c (s t+T ) = 0, which establishes the inequality in equation (3). Summing the inequality over households, we obtain: implying that 1 X r=1 X s t+r js t s t+r t [F (z(s t+r ); k(s t+r 1 ); n(s t+r )) k(s t+r )] v d 1 (s t ; ) < +1 which contradicts equation (9). Therefore, it follows that v wn (s t ; ) < +1 for all 2 Q s t (q; d).
Finally, to see that lim n!1 X s t+r js t s t+r t k(s t+r ) = 0 for all 2 Q s t (q; d), note that this directly follows from (a) in the k-economy. Further, to show that this is also the case in the e-economy, note that w(s t )n s t +d 1 (s t )+k(s t ) = F (z(s t ); k(s t 1 ); n(s t )).
Given this, we can use the same arguments as above to show that for some > 0, we have that w(s t )n s t [F (z(s t ); k(s t 1 ); n(s t ))]. This clearly implies that the …rst in…nite sum in equation (9) is …nite for every 2 Q s t (q; d). Therefore, for the total sum to be …nite, it must be the case that: 1 X r=1 X s t+r js t s t+r t [k(s t+r )] < +1 implying that lim r!1 P s t+r js t s t+r t k(s t+r ) = 0 for every 2 Q s t (q; d).

Proof of lemma 3.2
To prove the lemma, let b F i (s t ) and F i (s t ) be the set of budget feasible allocations at s t in the two production economies. Note …rst that b c i (s where we have substituted for the equilibrium values of b r(s t+1 ) = f k (z(s t+1 ); b k(s t ); b n s t ) + 1 .
Similarly, c i (s t ) 2 F i (s t ) if there exists a set of portfolio strategies (a l i ) l 1 such that, for all s t and all i 2 I: c i (s t ) + q 1 (s t )a 1 i (s t ) + X l 2 q l (s t )a l i (s t ) ! i (s t ) ! i (s t+1 ) = w i (s t+1 ) + [f k (z(s t+1 ); k(s t ); n s t ) + 1 ]q 1 (s t )a 1 i (s t )+ where we have used homogeneity of the production function and the fact that q 1 (s t ) = k(s t ) by lemma 3.1, implying that: q 1 (s t+1 ) + d 1 (s t+1 ) = (f k (z(s t+1 ); k(s t ); n s t ) + 1 )k(s t ) Let b c i (s t ) 2 b F i (s t ) and assume that the hypothesis of the lemma are satis…ed. We now show that a plan setting c i (s t ) = b c i (s t ) at each node is feasible in the e-economy. To see this, consider where the last equality holds by lemma 3.1. Further, we have used the fact that a 1 i generates market clearing in the e-economy. Given this, the two factor prices: w(s t ) = f n (z(s t ); k(s t 1 ); n s t ) b r(s t ) = R 1 (s t )=q 1 (s t 1 ) = (d 1 (s t ) + k(s t ))=k(s t 1 ) = f k (z(s t ); k(s t 1 ); n s t ) + 1 satisfy the …rm's optimality conditions in the k-economy, where we have substituted for the labor market clearing conditions and have again used lemma 3.1. Second, since (k 0 ; a 0 ; z 0 ; 0 ); , d a , B, q l l 2 and k are the same across the two production economies, lemma 3.2 implies that b F i (s t ) = F i (s t ) for all i 2 I and all s t 2 N . Thus, the fact that c i is optimal for each i 2 I in the e-economy implies that it is also optimal for each i 2 I in the k-economy. In addition, the portfolio strategies achieving this allocation, given by b a l i (s t ) = a l i (s t ) for l 2 and b k i (s t ) = q 1 (s t )a 1 (s t ), are optimal.