《金融投资学》教学资源（英文文献）Intertemporal Asset Pricing Theory.pdf_P11-P15

This special case of the “Kreps-Porteus utility” aggregates the role of the conditional distribution of future consumption through an “expected utility of next period’s utility.” If h and J are concave and increasing functions, then U is concave and increasing. If h(v) = v and if f(x, y) = u(x) + βy for some u : R+ → R and constant β > 0, then (for VT +1 = 0) we recover the special case of additive utility given by U(c) = E " X t βt u(ct) # . “Non-expected-utility” aggregation of future consumption utility can be based, for example, upon the local-expected-utility model of Machina [1982] and the betweenness-certainty-equivalent model of Chew [1983], Chew [1989], Dekel [1989], and Gul and Lantto [1990]. With recursive utility, as opposed to additive utility, it need not be the case that the degree of risk aversion is completely determined by the elasticity of intertemporal substitution. For the special case (7) of expected-utility aggregation, with differentiability throughout, we have the utility gradient representation πt = f1 (ct, Et[h(Vt+1)])Y s<t f2 (cs, Es[h(Vs+1)]) Es[h0 (Vs+1)], where fi denotes the partial derivative of f with respect to its i-th argument. Recursive utility allows for preference over early or late resolution of uncertainty (which have no impact on additive utility). This is relevant for asset prices, as for example in the context of remarks by Ross [1989], and as shown by Skiadas [1998] and Duffie, Schroder, and Skiadas [1997]. Grant, Kajii, and Polak [2000] have more to say on preferences for the resolution of information. The equilibrium state-price density associated with recursive utility is computed in a Markovian setting by Kan [1995].6 For further justification and properties of recursive utility, see Chew and Epstein [1991], Skiadas [1998], and Skiadas [1997]. For further implications for asset pricing, see Epstein [1988], Epstein [1992], Epstein and Zin [1999], and Giovannini and Weil [1989]. 6Kan [1993] further explored the utility gradient representation of recursive utility in this setting. 11

2.5 Equilibrium and Pareto Optimality Now,we explore the implications of multi-agent equilibrium for state prices. A key objective is to link state prices with important macro-economic vari- ables that are,hopefully,observable,such as total economy-wide consump- tion. Suppose there are m agents.Agent i is defined as above by a strictly increasing utility function Ui:L+R and an endowment process e()in L+.Given a dividend process 6 for N securities,an equilibrium is a collection ((1),...,(m),S),where S is a security-price process and,for each agent i, 0()is a trading strategy solving sup U:(e()+80), 0∈日 with∑e@-0. We define markets to be complete if,for each process x in L,there is some trading strategy 0 with 6o=xt,t>1.Complete markets thus means that any consumption process x can be obtained by investing some amount at time 0 in a trading strategy that,at each future period t,generates the dividend zt. The First Welfare Theorem is that complete-markets equilbria provide efficient consumption allocations.Specifically,an allocation (c),.cm) of consumption processes to the m agents is feasible if c()+.·+cm）≤ e(1)+...+e(m),and is Pareto optimal if there is no feasible allocation (6(1),...,b(m))such that U(b)U(c())for all i,with strict inequality for some i.Any equilibrium(a四，.，em）,S)has an associated feasible con- sumption allocation (c(),....c(m))defined by lettingc(e()be the dividend process generated by a(). First Welfare Theorem.Suppose (),...,0(m),S)is an equilibrium and markets are complete.Then the associated consumption allocation is Pareto optimal. An easy proof due to Arrow [1951 is obtained by contradiction.Suppose, with the objective of obtaining a contradiction,that (c),...,c(m))is the consumption allocation of a complete-markets equilibrium and that there is a feasible allocation ((1),...,b(m))such that U:(b(>U(c())for all i,with strict inequality for some i.Because of equilibrium,there is no arbitrage, and therefore a state-price density For any consumption process z,let 12

2.5 Equilibrium and Pareto Optimality Now, we explore the implications of multi-agent equilibrium for state prices. A key objective is to link state prices with important macro-economic variables that are, hopefully, observable, such as total economy-wide consumption. Suppose there are m agents. Agent i is defined as above by a strictly increasing utility function Ui : L+ → R and an endowment process e(i) in L+. Given a dividend process δ for N securities, an equilibrium is a collection (θ(1),...,θ(m) , S), where S is a security-price process and, for each agent i, θ(i) is a trading strategy solving sup θ∈Θ Ui(e(i) + δθ ), with Pm i=1 θ(i) = 0. We define markets to be complete if, for each process x in L, there is some trading strategy θ with δθ t = xt, t ≥ 1. Complete markets thus means that any consumption process x can be obtained by investing some amount at time 0 in a trading strategy that, at each future period t, generates the dividend xt. The First Welfare Theorem is that complete-markets equilbria provide efficient consumption allocations. Specifically, an allocation (c(1),...,c(m) ) of consumption processes to the m agents is feasible if c(1) + ··· + c(m) ≤ e(1) + ... + e(m) , and is Pareto optimal if there is no feasible allocation (b(1),...,b(m) ) such that Ui(b(i) ) ≥ Ui(c(i) ) for all i, with strict inequality for some i. Any equilibrium (θ(1),...,θ(m) , S) has an associated feasible consumption allocation (c(1),...,c(m) ) defined by letting c(i) −e(i) be the dividend process generated by θ(i) . First Welfare Theorem. Suppose (θ(1),...,θ(m) , S) is an equilibrium and markets are complete. Then the associated consumption allocation is Pareto optimal. An easy proof due to Arrow [1951] is obtained by contradiction. Suppose, with the objective of obtaining a contradiction, that (c(1),...,c(m) ) is the consumption allocation of a complete-markets equilibrium and that there is a feasible allocation (b(1),...,b(m) ) such that Ui(b(i) ) ≥ Ui(c(i) ) for all i, with strict inequality for some i. Because of equilibrium, there is no arbitrage, and therefore a state-price density π. For any consumption process x, let 12

T·x=E(∑TTt).We have·b≥T·c,for otherwise,given complete markets,the utility of c()can be increased strictly by some feasible trading strategy generating ()-e().Similarly,for at least some agent,we also have T.b@>T·c.Thus m∑b9>T∑c=T∑e, the equality from the market-clearing condition>()=0.This is impossi- ble,however,for feasibility implies that∑：bo≤∑：e.This contradiction implies the result Duffie and Huang [1985]characterize the number of securities necessary for complete markets.Roughly speaking,extending the spanning insight of Arrow [1953]to allow for dynamic spanning,it is necessary (and generically sufficient)that there are at least as many securities as the maximal number of mutually exclusive events of positive conditional probability that could be revealed between two dates.For example,if the information generated at each date is that of a coin toss,then complete markets requires a minimum of two securities,and almost any two will suffice.Cox,Ross,and Rubinstein [1979]provide the classical example in which one of the original securities has "binomial"returns and the other has riskless returns.That is,S=(Y,Z) is strictly positive,and,for all tT,we have 6=0,Yi+1/Y:a Bernoulli trial,and Z/Z a constant.More generally,however,to be assured of complete markets given the minimal number of securities,one must verify that the price process,which is endogenous,is not among the rare set that is associated with a reduced market span,a point emphasized by Hart [1975] and dealt with by Magill and Shafer [1990.In general,the dependence of the marketed subspace on endogenous security price processes makes the demonstration and calculation of an equilibrium problematic.Conditions for the generic existence of equilibrium in incomplete markets are given by Duffie and Shafer [1985]and Duffie and Shafer [1986].The literature on this topic is extensive. 7Bottazzi [1995]has a somewhat more advanced version of existence in single-period multiple-commodity version.Related existence topics are studied by Bottazzi and Hens [1996],Hens [1991],and Zhou [1997].The literature is reviewed in depth by Geanakoplos [1990].Alternative proofs of existence of equilibrium are given in the 2-period version of the model by Geanakoplos and Shafer [1990],Hirsch,Magill,and Mas-Colell [1990],and Husseini,Lasry,and Magill [1990];and in a T-period version by Florenzano and Gourdel [1994].If one defines security dividends in nominal terms,rather than in units of con- 13

π · x = E ( P t πtxt). We have π · b(i) ≥ π · c(i) , for otherwise, given complete markets, the utility of c(i) can be increased strictly by some feasible trading strategy generating b(i) −e(i) . Similarly, for at least some agent, we also have π · b(i) > π · c(i) . Thus π · X i b(i) > π · X i c(i) = π · X i e(i) , the equality from the market-clearing condition P i θ(i) = 0. This is impossible, however, for feasibility implies that P i b(i) ≤ P i e(i) . This contradiction implies the result. Duffie and Huang [1985] characterize the number of securities necessary for complete markets. Roughly speaking, extending the spanning insight of Arrow [1953] to allow for dynamic spanning, it is necessary (and generically sufficient) that there are at least as many securities as the maximal number of mutually exclusive events of positive conditional probability that could be revealed between two dates. For example, if the information generated at each date is that of a coin toss, then complete markets requires a minimum of two securities, and almost any two will suffice. Cox, Ross, and Rubinstein [1979] provide the classical example in which one of the original securities has “binomial” returns and the other has riskless returns. That is, S = (Y,Z) is strictly positive, and, for all t<T, we have δt = 0, Yt+1/Yt a Bernoulli trial, and Zt+1/Zt a constant. More generally, however, to be assured of complete markets given the minimal number of securities, one must verify that the price process, which is endogenous, is not among the rare set that is associated with a reduced market span, a point emphasized by Hart [1975] and dealt with by Magill and Shafer [1990]. In general, the dependence of the marketed subspace on endogenous security price processes makes the demonstration and calculation of an equilibrium problematic. Conditions for the generic existence of equilibrium in incomplete markets are given by Duffie and Shafer [1985] and Duffie and Shafer [1986]. The literature on this topic is extensive.7 7Bottazzi [1995] has a somewhat more advanced version of existence in single-period multiple-commodity version. Related existence topics are studied by Bottazzi and Hens [1996], Hens [1991], and Zhou [1997]. The literature is reviewed in depth by Geanakoplos [1990]. Alternative proofs of existence of equilibrium are given in the 2-period version of the model by Geanakoplos and Shafer [1990], Hirsch, Magill, and Mas-Colell [1990], and Husseini, Lasry, and Magill [1990]; and in a T -period version by Florenzano and Gourdel [1994]. If one defines security dividends in nominal terms, rather than in units of con- 13

Hahn [1994 raises some philosophical issues regarding the possibility of complete markets and efficiency,in a setting in which endogenous uncertainty may be of concern to investors.The Pareto inefficiency of incomplete markets equilibrium consumption allocations,and notions of constrained efficiency, are discussed by Hart [1975],Kreps [1979(and references therein),Citanna, Kajii,and Villanacci [1994],Citanna and Villanacci [1993],Pan [1993],and Pan[1995l. The optimality of individual portfolio and consumption choices in incom- plete markets in this setting is given a dual interpretation by He and Pages 1993.(Girotto and Ortu [1994 offer related remarks.)Methods for com- putation of equilibrium with incomplete markets are developed by Brown, DeMarzo,and Eaves 1996al,Brown,DeMarzo,and Eaves [1996b,Cuoco and He [1992],DeMarzo and Eaves [1996],and Dumas and Maenhout [2002] Kraus and Litzenberger [1975 and Stapleton and Subrahmanyam [1978 gave early parametric examples of equilibrium. 2.6 Equilibrium Asset Pricing We will review a representative-agent state-pricing model of Constantinides [1982.The idea is to deduce a state-price density from aggregate,rather than individual,consumption behavior.Among other advantages,this allows for a version of the consumption-based capital asset pricing model of Breeden [1979 in the special case of locally-quadratic utility. We define,for each vector A in Rm of "agent weights,"the utility function U:L+→Rby n Ua(x)=sup subject to c)+…+cm)≤x.(8) (c),c(m) i=1 Proposition.Suppose for all i that Ui is concave and strictly increasing. Suppose that (,...,m),S)is an equilibrium and that markets are com- sumption,then equilibria always exist under standard technical conditions on preferences and endowments,as shown by Cass [1984],Werner [1985],Duffie [1987],and Gottardi and Hens [1996],although equilibrium may be indeterminate,as shown by Cass [1989] and Geanakoplos and Mas-Colell [1989].On this point,see also Kydland and Prescott [1991],Mas-Colell [1991],and Cass [1991].Surveys of general equilibrium models in in- complete markets settings are given by Cass [1991],Duffie [1992],Geanakoplos [1990], Magill and Quinzii [1996],and Magill and Shafer [1991].Hindy and Huang [1993]show the implications of linear collateral constraints on security valuation. 14

Hahn [1994] raises some philosophical issues regarding the possibility of complete markets and efficiency, in a setting in which endogenous uncertainty may be of concern to investors. The Pareto inefficiency of incomplete markets equilibrium consumption allocations, and notions of constrained efficiency, are discussed by Hart [1975], Kreps [1979] (and references therein), Citanna, Kajii, and Villanacci [1994], Citanna and Villanacci [1993], Pan [1993], and Pan [1995]. The optimality of individual portfolio and consumption choices in incomplete markets in this setting is given a dual interpretation by He and Pag`es [1993]. (Girotto and Ortu [1994] offer related remarks.) Methods for computation of equilibrium with incomplete markets are developed by Brown, DeMarzo, and Eaves [1996a], Brown, DeMarzo, and Eaves [1996b], Cuoco and He [1992], DeMarzo and Eaves [1996], and Dumas and Maenhout [2002]. Kraus and Litzenberger [1975] and Stapleton and Subrahmanyam [1978] gave early parametric examples of equilibrium. 2.6 Equilibrium Asset Pricing We will review a representative-agent state-pricing model of Constantinides [1982]. The idea is to deduce a state-price density from aggregate, rather than individual, consumption behavior. Among other advantages, this allows for a version of the consumption-based capital asset pricing model of Breeden [1979] in the special case of locally-quadratic utility. We define, for each vector λ in Rm + of “agent weights,” the utility function Uλ : L+ → R by Uλ(x) = sup (c(1),...,c(m)) Xm i=1 λi Ui(ci ) subject to c(1) + ··· + c(m) ≤ x. (8) Proposition. Suppose for all i that Ui is concave and strictly increasing. Suppose that (θ(1),...,θ(m) , S) is an equilibrium and that markets are comsumption, then equilibria always exist under standard technical conditions on preferences and endowments, as shown by Cass [1984], Werner [1985], Duffie [1987], and Gottardi and Hens [1996], although equilibrium may be indeterminate, as shown by Cass [1989] and Geanakoplos and Mas-Colell [1989]. On this point, see also Kydland and Prescott [1991], Mas-Colell [1991], and Cass [1991]. Surveys of general equilibrium models in incomplete markets settings are given by Cass [1991], Duffie [1992], Geanakoplos [1990], Magill and Quinzii [1996], and Magill and Shafer [1991]. Hindy and Huang [1993] show the implications of linear collateral constraints on security valuation. 14