Two-Person Dynamic Equilibrium in the Capital Market

Two-Person Dynamic Equilibrium in the Capital Market Bernard Dumas University of Pennsylvania Centre HEC-ISA, National Bureau of Economic Research Wbe...
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Two-Person Dynamic Equilibrium in the Capital Market Bernard Dumas University of Pennsylvania Centre HEC-ISA, National Bureau of Economic Research Wben several investors with different risk aversions trade competitively in a capital market, the allocation of wealth fluctuates randomly among them and acts as a state variable against which each market participant will want to hedge. This hedging motive complicates the investors' portfolio choice and the equilibrium in the capital market. This article features two investors, with the same degree of impatience, one of them being logarithmic and the other having an isoelastic utility function. They face one risky constant-return-to-scale stationary production opportunity and they can borrow and lend to and from each other. The behaviors of the allocation of wealth and of the aggregate capital stock are characterized, along with the behavior of the rate of interest, the security market line, and the portfolio boldings. The two-investor equilibrium problem is as basic to financial economics as is the two-body problem to mechanics. Yet, to my knowledge, no complete description of the dynamic interaction between two investors This article is a revised version of NBER working paper 2016. Grant 82E1177 of the French Ministry of Research and Technology and the support of the A. Shoemaker Chair at the Wharton School are gratefully acknowledged. The author thanks the referee, George Constantinides, for his valuable suggestions. In addition, the following individual provided insightful comments for which the author is my thankful: Andy Abel, Simon Benninga, Michael Brennan, Ivan Brick, Daniel Cohen, Robert Cumby, Jean-Pierre Danthine. Philippe Dumas, Hamadi Hamadi, Jose Scheinkman, Qi Shen, Alan Stockman, Rene Stulz, and Charles Wyplosz. Jiang Wang provided valuable research assistance. Address reprint requests to Bernard Dumas, Professor of Finance, University of Pennsylvania, Wharton School, 3302 Steinberg Hall-Dietrich Hail, 3620 Locust Walk, Philadelphia, PA 19104-6367. 4

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exists in the literature. Related work includes capital asset pricing models (CAPM) in the presence of several investors [e.g., Constantinides (1982)] and proofs of the existence of the equilibrium in continuous time when investors have a finite horizon [Karatzas et al. (1987)). None of this work includes a description of equilibrium behavior: portfolio choices, process followed by the rate of interest, volume of trading, etc. In some cases equilibrium behavior is already known from past results. In specific situations outlined by Rubinstein (1974) HARA utility investors aggregate to one representative HARA utility investor;1 furthermore, optimal consumption-sharing rules between them are linear. If the menu of securities available includes a fixed-income consol and shares of stock, investors can use a constant portfolio of these securities to implement the optimal sharing rule. Investors simply hold these securities and live off the income. In this article we examine two investors who have access to a complete financial market [in the sense of Harrison and Kreps (1979)]. Here again aggregation is possible: There exists a welfare function, obtained as a weighted average (with constant weights) of the individual investors’ utility functions. This welfare function can be seen as the utility of a representative individual. But we analyze the general case where HARA utilities do not aggregate into a HARA welfare function and where consumptionsharing rules are not linear. Specifically, we deviate from the Rubinstein (1974) base case of equal cautiousness parameters by considering two isoelastic-utility investors with different degrees of relative risk aversion:2 One has a logarithmic utility, the other has an isoelastic utility that differs from the log. They have access to standardized securities such as shares of equity and instantaneous borrowing and lending. With such a menu of assets, the market, while complete, is only dynamically complete, which means that trading is needed.3 Our goals are l To derive the evolution of the distribution of wealth betweenthe two investors l To determine whether the process of capital accumulation settles down into a stationary probability distribution or wanders off forever4 l To characterize the two investors’ portfolio compositions and trading volumes at each point in time l To fully endogenize the stochastic process for the rate of interest

1

Rubinstein gave a list of conditions that were individually sufficient for hyperbolic absolute risk aversion (HARA) utilities to aggregate into HARA. Brennan and Kraus (1978) showed that these conditions were necessary. One of these conditions is that all investors share the same “cautiousness parameter.”

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For isoelastic utilities Rubinstein’s cautiousness parameter is equal to the relative risk aversion.

3

See Duffie and Huang (1985). An infinite amount of trading is needed.

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This part can be seen as a theory of optimal stochastic growth with a non-HARA welfare function, extending Merton (1975).

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l To examine equilibrium personal consumption and saving behaviors in relation to the rate of interest

The balance of the article is organized as follows. Section 1 describes the setting. Section 2 provides a proof of existence for the welfare optimum problem and establishes a correspondence between welfare optima and equilibria. Section 3 spells out the solution technique. The remaining sections (4 to 7) individually address the items listed above under “goals.” Section 8 concludes. 1. The Model The capital market of our model economy is populated with two investors, with infinite horizons, the same rate of impatience, but different risk aversions. The analysis is greatly simplified and does not lose its illustrative power if we restrict one investor to a logarithmic utility function, while the other investor exhibits any degree of risk aversion 1 - γ, where γ is the power of his isoelastic utility function.

His finite rate of consumption is The two investors consume a single good and have access to two investment opportunities: 1. They can buy shares in one constant-return-to-scale production activity, whose random output per unit of capital has a constant gaussian distribution with fixed parameters α and 2. They can borrow and lend to and from each other at the equilibrium riskless6 rate r, which varies over time in an endogenous fashion. Other notations are as follows:

5

Recall [Merton (1971)] that the logarithmic case an be obtained as the limit of the above case for γ - 0. Recall also that the log utility is bounded neither from above nor below. For γ > 0 the isoelastic utility is bounded from below but not above. For γ < 0 It is bounded from above but not below.

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Endogenous default resulting from unwillingness to pay Is left for future research. Inability to pay is ruled out (for as long as initial wealth is positive) by one well-known property of the isoelastic utility function: When consumption tends to zero, marginal utility tends to infinity.

7

Wealth and physical capital take the same value because of the assumption of constant returns to sale.

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(undiscounted) martingale pricing measure of Cox and Huang (1986). Given consumption behavior (6) the economic interpretation of Equation (9) is obvious: The first part of the integral is equal top, times the present value of the log investor’s consumption, which we can define as his initial wealth . The second part of integral (9) is p0 times the non-log investor’s wealth. Using Equation (6) it is seen that Equation (10) governs the evolution of the capital stock under the optimal consumption policy. 2.2 Existence of the welfare optimum Definition. The shadow rate of interest for the welfare problem is defined as

Given assumption (8) it is seen that the rate of interest of Equation (11) is bracketed as follows: The two extreme values for the rate of interest are those which would obtain if the log investor or the non-log investor held all the wealth. Lemma 1. For the solutions of Equations (9) and (10) to exist [i.e., for integral (9) to be convergent], it is sufficient that

The proof indicates that the part of integral (9) corresponding to the log investor’s consumption is unconditionally convergent, whereas the part corresponding to the non-log investor’s consumption converges under condition (13). Equation (13) therefore guarantees that the initial value p0 of marginal utility is well defined. Proposition 3. If condition (13) is satisfied and assumption (8) is met, the optimal value L(S) of the welfare function given by integral (4) is also well defined. Proof: See Appendix 1. ■ It is shown in the proof that condition (13), which ensures that the initial marginal utility p0 is well defined, also guarantees the convergence of the part of the welfare function (4) corresponding to the non-log investor’s utility. Convergence of the log investor’s part, however, is not guaranteed 162

unless the rate of interest remains bounded; hence the need for assumption (8), which implies (12).

Note that condition (14) is identical to the condition of existence that would have been appropriate had the isoelastic investor been alone [Merton (1971)].

2.3 Equilibrium The economy under study is subjected to only one Wiener process dz affecting output. Investors in the capital market have access to two nonperfectly correlated securities (the equity and the riskless asset). This implies that they face a complete financial market, in the sense of Harrison and Kreps (1979). In these circumstances it is well known [see, for example, Constantinides (1982)] that the welfare optimum can be replicated as a capital market equilibrium. Definition. An equilibrium is a stochastic process S and two functions W*(S) and r(S) governing the allocation of wealth between the two investors and the rate of interest, such that:

9

Propositions 4 and 5 state sufficient conditions for existence. One can safely believe that these conditions are also necessary, if one is not willing to place restrictions on the parameter λ which represents the relative weight of the two Investors. If one is willing to place such restrictions, however, it seems likely that the above conditions of existence can be relaxed. The study of this alternative type of welfare optima and equilibria will await future research.

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(undiscounted) martingale pricing measure of Cox and Huang (1986). Given consumption behavior (6) the economic interpretation of Equation (9) is obvious: The first part of the integral is equal top, times the present value of the log investor’s consumption, which we can define as his initial wealth . The second part of integral (9) is p0 times the non-log investor’s wealth. Using Equation (6) it is seen that Equation (10) governs the evolution of the capital stock under the optimal consumption policy. 2.2 Existence of the welfare optimum Definition. The shadow rate of interest for the welfare problem is defined as

Given assumption (8) it is seen that the rate of interest of Equation (11) is bracketed as follows: The two extreme values for the rate of interest are those which would obtain if the log investor or the non-log investor held all the wealth. Lemma 1. For the solutions of Equations (9) and (10) to exist [i.e., for integral (9) to be convergent], it is sufficient that

The proof indicates that the part of integral (9) corresponding to the log investor’s consumption is unconditionally convergent, whereas the part corresponding to the non-log investor’s consumption converges under condition (13). Equation (13) therefore guarantees that the initial value p0 of marginal utility is well defined. Proposition 3. If condition (13) is satisfied and assumption (8) is met, the optimal value L(S) of the welfare function given by integral (4) is also well defined. Proof: See Appendix 1. ■ It is shown in the proof that condition (13), which ensures that the initial marginal utility p0 is well defined, also guarantees the convergence of the part of the welfare function (4) corresponding to the non-log investor’s utility. Convergence of the log investor’s part, however, is not guaranteed 162

unless the rate of interest remains bounded; hence the need for assumption (8), which implies (12).

Note that condition (14) is identical to the condition of existence that would have been appropriate had the isoelastic investor been alone [Merton (1971)].

2.3 Equilibrium The economy under study is subjected to only one Wiener process dz affecting output. Investors in the capital market have access to two nonperfectly correlated securities (the equity and the riskless asset). This implies that they face a complete financial market, in the sense of Harrison and Kreps (1979). In these circumstances it is well known [see, for example, Constantinides (1982)] that the welfare optimum can be replicated as a capital market equilibrium. Definition. An equilibrium is a stochastic process S and two functions W*(S) and r(S) governing the allocation of wealth between the two investors and the rate of interest, such that:

9

Propositions 4 and 5 state sufficient conditions for existence. One can safely believe that these conditions are also necessary, if one is not willing to place restrictions on the parameter λ which represents the relative weight of the two Investors. If one is willing to place such restrictions, however, it seems likely that the above conditions of existence can be relaxed. The study of this alternative type of welfare optima and equilibria will await future research.

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Proposition 6. The shadow rate of interest (11) of the welfare problem supports an equilibrium. In this equilibrium, the rates of consumption by the two investors (and therefore the path of capital) are identical to what they are at the welfare optimum. Proof Available on request. ■ The assumption that one of the two investors has a logarithmic utility renders the mapping between equilibria and welfare optima particularly straightforward. To save space, we appeal to the known result [see Hakansson (1971) and Merton (1971)] that the optimal decisions of a log investor are

and that his wealth and his marginal indirect utility of wealth are inversely related. The mapping between equilibria and optima is therefore given by

Although the above approach, based on system (9) and (10), has been very helpful in establishing convergence conditions, there exists no technique to solve directly these equations when, as is the case here, they do not have a closed-form solution. For this reason we now revert to ordinary differential equations (ODE). These can be solved numerically, provided some appropriate boundary conditions can be established.

Proof. See Appendix 2. 164

The welfare optimum could be computed by solving the following ODE [which is equation (A2) in Appendix 1]:

This is an equation in the unknown function p(S).10 In solving this differential equation one imposes the inequality boundary condition (19) or (20). The initial condition is provided by the number S0. One must then search for numbers p0 and p'0 such that one can find a solution (i.e., satisfy the boundary condition throughout the domain). One very severe drawback of this procedure is the fact that the unknown function p(S) maps [0, + ∞] onto itself. The approach can be improved upon if we can find a change of variable and unknown function such that the new variable and the new unknown function evolve over bounded intervals, Propositions 7 and 8 imply that this double goal is achieved if one introduces, as a new independent variable, the quantity

and, as a new unknown function, the quantity

Equation (18) allows one to interpret ω as the distribution of wealth W/( W + W*): 0 ≤ ω ≤ 1. Similarly, the first-order condition (6) provides the interpretation of the quantity is equal to W/c, the inverse of the propensity to consume out of wealth of the non-log investor. It is not surprising that this rate should be bounded from above and below. Equations (22) and (23) define parametrically a functional relationship The change of variable and unknown function requires tedious algebraic calculations to provide the differential equation to be satisfied by on the basis of the ODE (21) for p(S):11

10

Actually, the inverse function S(p) satisfies a somewhat simpler quasilinear equation [for a similar observation see He and Pearson (1988)].

11

Observe that the parameter λ is no longer present in the ODE (24). In the market equilibrium version of this economy, the function is an invariant under a change of initial wealth distribution.

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While Equation (24) is highly nonlinear and considerably more complicated than the ODE (21) for p(S) , solving it numerically is easier because of the boundedness property. In fact, this property can serve to generate two natural boundary conditions which follow from the differential equation itself: Successively inserting ω = 0 and ω = 1 into the differential equation (24), and imposing that I be bounded, one gets

The solution of ODE (24) “subject to” boundary points (26) and (27) is a two-point boundary-value problem of the Dirichlet type, which lends itself nicely to numerical analysis. 4. The Equilibrium Wealth-Sharing Rule Along an equilibrium path the distribution of wealth and the aggregate capital stock fluctuate in tandem. We refer to the relationship as the equilibrium wealth-sharing rule. That the wealth of the individual investor should depend only on current aggregate wealth is the result of financial market completeness and the Markovian structure of the economy. In effect, when and if the capital stock increases, the more highly levered investor reaps a larger share of the increase than does the less levered investor. The manner in which aggregate wealth is dynamically distributed between the two investors is the direct product of the way in which aggregate consumption is distributed between them. The concept of a consumptionsharing rule was introduced by Wilson (1968) and Rubinstein (1974); it is designed precisely to describe the way in which consumption is allocated. In a complete (and therefore Pareto-optimal) market, marginal rates of substitution are equated across Individuals. This implies that, at any time, the levels of the marginal utilities of the various investors are proportional to each other. The proportionality relationship is valid both for the marginal utilities of consumption and for the marginal utilities of wealth, as the two are equated at the optimal level of consumption. The wealth-sharing function can be recovered from the Ι(ω) function by applying in the reverse the change of function and variable (19) and 166

(20). This yields p(S). Then the mapping of (16) provides W*(S). This function inherits from the consumption-sharing rule a number of properties which we now investigate by means of numerical analysis. Figure 1 provides a plot of the wealth-sharing rule for the two cases y > 0 and γ < 0 and for given initial wealths.12 It is apparent that whenever the aggregate capital stock S increases the wealth of the person with the smaller risk aversion increases more than proportionately. In fact, if S goes to infinity, the person with the smaller risk aversion ends up owning almost all the wealth, and if the aggregate capital stock goes to zero, the person with the larger risk aversion captures it almost entirely.

12

The two numerical illustrations we use throughout this article are for the following sets of parameter values: ● Numerical illustration A: averse than the log Investor)

(i.e., the non-log investor is less risk

● Numerical Illustration B: averse than the log Investor)

(i.e., the non-log investor is more risk

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The properties of the wealth-sharing function, as derived from extensive numerical analysis, are recorded in the following propositions. Proposition 9. When γ > 0 (γ < 0), the wealth-sharing function is an increasing concave (convex) function of the aggregate capital stock. Proposition 10. The distribution of wealth is a monotonic function of the aggregate capital stock:13 When γ > 0 (γ < 0), S and ω are positively (negatively) related. Proposition 11. Table 1 summarizes the variations and the asymptotic behavior of the wealth-sharing function and its derivatives. Propositions 9 to 11 and Figure 1 imply particular investment policies for the two investors. These can be stated in three equivalent ways. Proposition 12. 1. The investor with the smaller risk aversion holds the analog of a perpetual call on the risky asset (the market). 2. Equivalently (by virtue of put/call parity), he holds the risky asset, plus a put, minus a constant amount of the riskless asset. 3. Stating it yet another way (invoking dynamic replication of a call), he holds a variable fraction (marked in Figure 1)14 of the risky asset, and he borrows a positive variable amount (marked or in Figure 1) at the riskless rate. The second formulation of Proposition 12 demonstrates that a particular form of portfolio insurance is optimal. However, the put which the investor purchases to insure his holdings of the risky asset is not a standard put: It is a perpetual put which is a claim on the flow of output. Furthermore, it is customized to the two investors’ degree of risk aversion.15 The type of 13

Therefore, in what follows we can equivalently represent functional relations as functions of S or ω.

14

The log investor is the less risk-averse investor In the case γ - -1.

15

Observe that the less risk-averse investor is the one who “buys portfolio insurance.”

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portfolio insurance observed in practice (using a standard put on a stock, or replicating one by means of dynamic strategies) is at best an approximation.16 The third formulation of Proposition 12 describes the portfolio strategy of an investor who has access to a menu of securities comprising exclusively the risky asset and riskless short-term borrowing or lending. This is the setting described in Section 1. We return to that interpretation in Section 6. 5. Expanding vs. Contracting Economies and the Associated Behavior of the Distribution of Wealth The dynamics of the aggregate capital stock S = W + W* are written as Equation (2) above. The capital path is, of course, dictated by aggregate consumption behavior:

The drift term g in Equation (28) is henceforth called the expected rate of growth of the economy. Let us introduce some terminology. Whenever the expected rate of growth g is (strictly) larger than in a neighborhood of ω = 1 as well as in a neighborhood of ω = 0, we say that the economy is an expanding one. Whenever it is (strictly) smaller than in two such neighborhoods, we say that the economy is a contracting one. This terminology is justified by the following observations: l In circumstances in which the expected rate of growth is uniformly larger than σ2/2 for all values of S larger than some fixed value and for all values of S smaller than some fixed value, the aggregate capital stock has a positive probability of becoming infinitely large17 and a zero probability of reaching zero.18 16 17

18

See Brennan and Schwartz (forthcoming). In other words, + is an attracting boundary [see Karlin and Taylor (1981)]. Observe that the diffusion term In d ln S [Equation (29)] is a constant and therefore never vanishes. That is, zero is a nonattracting boundary.

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● Whereas when the expected rate of growth is smaller than σ2/2 uniformly in neighborhoods of 0 and the aggregate capital stock has a positive probability of reaching zero and a zero probability of becoming infinite. 19 These assertions can be verified by applying boundary classification techniques [Karlin and Taylor (1981)]. But they can be understood intuitively on the basis of Equation (26), which gives the behavior of the logarithm of the capital stock. Note that the diffusion coefficient in the stochastic differential equation for ln S is constant. In that case boundary and asymptotic behavior are properly understood on the basis of the drift term alone. Its sign determines whether the economy is expanding or contracting. Its functional form also will allow us to decide whether or not the economy reaches a steady state. Growth is the mirror image of aggregate consumption behavior. Consumption behavior is related to the equilibrium wealth-sharing rule in the following way [see Equations (6) and (18)]:

Proposition 13. The asymptotic behavior of consumption is summarized in Table 2. Proof. The values of c/W at ω = 0 and at ω - 1 are provided by the values of I (0) and I (1) [Equations (26) and (27)]. The values of (c + c*)/S follow by averaging over the two investors.20 ■ When ω = 0 (i.e., the logarithmic investor is alone, which happens either with γ > 0 and the expected rate of growth is seen to be equal to

When ω = 1 (i.e., the nonlogarithmic investor is alone, which happens either with γ > 0 and the expected rate of growth is found to be equal to

The numbers g (0) and g (1) are the expected rates of growth as they would be in the absence of trade, in two separate economies with identical production opportunity sets but with different investor-consumers. 19

20

In the current model, all these events occur, if at all, after an infinitely large expected time: The boundaries are not attainable [see Karlin and Taylor (1981)]. We call “propensities to consume” the three quantities

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and

.

Intuition suggests that, in all cases, the expected rate of growth of the economy g (ω) is a continuous function that reaches a finite number of maxima and minima between g (0) and g (1). In fact, numerical analysis indicates that the function g (ω) is typically monotonic between g (0) and g (1) although it is possible to construct examples [for instance, by choosing parameter values such that g (0) = g (1)] in which g (ω) has one maximum or minimum.21 The finite number of maxima and minima is enough to conclude that g cannot fluctuate indefinitely around σ2/2 as ω approaches 0 or 1. All the information needed to describe the behavior of our economy has now been gathered. Four cases have to be distinguished depending on 22 the values of the parameters. Three of these cases lead to an absence of stationary behavior while one of them produces stationarity.

The case definitions imply that the two investors, considered in isolation, are unanimous in their desire for expansion or contraction. As a result, the economy is unambiguously a contracting one in case 1 and an expanding

21

These properties evidently apply also to the function relating the aggregate propensity to consume (c + c*)/S to the distribution of wealth.

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I do not consider the borderline cases in which g (0) or g (1) equals σ2/2 because they are special cases arising for specific parameter configuration only. Furthermore, their analysts would require a finer knowledge of the behavior of g (ω) close to ω - 0 and ω - 1.

23

Recall that g (0) and g (1) are known cxplicitly [Equations (31) and (32)]. The case definitions are properly stated in terms of exogenous model parameters.

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one in case 2.24 There is no steady-state distribution for the capital stock or for any of the variables in this economy. No matter what the initial conditions may be, the probability density for the capital stock forever recedes toward zero (case 1) or infinity (case 2). The probability of reaching zero (case 1) or infinite (case 2) capital stock is positive (although that would happen in infinite expected time). Correspondingly, the probability is positive that the person with the higher (case 1) or lower (case 2) risk aversion will ultimately own all the wealth (i.e., under case 1, if γ > 0, ω = 0 is an attracting barrier and if γ < 0, ω = 1 is attracting; under case 2, if γ > 0, ω = 1 is an attracting barrier and if γ < 0, ω = 0 is attracting).

The two investors, considered in isolation, would disagree about whether the economy should be an expanding or a contracting one. As a result, the economy is neither a contracting nor an expanding one. It all depends on the current allocation of wealth which is itself driven by the capital stock. As is illustrated in Figure 2 (numerical illustration A), the case definition implies that the economy tends to expand when the capital stock is already large and to contract when the capital stock is small. While “instability” is a word which comes to mind to describe this situation, the following is a more rigorous rendition. There is no steady-state distribution for the capital stock or for any of the variables in this economy. No matter what the initial conditions may be, the probability density for the capital stock sometimes recedes toward infinity and sometimes recedes toward zero. Even after it has receded toward one boundary (zero or infinity) for some time, there is a positive probability that an appropriate succession of shocks will initiate a transition toward the other boundary. The probability of eventually reaching a zero capital stock (in infinite expected time) is always positive and so is the probability of reaching an infinite capital stock. Since the allocation of wealth ω is monotonically related to aggregate wealth, it exhibits a similar kind of behavior. 5.2 The case of stationarity

Again the two investors, considered in isolation, would disagree. The economy is neither a contracting nor an expanding one. Under γ > 0 (γ < 0),

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Observe, by examining Equations (28) and (29), that the direction of the inequality between g (0) and g (l) does not simply hinge on the sign off.

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Numerical illustration A falls under this subcategory: g (0) - 0.004. g (1) - 0.0055, σ2/2 - 0.005.

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Numerical illustration B falls under this subcategory: g (0) - 0.004, g (1) - 0.007, σ2/2 - 0.005.

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the case definition implies that the non-log investor desires an average rate of growth smaller (larger) than σ2/2 while the log investor wishes the opposite. The former investor is dominant (i.e., more wealthy) when the capital stock is comparatively large (small). Hence the economy now tends to expand when the capital stock is low and to contract when the capital stock is large. The probability of eventually reaching zero or infinite capital stock is always zero. This “stable” situation is illustrated in Figure 3 (numerical illustration B). Proposition 14. Under case 4, there exists a steady-state density for the capital stock and for all the variables of this economy, including the allocation of wealth. No matter what the initial conditions may be, the probability density for the capital stock converges to the stationary measure. One such stationary measure (for the logarithm of the capital stock) is displayed in Figure 4. The existence of a stationary density implies that empirical studies of the long-run behavior of the capital stock would be apt to find evidence of mean reversion.28 28

Fama and French (1988) and evidence of mean reversion in long holding-period returns. This is different from mean revision in the capitalization of the stock market that we have here. Under decreasing- or increasing-return technology, however, there would be a link between the two.

The allocation of wealth, like the aggregate capital stock to which it is monotonically related, also admits a stationary measure. In other words, there is no tendency for one person to concentrate all the wealth in his hands. A stationary capital stock process could have arisen neither under certainty with two investors nor under uncertainty with one investor only. Under certainty (σ = 0)) one of the two endpoints of the wealth distribution would necessarily be the long-run outcome: When the two investors have the same rate of impatience ρ, the one with the lower risk aversion29 gradually acquires all the wealth when the rate of impatience ρ is less than the earning rate a (expanding economy) and gradually relinquishes it all in the opposite case. 30 It is easy to see that cases 3 and 4 could not have arisen under certainty.” The case of uncertainty is therefore qualitatively different from the case of certainty. Similarly, if only one isoelastic investor (log or non-log) had been present, the economy would have been unambiguously either an expanding 29

Risk aversion would act then only as a measure of elasticity of substitution between periods.

30 31

When α - ρ, the long-run allocation of wealth is determined by the initial situation.

Refer to Equations (31) and (32) above: When σ - 0, g (1) and g (0) are either both positive or both negative.

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or a contracting one. The interaction of two individuals has generated new behavior patterns for the capital stock. 6. Equity Holdings, Savings, and Trading Flows Holdings of equity shares by the two investors are given by the values of and represent the shares of each investor’s wealth invested in the risky asset. However, the two investors’ shares of ownership in the risky production opportunity are equal to respectively. One property of the wealth-sharing rule, namely the fact that it is an increasing function, implies that neither investor is ever short the equity. Another property of this function, stated in Proposition 11, implies that 32 and that In other words: Proposition 15. The less risk-averse person levers himself in order to invest more than his wealth into equity. He is a perennial borrower.

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Shares of stock change hands in this capital market: n* and n are nonconstant functions of a fluctuating ω and changes in n and n* imply that one investor buys shares from and sells shares to the other as time passes.33 As an investor’s share of wealth fluctuates, so does his share of ownership of the risky asset; and, of course, his share of wealth does fluctuate because, as a result of different risk aversions, the two investors make up their portfolios differently. It should be clear that the volume of trading so generated is predicated on the particular menu of assets available. Considering that this menu of securities is sufficient for the capital market to be complete [in the sense of Harrison and Kreps (1979)], it is always possible to introduce another menu of securities with the same characteristic that will reduce trading down to zero. A single security with a payoff structure replicating the consumption-sharing rule (the perpetual call option described in Proposition 12) would reduce trading down to zero if it were available. The particular menu considered here (shares of stock and riskless asset) can be defended on the grounds that it involves standardized securities only. “Standardized securities” means securities whose contractual definition requires no knowledge of the distribution of risk aversions in the population of investors.34,35 Since this model features capital flows between investors it offers an opportunity to measure the current-account balance between them. Definition. A person's net rate of private savings is the flow (per unit of time) of dividends he receives,36 plus or minus interpersonal interest payments linked to borrowing and lending, minus his flow of consumption. (This flow is also equal to the value of shares purchased or sold plus or minus additional borrowing, per unit of time.) When summed across investors, this flow is identically equal to zero: We are referring here to private savings, to the exclusion of corporate savings 33

The present interpretation of the model features one centralized firm that issues shares of stock. The two investors will trade these shares because the aggregate dividend that is distributed is the one that is desirable in the aggregate. At any given time, this dividend is excessive for the consumption needs of one Investor and insufficient for the needs of the other. The former will then buy shares from the latter. In another interpretation, there could be two identical (and perfectly correlated) production units operated in the backyards of the two investors. Each one of them could then help himself to the amount of dividend he Individually desires and no trading of shares would be needed. The two interpretations are equivalent because of constant returns to scale, but the one presented here seems more natural. Whatever the Interpretation, there would be the same amount of trading in the short-term riskless asset: Backyard operations would not remove the need for borrowing and lending.

34

The possibility of borrowing and lending at the riskless rate is nonetheless predicated on the fact (assumed to be known by all parties) that all investors have isoelastic utilities. Only these utilities will guarantee that investors will never be in a situation in which they are unable to repay. Otherwise, a credit-rationing scheme would have to be superimposed on the debt-contracting activity to allow riskless borrowing and lending. Alternatively, the possibility of default could be introduced in the definition of debt instruments, but the menu of securities would then differ from the one we consider here.

35 36

See Duffie and Huang (1985). Unfortunately, an infinite amount of trading is needed. Recall that total dividend. in the one-firm interpretation of the model, is equal to total consumption. A person receives a dividend equal to total dividend times his share in the equity of the firm.

(i.e., the output reinvested by firms). A person’s private savings differs from the increase in his wealth dW by the amount of capital gains or losses on his shareholdings, themselves reflecting corporate savings. The sign of a person’s savings rate can be characterized. When γ > 0 (γ < 0) the non-log investor’s rate of savings is positive (negative). Proposition 16. The less risk-averse investor, who in terms of stocks is a borrower, in terms of flows is always a net private saver.

The first term in this equation is the increase in a person’s wealth as a 37 result of corporate savings (capital gains), the second term is due to his/ her private savings. As was seen in Section 4, the less risk-averse person, who is a borrower, also has a wealth function with a positive curvature. This establishes the proposition.38 ■ Define a person’s propensity to save as his rate of private savings per unit of time, divided by his wealth. The proof of Proposition 6 reveals that

We can illustrate the relationship between a person’s rate of private savings and the curvature of the wealth-sharing rule39 using Figure 1. In the case γ = - 1 the log investor is the less risk-averse; when the aggregate capital stock is equal to S0, he holds a fraction of the equity (risky asset) and is a borrower to the tune of Let a shock take the economy to the level of capital stock S1: He buys shares and increases his borrowings to His net saving is 40 a positive amount represented by the small, thickly drawn segment above S0. In the reverse, let the economy be in S1 and let a shock take it to S0. The log investor saves which is again a positive amount. 7. The Rate of Interest and the Equity Premium In this model the expected rate of return on equity is a given constant (a) and so is the risk (σ) attached to equity. The equity premium (the spread between the riskless rate of interest and the expected return on equity) 37

Recall that dW/dS = n is the fraction of shares held by the investor.

38

Under numerical illustration A, the non-log Investor is the lender and has a negative savings rate. Under numerical illustration B, the non-log Investor Is the borrower and has a positive savings rate.

39

That is, the “gamma” [see Cox and Rubinstein (1984)] of the option implied by the wealth-sharing rule (see Section 4).

40

In Merton’s (1971) accounting, additional shares

are purchased at the new prices S1.

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therefore serves to determine the riskless rate. As the allocation of wealth fluctuates between borrower and lender, the security market line of the traditional CAPM should be viewed as pivoting around one fixed point representing the risky production opportunity, while the variable slope of the line (the market price of risk) determines the current value of the riskless rate of interest.41 In terms of the wealth-sharing function W* (S), the market price of risk is equal to (S/W*) (dw*/dS) (also equal to the log investor’s portfolio share invested in the equity x* ), the equity premium is equal to (W*/ S)/(dW*/dS) σ2, and the rate of interest is

The properties of the rate of interest can therefore be derived from those of the wealth-sharing function (Propositions 9 and 11). Proposition 17. ● r is an increasing function of ω when γ > 0, a decreasing one otherwise; the opposite is true for the market price of risk and the equity premium. ● As a function of the accumulated capital stock S, the equilibrium rate of interest is always increasing (the market price of risk and the equity premium are decreasing functions). The instantaneous correlation between output and the rate of interest is equal to + 1.

The person with the smaller risk aversion being under all circumstances the one who borrows, a positive output shock shifts the wealth distribution toward him. At the next point in time, he will still be a borrower, and, because he is now richer, he will borrow more than before, thereby driving up the rate of interest. The behavior of the allocation of wealth is thus mirrored in the stochastic behaviors of the market price of risk, the equity premium, and the equilibrium riskless rate of interest r, which are monotonic functions of ω or S .42 Numerical analysis confirms assumption (8) and Equation (12) above: The market price of risk and the rate of interest admit two natural barriers, 41

In this article, we have considered only an economy with one risky investment opportunity. If there were several (imperfectly correlated) such opportunities, investors at equilibrium would choose mean-variance efficient portfolios and the Tobin separation theorem would hold, as it does in standard static financial theory. This is true despite the investors’ desire to hedge against fluctuations in the rate of interest; the reason it is true is that each investor’s wealth and the rate of interest are perfectly correlated. (However, their choice among mean-variance efficient portfolios would be made In accordance with an adjusted level of risk aversion reflecting the combined influence of wealth and the rate of Interest.) In a multiasset economy the equity premium (i.e., the expected excess rate of return on the market portfolio) would still be inversely related to the rate of interest, but In a nonlinear way.

42

In this model [see Equation (34)] the rate of interest and the expected excess return on equity are linked by a negative linear relationship. Campbell (1987) found evidence supporting such a relationship (see his table 2). Campbell’s work, however, like Fama and Schwert (1977), pertained to nominal rates of return.

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Figure 5 Behavior of rate of interest Numerical Illustration A: ρ - 0.106, γ - 0.5, α - 0.11, σ - 0.1. The process for the rate of interest is nonlinear autoregressive of order 1 (witness the drift function) and heteroskedastic (witness the diffusion function). Under case 3 the process for the rate of interest is nonstationary.

at 1 and 1 - γ for the market price of risk, and at α - σ2 and α (1 - γ)σ2 for the rate of interest. These values correspond to the endpoints ω = 0 and ω = 1, where one of the two investors would impose his risk aversion and his corresponding value of the rate of interest. For the purposes of empirical analysis, it is of interest to formulate the process for the rate of interest in an autoregressive form {the resulting formulation is necessarily autoregressive of order 1 [AR(l)] since every process in this economy is Markovian}. This is accomplished in continuous time by writing the stochastic differential equation for r, which contains a drift term and a diffusion term. Examples of the resulting drift and diffusion functions are displayed as Figures 5 and 6. The drift function is of necessity nonlinear because it is zero at the boundary points; in addition, it can change sign and admit one maximum and one minimum. This is a highly nonlinear AR(l) process. The diffusion term of the rate of interest is also by no means constant :43 It is zero at the two barriers and exhibits a maximum somewhere between them. Hence this is a heteroskedastic AR(1) process. Proposition 18. The process for the rate of interest is a nonlinear heteroskedastic AR(1) process, which maintains the rate within a corridor and admits a stationary measure only under case 4 of Section 5.

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Behavior of rate of interest Numerical illustration B: ρ - 0.106, γ - -1, α - 0.11, σ - 0.1. The process for the rate of interest is nonlinear autoregressive of order 1 (witness the drift function) and heteroskedastic (witness the diffusion function). Under case 4 the process for the rate of interest is stationary.

It follows from the analysis of Section 5 that the rate of interest may not possess a stationary distribution, There are four possible long-run behaviors of the rate of interest, depending on the case situation at hand: in cases 1 to 3 one or both of the two boundaries are attracting but not attainable, while in case 4 the interior region is stable in the stochastic sense. Whenever there exists a stable interior distribution of the allocation of wealth, there is also one for the rate of interest, which wanders between the two extreme values while tending to return to the stable interior region. Although the present general equilibrium model is about the simplest one can conceive that still exhibits a variable rate of interest, the process so obtained is much more complex than any of those which previously have been utilized to model interest rate behavior [e.g., the Ornstein-Uhlenbeck process used by Vasicek (1977)].44 The short-term rate of interest may be the most readily observable of all financial variables, although the problem of inferring the real rate is by no means trivial.45 One might expect to generate a number of empirically testable propositions by relating other perfectly or imperfectly observable 44

45

Also, Vasicek (1977). Brennan and Schwartz (1982), and others assume that the market price of (interest rate) risk Is a constant. This assumption would be incompatible with the current general equilibrium setting. See, for example, Mishkin (1984).

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variables to the rate of interest. A distinction must be drawn in this regard between three sets of variables: stock variables, aggregate flow variables, and disaggregated flow variables. Generally the rate of interest is monotonically related to all the stock variables of the economy. More specifically, it is related, via an increasing function, to the aggregate capital stock (measured, for example, by the level of the stock market) and to the wealth of each and every consumerinvestor. When functional relationships are monotonic, albeit nonlinear, they can be tested by means of correlation analysis: Any increase in S should be associated with an increase in the rate of interest. Aggregate flow variables, which are more easily observable than stock variables, offer a slightly weaker prospect for empirical testing of the theory. We saw in Section S how the conditionally expected rate of growth of the capital stock g was related to the distribution of wealth: When g (1) > g (0),46 g (ω) is typically an increasing function of ω, a decreasing one otherwise. Since the distribution of wealth and the rate of interest are monotonically related to each other (Proposition 17), it follows that g and rare typically monotonically related as well. The sign of the relationship can be easily deduced. For instance, when g (1) > g (0) and γ > 0 or when g (1) < g (0) and γ < 0, g and r are directly related, inversely related otherwise. Similar statements with the opposite sign can be made regarding the aggregate propensity to consume [since (c + c*)/S = α - g ] and generally regarding any aggregate flow variable. We indicated, however, that it is possible to find counterexamples to these statements, that is, parameter combinations for which these relationships are not monotonic and therefore harder to test. The regions of the parameter space which produce such unpleasant results are narrow, but it has not been possible so far to isolate them. Finally, one should not generally expect to find any monotonic relationship between disaggregated flows and the rate of interest. In some cases (as when there are several countries) it is possible to measure the current account balance (the capital flow) between. two subpopulations, a quantity that we labeled the rate of private savings. It would be desirable to characterize the behavior of the current account in equilibrium. But Figure 7 (drawn for numerical illustration B) illustrates, that a person’s rate of private savings is not generally monotonically related to the rate of interest. 8. conclusion The following has been accomplished based on two investor types, differing in their degree of risk aversion: ● The role of the distribution of wealth in the determination of the equilibrium prices of assets has been clarified. Shifts in the distribution of

46

Refer to Section 5 for the expressions of g (0) and g (1) in terms of exogenous parameter values.

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wealth affect the fluctuating rate of interest. For this reason, the distribution of wealth plays the role of an additional state variable against which investors want to hedge. But in this economy with only one source of shocks the distribution of wealth is, at equilibrium, functionally (and monotonically) related to the level of the capital stock. l In regard to the process of capital accumulation, it has been possible to classify parameter values into four possible cases. In three of these cases there exists no stationary probability distribution for the capital stock, as would be true with a single investor population. In one of the four cases, however, the capital stock reaches a stationary distribution. This new possibility arises from the fluctuating distribution of wealth. Generally the two investor populations disagree as to the optimal rate of growth of the economy. If the choice of parameters is such that the persons desiring a higher rate of growth gain wealth whenever the capital stock falls and lose wealth when it rises, a stabilizing force is generated and a stationary probability ensues. ● The dynamic portfolio choices of the two investor categories has been exactly described via their wealth-sharing rule, which reflects a form of optimal portfolio insurance. With a simple menu of assets, the financial market is dynamically complete but not statically so. An infinite volume of trading results from the presence of Brownian shocks. At all times, the person with the lower risk aversion is, as far as stocks are concerned, a borrower, and, as far as flows are concerned, a net saver. 182

● The net saving of one investor group is the other group’s dissaving, a flow which is commonly referred to as the current account in the international economics literature. We have derived exactly the current account from the equilibrium wealth-sharing rule and indicated how it varies in relation to other variables in the economy. ● The process for the rate of interest has been identified: It has been found to be highly nonlinear and heteroskedastic. Under three of four parameter combinations, the process for the rate of interest does not admit a stationary probability distribution. With isoelastic utilities a single investor population would produce a nonstationary capital stock but a constant rate of interest. When there are two investor populations, the capital stock and the rate of interest are, under all cases, positively functionally related: They rise or fall and are or are not stationary together.

Avenues for further research include ● Extensions of the two-investor problem to other types of utilities {preferences of the type defined by Epstein and Zin (1988); time-nonseparable preferences [Sundaresan (1989), Constantinides (1988)]} ● Extensions to incomplete financial markets ● Extensions to international situations [Uppal (1988)] leading to realistic theories of a country’s current account ● Extensions to situations where diversity of investor population is deemed essential (as when they, for instance, act on the basis of different information)

Finally, an investigation of the three-investor problem seems warranted: It may lead to equilibrium behavior as critically different from the twoinvestor situation as the latter was from the single-investor case. A precedent is to be found in mechanics in regard to the three-body problem.47

Appendix 1: Proofs of Propositions 2 to 5 Substitute the optimal value of c and c* from Equation (6) into the Hamilton-Jacobi Equation (5) and differentiate with respect to the state variable S to obtain

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With these notations, the necessary condition (A1) can be restated as follows:

A straightforward application of Itô's lemma (in the reverse) then allows one to rewrite the ordinary differential equation (A3) and other necessary conditions of optimality into a system of stochastic differential equations as expressed in the following lemma: Lemma A1. A necessary condition of optimality is that there exist a number p0 (the value of p at time 0) and a process r such that

Proof [See Bismut (1973, 1975)]. By virtue of Itô's lemma, the right-hand side of Equation (A3) is the conditionally expected change in p. The drift term of stochastic differential equation (A4) reflects the fact that this expected change is equal to ( ρ - r)p. As for the diffusion term of stochastic differential equation (A4), it is provided by the definition of r and the diffusion term of S. (A5) and (A6) are restatements of Equations (2) and (7), under the optimal consumption policy (6). ■ Proof of Proposition 2 From (A4) and (A5) obtain the stochastic differential equation for pS:

Then integrate this equation “forward,” subject to the boundary condition (A6). ■ Proof of Lemma 1 One part of integral (9) can be integrated explicitly:

The other part

converges if the rate of discount ρ is strictly larger than the conditionally expected rate of increase of at all times with probability 1. The expected rate of increase of

is obtained from (A4) by the application of Itô's lemma:

which is condition (13) in the text. ■ Proof of Proposition 3 The expected discounted utility (4) is in three parts. The first one is

The conditionally expected change in ln (pt) obtained from (A4) is

The quantity (A11) is evidently bounded from above. If the rate of interest 185

remains bounded, as assumed [see (12)], (A11) is bounded from below as well, and convergence is guaranteed. The second part

is proportional to (A8), which we know (Lemma 1) is well defined under condition (13). The third part is

Proof of Proposition 4 The right-hand side of condition (13) is a quadratic function of r. When γ < 0, this function admits a maximum equal to

Choosing

ρ greater than this maximum guarantees that the sufficient condition (13) is satisfied, no matter what value r may take. ■ Proof of Proposition 5 We now consider the case γ > 0. As in the proof of Proposition 4, the right-hand side of condition (13) is a quadratic function of r but one which now admits a minimum. An upper bound for this right-hand side can nonetheless be obtained because the rate of interest is at all times bracketed, as in Equation (12). ■ Appendix 2: Proof of Bounds Given in Propositions 7 and 8 Proof of Proposition 7 ( γ < 0) The value of pS is, at all times, given by Equation (9). The proposition is therefore equivalent to saying that the integral (A8) is larger than zero and smaller than the second term in the right-hand side of (20). That it is larger than zero follows from the fact that p is positive with probability 1. As for the upper bound, observe that the conditionally expected rate of growth of is given as the drift term in stochastic differential equation (A9), a term which is quadratic in r. When γ < 0, the largest possible value of this conditionally expected growth rate is

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Multiplying numerator and denominator by 1 - γ we get the desired result. n Proof of Proposition 8 (γ > 0) As in the proof of Proposition 7, the conditionally expected rate of growth of is a quadratic function of r, but one which admits a minimum rather than a maximum. An upper bound for this rate of growth can nonetheless be established, as r takes values within a bounded interval (Equation 12). n References Bismut, J.-M., 1973, “Conjugate Convex Functions in Optimal Stochastic Control,” Journal of Mathematics Analysis and Applications, 44,384-404. Bismut, J.-M., 1975, “Growth and Optimal Intertemporal Allocation of Risks,” Journal of Economic Theory, 10,230-257. Brennan, M.J., and A. Kraus, 1978, Necessary conditions for Aggregation in Securities,” Journal of Financial and Quantitative Analysis, Sep, 407-418. Brennan, M.J., and E.S. Schwartz, 1982, “An Equilibrium Model of Bond Pricing and a Test of Market Efficiency,” Journal of Financial and Quantitative Analysis, XVII(3), 301-331. Brennan, M.J., and E. S. Schwartz, “Portfolio Insurance and Financial Market Equilibrium,” Journal of Business, forthcoming. Campbell, J.Y., 1987, “Stock Returns and the Term Structure.” Journal of Financial Economics, 18(2), 373-400. Constantinides, G.M., 1982, “Intertemporal Asset Pricing with Heterogeneous Consumer and without Demand Aggregation,” Journal of Business, 55(2), 253-267. Constantinides, G.M., 1988, “Habit Formation: A Resolution of the Equity Premium Puzzle,” working paper, University of Chicago. Cox, J. C., and C. P. Huang, 1986, “A Variational Problem Arising in Financial Economics with an Application to a Portfolio Turnpike Theorem,” Working Paper 1751-86, Massachusetts Institute of Technology. Cox, J. C., and M. Rubinstein, 1984, Option Markets, Prentice-Hall, Englewood Cliffs, NJ. Duffie, D., and C. Huang, 1985, “Implementing Arrow-Debreu Equilibria by Continuous Trading of Few Long-Lived Securities,” Econometrica, 53,1337-1356. Epstein, L. G., and S. E. Zin, 1988, “Substitution, Risk Aversion and the Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework,” mimeo, University of Toronto and Queens University. Fama, E.P., and K.R. French, 1988, “Permanent and Temporary Components of Stock Prices,” Journal of Political Economy, 96, 246-273. Fama, E. P., and G. W. Schwert, 1977, “Asset Returns and Inflation,” Journal of Financial Economics, 5, 115-146. Hakansson, N. H., 1971, “On Optimal Myopic Portfolio Policies, with and without Serial Correlation of Yields,” The Journal of Business of the University of Chicago, 44,324-334. Harrison, M., and D. Kreps, 1979, “Martingales and Multiperiod Securities Markets,” Journal of Economic Theory, 20,381-408. He, H., and N. D. Pearson, 1988, “Consumption and Portfolio Policies with Incomplete Markets and ShortSale Constraints: The Infinite Dimensional Case,” working paper, Massachusetts Institute of Technology. Hénon, M., 1976, “A Two-Dimensional Mapping with a Strange Attractor,” Communications in Mathematical Physics, 50,69-77.

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Karatzas, I., J. P. Lehocszky, and S. E. Shreve, 1987, “Existence and Uniqueness of Multi-Agent Equilibrium in a Stochastic Dynamic Consumption/Investment Model,” working paper, Columbia University. Karlin, S., and H. M. Taylor, 1981, A Second Course in Stochastic Processes, Academic Press, New York. Merton, R. C.. 1971, “Optimum Consumption and Portfolio Rules In a Continuous-Time Model,” Journal of Economic Theory, 3,373-413. Merton, R. C., 1975, “An Asymptotic Theory of Growth under Uncertainty,” Review of Economic Studies, XLII (3), 375-393. Mishkin, F. S., 1984, “Are Real Interest Rates Equal Across Countries? An Empirical Investigation of International Parity Conditions,” Journal of Finance, 39, 1345-1357. Rubinstein, M.. 1974, “An Aggregation Theorem for Securities Markets,” Journal of Financial Economics, 1(3), 201-224. Sundaresan, S. M., 1989, “Intertemporally Dependent Preferences and the Volatility of Consumption and Wealth,” Review of Financial Studies, 2,73-89. Uppal, R., 1988, “Deviations from Purchasing Power Parity and Capital Market Equilibrium,” working paper, Wharton School, Philadelphia. Vasicek, O., 1977. “An Equilibrium Characterization of the Term Structure,” Journal of Financial Economics, 5(2), 177-188. Wilson, R., 1968, “The Theory of Syndicates.” Econometrica, 36(1), 119-132.

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