Equilibrium portfolio choices and savings with consumption externalities Christian Gollier1 University of Toulouse July 7, 2000

1 This

is a very preliminary paper. Comments welcomed: [email protected].

Abstract We examine the equilibrium consumption and portfolio strategies in an economy with consumption externalities. We consider a model where the von Neumann Morgenstern utility is a function of one's own consumption and of the average consumption in the economy in the corresponding state or date. We show that, under some conditions on the degree of conformism in the economy, the optimal portfolio and consumption choices observed at equilibrium in the economy with consumption externalities are equivalent to those that are optimal without any externality, but with an adjusted degree of risk aversion. If these conditions are not ful¯lled, taking zero-mean risks may be optimal for risk-averse agents who are very envious. Also, individual consumptions may comove negatively at equilibrium, thereby showing that not all diversi¯able risks are washed away at equilibrium. Finally, we link the degree of conformism in the economy to its degree of risk aversion. Keywords: Consumption externalities, conformism, envy, portfolio choice, mutuality principle.

1

Introduction

Being unlucky when others are unlucky has a smaller adverse e®ect on one's own welfare than when others are lucky. My young twin boys know that very well. On my part, getting a rejection letter from an editor is harder to swallow in a good department with a high publication rate than in a department with a lower one. This means, my twins and I are conformist. Conformism can be observed in many activities in life. It can be a genetic characteristic of human being, or it can be induced by speci¯c incentive schemes, as portfolio managers who get bonuses based on a function of their performance and of the average performance of other managers. In this paper, we examine the relationship between conformism and risk-taking. Because conformist consumers su®er less from a loss when others su®er the same loss, conformism should induce more risk-taking. We formally de¯ne a conformist agent as someone whose marginal utility of consumption is increasing with the average consumption in the economy. Our benchmark individual decision problem is a static Arrow-Debreu portfolio selection. Anti-conformism is allowed. Each investor selects the portfolio that maximizes her expected utility by making rational expectation about the optimal portfolio selected by others. Our model generalizes the model by Gali (1994) who considers the special case of multiplicative consumption externalities and a power "outer" utility function in an economy with identical agents. Our main ¯ndings can be summarized as follows. We ¯rst show that investors with a su±ciently low degree of conformism never want to accept zero-mean gambles. Second, if in addition all investors are conformists, then all individual consumption are comonotone, i.e., all agents are lucky or unlucky at the same time. This implies that there is no way ¯nd risk-sharing contracts at equilibrium that reduce risk on individual consumptions. There is no diversi¯able risk at equilibrium. Under these conditions, the equilibrium portfolios are not di®erent from those that we would observe in an economy without consumption externalities, but with adjusted degrees of risk aversion. None of these results are true in general. In other words, it is possible to ¯nd some speci¯cations where risk-averse, but highly conformist agents accept zero-mean lotteries. It is also possible for highly anti-conformists agents to purchase assets portfolio with a negative beta. Finally, we examine the special case of identical agents. This allows us to link the degree of con1

formism to the dynamism of the economy. We also examine the equivalent consumption-saving problem under certainty.

2

Description of the economy

There is a continuum of agents that are indexed by µ 2 £: Agents of type µ have a von Neumann Morgenstern utility function U (c; C; µ); where c is the agent's consumption and C is the average consumption in the group. We assume that U11 is negative: agents are risk-averse towards their own consumption risk when C is certain. The model is a standard static Arrow-Debreu portfolio choice. There are two dates. At date 0, agents are endowed with initial wealth z(µ) that is used for investment. Consumption takes place only at date 1. There is some uncertainty about the state s that will prevail at that date. The set of investment opportunities with constant return to scale1 is assumed to be complete. The exogenous randomness of the productivity of capital describes a technological uncertainty in the economy. The problem of agent µ is to allocate his initial wealth into various investment projects in order to maximize his expected utility at date 1, given his expectation about the average consumption in the economy: max c(:;µ)

s:t:

EU (c(~ s; µ); C(~ s); µ)

(1)

E¼(~ s)c(~ s; µ) = z(µ);

(2)

where ¼(s) is the state price per unit of probability of state s. In fact, this problem is formally equivalent to a lottery game where each agent can bet on the occurrence of various events s~ where each dollar at stake in state s yields a payo® (¼(s))¡1 dollars per unit of probability if state s occurs, and nothing otherwise. The ¯rst-order condition is written as U1 (c(s; µ); C(s); µ) = ¸(µ)¼(s)

8s:

(3)

Because we assumed that U is concave in c, this condition together with the budget constraint (2) is necessary and su±cient to characterize the optimal portfolio strategy, for a given distribution of C(~ s). 1

This assumption implies that the equilibrium state prices are ¯xed. We could have considered the case of an exchange economy with state-dependent endowments and complete risk-sharing markets. Our qualitative results would be unchanged.

2

We assume rational expectation. Thus, the equilibrium condition is that e C(s) = Ec(s; µ)

8s;

(4a)

where µe is the random variable characterizing the distribution of types in the group. Solving the system of equations (2), (3) and (4a) yields the equilibrium allocation of risk in the economy. Let us go back to the characteristics of the utility function U: In addition to the assumption that the utility is increasing and concave with respect to one's own consumption level, the model allows for either envy (U2 < 0) or altruism (U2 > 0). The agent whose own consumption is risk free is averse to the group's average risk of U is concave with respect to C: Finally, the supermodularity of U with respect to (c; C);i.e., U12 > 0 means that an increase in average consumption in the group increases the marginal utility of one's own consumption. Agents want to "keep up with the Joneses" in that case. Following Gali (1994), there is a positive consumption externality. Conformism is another word used in this context. We can de¯ne a local index of conformism as follows: ¯ dc ¯¯ U12 (c; C; µ) ¡(c; C; µ) = : (5) =¡ ¯ dC U1 U11 (c; C; µ) ¡ is the increase in one's own consumption that left marginal utility unchanged after a unit increase of the mean consumption in the economy. Anti-conformists are characterized by ¡ < 0.

3

Optimal risk taking without consumption externalities

It is useful as a benchmark to recall the characteristics of the solution when there is no consumption externalities, i.e., when U12 (c; C; µ) = 0 for all (c; C; µ). The problem for a speci¯c agent with externality-free utility function u(:); such that u0 (c) = U1 (c; C; µ); and initial wealth k is written as Á(:; k) = arg max c(:)

Eu(c(~ s))

s:t:

E¼(~ s)c(~ s) = k:

(6)

The ¯rst-order condition is written as u0 (c(s)) = ¸¼(s): 3

(7)

The characteristics of the solution c(:) = Á(:; k) are well-known.2 First, if there are two states such that the state prices per unit of probability are the same, then all consumers must consume the same amount of the good in the two states: ¼(s) = ¼(t) =) c(s) = c(t): (8) It is easy to show that doing otherwise would yield a mean-preserving spread of ¯nal consumption (Gollier (2000)). Under risk aversion, this would be disliked. This result means in particular that if there is no technological uncertainty, i.e., if ¼(s) = ¼ for all s, consumers will not take any risk. In a risk free economy, betting on states is never optimal. More generally, property (8) means that consumption depends upon the state only through the corresponding state price per unit of probability. >From (7), we also see that c is decreasing in ¼ under risk aversion. All consumption plans comove negatively with ¼: This principle is often called the mutuality principle, since it states that agents wash out all diversi¯able risks at equilibrium. In other words, at equilibrium, agents will never play zero-sum gambles against each others. In fact, as used for example by Dybvig (1988), this property of optimal portfolios characterizes expected utility: any allocation c(:) that covaries negatively with ¼(:) is optimal for some riskaverse expected-utility maximizer. A third property of optimal portfolios in the absence of consumption externalities is that more risk-averse agents purchase safer portfolios. This is well-known for the one-safe-one-risky-asset model since Pratt (1964) and Arrow (1965). Using a single-crossing condition for the consumption plans to de¯ne the notion of a riskier portfolio in the complete market setting, this comparative statics property is proven in Gollier (1999). It implies in particular that poorer investors select a safer portfolio if absolute risk aversion is decreasing. The traditional case where the optimal portfolio can be analytically derived is when the utility function is HARA, i.e., when the inverse of absolute risk aversion is linear with consumption. Let u0 (c) = (´ + (c=°))¡° ; for some parameters ´ and ° guaranteeing that u0 is positive and decreasing in the relevant domain. Then it is easily veri¯ed that the optimal solution to (2) and (7) is Á(s; k) = b(s)°´ + a(s)k; (9) 2

See for example Ingersoll (1987).

4

with a(s) = [¼(s)]¡1=° =E [¼(~ s)](°¡1)=° and b(s) = a(s)E¼(~ s) ¡ 1: Statecontingent consumptions are linear in wealth.

4

Risk-taking in a technologically risk free economy

We now examine the characteristics of the equilibrium risk-taking behaviors in the presence of consumption externalities. Our ¯rst result states that, as in the classical case, consumers will not take any risk in an economy without any technological uncertainty. Proposition 1 Suppose that the degree of conformism is uniformly less than unity: ¡(c; C; µ) < 1 for all (c; C; µ). Then, all equilibrium consumptions depend upon the state of nature only through the state price per unit of probability: ¼(s) = ¼(t) =) c(s; µ) = c(t; µ); for all µ: The same result holds if the degree of conformism ¡ is uniformly larger than unity. Proof: If ¼(s) = ¼(t) = ¼; then , from condition (3), we must have that U1 (c(s; µ); C(s); µ) = U1 (c(t; µ); C(t); µ) = ¸¼

(10)

for all µ. Suppose that C(s) 6= C(t). De¯ne function Ã(C; µ) in such a way that U1 (Ã(C; µ); C; µ) = ¸¼. Because ¯ U12 (Ã; C; µ) dà ¯¯ =¡ < 1; (11) ¯ dC U1 U11 (Ã; C; µ) we obtain that

c(s; µ) ¡ c(t; µ) = Ã(C(s); µ) ¡ Ã(C(t); µ) R C(s) U12 (Ã(C; µ); C; µ) = C(t) ¡ dC U11 (Ã(C; µ); C; µ) < C(s) ¡ C(t): Taking the expectation with respect to µe yields h i e ¡ c(t; µ) e < C(s) ¡ C(t); C(s) ¡ C(t) = E c(s; µ) 5

(12)

(13)

a contradiction. We must thus have that C(s) = C(t). Because U1 is strictly decreasing in c, condition (10) yields the result. A symmetric argument can be used for the case U11 + U12 > 0 for all µ:¥ Proposition 1 states that no one will take any risk in an economy without any technological uncertainty (¼(s) = ¼ for all s). The same result holds if all agents have a degree of conformism that is larger than unity. However, this result does not need to hold when some agents have a low degree of conformism (¡ < 1), and others have a large degree of conformism (¡ > 1). This is con¯rmed by the following counterexample. Counterexample: Following Gali (1994) and others, consider the following set of utility functions U (c; C; µ) = u(cC ¡µ ) (14) for the increasing and concave outer utility function u(z) = z (1¡°) =(1 ¡ °): It yields µ(° ¡ 1) c : (15) ¡(c; C; µ) = ° C Assume that µ has its support in R+ . Thus, agents are conformist (¡ > 0) if relative risk aversion is larger than unity. Under this condition, the degree of C °¡1 conformism is less than unity only if is larger than µ : Our numerical c ° counterexample is then as follows: we assume that ° = 2 and z(µ1 ) = z(µ2 ) = 1. There are two types of equal size. The ¯rst type is neutral, with µ1 = 0. The second type is conformist with µ2 = 5. There are two equally likely states, s and t, with ¼(s) = ¼(t) = 1. Because of risk aversion, the neutral type does not take any risk at equilibrium, with c(s; µ1 ) = c(t; µ1 ) = 1: The ¯rst-order condition for type µ2 is written as · ¸5 · ¸5 ¡2 1 + c ¡2 3 ¡ c ¡ (2 ¡ c) = 0; (16) f (c) = c 2 2 where c(s; µ2 ) = c and c(t; µ2 ) = 2¡c: Function f is depicted in Figure 1. We see that equation (16) has three solutions. The ¯rst one is c = 1; which corresponds to the classical solution with no risk taking. The other two symmetric solutions correspond to an equilibrium with risk-taking: c(s; µ2 ) = 0; 2357 and c(t; µ2 ) = 1:7643. In spite of the aversion to risk on their own consumption, agents of type 2 are willing to take zero-mean risks at equilibrium.

6

This is because they anticipate that others will do the same, with an average consumption in states s and t equaling respectively C(s) = 0:6179 and C(t) = 1:3821: Their risk aversion is dominated by their high degree of conformism, which induces them to consume much more in state t than in state s.

The two equilibria of the risk free economy.

5

Existence of diversi¯able risks at equilibrium

Proposition 1 is an important ¯rst step in providing conditions guaranteeing that all consumers in the economy with consumption externalities behave as if they would do in a classical economy, but with an adjusted degree of risk aversion. To prove this, we also need to guarantee that all consumptions covary negatively with ¼. We start with the following result. Proposition 2 Suppose that ¡(c; C; µ) < 1 for all (c; C; µ): Then the average consumption in the group is decreasing in the state price per unit of probability.: C 0 (¼) < 0. It is increasing if ¡ > 1 for all (c; C; µ): Proof: From Proposition 1, because ¡ < 1, we assume without loss of generality that ¼(s) = s for all s. Fully di®erentiating condition U1 (c(¼; µ); C(¼); µ) = ¸¼ yields dc ¸ U12 (c; C; µ) dC = ¡ : (17) d¼ U11 (c; C; µ) U11 (c; C; µ) d¼ 7

Taking the expectation with respect to µe yields in turn # # " " e ¸ dC U12 (c; C; µ) dC =E ¡ E ; e e d¼ d¼ U11 (c; C; µ) U11 (c; C; µ)

(18)

or, equivalently,

¸E [(U11 )¡1 ] dC ¸: = · U11 + U12 d¼ E U11

(19)

Under our assumptions U11 < 0 and U11 + U12 < 0; this is negative. ¥ >From the previous two Propositions, we conclude that an external observer looking at aggregate consumption data could not distinguish this economy from a classical economy without any consumption externalities. Can we do better by using individual data? Proposition 3 If 0 · ¡(c; C; µ) < 1 for all (c; C; µ); then individual con@c (¼; µ) < sumptions are decreasing in the state price per unit of probability: @¼ 0: There is no diversi¯able risk at equilibrium, and the economy is observationally equivalent to a classical economy with no consumption externalities. Proof: Using equations (17) and (19), we obtain that 1 U12 (c; C; µ) 1 dc = ¡ ¸ d¼ U11 (c; C; µ) U11 (c; C; µ)

E [(U11 )¡1 ] · ¸; U11 + U12 E U11

(20)

e >From our assumptions, where the expectation operator is with respect to µ. this is unambiguously negative.¥ Thus, under the conditions of this Proposition, all consumers behave with respect to risk as if their utility function would be independent of their neighbors' level of consumption. There is no way to test for the presence of consumption externalities in such an economy. It is important to notice that we have introduced an additional restriction on preferences to get this result. Namely, we assumed that one's own consumption and neighbors' consumption are complement (conformism): ¡ ¸ 0. If they are substitutes, 8

it may be possible to ¯nd some consumers whose consumption would covary positively with state prices, in spite of their risk aversion! Counterexample: The case of additive consumption externalities provides a simple example in which diversi¯able risks may prevail at equilibrium when the conditions of Proposition 3 are not satis¯ed. Let us assume that U(c; C; µ) = u(c ¡ µC):

(21)

Function u is the "outer" utility function, which is assumed to be increasing, concave, and identical for all consumers. This is a case where the degree of conformism is constant, with ¡(c; C; µ) = µ: (22) ¯ dc ¯¯ Notice that we also have that µ = ; which means that envy and condC ¯U formism are equivalent in this speci¯cation: The solution for this additive case is easily obtained by relying on the solution of the classical case without externalities. Because of the budget constraint (2) can be rewritten as e the optimal consumption plan of agent µ must e = z(µ) ¡ µEz(µ); Ee ¼ (e c ¡ µC) be equal to e + µC(¼), c(¼; µ) = Á(¼; z(µ) ¡ µEz(µ)) (23)

where function Á is de¯ned by condition (6). From this condition, we see that the consumption externality has a substitution e®ect and a wealth e®ect on the demand for risky assets. The substitution e®ect is described by the second term of the right-hand side of equality (23): the agent compensates the externality by a corresponding increase in the demand for contingent claim. But this cannot be done in all states without violating the budget constraint. Therefore, the agent uniformly reduces his demand for all assets e This is the wealth e®ect as if his initial wealth would be reduced by µEz(µ). of the consumption externalities. Equation (23) is not a closed form solution of the equilibrium, since C(:) = e is endogenous. We can solve the model completely if we assume E(c(:; µ)) that u is HARA, with u0 (c) = (´ + (c=°))¡° . From condition (9), we know that Á is linear in its second argument, i.e., Á(¼; k) = b(¼)´° + a(¼)k. It implies that Á(¼; (1 ¡ µ)z) b(¼)´° C(¼) = = + a(¼)z; (24) 1¡µ 1¡µ 9

and c(¼; µ) =

1+µ¡µ b(¼)´° + a(¼)z(µ); 1¡µ

(25)

e are respectively the mean degree of conformism where µ = E µe and z = Ez(µ) and the mean initial wealth in the economy. We can now use condition (25) to show that the all consumptions do not necessarily covary negatively with state prices. Using the fact that b(¼) = a(¼)Ee ¼ ¡ 1; equation (25) is rewritten as · ¸ 1+µ¡µ 1+µ¡µ + a(¼) z(µ) + °´Ee ¼ : (26) c(¼; µ) = ¡°´ 1¡µ 1¡µ If the bracketed term of the above equality is negative, the corresponding agent µ has a larger consumption level in more extensive states, since a0 < 0. To illustrate, consider the following example, with ° = ´ = z(µ) = Ee ¼ = 1: Suppose also that there are two types of agents in the group. There is a subgroup of anti-conformist agents with µ = µ1 = ¡0:75: The other subgroup gathers envious agents with a µ = µ2 = 0:85. The proportion of anti-conformist agents is 1=16, which implies that µ = 0:75: It is then easy to compute the equilibrium allocation of risks, which is c(¼; µ1 ) = 2 ¡ ¼¡1 and c(¼; µ2 ) = ¡4:4 + 5:4¼ ¡1 . These equilibrium consumption plans are represented in Figure 2. Observe here that the mutuality principle does not hold. At equilibrium, there exist unexploited feasible risk-sharing contracts that would induce a mean-preserving reduction in risk on consumption for the two types. Not all diversi¯able risks are washed out at equilibrium. Observe that condition U11 + U12 is negative for all µ in this economy. It implies that the properties of Propositions 1 and 2 hold . First, without any uncertainty (¼(~ s) = 1 almost surely), no one takes any risk: c(1; µ) = 1: Second, the mean consumption in the economy is decreasing in ¼. In this numerical example, we have C(¼) = ¡4 + 5¼ ¡1 .

6

The case of an homogenous economy

The search for an equilibrium is much simpli¯ed if we assume that agents have identical preferences and identical initial wealth: U3 (c; C; µ) = 0 = z 0 (µ). Under this assumption of Proposition 1, there is a symmetric equilibrium 10

Figure 1: which is characterized as follows: v 0 (c) = ¸¼

8¼;

(27)

where function v(:) is de¯ned as v 0 (c) = U1 (c; c):

(28)

If v 00 (c) = U11 (c; c) + U12 (c; c) is negative for all c, function v is concave. In that case, the symmetric risk-taking equilibrium of the homogenous economy with consumption externalities is equivalent to the optimal risk taking in an externality-free economy whose agents have concave utility function v. Proposition 4 Consider an homogenous economy with ¡ < 1 . Then, the unique equilibrium risk allocation is optimal for an externality-free risk-averse investor whose concave utility function v is characterized by condition v 0 (c) = U11 (c; c). There are two simpli¯cations due to the homogeneity assumption. First, contrary to Proposition 3, we don't need to assume that c and C are complements to guarantee that all agents behave in a risk-averse way at equilibrium. Second, we immediately characterize the observationally equivalent 11

(marginal) utility function v 0 (c) = U1 (c; c): The degree of concavity of the v function characterizes the equilibrium degree of risk aversion of the homogenous economy as a whole. We have that ¡

U11 (c; c) v00 (c) =¡ [1 ¡ ¡(c; c)] : 0 v (c) U1 (c; c)

(29)

Term ¡U11 =U1 of the right-hand side of (29) is the local aversion to risk on one's own consumption for an agent who doesn't take account of consumption externalities. More precisely, consider an economy where all agents consume c with certainty. Agents are considering the possibility to take a small (perfectly correlated) risk "e around c. ¡U11 (c; c)=U1 (c; c) is the relevant degree of risk aversion to solve the decision problem of an agent who believes he is the only one to be o®ered lottery "e. The multiplicative term 1 ¡ ¡ in (29) is the adjustment to take into account of the consumption externality. Each conformist agent realizes that his neighbors consume much in states where he already consumes much. Assuming ¡ > 0, he reacts by increasing his consumption in those states. Ex ante, that means taking more risk. The adjustment term in the right-hand side of equation (29) represents this interaction e®ect. Observe in particular that an increase in the degree of conformism ¡(c; c) induces all agents to become more risk-prone. In short, more conformist societies undertake more risky projects, thereby ending up with a larger expected growth. When ¡ > 1 is positive, condition (27) still characterizes the symmetric equilibrium. But in this case, the equilibrium consumption is increasing in ¼. Such a solution cannot be duplicated by an externality-free economy. Notice in particular that if the representative agent of an externality-free economy would have the convex utility function v, he would select an unbounded consumption plan. We hereafter discuss the two particular speci¯cations that we already introduced in the paper.

6.1

The case of multiplicative consumption externalities

Suppose that U (c; C; µ) is de¯ned by condition (14) for some µ. Gali (1994) considers the special case where u is a power utility function. The equilibrium risk-taking behavior with an homogenous multiplicative consumption 12

externality is characterized by v(c) = u(c1¡µ ):

(30)

Notice that, as observed by Ross (1999), v is not necessarily more concave than u in the sense of Arrow-Pratt, even when µ 2 [0; 1] : In fact, we have that Rv (c) = µ + (1 ¡ µ)Ru (c1¡µ ); (31)

where Rh (z) = ¡zh00 (z)=h0 (z) is the relative risk aversion of function h measured at z. If µ 2 [0; 1] ; the adjusted relative risk aversion (Rv ) is a weighted average of 1 and the relative risk aversion of the outer utility function (Ru ). In particular, we obtain for the multiplicative case that ² the consumption externality has no e®ect on equilibrium risk-taking attitude if and only if the outer utility function is logarithmic (Ru ´ 1); ² under the constant relative risk aversion of u, the adjusted relative risk aversion Rv is larger than Ru if and only if the latter is smaller than unity.

6.2

The case of additive consumption externalities

Suppose alternatively that function U (c; C; µ) belongs to the class de¯ned by condition (21). It yields an equivalent externality-free utility function v such that v(c) = u((1 ¡ µ)c): (32)

The absolute risk aversion of v equals ¡

¡u00 ((1 ¡ µ)c) v00 (c) = (1 ¡ µ) : v 0 (c) u0 ((1 ¡ µ)c)

(33)

In words, the absolute risk aversion of the equivalent utility function is a weighted average of zero and the absolute risk aversion of the outer utility function.3 Using absolute risk tolerance Th (:) = ¡u0 (:)=u00 (:); this condition is rewritten as Tv (c) = (1 ¡ µ)¡1 Tu ((1 ¡ µ)c): (34) 3

The analogy between the additive and the multiplicative cases comes from the fact b with u b = ln C: Observe ¯nally b(b b(b c ¡ µC); z ) = u(exp zb); b c = ln c and C that u(cC ¡µ ) = u z ) = ¡1 + Ru (exp zb): that Aub (b

13

We can determine how do consumption externalities a®ect by re¯ning to the notion of superhomogeneity. A function g is superhomogeneous. (resp. subhomogeneous.) if g(kz) ·(resp. ¸)kg(z) for all z and all k · 1: Condition (34) directly yields the following result: Proposition 5 Consider an homogenous group with additive consumption externalities U (c; C) = u(c ¡ ¡C). Suppose that ¡ 2 [0; 1] : Then the equilibrium exposure to risk is smaller (resp. larger) than in the externality free group if the absolute risk tolerance of u is superhomogeneous. (resp. subhomogeneous.). These results are reversed if ¡ is negative. In particular, in the HARA case (Tu (c) = ´ + c=°), the introduction of envy in the economy reduces the equilibrium risk exposure in the economy if the minimum level of subsistence is positive (¡´ > 0):

7

Another interpretation of the model: the consumption-saving problem

These results can be reinterpreted in the framework of the consumptionsaving problem. Consider an economy where people live from date t = 0 to t = T . The felicity of agent µ at date t is a function U (ct ; Ct ; µ) of his own consumption, and of the average consumption in the economy at that date. In this framework, the concavity of U with respect to c or C is representative of preferences for the smoothing of the corresponding variable through time. There is a technology producing ½ units of the good at date t for each unit invested at date t ¡ 1. At this stage, there is no uncertainty about ½. Agent µ selects the consumption plan which maximizes the discounted value of his felicity, given his expectation about the average consumption plan in the economy: T X max ¯ t U(ct ; Ct ; µ) (35) t=0

s:t:

T X ct = z(µ); ½t t=0

14

(36)

where ¯ is the discount factor and z(µ) is the initial wealth of the agent. The e equals Ct for t = 0; :::; T , where µe is solution ct (µ) is an equilibrium if Ect (µ) the distribution of types in the economy. This problem is formally equivalent to the one that we examined earlier in this paper. Consider a discrete random variable s~Pwith support f0; 1; :::; T g : The probability of s~ = t is pt = ¯ t =b with b = Tt=1 ¯ t . Let also ¼(s) be equal to k(¯½)¡t . For this speci¯cation of s~ and ¼(:), program (1) (2) is equivalent to (35) (36) with ct (µ) = c(t; µ). Without consumption externalities, the characteristics of the optimal consumption path are well-known: If ¯½ = 1, i.e., if the risk free rate of the economy equals the rate of pure preference for the present, the optimal consumption is constant through time under the assumption that U11 < 0. More generally, consumption is increasing (decreasing) with time if ¯½ is larger (less) than unity. Finally, under the condition that ¯½ > 1; an increase in the concavity of U increases c0 (µ). From the above-mentioned analogy, we immediately obtain the following results: ² Under the assumption the degree of conformism is less than unity, all equilibrium consumptions are constant through time if ¯½ = 1. This result does not need to hold when ¡ is larger than unity. ² Let us hereafter assume that ¯½ is larger than unity. Under the assumption that the degree of conformism is less than unity, the average consumption in the economy is increasing with time. If in addition, ¡ is positive, then all individual consumptions are increasing with time. This latter result needs not to hold for anti-conformist agents. Analytical solutions for the additive and multiplicative consumption externalities can easily be derived from our earlier derivations for the portfolio problem. Finally, it is straightforward to combine the saving problem and the portfolio problem into a single dynamic saving-portfolio problem µa la Arrow-Debreu.

8

Concluding remark

Under some technical conditions, equilibrium saving and portfolio behaviors in an economy with consumption externalities are observationally equivalent to the behaviors of externality-free agents with an adjusted degree of risk 15

aversion. An important consequence of this result is that the measurement of the risk aversion is context-dependent. The standard experiment to measure risk aversion of subjects is based on the elicitation of the certainty-equivalent Á of a subject-speci¯c lottery x e. In the additive case (21), it is de¯ned as e = Eu(w0 + x e Eu(w0 + Á ¡ µ C) e ¡ µC)

(37)

e is the mean payo® of other subjects. If we also assume that u is where C CARA,4 this is equivalent to u(w0 + Á) = Eu(w0 + x e). The observation of Á thus provides an estimation of the concavity of the outer utility function u; since consumption externalities have no e®ect on Á in that case. As explained in this paper, the solution would be completely di®erent if all subjects would be o®ered the same lottery, with a single draw applied to all players.

e in the argument of the outer utility function u If u is not CARA, the presence of ¡µC plays the role of an independent background risk. Its e®ect of the certainty equivalent Á is examined in Gollier and Pratt (1996). 4

16

REFERENCES Arrow, K.J., (1965), Yrjo Jahnsson Lecture Notes, Helsinki. Reprinted in Arrow (1971). Dybvig, P.H., (1988), Ine±cient dynamic portfolio strategies or How to throw away a million dollars in the stock market?, The Review of Financial Studies, 1, 67-88. Gali, J., (1994), Keeping up with the Joneses: Consumption externalities, portfolio choice, and asset prices, Journal of Money, Credit, and Banking, 26, 1-8. Gollier, C., (2000), Optimal insurance design: what can we do with and without expected utility?, Contributions to Insurance Economics, G. Dionne editor, forthcoming. Gollier, C. and J.W. Pratt, (1996), Risk vulnerability and the tempering e®ect of background risk, Econometrica, 64, 11091124. Ingersoll, J.E., (1987), Theory of ¯nancial decision making, Rowman and Little¯eld Publishers. Pratt, J., (1964), Risk aversion in the small and in the large, Econometrica, 32, 122-136.

17