Shareholders Unanimity With Incomplete Markets

Macroeconomic models with heterogeneous agents and incomplete markets (e.g. Krusell and Smith, 1998) usually assume that consumers, rather than firms, own and accumulate physical capital. This assumption, while convenient, is without loss of generality only if the asset market is complete. When financial markets are incomplete, shareholders will in general disagree on the optimal level of investment to be undertaken by the firm. This paper derives conditions under which shareholders unanimity obtains in equilibrium despite the incompleteness of the asset market. In the general equilibrium economy analyzed here consumers face idiosyncratic labor income risk and trade firms' shares in the stock market. A firm's shareholders decide how much of its earnings to invest in physical capital and how much to distribute as dividends. The return on a firm's capital investment is affected by an aggregate productivity shock. The paper contains two main results. First, if the production function exhibits constant returns to scale and short-sales constraints are not binding, then in a competitive equilibrium a firm's shareholders will unanimously agree on the optimal level of investment. Thus, the allocation of resources in this economy is the same as in an economy where consumers accumulate physical capital directly. Second, when short-sales constraints are binding, instead, the unanimity result breaks down. In this case, constrained shareholders prefer a higher level of investment than unconstrained ones.

resolved using majority voting. In this case, provided that constrained shareholders own a minority of the Þrm's shares, the equilibrium stock price of the Þrm is always equal to its capital stock and the allocation is again the same as in the version of the economy in which consumers accumulate capital directly. Third, the paper extends the unanimity result to a multiperiod economy.
Our companion paper (Carceles-Poveda and Coen-Pirani, 2004) provides a different, though complementary, approach to the problem of the Þrm's investment under incomplete markets. In the latter we also compare a setting where Þrms maximize period-by-period proÞts (i.e. the standard setting) with a setting where Þrms make intertemporal investment decisions and households hold their stock. However, instead of directly addressing the question of unanimity among shareholders, we show that there exists a particular objective for the dynamic Þrm that implies the same equilibrium allocation as in the standard setting. This objective corresponds to discounting the cash ßows of the Þrm with any present value price that does not allow for arbitrage opportunities. Differently from the present paper, this result is valid even if portfolio restrictions are binding.
While the macroeconomic literature with incomplete markets has mostly assumed away the problem of joint ownership of the Þrm, this problem has received plenty of attention in the theory literature, starting from Diamond's (1967) classic paper. The economies considered in the latter literature bear some similarity with the one usually considered by macroeconomists, with some important differences.
First, the theory literature generally consider models in which capital is the only input in production and the production function exhibits decreasing returns to scale (see, e.g., Magill and Quinzii, 1996, page 378). Macroeconomists, instead, typically assume constant returns to scale production technologies that use as inputs capital and labor. The assumption of constant returns to scale is crucial for the unanimity result of this paper.
Second, in the theory literature, borrowing or short-sale constraints are generally assumed not to be binding for shareholders. 2 Binding borrowing constraints, instead, play a more central role in the macroeconomic literature (see e.g. Krusell and Smith, 1998), so it is important to investigate their effect on the investment decision of the Þrm. This paper shows that the unanimity result derived here breaks down when short-sales constraints are binding, and that in this situation constrained shareholders would like the Þrm to purchase more capital than unconstrained ones.
Last, the theory literature typically considers two-period models, while the macroeconomic literature considers inÞnite horizon economies. It turns out that the main intuitions of this paper can be presented in a two period setting. The two-period model also makes it easier to relate the results of this paper with the classic contributions of Diamond (1967), Grossman and Hart (1979), and Ekern and Wilson (1974). This is done in section 3.3. The generalization to multiperiod and inÞnite horizon economies is introduced in the last section of the paper.
The rest of the paper is organized as follows. Section 2 introduces a standard two-period model economy with incomplete markets and idiosyncratic risk. Section 3 studies the equilibrium of this economy under the assumption that consumers' borrowing constraints are not binding. Section 4 considers the case of binding borrowing constraints. Section 5 analyzes the multiperiod case. Section 6 summarizes the results.

Model Economy
In this section we introduce the model economy. Since all relevant intuition can be obtained in a two-period version of the benchmark incomplete markets model used in macroeconomics (see Ayiagari, 1994 and Krusell and Smith, 1998), we will start from this case. 3 Firms produce an homogeneous good that can be either consumed or invested. Each Þrm operates the production function y = F (k, l; z) , where k denotes the physical capital input, l the labor input, and z is a random variable. The production function F is assumed to be twice differentiable with respect to k and l and display constant returns to scale: F (µk, µl; z) = µF (k, l; z) , for all µ > 0. The constant returns to scale assumption will play a key role in the analysis.
The economy is characterized by both aggregate and individual uncertainty. The former is captured by the random variable z, that is common to all Þrms. At time zero the value of z is known and equal to z 0 . In the second period, instead, z 1 is random and can take a Þnite number of values in the set Z = © z 1 , z 2 , ..., z N ª .
Let s 0 = {z 0 } and s 1 = {z 0 , z 1 } denote the length-0 and length-1 histories of aggregate shocks, and π z (s t ) the unconditional probability of history s t , with π z ¡ s 0 ¢ = 1. Also, denote by S 1 the set Z × Z.
The initial capital level of each Þrm at time zero is denoted by k j 0 . This is exogenously given to each Þrm and can be different across Þrms. At time zero the Þrm decides on the labor input l j Individual uncertainty refers to the fact that a consumer's labor endowment at t = 1 is random. At t = 0 a consumer i is endowed with x i 0 units of labor, where x i 0 is non-random and can differ across consumers. 3 In Section 5, I extend the model to the multiperiod case.

4
Let L 0 denote the aggregate labor endowment at time zero: denote the histories of endowment shocks for consumer i. The individual shocks might be correlated with the aggregate one, so that the probability of history s i1 might depend on s 1 . Denoting by π x ¡ s i1 |s 1 ¢ this conditional probability, let π ¡ s i1 , indicate the joint probability of s i1 and s 1 . We assume that the only aggregate source of uncertainty is the productivity shock z 1 . Individual uncertainty is assumed to disappear in the aggregate due to a law of large numbers. In particular, the aggregate units of labor available for production at time 1 following aggregate history s 1 are equal to: Each consumer i is endowed with the following utility function: where c 0 denotes consumption at time zero and c 1 consumption at time one. The momentary utility function U is assumed to be twice differentiable with U 0 > 0 and U 00 < 0.
The consumer also derives income from trading shares of the Þrms. At time zero a consumer i is endowed where θ > 0.
where w 0 ¡ s 0 ¢ and w 1 ¡ s 1 ¢ denote the time 0 and time 1 wages per unit of labor endowment.
The measure of each Þrm's shares outstanding is normalized to one in both periods:

Equilibrium
The key difficulty in solving this model is represented by the fact that each Þrm is, in general, owned by many consumers-shareholders, and the latter might disagree on its optimal level of investment. Here we assume that initial, rather than Þnal, shareholders make the investment decision. 5 Given this timing, a Þrm's initial shareholder has to form an expectation on how the Þrm's stock price is going to change with different levels of investment.
The theoretical literature concerning the objectives of the Þrm under incomplete markets has mostly proceeded under the assumption of competitive price perceptions (from now on CPP), originally introduced by Grossman and Hart in their seminal 1979 paper (see Magill and Quinzii, 1996 for a discussion of this approach). With CPP each shareholder forms expectations about the effects of investment on the Þrm's stock price using his own state prices. As a result, initial shareholders will unanimously agree to maximize the Þrm's net value p 0 + d 0 , while potentially disagreeing on the best way to achieve this result. In turn, this conßict can be either resolved by allowing transfers among shareholders, as suggested by Grossman and Hart (1979) or by some voting procedure (see e.g. DeMarzo, 1993). One problem with the CPP approach is that it is only applicable when there are no binding restrictions on short-sales of a Þrm's stock. 6 Moreover, the CPP approach implies that in an incomplete markets equilibrium shareholders have different opinions 5 The results of this section easily apply to the situation in which Þnal shareholders make the investment choice. The assumption that initial shareholders make the investment decision implies that the latter have to form expectations about the effect of investment on the Þrm's stock price. This would be always the case, independently of whether initial or Þnal shareholders decide on investment, in a model with more than two periods. 6 See Grossman and Hart (1979, page 299, footnote 5) for a discussion of this point. In case an initial shareholder faces binding short-sale restrictions, his state prices do not contain sufficient information on the change in the Þrm's stock market price following a change in its investment. about the sensitivity of stock prices to the level of investment. 7 In this paper, instead, we adopt the standard rational expectations assumption to derive shareholders' expectations on the effects of different investment levels on the Þrm's stock price. Among other things, this assumption implies that shareholders will always agree on the effect of a change in investment on the Þrm's stock price, while possibly disagreeing on the optimal level of investment to be undertaken by the Þrm.
In what follows, the plan is to Þrst deÞne an exchange equilibrium for the economy described above by taking the production and investment plans of each Þrm as given. This exchange equilibrium determines the relationship between Þrm j's stock price p j 0 and its capital stock in period 1, k j 1 , for given capital chosen by the other Þrms in the economy. The rational expectations approach to solving the investment decision problem of a Þrm consists of deriving the sensitivity of a Þrm's stock price to variations in k j 1 from this relationship, rather than from the agent's individual state prices. Given this pricing function we can then turn to the issue of determining the optimal investment level for Þrm j. It turns out that in this economy, under constant returns to scale in production, the set of unconstrained initial shareholders of Þrm j will unanimously agree on its optimal investment k j 1 . 8

Exchange Equilibrium
Let a Þrm's dividends paid in periods 0 and 1 be deÞned as: Before deÞning an exchange equilibrium for this economy, it is important to point out that the assumption of constant returns to scale in production implies that To see why this is the case, it is convenient to consider a Þrm's labor demand. Since this is a static choice, its shareholders will always agree about hiring labor up to the point where its marginal product is 7 See section 3.3 for further discussion on the CPP approach. 8 Final shareholders will agree as well, but they are not the ones that are assumed to make the investment decision.
equal to the wage: Therefore, since under constant returns to scale F L (k, l; z) is homogeneous of degree zero in k and l, all Þrms will have the same capital-labor ratio. In particular, where g is a function of w 1 ¡ s 1 ¢ and z 1 only. Replacing this expression into (7) and using again the constant returns to scale assumption to collect k j 1 ¡ s 0 ¢ yields equation (8). An important implication of this equation any two Þrms in period 1 are always linearly dependent.
An exchange equilibrium for this economy is deÞned as follows.
DeÞnition (Exchange Equilibrium). Given the dividends ((d j t (s t )) t=0,1 ) j∈J paid by Þrms and the initial distribution of shares ((θ i 0j ) j∈J ) i∈I , an exchange equilibrium is a represented by a collection of stock such that: ) j∈J are optimal for consumer i ∈ I, i.e., they maximize (1) subject to (3), (4) and (2).
) j∈J is such that the market for the shares of each Þrm clears, i.e., (5) holds.
A key property of an exchange equilibrium of this economy is summarized in the following proposition: In an exchange equilibrium, the rates of return on the stocks of all Þrms are equalized for all possible realizations of the aggregate shock z 1 in period 1. Formally: for all j, j 0 , and s 1 .
To see this, suppose that there existed a couple of Þrms j and j 0 and a state of the world z 1 such that equation (12) did not hold. Then, using the expression (8) for a Þrm's dividend, it must be the case that, so that the equality condition is in fact violated for all possible histories s 1 . Thus, equation (13) implies that the rate of return on Þrm j's stock is always higher than the rate of return on Þrm j 0 's stock, for all possible realizations of z 1 . Then, clearly, there would be an incentive for a consumer holding Þrm j 0 's stock to sell it and buy instead Þrm j's stock. This in turn would imply that the economy is not at an exchange equilibrium. Therefore, equation (12) must hold.
This condition is important because it provides us with a way to compute the effects of a variation in Þrm j's investment on its stock price. In particular, rearranging (12) and using (8) to replace d j When deciding on k j 1 ¡ s 0 ¢ , Þrm j's shareholders take as given the stock price and the investment decision made by all other Þrms. Thus, equation (14) provides them with information about the effect of different investment levels on their Þrm's stock price. Using this equation we obtain: Notice that with an arbitrary production function, changes in the level of capital k j We conclude this section by stating the Þrst order condition associated with a consumer's optimal portfolio choice in an exchange equilibrium: where Equation (16) holds with equality if for consumer i: Notice that, because of equation (12), if condition (16) holds with equality for Þrm j, then it must hold with equality for all other Þrms j 0 6 = j.
In what follows we will separately consider the case in which the constraint (2) is never binding for any agent from the case in which there is a positive measure of agents for which it is binding. The distinction is important because in the former case all initial shareholders of a Þrm will unanimously agree on the amount of capital to be invested. In the latter case, instead, there will be disagreement among initial shareholders, and it will be necessary to specify a mechanism to aggregate these heterogeneous preferences into one investment decision.

A Firm's Investment Decision When Short-Sales Constraints are Not Binding
In this section we consider the case where the short-sales constraint (2) is never binding for any agent i. 9 Under this assumption let's now turn to the decision problem faced by a Þrm in this economy. A Þrm in this model makes three decisions. It chooses the labor input at time 0 and at time 1 in each aggregate state of the world. It also chooses a level of physical capital k j 1 (s 0 ) to be used as an input in production in period 1. The capital stock depreciates at the rate δ after production.
The static nature of the labor input decision and its timing imply that all shareholders will agree on the optimal quantity of labor to hire in a given period. The dynamic nature of the investment decision in this incomplete markets setting, instead, suggests that there might be scope for disagreement among initial shareholders on the optimal level of investment. The main result of this section is that, under constant returns to scale in production and no binding short-sales constraints, there will be unanimity among shareholders.
Consider the investment decision of Þrm j. Investment is assumed to be Þnanced entirely out of the Þrm's retained earnings at time zero. 10 An initial shareholder i of the Þrm would choose k j 1 ¡ s 0 ¢ in order to maximize the utility function V i subject to (6) and (7), taking as given the price of labor and the investment decisions of other Þrms. 11 The Þrst order condition for his problem is: where Equation (18) is the key to understanding the effect of incomplete markets on shareholders' investment 9 Sufficient conditions for this to be the case are: i) θ = 0; ii) the minimum possible labor endowment realization in period 1 is zero, i.e., x 1 = 0; iii) the marginal utility of consumption at zero is inÞnite, i.e. U 0 (0) = +∞. These assumptions imply that a consumer will never choose θ i 1j ¡ s i0 , s 0 ¢ = 0 for any j. Suppose, in fact, that for some Þrm j, θ i 1j ¡ s i0 , s 0 ¢ = 0 and that equation (16) holds as an inequality. Then, equation (12) would imply that the Þrst order condition (16) holds as an inequality for all Þrms j ∈ J. Thus, θ i 1j ¡ s i0 , s 0 ¢ = 0 for all j ∈ J. In this case, since x 1 = 0, a consumer will experience zero consumption with positive probability. The assumption that U 0 (0) = +∞ is sufficient to rule this case out. Thus, the short-sale constraint will never bind. 10 It is straightforward to add bonds to this economy without altering the unanimity result of the paper. As long as all initial shareholders are unconstrained in their portfolio choice, the Modigliani-Miller theorem holds in this setup and the Þnancial structure of the Þrm is indeterminate. 11 By assumption, agents for whom θ i 0j < 0 do not participate in the investment decision of the Þrm. Their goal would be to minimize rather than maximize the Þrm's value.
decisions. As initial owners of the Þrm, shareholders care about the way investment in physical capital affects the net stock market value of the Þrm. This effect is captured by the term multiplying θ i 0j in (18 When the asset market is incomplete, instead, state prices are not necessarily equal among shareholders. In this circumstance, the assumption of constant returns to scale in production guarantees that trading in the stock market is sufficient to make shareholders agree on the expected discounted value of an extra unit of investment. To see this, recall that from equation (8) ∂d j , and from equation (15): Replacing these two equations into (18), this Þrst order condition for investment becomes The expression in parenthesis multiplying θ i 1j ¡ s i0 , s 0 ¢ is equal to zero because stock market trading equalizes Þnal shareholders' valuations of future dividends, i.e. equation (16) holds with equality. This observation leads us to the following proposition: Proposition 2. The unconstrained initial shareholders of a Þrm j will unanimously agree in setting its level 12 With complete markets the Þrst order condition for holding stock is: where m 0 ¡ s 1 ¢ denotes the price at time zero of a security that pays one unit of consumption in state s 1 and zero in all other states. Given that a Þrm takes these prices as given, this equation can be used to obtain: . This condition implies that the second term in (18) is equal to zero for all shareholders, independently of whether the production function displays constant returns to scale or not.
of capital for period 1 according to the condition where the derivative on the left hand side of this equation is deÞned in equation (19).
It follows that for all initial shareholders the optimal investment strategy for each Þrm j ∈ J is such that Notice that all Þrms make the same decisions regarding period 1 capital stock, k j independently of their initial capital stock k j 0 . In fact, the decision about k j 1 ¡ s 0 ¢ depends only on the cost of investment, which is equal to one, and the present discounted value of period 1 dividends, which is independent of a Þrm's initial capital.
A competitive production-exchange equilibrium for this economy is therefore deÞned as follows: Production-Exchange Equilibrium. A production-exchange equilibrium for this economy is represented 1 ) i∈I constitute an exchange equilibrium for given dividends (d t (s t )) t=0,1 .
ii) Shareholders of all Þrms unanimously agree to set k 1 iii) The labor market clears: if wages (w t (s t )) t=0,1 are given by (9) and (10), aggregate labor demand equals aggregate labor supply in t = 0, 1.
As a corollary to Proposition 2, we have the following result: Corollary 1. Unanimity among initial shareholders implies that the equilibrium allocation of consumption and capital in this incomplete markets economy is the same as in an economy where consumers, instead of Þrms, accumulate physical capital directly.
To show this point formally, consider a version of the economy where consumers, instead of Þrms, accumulate physical capital. In this version the budget constraint of consumer i reads: where r t (s t ) is the rental rate of the stock of capital owned by consumer i in period t. Letting aggregate capital be deÞned as: the rental rates are equal to the marginal products of capital in the two periods: Further, each consumer accumulates physical capital according to the marginal condition It is easy to see that the equations that characterize the equilibrium of this economy are the same as the conditions that characterize a production-exchange equilibrium. First, the Þrst order condition (23) is the same as (16) because , and noticing that

Discussion and Relationship with the Literature
The key assumptions for the unanimity result obtained above are that the production function F displays constant returns to scale and that short-sales constraints are not binding. These assumptions jointly guarantee that the term in parenthesis multiplying θ i 1j ¡ s i0 , s 0 ¢ in equation (18) is equal to zero: 1) The constant returns to scale assumption guarantees that the latter term can be written as in equation (20); 2) The fact that the portfolio decision is interior guarantees that the Þrst order condition for agent i's portfolio choice can be used to set this term to zero.
It is easy to show that this unanimity result is robust to extending the model to consider: 1) preferences 13 I am abstracting from the Þrm sub-index j here because in a production-exchange equilibrium there is symmetry across all Þrms. characterized by non-expected utility; 2) preference heterogeneity among consumers; 3) different "opinions" among consumers about the likelihood of a given state of the world; 4) adjustment costs in the installation of new capital, as long as the adjustment cost function is homogeneous of degree one in current capital and investment (as in Hayashi, 1982). 14 In what follows we discuss how the unanimity result obtained here relates to some of the main contributions to the theoretical literature on the objectives of the Þrm under incomplete markets. Consider, Þrst, Diamond (1967)'s classic paper. Diamond focused his attention on the case where: 1) uncertainty faced by Þrms takes a multiplicative form; 2) capital is the only input used in production; 3) production occurs under decreasing returns to scale according to a production function of the form zG (k) , where G 0 > 0 and G 00 < 0; and 4) capital fully depreciates within a period, i.e., δ = 1. Under these assumptions a Þrm's period Given that aggregate uncertainty affects dividends multiplicatively, an argument analogous to the one developed in section 3.1 implies the equalization of ex-post rates of return among Þrms (equation 12). Thus the equivalent of equation (14) is for all j, j 0 .

It follows that
Replacing (24) and the interior version of the Þrst order condition (16) in (18) yields the equation that determines the optimal choice of capital for shareholder i: Therefore shareholders will be unanimous in their choice of k j 1 ¡ s 0 ¢ . Now, consider the relationship between this result and the one obtained in the previous section. To do this, consider a possible extension of Diamond's result to an economy where production occurs using capital and labor while assumptions 1) and 3)-4) are preserved. Denote the production function by zG (k, l) , where G does not need to exhibit constant returns to scale. The dividend in period 1 is 14 This homogeneity guarantees that period 1 dividends can still be written as k j 1 ¡ s 0 ¢ times a term that depends only on aggregate variables.
, the latter can be written as: where h and e q are two functions. If this were the case then we could apply Diamond's argument summarized above. The problem with this argument is that, even if uncertainty enters multiplicative in the original production function, there is no guarantee that the decomposition (26) applies. Consider for example the decreasing returns to scale production function: Solving the labor demand problem and plugging the optimal l j 1 ¡ s 1 ¢ back into (25) yields: , which clearly does not take the form (26). 15 In summary, in the model considered here, Diamond's assumption of multiplicative uncertainty does not, in general, give rise to unanimity among shareholders if the production function exhibits decreasing returns to scale in capital and labor. The assumption of constant returns to scale in production, instead, is sufficient to obtain unanimity. 16 Of course, as the example of footnote (15) indicates, the assumption of constant returns to scale in production is not necessary in order to obtain the unanimity result.
It is interesting to contrast the approach taken here in solving the model, with the one pursued by Grossman and Hart in their seminal (1979) paper. They do not assume that F is constant returns to scale.
Instead, they postulate that consumers have CPP, i.e., that they use their own state prices m i 0 ¡ s i1 , s 1 ¢ to evaluate the effect of a change in k j 1 ¡ s 0 ¢ on the Þrm's stock price: 15 Notice, however, that when the production function takes the commonly used Cobb-Douglas form this problem does not arise. For example, if G (k, l) = k α l σ , with α + σ < 1, then: which satisÞes (26). 16 Notice that if the production function displays constant returns to scale, the aggregate shock z must affect this function multiplicatively.
where the subindex i on the left-hand side indicates that the perception of this derivative varies across shareholders. This amounts to setting the second term in equation (18) equal to zero by assumption. Thus, while shareholders will be unanimous in their desire to maximize the Þrm's net stock market value, by setting they will, in general, disagree on the choice of k j 1 ¡ s 0 ¢ . Grossman and Hart resolve this conßict by allowing for income transfers among a Þrm j's initial shareholders at time zero. In their approach, an optimal investment plan for the Þrm is such that there is no other investment plan and set of income transfers among shareholders such that all shareholders are better off.
In the model considered here, even if shareholders use their own state prices to evaluate ∂p j they still agree on the magnitude of this derivative. This is because trading in the stock market and constant returns to scale in production guarantee that the right-hand side of equation (27) is the same among all shareholders. An advantage of not postulating CPP is that we can extend our approach to the case where short-sales constraints are binding (see section 4). In this case the CPP approach is not applicable because the right-hand side of (27) does not measure the marginal beneÞt of higher investment by the Þrm for a constrained shareholder.
The unanimity result obtained here can also be interpreted as a special case of the "spanning" result derived by Ekern and Wilson (1974). They showed that if the asset market is incomplete shareholders will be unanimous in approving investment plans generating vectors of dividends that are spanned by the payoffs of existing securities. In this model this condition is satisÞed. Each Þrm j's period 1 dividend vector (d j is just a multiple of the dividend vector of any other Þrm j 0 because d j This is because the labor input in period 1 is chosen after the capital stock is already in place, and, by constant returns to scale, the optimal labor input in period 1 is a linear function of capital (see equation 11).

Binding Short-Sales Constraints
In this section we analyze the case where the short-sales constraint (2) binds for some initial shareholders of a Þrm. In this case, for these initial shareholders the term multiplying θ i 1j ¡ s i0 , s 0 ¢ in equation (18) is not equal to zero. Except for the case in which the binding short-sales constraint is θ= 0, in which case the second term in (18) disappears, this situation introduces the possibility of disagreement among initial shareholders about the optimal size of investment to be undertaken by the Þrm.
Consider an initial shareholder i for whom the short-sales constraint (2) binds. As noticed above, if this constraint binds for one Þrm j, then it must also bind for all Þrms. The introduction of a short-sale constraint of the kind (2) Replacing this into the Þrst order condition for the shareholder i's preferred investment level (equation

18), taking into account equation (19) and the constraint θ
It follows that: Proposition 4. Constrained initial shareholders of Þrm j prefer a higher level of period 1 capital than unconstrained shareholders. In particular, constrained shareholder i's preferred investment is given by: while all unconstrained shareholders would want to choose k j A constrained initial shareholder would like to set the Þrm's investment to the point where the derivative When initial constrained shareholders own more than Þfty percent of the Þrm's shares the analysis becomes more complex. In a majority voting equilibrium, provided it exists, period 1 capital stock tends to be higher than a Þrm's stock price: k j 1 ¡ s 0 ¢ > p j 0 ¡ s 0 ¢ , as preferred by constrained shareholders. While interesting, this case seems unlikely to occur in the class of incomplete markets models analyzed in the macroeconomic literature. In the latter, in fact the measure of constrained agents is usually negligible and their share of the aggregate capital stock (the equivalent of the initial shares θ i 0j in an economy where consumers accumulate capital) even smaller.

Extension to Multiple Periods
In this section we show how the unanimity result derived in section 3.2 extends to a multiperiod, possibly inÞnite horizon, economy. The additional feature of the model in this context is the fact that stock returns depend not only on future dividends but also on future stock prices.
Consider Þrst the case where T is Þnite. Denote by s it the length-t history of individual shocks for agent i and by s t the history of aggregate shocks. Each consumer maximizes the following utility function where π ¡ s it , s t ¢ denotes the probability, as of time zero, of histories ¡ s it , s t ¢ . As a consumer, agent i faces the sequence of budget constraints: where the notation is analogous to the one of section 3. For simplicity, in this section we rule out short-selling: 17 scale in production imply that where the lack of a terminal point prevents us from directly replacing p j t (s t ) instead of k j t+1 (s t ) on the left hand side of this equation (as in equation 33). The analysis in this case must proceed by Þrst guessing that in fact p j t (s t ) = k j t+1 (s t ) , and then verifying that this guess is valid. Showing that the guess is valid amounts to showing that there is unanimity among initial shareholders regarding the investment decision of every Þrm. The latter point can be proved exactly as above for the Þnite horizon economy.

Summary
In this paper we have studied versions of the standard incomplete markets economy introduced by Aiyagari (1994) and Krusell and Smith (1998) under the assumption that Þrms, rather than consumers, accumulate physical capital. If borrowing constraints are not binding and production occurs under constant returns to scale, the equilibrium allocation of this economy is shown to coincide with the one that characterizes the standard model. In particular, a Þrm's shareholders will unanimously agree on the Þrm's optimal level of investment.
In the case in which borrowing constraints are binding, instead, the unanimity result breaks down.
Constrained initial shareholders would like the Þrm to invest more than unconstrained ones. Given that in a typical macroeconomic model with incomplete market (Krusell and Smith, 1998), the measure of agents for whom short-sale constraints are binding is negligible, it is unlikely that taking this disagreement explicitly into account would have a signiÞcant effect on its quantitative properties. Therefore, taken as a whole, this paper suggests that, in practice, there might not be any signiÞcant loss of generality in focusing on incomplete markets models a la Aiyagari-Krusell-Smith where consumers, rather than Þrms, undertake the capital accumulation decision.