Short-horizon Duesenberry Equilibrium

SHOR T-HORIZON DUESENBERR Y EQUILIBRIUM JAIME A. LONDO ˜ NO Abstract. W e develop a contin uous-time general equilibrium framew ork for economies with a heterogeneous population—mo deled as a con tin uum—that repeatedly optimizes ov er short horizons under relativ e-income (Duesenberry- type) criteria. The cross-section evolv es through a Brownian ﬂo w on a type space, transporting an eﬀective impatience ﬁeld that captures time v ariation in preferences induced by demographic changes, aging, and broader social shifts. W e establish three main results. First, w e prov e an optimal consumption– inv estment theorem for inﬁnite heterogeneous p opulations in this Bro wnian- ﬂow setting. Second, we deﬁne a short-horizon Duesenb erry e quilibrium —a sequential-trading (Radner-type) equilibrium in which agen ts repeatedly solve v anishing-horizon problems under a relative-income criterion—and provide a complete c haracterization and existence proof under mild regularity conditions; notably , market completeness and absence of (state-tame) arbitrage emerge endogenously from the market clearing, rather than being imp osed as hypothe- ses. Third, we derive sharp asset-pricing impl ications: in equilibrium, the market price of risk is pinned down b y the v olatility of aggregate total wealth (ﬁnancial plus human capital), implying that the equity premium is go verned by the magnitudes and correlations of w ealth and equit y volatilities rather than by consumption volatilit y alone. This shifts the equit y premium puzzle from an implausibly low consumption volatilit y to a question ab out the volatilit y of aggregate total w ealth. The framew ork deliv ers explicit decompositions of the risk-free rate that are consistent with macro-ﬁnance stylized facts. All equilibrium quantities are expressed in terms of market primitives. 1. Introduction This pap er inv estigates intertemporal equilibrium in economies in whic h agents optimize ov er short time horizons using criteria grounded in relative income. This pap er is a companion to Londo ˜ no [45] , whic h ﬁrst prop osed the short-horizon optimization mec hanism (see Londo ˜ no [45 , Section 5.2 ] ). The framew ork accomm o- dates heterogeneous agents, including a contin uum of agent types, and provides a c haracterization of the equilibrium along with a pro of of its existence. The proposed b eha vior of consumers’ decisions is based on the limiting behavior of a consumer who, o v er eac h short perio d, optimizes a dynamic, time-v arying, iso elastic preference structure ov er discounted consumption and wealth (with resp ect to the state price pro cess). Moreov er, the dynamic preference structure may reﬂect c hanges in agents’ tastes. These changes can arise from aging or demographic shifts. Date : March 19, 2026. 2020 Mathematics Subject Classiﬁc ation. 91B50, 91B55, 91G10, 60H10. Key words and phr ases. Duesen berry equilibrium, Brownian ﬂow, State-dependent utility . This research did not receive any sp eciﬁc grant from funding agencies in the public, commercial, or not-for-proﬁt sectors. Declarations of interest: none. 1 2 JAIME A. LONDO ˜ NO The motiv ation for developing a new approach to equilibrium in markets based on relativ e income arises from the misalignmen t b etw een classical consumption theories, equilibrium mo dels of ﬁnancial markets, and empirical data. This misalignmen t is evidenced b y several well-kno wn puzzles, suc h as the “equity premium puzzle” (Mehra and Prescott [50] ), the “risk-free rate puzzle” (W eil [65] ), and the “risk- a v ersion puzzle” (Jac kw erth [30] ). F or a discussion of the theoretical problems with the approaches to the existence of equilibrium, see Anderson and Raimondo [3] , Hugonnier et al. [29] , Kramko v and Predoiu [36] , Riedel and Herzb erg [60] and Raimondo [58]. T o address these inconsistencies, Londo ˜ no [43 , 45] prop osed an approach for opti- mal consumption and in v estmen t based on optimizing a functional on consumption and wealth discounted by the state-price pro cess. In this pap er, we generalize this solution to inﬁnite p opulations, incorp orating p opulation growth (see Theorem 1). A p osteriori , the optimal aggregate wealth and consumption can b e in terpreted as a comparison of p ersonal consumption with so ciet y’s consumption (or wealth), or with that of other consumers (see Londo ˜ no [45, Remark 17] and Remark 5). The use of utilities that compare one’s own consumption to that of others is not new; references date back to V eblen [64] and Duesenberry [15] and include Gal ´ ı [27] , Ab el [1] , F rank [23] , F rank [24] , F rey and Stutzer [25] , Scito vsky [62] , Sen [63] , and Schor [61] to cite a few. Some works that sho w evidence ab out relative income to address problems of consumption and inv estment are Easterlin [16] , F rey and Stutzer [25] and Easterlin et al. [17]. The solution obtained by optimizing relative wealth and consumption do es not, b y itself, resolve the risk-free rate puzzle or the equity premium puzzle. T o address this shortcoming, Londo ˜ no [45 , Section 5.2 ] suggested an approac h that retains the optimal b eha vior describ ed ab ov e, but allows consumers to solv e their consumption and inv estmen t problem ov er short horizons, and to reset their optimization contin u- ously as preferences and tastes change. As shown by Londo ˜ no [45 , Section 5.2 ] , this mec hanism pro vides a satisfactory explanation for b oth puzzles. The present study dev elops and generalizes the prop osed approach to a setting with inﬁnitely man y heterogeneous agents and p opulation gro wth, obtaining a solution to the short- horizon optimization problem (see Theorem 2) and establishing the c haracterization and existence of equilibrium under weak hypotheses (see Theorem 3). Indeed, the relative-income, short-horizon optimization mechanism prop osed here breaks the tight link b etw een aggregate consumption volatilit y and asset returns. By allo wing preferences to mov e with demographics, the mo del pro duces low risk-free rates together with sizable equity premia, thereby oﬀering a uniﬁed framework that translates the classic puzzles into a question ab out the volatilit y of aggregate total w ealth—a quantit y whose magnitude is empirically plausible (see Remark 8)—and pro vides a plausible explanation of stylized facts related to p opulation growth, inﬂation, and interest rates (see Remark 10). This work develops a framework to address state spaces characterizing ﬁnite and inﬁnite p opulations of heterogeneous consumers. W e assume a state space (an op en set D ⊂ R d ), a Brownian ﬂo w of tw o-parameter pro cesses φ s,t ( x ) deﬁned on D ⊂ R d whic h mo dels the sto chastic b ehavior of the state v ariables, and provides a natural framework for mo deling inﬁnite p opulations. Indeed, our approac h is an extension of the classical framew ork where the relev an t v ariables in the economy are SHOR T-HORIZON DUESENBERR Y EQUILIBRIUM 3 of the form R U f ( t, φ t ( x )) dµ ( x ) where φ t is an Itˆ o pro cess describing the evolution of state v ariables, f is a diﬀerentiable function describing the relation b etw een the state v ariables and the quan tities of in terest, and µ is the initial distribution on the state v ariables at time 0. In the context of mean ﬁeld games and optimal p ortfolio management under comp etition and relative p erformance criteria, Lack er and Zariphop oulou [38] address the problem of mean ﬁeld games for portfolio optimization with inﬁnite p opulations. W e use the pricing theory based on state-tameness (see Londo ˜ no [41 , 42 , 45] ), whic h do es not require completeness or a priori conditions on the volatilit y of the price pro cess—a feature that constitutes a ma jor obstacle in the theory of the existence of intertemporal equilibrium (see Londo ˜ no [45] and references therein). A distinctive feature of the short-horizon Duesenberry equilibrium framework is that mark et completeness (in the sense of no state-tame arbitrage and well-deﬁned pricing) emerges endo genously from the equilibrium conditions, rather than b eing imp osed as an exogenous assumption. This contrasts sharply with the classical literature (Anderson and Raimondo [3] , Hugonnier et al. [29] , Kramko v and Predoiu [36] , Riedel and Herzb erg [60] ), where “p otentially dynamically complete markets” is a condition with no direct economic interpretation and must therefore b e assumed a priori . In our setting, the condition that implies completeness (the Smo oth Market Condition 1) is merely a regularity requirement (contin uit y of κ t ), not a structural restriction on the market architecture. The existence of κ t satisfying κ t σ t = ϑ t , where σ t is the volatilit y of the market price and ϑ t is the market price of risk, is guaran teed for an y ﬁnancial mark et; only its con tin uit y is assumed. The no-arbitrage and completeness prop erty then follows as a c onse quenc e of equilibrium, not as a prerequisite for its existence. On the other hand, for the results studied here, w e only require that the income of any agent b e hedgeable or insurable in a rich enough actuarial market (see Remark 3), and we do not require the hedgeability of an y other ﬁnancial instruments. Also, we fo cus on the behavior of the aggregate mark et price rather than on that of individual sto c ks. As shown by Londo ˜ no [45] , Duesen b erry equilibria do not imply any particular b ehavior for individual securities. Instead, they imply the absence of arbitrage opp ortunities when hedging with the mark et p ortfolio and the b ond. T o mo del changing preferences for relativ e consumption and wealth ov er time, w e deﬁne sto c hastic, consistent iso elastic preference structures for a p opulation. Ho w ev er, w e point out that most deﬁnitions and results can b e extended to n umerous t yp es of utilities, including logarithmic utility functions and homogeneous preference structures (see [ 43 , Condition 1]). W e deﬁne sto c hastic dynamic preference structures in tw o steps. The ﬁrst step is to deﬁne a time-c onsistent state-pr efer enc e structur e : a family of pairs of utilities for consumption and for w ealth discounted b y the state-price process. Time consistency means that the same optimal consumption and in v estmen t solution is obtained across all common inv estmen t horizons (see Deﬁnition 9), allo wing us to naturally address the problem of an inﬁnitely liv ed age n t. The second step mo dels the preference of each t yp e of individual in the so ciety (represen ted by a p oint x ∈ D ) as a pro cess with v alues in a family of time-consistent preference structures U s , whose realization U s ( ω )( x ) describ es the preference of the agen t at time s in state ω . This sto c hastic dynamic preference structure captures 4 JAIME A. LONDO ˜ NO the p ossibilit y that individuals can c hange their preferences ov er time, for instance, due to aging. Some references on progressive, dynamic utilities to mo del c hanges ov er time of preference b ehavior are Musiela and Zariphop oulou [55 , 56] and Karoui et al. [33] , among others. As a preliminary step tow ard our prop osed approach, we provide an optimiza- tion result for the optimal consumption and inv estmen t (see Theorem 1), which a p osteriori implies a maximization behavior on relative income (see Remark 5). In v estmen t in markets might allow arbitrage opp ortunities, incompleteness, and a large heterogeneous p opulation. Theorem 1 assumes that preferences remain un- c hanged as time evolv es. This optimization result is a straightforw ard generalization to a large (heterogeneous) p opulation of Londo ˜ no [45, Theorem 1]. As a natural step to address equilibrium, we include a second group of results con- sisting of Theorem 2 and Theorem 3. F rom these tw o results, we deriv e consumption, in v estmen t, and p ortfolios as the limiting outcomes of the optimal pro cesses ov er time in terv als in which each agent solves the optimization problem of Theorem 1. Finally , as a result of the aggregate consumption and wealth c haracterization, we obtain a c haracterization and existence theorem (Theorem 3) for the implied intertemporal equilibrium under weak conditions. Corollary 3 synthesizes the macro economic consequences: the state-price pro cess factorizes as the ratio of the Duesen b erry loading to aggregate income, and b oth the equity premium and the short interest rate admit explicit decomp ositions into a classical consumption-risk comp onen t and an impatience-risk comp onen t driven by demographic heterogeneity . Remark 8 sho ws that this decomp osition provides a satisfactory quantitativ e resolution of the equit y premium puzzle and the risk-free rate puzzle using observ able data, while Remark 10 discusses qualitative implications for inﬂation pass-through and the secular decline of interest rates asso ciated with demographic trends. Also, w e review the leading approaches to the equity premium puzzle—habit formation, long-run risk, and idiosyncratic-income heterogeneit y—and argue that eac h relies on at least one latent state v ariable or p o orly identiﬁed preference parameter to amplify the sto chastic discoun t factor (see Remark 9). The structure of the pap er is as follows. Section 2 reviews the mo dels and deﬁnitions presented in Londo ˜ no [43 , 42 , 45] in the con text of markets with inﬁnitely man y agen ts and deﬁnes the extension of the theory of Brownian ﬂows that is appropriate to address the problems in the pap er. In Section 3, w e deﬁne consisten t state preference structures and sto chastic consisten t preference structures; we also extend a result from Londo ˜ no [43 , 44 , 45] on the optimal consumption and inv estment theorem for utilities on state-price discoun ted v alues for inﬁnite p opulations. In Section 4, we establish Theorem 2, whic h characterizes the optimal consumption and inv estmen t strategy for consumers who contin uously reset their optimization problem, optimizing ov er state-price-discounted consumption and wealth. Finally , w e reach a characterization theorem on an intertemporal equilibrium where agents use the lo cally optimal b ehavior of Theorem 2 to determine their consumption and in v estmen t decisions. In Section 5, w e dev elop tw o examples that illustrate tw o complementary parametrizations: one treating the present v alue of human wealth as the primitiv e, the other starting from the lab or-income pro cess itself. SHOR T-HORIZON DUESENBERR Y EQUILIBRIUM 5 Finally , Section 6 presents several conclusions and p ossible extensions of the framew ork prop osed in this pap er. Among the most promising extensions are the application of the equilibrium framework to the term structure of in terest rates (ﬁxed-income pricing), its generalization to pro duction economies in which the aggregate income pro cess is endogenously determined by pro duction technology , and the developmen t of Duesenb erry e quilibria with endo genous b orr owing pr emia , in which state-dep endent household b orrowing rates generate large ﬂuctuations in total wealth without altering the aggregate pricing k ernel. T echnical results on aggregation and semimartingale prop erties of p opulation- w eigh ted pro cesses are collected in App endix A. 2. Ma thema tical Framework 2.1. Stochastic Flo ws and Consisten t Pro cesses. Let (Ω , F , ( F t ) t ≥ 0 , P ) b e the canonical ﬁltered Wiener space, where Ω = C ([0 , ∞ ) , R n ), W ( t )( ω ) = ω ( t ) is the co ordinate pro cess, P is Wiener measure, ( F t ) t ≥ 0 is the usual augmen tation of σ ( W ( u ) : 0 ≤ u ≤ t ) by null sets, and F = F ∞ ≡ σ ( ∪ t ≥ 0 F t ) ∪ N where N is the collection of P -n ull subsets of σ ( ∪ t ≥ 0 F t ). F or eac h T > 0, let ( F T s,t ) =  F T s,t , 0 ≤ s ≤ t ≤ T  b e the t wo-parameter ﬁltration where F T s,t is the smallest sub σ -ﬁeld containing all null sets and σ ( W s ( u ) | s ≤ u ≤ t ), where W s ( u ) ≡ W ( u ) − W ( s ). Assume open sets D ⊂ R d , D ′ ⊂ R d ′ and for eac h T > 0 let φ s,t ( x, ω ), 0 ≤ s ≤ t ≤ T , x ∈ D , b e a R d -v alued c ontinuous C m,χ ( D : D ′ ) -semimartingale with 0 ≤ χ ≤ 1. W e remind the reader that φ s,t ( x, ω ) is a C m,χ ( D : D ′ ) pro cess if φ s,t ( x, · ) is a F measurable contin uous random ﬁeld (almost everywhere with the exceptional s et b eing in F ) taking v alues in D ′ whic h is m -times contin uously diﬀerentiable with resp ect to the spatial v ariable x . Moreov er, following Kunita [37 , Section 3.1 ] , it is assumed that for ω outside the exceptional set, ∂ α φ s,t ( x, ω ) /∂ x α is lo cally H¨ older con tin uous with exp onen t χ . Here, α = ( α 1 , · · · , α d ) with P i α i = m . Additionally , w e say that φ s,t ( x ) is a contin uous semimartingale if it can b e decomp osed as φ s,t ( x ) : = M s,t ( x ) + B s,t ( x ). In this case, M s,t ( x ) and B s,t ( x ) are C m,χ ( D : D ′ ) pro cesses. W e assume that for s < T , t → M s,t ( x, · ) is a contin uous ( F T s,t ) local- martingale. Additionally , t → ∂ β B s,t ( x, ω ) /∂ x β is a contin uous ( F T s,t ) adapted pro cess of b ounded v ariation for β = ( β 1 , · · · , β d ) with P i β i ≤ m . F or χ = 0, this is a semimartingale of class C m . Next, w e describ e several t yp es of consistencies. Assume that for each s and T > 0, ψ s,t ( x ), s ≤ t ≤ T , and x ∈ D is a contin uous C ( D : D ′ )-semimartingale (adapted to ( F T t )). Also, φ s,t ( x ), 0 ≤ s ≤ t ≤ T , x ∈ D is a con tin uous C ( D : D )- semimartingale. W e say that the family ψ is φ -c onsistent if for each T > 0 there exists a set N T ∈ F T T with P ( N T ) = 0, such that for all s ≤ s ′ ≤ t ≤ T ω / ∈ N T ψ s ′ ,t ( φ s,s ′ ( x )) = ψ s,t ( x ) for all x ∈ D . (1) In the case ψ = φ , we say φ is consisten t. F or consistent pro cesses, we fo cus on cases where for all T > 0, the mapping φ s,t ( x ), with s, t ∈ [0 , T ], forms a Brownian ﬂow of C m diﬀeomorphisms (for some m ≥ 0). In particular, such ﬂows often result from solutions to non-explosiv e sto c hastic diﬀerential equations, which w e discuss next. As examples of Brownian ﬂows, assume that ρ : D → R d , and ϱ : D → L ( R n : R d ), are functions of class C m,χ for an op en set D where m = 0 and χ = 1 or m ≥ 1, 6 JAIME A. LONDO ˜ NO and χ > 0. In this pap er, L ( R n : R d ) stands for the set of d × n matrices. Then it is known that there exists a unique lo cal (maximal) solution φ s,t ( x ) of dφ s,t ( x ) = ρ ( φ s,t ( x )) dt + ϱ ( φ s,t ( x )) dW s ( t ) , φ s,s ( x ) = x (2) F or x ∈ D , ϱ ( x ) is a contin uous function. See Kunita [37 , Theorem 4.7.1, and Theorem 4.7.2 ] for a deﬁnition of lo cal solutions and similar results. If D = R d and ϱ ( x ) is uniformly Lipschitz contin uous in x , or if the eigenv alues of ϱ ( x ) ϱ ⊺ ( x ) are uniformly aw a y from zero, then there exists a version of the solution φ s,t ( x ) which is a contin uous C m,χ ( R d : R d )-semimartingale (see [ 37 , Theorem 4.2.7]). A similar result can b e obtained if w e assume that ϱ ( x ) is twice con tin uously diﬀerentiable and P k ∂ 2 ϱ i,k ( x ) ϱ k,i ( x ) /∂ x 2 i is b ounded for all i (see [ 37 , Theorem 4.2.7]). In tw o of the cases mentioned ab o v e, it is p ossible to choose a version of φ s,t ( x ) that is a forw ard Brownian ﬂow. Examples of families of pro cesses ψ s,t ( x ), 0 ≤ s ≤ t , and x ∈ D , which are φ - consisten t, are pro cesses ψ s,t ( x ) = f ( φ s,t ( x )) where f is some smo oth or contin uous function, and φ is a consistent semimartingale pro cess. Moreo v er, in some applications, it is p ossible to prov e that lo cal solutions can b e global (i.e., there is no explosion in ﬁnite time). Some cases of this latter type include those with an explicit solution of the SDE or those where a particular tec hnique applies. Examples include pro cesses studied by Londo ˜ no and Sandov al [46] or reducible SDE’s (Klo eden and Platen [34] ). Most applications in ﬁnance in v olv e solutions of SDEs on a subset of R d . F or instance, most price pro cess examples are deﬁned on the p ositiv e cone of R d . In this pap er, w e assume that the latter is the case and that φ s,t ( x ) is the lo cal solution of class C m,ϵ for an y 0 ≤ ϵ < δ . See Kunita [37 , Theorem 4.7.1 and Theorem 4.7.2 ] . If for any x ∈ D , φ s,t ( x ) has a non-explosive solution with v alues in D , we will say that φ s,t ( x ), x ∈ D is the non-explosive solution to the sto c hastic diﬀerential equation on D . Theoretical conditions for the existence of non- explosiv e solutions on R for Equation (2) are given by Kunita [37 , Theorem 4.7.6 ] . F or this latter suﬃcient condition, let c ( x ) = exp (2 R x 0 ρ ( y ) /ϱ 2 ( y )) dy , where ρ ( x ), and ϱ ( x )  = 0 are contin uous functions. It follows by [ 37 , Theorem 4.7.6] that a suﬃcient condition for non-explosion on the solutions of Equation (2) is that K ( ∞ ) = K ( −∞ ) = ∞ , where K ( x ) = R x 0 2 c − 1 ( z ) R z 0 c ( y ) /ϱ 2 ( y ) dy dz . Without loss of generality , we can assume that φ s,t ( x )( ω ) = φ 0 ,t − s ( x )( θ s ( ω )) for ω ∈ Ω where θ : [0 , ∞ ) × Ω → Ω is the P -preserving ﬂow on Ω deﬁned by θ ( t, ω ) ≡ W · − W t . The latter approach ﬁts in the framework of the theory of Random Dynamical Systems introduced by L. Arnold and his school (see Arnold [4] ), where ( φ, θ ) is a p erfect co cycle. More details can b e found in Mohammed and Sc heutzo w [54, Theorem 2.1]. In this pap er, we assume an op en set D ⊂ R d and assume that φ is a consistent pro cess that, for the sak e of concreteness, is assumed to b e a d -dimensional temp orally homogeneous Itˆ o pro cess φ = ( φ s,t ( x ) , 0 ≤ s ≤ t ≤ T , x ∈ D ), of tw o parameters with v alues in C ( D : D ). W e assume that φ is the solution of dφ i s,t ( x ) = ρ i ( φ s,t ( x )) dt + X 1 ≤ j ≤ n ϱ i,j ( φ s,t ( x )) dW j s ( t ) φ i s,s ( x ) = x i , i = 1 , · · · , d, (3) SHOR T-HORIZON DUESENBERR Y EQUILIBRIUM 7 where x = ( x 1 , · · · , x d ) ⊺ . W e notice that the assumptions on φ imply that the distributions of φ s,t ( x ) and φ 0 ,t − s ( x ) are identical. Assumption 1 (Lo cal well-posedness, non-explosion, and diﬀeomorphic Brownian ﬂo w) . L et D ⊂ R d b e op en and assume a Br ownian ﬂow φ on D , which satisﬁes Equation (2) . The drift ρ and diﬀusion c o eﬃcient ϱ ar e lo cally Lipschitz in x on D and (jointly) c ontinuous functions, with non-explosion / no b oundary hitting in ﬁnite time. Mor e over, ρ and ϱ ar e suﬃciently smo oth in x (e.g. C 2 , 0+ with lo c al ly Lipschitz derivatives) so that the solution map ( s, t, x ) 7→ φ s,t ( x ) deﬁnes a Br ownian ﬂow of C 2 –diﬀe omorphisms on D , as in Kunita [37]. W e assume that ρ i , ϱ i,j for j = 1 , · · · , n and i = 1 , · · · , d are (join tly) contin uous functions which are lo cally Lipschitz contin uous in the spatial v ariable, for which global solutions to the sto chastic diﬀeren tial equation (3) exist. Let φ s,t ( x ) b e a contin uous, D -v alued Brownian ﬂow of homeomorphisms as ab o v e. Deﬁnition 1. Let M ( D ) denote the set of Borel measures on D . F or µ ∈ M ( D ) and 0 ≤ s ≤ t , we write φ s,t ( µ )( ω ) for the image of µ under the map φ s,t ( · )( ω ), namely φ s,t ( µ )( F ) = µ  φ − 1 s,t ( F )  , F ∈ B ( D ) . R emark 1 . Throughout the pap er, we present the mo del with types in an op en set D ⊂ R d , follo wing the classical form ulation of sto c hastic ﬂows and their induced transp ort of measures. This choice is only for notational conv enience. All deﬁni- tions and results extend verbatim when the type space is a smo oth manifold M and the common idiosyncratic mobility is generated b y a C 2 sto c hastic ﬂow of diﬀeomorphisms ( φ s,t ) 0 ≤ s ≤ t on M . 2.2. P opulation Dynamics and Aggregation. The semimartingale prop erties of p opulation-weigh ted aggregates are fundamental to the equilibrium analysis. W e establish these prop erties in App endix A, where we prov e that under suitable in tegrabilit y conditions, the aggregate process ψ µ s,t inherits the semimartingale structure from the individual pro cesses. The key results are: • Prop osition 1 establishes a general Itˆ o formula for aggregated pro cesses; • Lemma 1 develops the co cycle prop erties of p opulation weigh ts that ensure consistency of aggregation across time; • Prop ositions 2 and 3 sp ecialize to discrete and contin uous p opulations, re- sp ectiv ely . In this pap er, we assume tw o types of p opulation structures: discrete (Assump- tion 3) and con tin uous (Assumption 4). Both cases admit so cial mobility , immigra- tion, emigration, or extinction of families within types. Under either assumption, the p opulation aggregate is a con tin uous semimartingale (Corollary 4). Next, we need the concept of aggregation: Deﬁnition 2 (Population-w eigh ted aggregation) . Let φ b e a Brownian ﬂow sat- isfying Assumption 1, and let h : D → R b e a deterministic p opulation growth rate, b ounded from ab ov e and b elow, of class C 2 , 0+ . Let P φ ⊂ P ( D ) b e a fam ily of probabilit y measures closed under transp ort by φ , that is, for every 0 ≤ s ≤ t and ev ery ν ∈ P φ , φ s,t ( ν ) ∈ P φ , 8 JAIME A. LONDO ˜ NO where φ s,t ( ν ) denotes the image of ν under φ s,t as in Deﬁnition 1. Deﬁne the p opulation weigh t Λ s,t ( x ) : = exp  Z t s h ( φ s,u ( x )) du  , 0 ≤ s ≤ t, x ∈ D , and the measure-v alued kernel ν s,t ( x, · ) ∈ M ( D ) by ν s,t ( x, B ) : = Λ s,t ( x ) δ φ s,t ( x ) ( B ) , B ∈ B ( D ) . F or any ﬁnite Borel measure µ on D , deﬁne the action of the kernel ν s,t on µ b y ( µν s,t )( B ) : = Z D ν s,t ( x, B ) dµ ( x ) , B ∈ B ( D ) . A p opulation structur e after time s is a triple µ = ( f µ , ν s,t , ¯ µ s,t ), where f µ is an F s -measurable random v ariable such that f µ ≥ 0 and E [ | f µ | ] < ∞ , and µ : = f µ ¯ µ s,s , ¯ µ s,s ∈ P φ ( D ) . W e then deﬁne the structural no-growth comp osition by ¯ µ s,t : = φ s,t ( ¯ µ s,s ) , s ≤ t, and the asso ciated size-weigh ted p opulation measure b y µ s,t : = µν s,t , that is, µ s,t ( B ) = Z D 1 { φ s,t ( x ) ∈ B } Λ s,t ( x ) dµ ( x ) . Assume ¯ ψ = ( ¯ ψ s,t ( x )) is a contin uous C ( D ; R ) semimartingale random ﬁeld. The p opulation-weighte d aggr e gation of ¯ ψ is the φ -consistent family deﬁned by ψ µ s,t : = Z D ¯ ψ s,t ( x ) Λ s,t ( x ) dµ ( x ) = Z D ψ s,t ( x ) dµ ( x ) , (4) whenev er the right-hand side is a well-deﬁned semimartingale, where ψ s,t ( x ) : = ¯ ψ s,t ( x )Λ s,t ( x ) is the mass-weigh ted typewise total ﬁeld. Convention 1 . Throughout the pap er, unless stated otherwise, we work with mass- w eigh ted (t yp e-wise total) ﬁelds X s,t ( x ). Their p er-capita coun terparts are denoted b y ¯ X s,t ( x ) = X s,t ( x ) / Λ s,t ( x ). F or an y type-level (p er-capita) quantit y ¯ X s,t ( x ) or ¯ X t ( x ) : = ¯ X 0 ,t ( x ), we write X µ s,t or X µ t : = X µ 0 ,t for its corresp onding p opulation- w eigh ted aggregate. A similar con v en tion is adopted for the mass-w eighted (type-wise total) ﬁelds X s,t ( x ) and X t ( x ) = X 0 ,t ( x ) for its corresp onding p opulation-weigh ted aggregate X µ s,t and X µ t . This sup erscript notation is used only whenever a type-level coun terpart exists, or when emphasizing the dep endence on the p opulation measure µ ; it may b e omitted when the measure is clear from the con text. 2.3. Financial Market Structure. In this pap er, w e fo cus on the b ehavior of the aggregate market price rather than on that of individual sto c ks. As shown b y Londo ˜ no [45] , Duesenberry equilibria do not imply an y particular b eha vior for individual securities. Since we assume that markets result from aggregated decisions, we require that the price pro cesses and all their co eﬃcien ts satisfy the fundamental consistency condition given by the following deﬁnition (for a motiv ation, see Corollary 4). SHOR T-HORIZON DUESENBERR Y EQUILIBRIUM 9 Deﬁnition 3. Assume a p opulation structure µ with initial measure µ after time 0 (see Deﬁnition 2). F or each random measure µ t = µ 0 ,t , t ≥ 0 w e assume a family of (random) p ositive contin uous F t + s -adapted semimartingale pro cesses X µ t = ( X µ t s ) for 0 ≤ s < ∞ , with the prop erty that X µ t u = X µ t + u . (5) W e say that the family with the ab ov e prop erties is a family of pr o c esses c onsistent with the p opulation structur e µ . W e notice that a family as ab o v e is uniquely c haracterized b y the pro cess X µ t for t ≥ 0. In this case we use interc hangeably the family X µ and the pro cess X µ t . The meaning of the pro cess X µ t s , for s ≥ 0, is the ev olution of pro cess after time t , for s units of time after, time t , where the aggregate of the p opulation is giv en by the random measure µ t . Throughout this pap er, we adopt the notation X µ t , for a family of pro cesses satisfying this deﬁnition for a p opulation structure µ , with initial measure µ at time 0. Next, we describ e a ﬁnancial mark et. Deﬁnition 4 (Financial market) . Let µ b e a p opulation structure with initial measure µ after time 0 satisfying Deﬁnition 2. A ﬁnancial market with p opulation structure µ is a structure M = ( P µ , µ , b µ , σ µ , δ µ , ϑ µ , r µ ) where eac h comp onen t is a contin uous semimartingale pro cess consistent with µ in the sense of Deﬁnition 3, satisfying the follo wing: (i) Price pro cess. The market pric e pr o c ess P µ = ( P µ t ) t ≥ 0 is a p ositiv e contin- uous semimartingale with decomp osition dP µ t = P µ t b µ t dt + P µ t ( σ µ t ) ⊺ dW ( t ) P µ 0 > 0 , (6) where the r eturn pr o c ess b µ = ( b µ t ) and the volatility matrix (c olumn ve ctor) pr o c ess σ µ = (( σ µ ) j t ) 1 ≤ j ≤ n are contin uous pro cesses consisten t with µ , and dW ( t ) = ( dW 1 ( t ) · · · , dW n ( t )) ⊺ . (ii) Dividend pro cess. The dividend r ate pr o c ess δ µ = ( δ µ t ) is a contin uous pro cess consistent with µ . (iii) In terest rate and b ond. The inter est r ate pr o c ess r µ = ( r µ t ) is a contin uous pro cess consistent with µ , and the b ond pric e pr o c ess B µ = ( B µ t ) is the p ositiv e contin uous semimartingale satisfying dB µ t = r µ t B µ t dt, B µ 0 = 1 , t ≥ 0 . (7) (iv) Mark et price of risk. The market pric e of risk pr o c ess ϑ µ = ( ϑ µ t ) is a con tin uous R n -v alued (as column vector) pro cess consistent with µ , with ϑ µ t ∈  k er( σ µ t )  ⊥ for all t ≥ 0, satisfying the no-arbitrage relation b µ t + δ µ t − r µ t − Pro j ker( σ µ t ) ⊺  b µ t + δ µ t − r µ t  = ( σ µ t ) ⊺ ϑ µ t . (8) Giv en such a ﬁnancial mark et M , the state pric e pr oc ess H µ = ( H µ t ) is deﬁned as the p ositive contin uous semimartingale H µ t : = Z µ t B µ t , t ≥ 0 , (9) where Z µ t : = exp  − Z t 0 ( ϑ µ u ) ⊺ dW ( u ) − 1 2 Z t 0 ∥ ϑ µ u ∥ 2 du  . (10) 10 JAIME A. LONDO ˜ NO By Itˆ o’s Lemma, the inv erse state price pro cess ( H µ t ) − 1 satisﬁes d ( H µ t ) − 1 = ( H µ t ) − 1  r µ t + ∥ ϑ µ t ∥ 2  dt + ( H µ t ) − 1 ( ϑ µ t ) ⊺ dW ( t ) , ( H µ 0 ) − 1 = 1 . (11) Let us deﬁne H µ s,t = H µ t /H µ s for s ≤ t . It follows that for s ≤ u ≤ t , H µ u,t H µ s,u = H µ s,t and d ( H µ s,t ) − 1 = ( H µ s,t ) − 1  r µ t + ∥ ϑ µ t ∥ 2  dt + ( H µ s,t ) − 1 ( ϑ µ t ) ⊺ dW ( t ) , ( H µ s,s ) − 1 = 1 for s ≤ t ≤ T , so H µ s,t for s ≤ t is the state price pro cess of the mark et reset at s . Throughout this pap er, we assume that the market satisﬁes Condition 1 giv en b elo w, whic h we call the smo oth m ark et condition. W e notice that since ϑ µ t ∈ k er ⊥ ( σ µ t ) = Im (( σ µ t ) ⊺ ) the existence of a progressive measurable function κ with the prop ert y expressed in Equation (12) follo ws for any ﬁnancial market (see Londo ˜ no [41] ). Hence, the condition b elo w is a weak condition on the smo othness of the aforemen tioned prop erty . Condition 1 (Smo oth Market Condition) . Ther e exists a contin uous (sc alar) semimartingale pr o c ess κ µ t c onsistent with µ in the sense of Deﬁnition 3, taking values in span(( σ µ ) 1 t , · · · , ( σ µ ) n t ) , such that for al l t ≥ 0 , κ µ t σ µ t = ϑ µ t . (12) R emark 2 . In general, the notion of Du esenberry equilibrium implies the dynamics of the mark et index (the whole mark et) and the dynamics of the b ond prices (Londo ˜ no [45 , Theorem 16 ] ). Ho w ev er, the Duesen b erry equilibrium implies no restriction on the dynamics of individual sto cks. Therefore, we assume that w e can only trade b onds and the market index. W e p oin t out that unless Pro j ker(( σ µ t ) ⊺ ) ( b µ t + δ µ t − r µ t ) = 0, there are state-tame arbitrage opportunities. Therefore, in this pap er, we do not impose the absence of arbitrage at the aggregate level a priori . See Londo˜ no [41] for deﬁnitions and c haracterization of state arbitrage opp ortunities, and state-tame p ortfolios. Next, w e review and extend some deﬁnitions from Londo˜ no [43] needed to describ e the Duesenberry equilibrium in the setting of this pap er. These extensions are natural adaptations of the classical theory of consumption and inv estmen t (Karatzas and Shreve [32] ) to the new setting prop osed herein. F or a detailed description of consisten t and related pro cesses, see Londo ˜ no [43] and Londo ˜ no [42]. Deﬁnition 5. Assume a market M with a p opulation structure µ . Assume con- tin uous, consisten t with the population structure µ , real-v alued random ﬁelds ξ µ : = ( ξ µ ( x )), c µ : = ( c µ ( x )), Q µ : = ( Q µ ), π µ : = ( π µ ( x )) and L µ : = ( L µ ( x )), with: Rate of consumption, Rate of endowmen t, W ealth, Portfolio and hedgeability: Assume non-negative pro cesses as ab o v e c µ t ( x ) and Q µ t ( x ) such that for all x ∈ D , E  R ∞ 0 H µ t c µ t ( x ) dt  < ∞ and E  R ∞ 0 H µ t Q µ t ( x ) dt  < ∞ . W e sa y that a con tin uous random ﬁeld π µ t ( x ) is a p ortfolio that ﬁnances ξ µ t ( x ) with he dge able r ate of c onsumption and r ate of endowment structur e if the family of pro cesses ξ µ t ( x ) can b e ﬁnanced by π µ t ( x ) using the endowmen t Q µ t ( x ) and consumption c µ t ( x ), namely: ( B µ t ) − 1 ξ µ t ( x ) = ξ µ 0 ( x ) + Z t 0 ( B µ u ) − 1 ( Q µ u ( x ) − c µ u ( x )) du + Z t 0 ( B µ u ) − 1 π µ u ( x )( σ µ u ) ⊺ dW ( u ) + Z t 0 ( B µ u ) − 1 π µ u ( x )( b µ u + δ µ u − r µ u ) du (13) SHOR T-HORIZON DUESENBERR Y EQUILIBRIUM 11 for all x ∈ D and 0 ≤ t < ∞ , and sa y that the family ( ξ µ , c µ , Q µ ) as ab ov e is a wealth-consumption-and-income structure, and ξ µ is a wealth pr o c ess structur e , c µ is termed the r ate of c onsumption structur e , Q µ is called a r ate of endowment structur e . Also, δ µ t π µ t ( x ) is known as the return on equit y . W e say that π µ t ( x ) is a non-arbitrage p ortfolio if for each T > 0, and 0 ≤ t ≤ T , H µ t G µ t ( x ) is uniformly b ounded b elo w, where the b ound dep ends on eac h x and T . Here G µ t ( x ) denotes the gain-in-excess pro cess (see [ 42 , Remark 1]) and it is deﬁned by G µ t ( x ) = B µ t Z t 0 ( B µ u ) − 1 π µ u ( x )( σ µ u ) ⊺ dW ( u )+ B µ t Z t 0 ( B µ u ) − 1 π µ u ( x )( b µ u + δ µ u − r µ u ) du. If π µ t ( x ) is a non-arbitrage p ortfolio, it follows (see Londo ˜ no [45 , Remark 6 ] ) that H µ t ξ µ t ( x ) + Z t 0 H µ u ( c µ u ( x ) − Q µ u ( x )) du = ξ µ 0 ( x ) + Z t 0 H µ u  π µ u ( x )( σ µ u ) ⊺ − ξ µ u ( x )( ϑ µ u ) ⊺  dW ( u ) . (14) If a wealth pro cess ( ξ µ , 0 , Q µ ) is a hedgeable rate of consumption and rate of endo wmen t structure, w e sa y that ( ξ µ , Q µ ) is a hedgeable w ealth and rate of endowmen t structure. Admissibilit y , and Subsistence random ﬁelds: A subsistenc e r andom ﬁeld structur e L µ = ( L µ t ( x )) x ∈ D for the market M is a wealth pro cess structure where for each T > 0, L µ t ( x ) H µ t , 0 ≤ t ≤ T is uniformly b ounded b elo w (where the b ound might dep end on x , and T ) such that E [ H µ t L µ t ( x )] < ∞ for all t and x ∈ D . W e sa y that the couple ( π µ , c µ ) of p ortfolio on sto c ks structure and rate of consumption structure, is admissible for ( L µ , Q µ ), and write ( π µ , c µ ) ∈ A ( L µ , Q µ ) if ξ µ t ( x ) ≥ L µ t ( x ) for all 0 ≤ t < ∞ , x ∈ D . This inequality means that ﬁnancial wealth cannot fall b elo w the (p ossibly negativ e) subsistence level, interpreted as the current v alue of future lab or income (or its insured comp onent). The pro cesses c µ t = R D c µ t ( x ) dµ ( x ), Q µ t = R D Q µ t ( x ) dµ ( x ), ξ µ t = R D ξ µ t ( x ) dµ ( x ), π µ t = R D π µ t ( x ) dµ ( x ), and L µ t = R D L µ t ( x ) dµ ( x ) are the aggr e gate r ate of c onsumption pr o c ess , aggr e gate r ate of endowment pr oc ess , aggr e gate we alth pr o c ess , aggr e gate p ortfolio of sto cks , and aggr e gate subsistenc e pr o c ess , where µ denotes the p opulation measure at time 0 introduced in Deﬁnition 2. Next, we deﬁne a hedgeable rate of endo wmen t structure: Deﬁnition 6. Assume a market M with p opulation structure µ . Assume an endo wmen t pro cess structure Q µ , with the current v alue of future endo wmen ts 12 JAIME A. LONDO ˜ NO structure L µ L µ t ( x ) = − 1 H µ t E  Z ∞ t H µ u Q µ u ( x ) du | F t  = lim T →∞ − 1 H µ t E " Z T t H µ u Q µ u ( x ) du | F t # If ( L µ , Q µ ) is a he dge able r ate of endowment structur e with a p ortfolio π µ Q w e say that Q µ is a hedgeable rate of endowmen t structure. In fact, the p ortfolio π µ Q is indeed a non-arbitrage p ortfolio (see Londo ˜ no [45, Theorem 11]). R emark 3 . In this pap er, insur ability of lab or income refers only to the insur able (non-macr o) comp onents of lab or-income risk. Sp eciﬁcally , we mean idiosyncratic mortalit y and disabilit y risk, as well as idiosyncratic employmen t and wage risk, i.e. job-loss even ts and wage sho cks that are not driven by aggregate macro economic factors. Macro economic sources of lab or-income v ariation (aggregate pro ductivit y , business-cycle conditions, etc.) are instead captured endogenously b y the equilibrium mo del through traded aggregate risks and the state-price pro cess, and therefore, they are not part of the “insurable comp onen ts” discussed herein. As sho wn in Londo ˜ no [47] , in the presence of an actuarial mark et satisfying life-insurance completeness, mortality , and disability risks are hedgeable through insurance contracts. Similarly , unemploymen t or wage-insurance contracts may hedge idiosyncratic job loss risk. In such markets, total wealth equals ﬁnancial w ealth plus the actuarially priced v alue of insured lab or income. Th us, we do not lose generality assuming hedgeability of lab or income, as long as there is a suﬃciently rich actuarial market in the sense of (Londo ˜ no [47]). 3. Optimal Consumption and Investment Throughout this pap er, w e are mainly interested in p ortfolio ev olution structures that we obtain as a result of the optimal b ehavior of consumers using iso elastic utilities: Deﬁnition 7. Consider a function U : (0 , ∞ ) 7→ R con tin uous, strictly increasing, strictly concav e, and contin uously diﬀerentiable, with U ′ ( ∞ ) = lim x →∞ U ′ ( x ) = 0 and U ′ (0+) : = lim x ↓ 0 U ′ ( x ) = ∞ . Such a function will b e called a utility function. F or ev ery utilit y function U ( · ), w e denote b y I ( · ) the in v erse of the deriv ative U ′ ( · ); b oth of these functions are contin uous, strictly decreasing and map (0 , ∞ ) onto itself with I (0+) = U ′ (0+) = lim x → 0 + U ′ ( x ) = ∞ , I ( ∞ ) = lim x →∞ I ( x ) = U ′ ( ∞ ) = 0. W e extend U b y U (0) = U (0 + ), and keep the same notation for the extension to [0 , ∞ ) of U hoping that it will b e clear to the reader to which function we are referring. It is a well-kno wn result that max 0 0, 0 < α < 1, and 0 < β < 1. Deﬁne U 1 ( t, y ) = ce − β t y α and U 2 ( t, y ) = de − β t y α , where d = c ((1 − α ) /β ) 1 − α . It is straightforw ard to see that U 1 , U 2 : [0 , ∞ ) × (0 , ∞ ) → (0 , ∞ ) deﬁne a time-consisten t preference structure with in tegrable inv erse marginal utility . Deﬁnition 10. W e say that a consistent iso elastic preference structure is a family of utility functions of the form U 1 ( t, y ) = ce − β t y α , U 2 ( t, y ) = c  1 − α β  1 − α e − β t y α , for 0 ≤ t < ∞ with x ∈ D , 0 < α < 1, 1 > β > 0, and c > 0. A preference structure for a p opulation is deﬁned as a parametric family of consistent iso elastic preference structures U ( x ) = ( U 1 ( x ) , U 2 ( x )) where U 1 ( x ) = ( U 1 ( t, · )( x ) : 0 ≤ t ) and U 2 ( x ) = ( U 2 ( t, · )( x ) : 0 ≤ t ) is a state preference structure for eac h x ∈ D . Moreo v er, we say that the preference structure for a p opulation is an iso elastic preference structure if for each x , U ( x ) is an iso elastic preference structure where w e assume that the eﬀe ctive imp atienc e r ate γ ( x ) = β ( x ) / (1 − α ( x )) is a smo oth function of x , where β ( x ), and α ( x ) corresp ond to the exp onen t parameter and time discoun t factor, resp ectiv ely , for the iso elastic preference structure ( U 1 ( x ) , U 2 ( x )). The interpretation is the following: the preference structure U ( x ) represents the 14 JAIME A. LONDO ˜ NO preference b ehavior tow ard intermediate consumption and wealth for an agent c haracterized by the state v alue x for future consumption. An iso elastic c onsistent sto chastic pr efer enc e structur e for a p opulation is a pro cess U s,t : Ω → U for 0 ≤ s ≤ t where U is a class of iso elastic preference structures for a p opulation, where the corresp onding eﬀe ctive imp atienc e ﬁeld γ s,t ( ω )( x ) of U s,t ( ω )( x ), is γ ( φ s,t ( x )( ω )) whic h is a con tin uous random ﬁeld, where γ ( · ) is a p ositiv e function of type C 2 ( D ). W e sa y that the gener ator of the iso elastic c onsistent sto chastic pr efer enc e structur e for a p opulation is the family U t = U 0 ,t with eﬀe ctive imp atienc e r ate γ ( φ t ( x )( ω )). T o av oid any unnecessary technicalit y , w e do not assume any measurability of U s,t but instead, we assume contin uit y of γ . A natural in terpretation of an iso elastic sto chastic preference structure U s,t is as follo ws: An agent is characterized by x ∈ D at time s , and sto chastic evolution of the state v alue giv en by φ s,t ; Moreov er, the v alue of utility for consumption and (ﬁnal) wealth after time t , given that at time s its state v alue is x , is the iso elastic preference structure with eﬀective impatience rate γ ( φ s,t ( x )( ω )). T o study the optimal aggregate b eha vior of a population with a time-homogeneous p opulation growth rate, we need to imp ose some conditions on the p opulation structure: Assumption 2. µ is a p opulation structur e with a deterministic p opulation gr owth r ate h , as in Deﬁnition 2. Assume c onsumers have a c onsistent iso elastic pr efer enc e structur e for a p opulation U ( x ) for x ∈ D , w her e U ( x ) = ( U 1 ( x ) , U 2 ( x )) . Assume that the eﬀe ctive imp atienc e r ate γ ( x ) is a b ounded (fr om b elow and ab ove) C 2 ( D ) function satisfying γ ( x ) ≥ γ for some γ > 0 . W e present b elow a theorem that describ es the optimal b ehavior of consumers (Theorem 1). In the following theorem, which assumes no changes in tastes, we do not assume that the market is free of (state) arbitrage or (state) complete. Theorem 1. Assume a p opulation structur e µ after time 0 as in Deﬁnition 2, satisfying Assumption 2. L et M = ( P , µ , b, σ , δ, ϑ, r ) b e a ﬁnancial market that satisﬁes the Smo oth Market Condition (Condition 1), wher e κ t is the (c ontinuous) pr o c ess taking values in span(( σ ) 1 t , · · · , ( σ ) n t ) with κ t σ t = ϑ t . Also, assume c onsumers with a c onsistent iso elastic pr efer enc e structur e for a p opulation, U ( x ) for x ∈ D with r esp e ct to φ t ( x ) that enc o des the r andom dynamic of al l state variables, wher e U ( x ) = ( U 1 ( x ) , U 2 ( x )) and C 2 ( D ) p ar ameter function γ ( x ) wher e γ ( φ t ( x )) = γ ( x ) is c onstant along the evolution of the ﬂow. In other wor ds, φ t ( x ) only enc o des the p opulation dynamics, and on e ach typ e x , the pr eferenc e r emains the same at al l times. Assume a he dge able r ate of endowment (p opulation- sc ale d) structur e Q t ( x ) (with he dging p ortfolio on the sto cks ϖ t ( x ) that is a non- arbitr age p ortfolio), and the curr ent value of (p opulation-sc ale d) futur e endowments structur e L t ( x ) , as in Deﬁnition 6. L et ξ b e deﬁne d by ξ t ( x ) : = L t ( x ) + e − tγ ( x ) H − 1 t ( ξ 0 ( x ) − L 0 ( x )) and c deﬁne d by c t ( x ) : = γ ( x ) e − tγ ( x ) H − 1 t ( ξ 0 ( x ) − L 0 ( x )) , with a p ortfolio of sto cks given by π t ( x ) : = e − tγ ( x ) H − 1 t ( ξ 0 ( x ) − L 0 ( x )) κ t − ϖ t ( x ) SHOR T-HORIZON DUESENBERR Y EQUILIBRIUM 15 for e ach t ≥ 0 . Then, ( ξ t , c t , Q t ) is a he dgeable cumulative c onsumption and endowment structur e and non-arbitr age p ortfolio ( π t , c t ) ∈ A ( L t , Q t ) that is optimal for the pr oblem of optimal c onsumption and investment, wher e L t is the p opulation- sc ale d curr ent value of futur e endowments. The optimality is in the sense that for al l T > 0 , E [ Z T 0 U 1 ( t, H t c t ( x ))( x ) dt + U 2 ( T , H T ξ T ( x ))( x )] ≥ E [ Z T 0 U 1 ( t, H t ˜ c t ( x ))( x ) dt + U 2 ( T , H T ˜ ξ T ( x ))( x )] (17) for al l x ∈ D wher e ( ξ t , c t , Q t ) is any other p opulation-sc ale d he dge able cumulative c onsumption and endowment structur e. Pr o of. If ( ξ , c, Q ) is deﬁned as ab ov e, it follows that Equation (14) holds. Using Itˆ o’s formula it follo ws that ξ ( x ) can b e ﬁnanced using the endowmen t Q t ( x ) and consumption c t ( x ) (see Equation (13) ), since ϖ t ( x ) is a non-arbitrage p ortfolio for Q t . The pro of follows the lines of the pro of of Londo ˜ no [45 , Theorem 11 ] and Londo ˜ no [42 , Theorem 2 ] with the appropriate mo diﬁcations. W e emphasize that the smo oth mark et condition (Condition 1) is used in the pro of of this theorem. □ Corollary 1. Assume the c onditions of The or em 1 and assume Assumption 2. Assume that ( ξ , c, Q ) is a we alth-c onsumption-and-inc ome structur e. We deﬁne the p opulation-w eigh ted eﬀective aggregate pro cess η t = η µ t as fol lows: η t : = Z D e − tγ ( x )  ξ 0 ( x ) − L 0 ( x )  dµ ( x ) , t ≥ 0 . Then the (p opulation-weighte d) aggr e gate optimal we alth pr o c ess satisﬁes ξ µ t − L µ t : = Z D  ξ t ( x ) − L t ( x )  dµ ( x ) = 1 H t η t , and c µ t : = Z D c t ( x ) dµ ( x ) = 1 H t ( − ∂ t η t ) , for al l t > 0 , wher e − ∂ t η t = Z D γ ( x ) e − tγ ( x )  ξ 0 ( x ) − L 0 ( x )  dµ ( x ) , t ≥ 0 , wher e L µ t is the aggr e gate of the curr ent value of futur e endowments L t ( x ) , namely L µ t = Z D L t ( x ) dµ ( x ) . Mor e over, the aggr e gate optimal p ortfolio in sto cks is given by π µ t : = Z D π t ( x ) dµ ( x ) = ( H t ) − 1 η t κ t − ϖ µ t , wher e ϖ µ t is the aggr e gate he dging p ortfolio pr o c ess asso ciated with the he dging p ortfolio ϖ t ( x ) (of the endowment Q t ( x ) ), ϖ µ t = R D ϖ t ( x ) dµ ( x ) . R emark 5 . The criterion in Theorem 1 oﬀers conceptual simplicity , yet a natural ob jection is that individual agents cannot directly observe the ﬁnancial state-price deﬂator H t . This apparent diﬃculty dissolv es once we examine the ex p ost structure of optimal b ehavior. 16 JAIME A. LONDO ˜ NO F rom Theorem 1 and Corollary 1, the comparison criterion (17) can b e represen ted as follows: F or an y reference type y ∈ D , deﬁne its relative total-w ealth gain G t ( y ) : = ξ t ( y ) − L t ( y ) ξ 0 ( y ) − L 0 ( y ) . Then the optimization criterion used in Theorem 1 is based on the maximization of E h Z T 0 U 1  t, H t c t ( x )  ( x ) dt + U 2  T , H T ξ T ( x )  ( x ) i = E h Z T 0 U 1  t, c t ( x ) e − tγ ( y ) G t ( y )  ( x ) dt + U 2  T , ξ T ( x ) e − T γ ( y ) G T ( y )  ( x ) i = E h Z T 0 U 1  t, c t ( x ) η t ξ µ t − L µ t  ( x ) dt + U 2  T , ξ T ( x ) η T ξ µ T − L µ T  ( x ) i , where η t : = R D e − tγ ( x ) ( ξ 0 ( x ) − L 0 ( x )) dµ ( x ), and the parameter γ enco des impatience. This representation reveals a fundamental insight: ex p ost , agents are not max- imizing wealth in the traditional sense but rather maximizing utility ov er rela- tiv e income—sp eciﬁcally , their consumption and terminal wealth relativ e to the p opulation-w eigh ted total w ealth ( ξ µ t − L µ t ). The p eer-group representation (ﬁrst iden tit y) shows equiv alently that agents compare themselves against the wealth gro wth G t ( y ) of a reference cohort. This relative income maximization—rather than absolute wealth maximization—is the deﬁning characteristic that justiﬁes calling our framew ork a Duesenb erry e quilibrium , connecting it directly to Duesenberry [15]’s relative income hypothesis. F rom a practical p ersp ective, a typical agent do es not need to observ e H t directly . Instead, b y trac king the w ealth dynamics of a reference p eer group—information that is plausibly a v ailable through ordinary so cial and economic observ ation—the agent obtains suﬃcient information to make consumption and inv estmen t decisions ov er short horizons. The ratio G t ( y ) summarizes the growth of total wealth (ﬁnancial plus human) for the reference cohort and deliv ers an op erational proxy for H t through relative comparisons. The last identit y conﬁrms that, in equilibrium, this p eer-group proxy coincides with the aggregate b enchmark. It is imp ortan t to emphasize that this relative comparison criterion op erates only for short-horizon decisions. Agents use p eer comparisons to determine their optimal consumption and inv estmen t ov er each short interv al; the full dynamic solution emerges by rolling forward these local decisions, as we explain in Deﬁnition 11, Theorem 2, and Theorem 3. Bey ond the short-horizon optimization step, the in terpretation of “keeping up with the Joneses” no longer applies—the rolling pro cedure is purely mechanical. R emark 6 . A p oten tial source of confusion is that decisions are made b y individual households (or families), not by p opulation aggregates. The p oint is that under CRRA (iso elastic) preferences and inﬁnite divisibility , a household with time-v arying mass can b e represented b y a typ e blo ck whose natural quantities are mass-s caled. F or each type x , w e consider p er-capita wealth, consumption, and endowmen t ¯ ξ t ( x ) := Λ − 1 t ( x ) ξ t ( x ), ¯ c t ( x ) := Λ − 1 t ( x ) c t ( x ), ¯ Q t ( x ) := Λ − 1 t ( x ) Q t ( x ), ¯ π t ( x ) := Λ − 1 t ( x ) π t ( x ). Population growth changes Λ t ( x ) by introducing new mass (new en tran ts) and hence new aggregate resources; it is therefore not economically meaningful to require a p er-capita self-ﬁnancing iden tit y for ( ¯ ξ t ( x ) , ¯ c t ( x ) , ¯ Q t ( x )), SHOR T-HORIZON DUESENBERR Y EQUILIBRIUM 17 since this would implicitly treat the inﬂow due to mass growth as a cash ﬂow replicable by trading. F easibility/admissibilit y is instead imp osed on the blo ck-lev el pro cesses ( ξ , c, Q ), for which the standard deﬂated budget identit y holds under the mark et deﬂator H . Equiv alently , one may view the mo del through a short-horizon (rolling) lens: ov er a small in terv al [ t, t + ∆] the mass is approximately constant, so each household solv es a standard consumption–inv estmen t problem; the up date from t to t + ∆, then incorp orates the net entry of mass at rate h , and the contin uous-time limit yields precisely the blo ck-lev el feasibility used in Theorem 1. 4. Shor t-horizon Duesenberr y Equilibrium: Chara cteriza tion and Existence Next, we describ e the setting of the mo del we prop ose for the equilibrium. Deﬁnition 11. Assume a ﬁnancial market M = ( P , µ , b, σ , δ, ϑ, r ) that satisﬁes the Smo oth Mark et Condition (Condition 1). Suppose an iso elastic consistent preference structure for a p opulation U with corresp onding parameter γ ( φ t ( x )( ω )) of U t ( ω )( x ), where γ ( · ) is a function of t yp e C 2 ( D ), satisfying Assumption 2. Assume an aggregate hedgeable (by state-tame p ortfolios) rate of consumption and rate of endowmen t structures ( ξ , c, Q ) and non-arbitrage p ortfolio π . Let Γ = { s 0 = 0 < s 1 < · · · < s m = T } b e a partition of the interv al [0 , T ]. The endowment and c onsumption structur e asso ciate d with Γ and pr efer enc e structur e U satisfying Assumption 2, are the endowmen t and consumption pro cesses deﬁned b y the wealth-consumption-and-income structure ( ξ Γ , c Γ , Q ) explained b elo w where ξ Γ = ( ξ Γ t ( x )) is a contin uous wealth process and c Γ = ( c Γ t ( x )) is the piecewise con tin uous pro cess, where on each interv al [ s i , s i +1 ] they are the optimal pro cesses describ ed in Theorem 1. In other words, for any x ∈ D , s k ≤ t < s k +1 and we denote s + k = t and s + i = s i +1 for 0 ≤ i ≤ k − 1: ξ Γ t ( x ) − L t ( x ) : = e − ( t − s k ) γ ( φ s k ( x )) H − 1 s k ,t ( ξ Γ s k ( x ) − L s k ( x )) = ( ξ 0 ( x ) − L 0 ( x )) i = k Y i =0 e − ( s + i − s i ) γ ( φ s i ( x )) H − 1 s i ,s + i , (18) and c Γ is given by c Γ t ( x ) : = γ ( φ s k ( x )) e − γ ( φ s k ( x ))( t − s k ) H − 1 s k ,t ( ξ Γ s k ( x ) − L s k ( x )) = γ ( φ s k ( x ))  ξ Γ t ( x ) − L t ( x )  , (19) and non-arbitrage p ortfolio π Γ t = ( π Γ t ( x )) deﬁned by: π Γ t ( x ) : = e − ( t − s k ) γ ( φ s k ( x )) H − 1 s k ,t ( ξ Γ s k ( x ) − L s k ( x )) κ t − ϖ t ( x ) =  ξ Γ t ( x ) − L t ( x )  κ t − ϖ t ( x ) , (20) where ϖ t ( x ) is the p ortfolio that hedges the rate of endo wmen t structure Q , and L t ( x ) is the current v alue of the future endowmen t structure. P assing to the limit as ∥ Γ n ∥ → 0 in (18)–(20) yields the follo wing theorem. Theorem 2. Assume a ﬁnancial market M = ( P , µ , b, σ , δ, ϑ, r ) that satisﬁes the Smo oth Market Condition (Condition 1). Assume an iso elastic c onsistent pr efer enc e structur e for a p opulation U with eﬀe ctive imp atienc e ﬁeld γ ( φ t ( x )( ω )) of U t ( ω )( x ) , 18 JAIME A. LONDO ˜ NO wher e γ ( · ) is a function of typ e C 2 ( D ) , satisfying Assumption 2. L et ξ b e the family of c ontinuous semimartingales ξ t ( x ) : = L t ( x ) + e − R t 0 γ ( φ u ( x )) du H − 1 t ( ξ 0 ( x ) − L 0 ( x )) and c the family of c ontinuous semimartingales given by c t ( x ) : = γ ( φ t ( x )) e − R t 0 γ ( φ u ( x )) du H − 1 t ( ξ 0 ( x ) − L 0 ( x )) , and non-arbitr age p ortfolio on sto cks π with gener ator π t ( x ) given by e − R t 0 γ ( φ u ( x )) du H − 1 t ( ξ 0 ( x ) − L 0 ( x )) κ t − ϖ t ( x ) for any x ∈ D , wher e ϖ t ( x ) is the p ortfolio that he dges the r ate of endowment structur e Q . Then, ( ξ , c, Q ) is a he dge able cumulative c onsumption and endowment structur e, and ( π , c ) ∈ A ( L, Q ) is a non-arbitr age p ortfolio that arises as the p oint- wise limit of the se quenc e ( ξ Γ n , c Γ n , Q ) asso ciate d with p artitions Γ n , wher e e ach ( ξ Γ n , c Γ n , Q ) is optimal on e ach subinterval of Γ n in the sense of The or em 1 (cf. Deﬁnition 11), and L t ( x ) is the curr ent value of futur e endowments for e ach typ e x ∈ D . Pr o of. Fix x ∈ D and T > 0. F or any partition Γ = { 0 = s 0 < · · · < s m = T } , deﬁne ξ Γ , c Γ , and π Γ b y rolling forward the optimal policy of Theorem 1 on each in terv al [ s i , s i +1 ] with frozen impatience γ ( φ s i ( x )) and initial wealth ξ Γ s i ( x ). This yields the explicit recursions (18) – (20) , and in particular  ξ Γ , c Γ , Q  is hedgeable and  π Γ , c Γ  is a state-tame (non-arbitrage) p ortfolio for every Γ. Using the co cycle prop erties H s,t = H t /H s (and the ﬂow/w eigh t co cycle, cf. App endix A), the pro duct form in (18) can b e written as ξ Γ t ( x ) − L t ( x ) = H − 1 t ( ξ 0 ( x ) − L 0 ( x )) exp  − X i : s i 0 for all 0 ≤ t a.e., where D µ t = δ µ t P µ t . Clearing of the sto ck mark et: π µ t = P µ t almost everywhere, where ξ µ t , π µ t , c µ t , Q µ t , and L µ t are the aggregate pro cesses asso ciated with ξ t ( x ), π t ( x ), c t ( x ), Q t ( x ) and L t ( x ) resp ectively as in Deﬁnition 2. Deﬁnition 13. Assume an economy E with underlying market M as deﬁned by Theorem 2, a p opulation structure µ (with initial distribution µ ) and he dge able (by state-tame p ortfolios) r ate of c onsumption and r ate of endowment structur es ( ξ , c, Q ), curren t v alue of future endo wmen ts L and non-arbitrage p ortfolio π deﬁne d by The or em 2 . W e say that E is a short-horizon Duesenb erry e quilibrium if the 20 JAIME A. LONDO ˜ NO equations and conditions deﬁning the clearing of the money market, the clearing of the commo dity market and the clearing of the sto ck mark et of Deﬁnition 12 hold. Theorem 3. L et φ b e a C ( D : D ) -value d Br ownian ﬂow of C 2 ( D ) diﬀe omorphisms, and let µ b e a p opulation structur e on D , with µ initial p opulation me asur e (se e Deﬁnition 2). Assume a dynamic, c onsistent iso elastic pr efer enc e structur e for a p opulation U s,t ( x ) as in Deﬁnition 10, with c orr esp onding eﬀe ctive imp atienc e r ate γ ( x ) of typ e C 2 ( D ) satisfying Assumption 2. (a) (Char acterization.) Supp ose E is a short-horizon Duesenb erry e quilibrium e c onomy with underlying market M with p opulation structur e µ that satisﬁes the smo oth market c ondition (Condition 1) with aggregate hedgeable (by state-tame p ortfolios) rate of consumption and rate of endowmen t structures ( ξ µ , c µ , Q µ ) and non-arbitr age p ortfolio π µ deﬁned b y Theorem 2 . Deﬁne the p opulation-weighte d eﬀe ctive aggr e gate pr o c ess η t = η µ t η t : = Z D exp  − Z t 0 γ ( φ u ( x )) du  y ( x ) dµ ( x ) , (24) and the aggr e gate Duesenb erry lo ading − ∂ t η t = Z D γ ( φ t ( x )) exp  − Z t 0 γ  φ u ( x )  du  y ( x ) dµ ( x ) , t ≥ 0 . (25) (sinc e γ is b ounde d ab ove). Then, ther e exists a C 2 function y ( x ) with Z D y ( x ) γ ( x ) dµ ( x ) = I µ 0 = Q µ 0 + D µ 0 > 0 , (26) wher e D µ t = δ µ t P µ t , and I µ t = Q µ t + D µ t is the aggr e gate inc ome pr o c ess. Mor e over, the state pric e pr o c ess satisﬁes H µ t = − ∂ t η t I µ t , (27) and P µ t = I µ t ∂ t η t Z ∞ t Ψ µ t s ds − I µ t ∂ t η t η t = I µ t ∂ t η t  Z ∞ t Ψ µ t s ds − η t  , (28) wher e Q µ t = Z D Q µ t ( x ) dµ ( x ) , and Ψ µ t is deﬁne d as Ψ µ t = − E  Q µ t I µ t ∂ t η t  = Z D Ψ µ t ( x ) dµ ( x ) , (29) with Ψ µ t ( x ) = − E  Q µ t ( x ) I µ t ∂ t η t  = E [ H µ t Q µ t ( x )] . (b) (Existenc e.) Assume non-ne gative (semimartingale) r andom ﬁelds Q µ t ( x ) and I µ t ( x ) of typ e C 2 ( D ) c onsistent with the p opulation structur e µ , satisfying (either) Assumption 3 or Assumption 4. Assume 0 ≤ Q µ t ( x ) < I µ t ( x ) with I µ t ( x ) > 0 for al l x ∈ D , and t ≥ 0 . Mor e over, assume that ther e exists a C 2 function y ( x ) > 0 for al l x with 0 < Z D y ( x ) γ ( x ) dµ ( x ) = Z D I µ 0 ( x ) dµ ( x ) < ∞ , (30) SHOR T-HORIZON DUESENBERR Y EQUILIBRIUM 21 F urthermor e, we deﬁne Ψ µ t ( x ) and Ψ µ t by the fol lowing e quations: Ψ µ t ( x ) = − E  Q µ t ( x ) I µ t ∂ t η t  , Ψ µ t = Z D Ψ µ t ( x ) dµ ( x ) (31) wher e Z ∞ 0 Z D Ψ µ t ( x ) dµ ( x ) dt < ∞ and Z ∞ 0 Ψ µ t ( x ) dt < ∞ , ∀ x ∈ D . (32) We deﬁne the market M = ( P µ , µ , b µ , σ µ , δ µ , ϑ µ , r µ ) by the fol lowing e qua- tions: H µ t = − ∂ t η t I µ t , (33) and P µ t = I µ t ∂ t η t  Z ∞ t Ψ µ t s ds − η t  , δ µ t P µ t = D µ t = I µ t − Q µ t , (34) wher e I µ t and Q µ t ar e the aggr e gate pr o c esses of I µ t ( x ) , and Q µ t ( x ) r esp e ctively, and it is assume d that ther e exists a contin uous semimartingale pr o c ess κ µ t , c onsistent with the p opulation structur e µ , with κ µ t σ µ t = ϑ µ t . (35) Then, ther e exists a short-horizon Duesenb erry e quilibrium E with underly- ing market M with p opulation structur e µ and a dynamic, c onsistent iso elastic pr efer enc e structur e for a p opulation with eﬀe ctive imp atienc e r ate γ ( x ) with typ e sp ac e D . (c) In either c ase, the underlying market is fr e e of state-tame arbitr age. Pr o of of The or em 3. W e divide the argument into tw o parts. First, we prov e the char acterization : starting from a short-horizon Duesenberry equilibrium, we obtain the identities (24) – (26) and the pricing form ulas (27) – (28) . Then, we prov e the existenc e direction by reversing the construction. Finally , we pro v e non-arbitrage of the market in b oth cases. Characterization. Assume a short-horizon Duesenberry equilibrium E with under- lying market M that satisﬁes the Smo oth Market Condition (Condition 1). Under the hypotheses of Theorem 2, for each agent of (initial) type x ∈ D , the lo cally optimal wealth, consumption, and p ortfolio pro cesses are given by ( ξ µ , c µ , Q µ ) and π µ as follows: ξ µ t ( x ) = L µ t ( x ) + e − R t 0 γ ( φ u ( x )) du ( H µ t ) − 1 y ( x ) (36) where the initial net we alth y ( x ) = ξ µ 0 ( x ) − L µ 0 ( x ) is a function of class C 2 and c µ t ( x ) = γ ( φ t ( x )) e − R t 0 γ ( φ u ( x )) du ( H µ t ) − 1 y ( x ) (37) and a state-tame (no-arbitrage) p ortfolio in the sense of Theorem 2 and Remark 1 giv en by: π µ t ( x ) = e − R t 0 γ ( φ u ( x )) du ( H µ t ) − 1 y ( x ) κ µ t − ϖ µ t ( x ) . (38) No w, we aggregate ov er the (random) p opulation at time t the second term of the right-hand side of Equation (36) . This yields Equation (24) b y Corollary 2. Similarly , integrating equations (36) and (37) o v er the (time- t ) distribution µ t giv es the aggr e gate pro cesses ξ µ t = L µ t +  H µ t  − 1 η t (39) 22 JAIME A. LONDO ˜ NO and − ∂ t η t = Z D γ  φ t ( x )  exp  − Z t 0 γ  φ u ( x )  du  y ( x ) dµ ( x ) . (40) W e notice that t 7→ exp ( − R t 0 γ ( φ u ( x )) du ) is non-increasing for each x , hence, so is η t . Also, by Corollary 4 η t , and − ∂ t η t are consiste n t semimartingales with resp ect to the p opulation structure µ . Moreov er, by the dominated con v ergence theorem η t → 0 a.s. Aggregating (37) against the p opulation structure gives c µ t = −  H µ t  − 1 ∂ t η t . (41) By the clearing of the commo dit y market c µ t = I µ t > 0 for all t ≥ 0 (a.e.), w e obtain H µ t = − ∂ t η t I µ t , (42) whic h is Equation (27) , where Q µ t is the aggregate (hedgeable) endowmen t rate and D µ t = δ µ t P µ t is the total dividends of the aggregate sto c k. Using the clearing of the commo dit y market at t = 0 and using (40) we obtain Equation (26). W e notice that by Equation (42) , the dominated conv ergence theorem, Assump- tion 2, and Equation (26), imply that 0 ≤ Z ∞ 0 H µ t Q µ t dt = Z ∞ 0 Q µ t I µ t ( − ∂ t η t ) dt ≤ Z ∞ 0 ( − ∂ t η t ) dt = η 0 < ∞ (43) By Equation (43) , since ( L µ , Q µ ) is a hedgeable rate of endowmen t (with non- arbitr age p ortfolio π µ Q ), using F ubini’s theorem and the co cycle prop erty , it follows that L µ t = Z D L t ( x ) dµ ( x ) = − ( H µ t ) − 1 E  Z ∞ t H µ u Q µ u du    F t  = − ( H µ t ) − 1 Z ∞ t Ψ µ t u du (44) Equation (28) is a direct consequence of Corollary 2, and Equation (44). This completes the deriv ation of (28) from the market-clearing conditions together with (27). On the other hand, by Itˆ o’s Lemma, Equation (27) , Equation (28) , and since η t → 0 it follows that E h Z ∞ 0  1 − Q µ u I µ u  ( − ∂ u η u ) du | F t i = H µ t P µ t + Z t 0 H µ u D µ u du = P µ 0 + Z t 0 H µ u P µ u  b µ u + δ µ u − r µ u − ( σ µ u ) ⊺ ϑ µ u  du + Z t 0 H µ u P µ u (( σ µ u ) ⊺ − ( ϑ µ u ) ⊺ ) dW ( u ) It follows that b µ u + δ µ u − r µ u − ( σ µ u ) ⊺ ϑ µ u = 0, proving that there are no state-tame arbitrage opp ortunities. Existence. Assume non-negative random ﬁelds Q µ ( x ) and I µ t ( x ) of type C 2 ( D ), with the prop ert y that there exist a C 2 function y ( x ) > 0 for all x satisfying Equation (26) . Deﬁne Ψ µ t ( x ), and Ψ µ t b y Equation (31) , assuming Equation (32) . Deﬁne, the eﬀective aggregate pro cess η t > 0 by Equation (24) , and deﬁne H µ t b y Equation (33). By (26) and (24), Z D γ ( x ) y ( x ) dµ ( x ) = I µ 0 , SHOR T-HORIZON DUESENBERR Y EQUILIBRIUM 23 hence H µ 0 = 1 by Equation (27) . Since γ ( x ) > 0, y ( x ) > 0 and I µ t ( x ) > 0 for all x , H µ t > 0. Let ( r µ , ϑ µ ) b e the co eﬃcien ts obtained from the Itˆ o decomp ositions of H µ . Also, Z ∞ t Ψ µ s ds = E  Z ∞ 0 Z D Q µ u ( x ) I µ u ( − ∂ u η u ) dµ ( x ) du i < E h Z ∞ 0  − ∂ u η u  du i = E [ η 0 ] < ∞ Moreo v er, Z ∞ t Ψ µ t s ds − η t = E h Z ∞ t Q µ u I µ u ( − ∂ u η u ) du − Z ∞ t ( − ∂ u η u ) du | F t i < 0 (45) It follows from Equation (34), and Equation (45): P µ t = I µ t ∂ t η t  Z ∞ t Ψ µ t s ds − η t  > 0 . Deﬁne ( b µ , σ µ ) the co eﬃcients obtained satisfying Assumption 3 or 4 according to the underlying hypothesis of the theorem, and deﬁne δ µ t : = D µ t /P µ t as the aggregate dividend yield, where D µ t = I µ t − Q µ t > 0 is the aggregate dividend income. Assume that (35) holds with a consistent (relative to the p opulation structure µ ) pro cess κ µ t suc h that κ µ t σ µ t = ϑ µ t . In particular the market deﬁned b y P µ t and H µ t ab o v e satisﬁes the smo oth market condition, and therefore Theorem 2 and Corollary 2 apply . Con- sider the consumer problem in the market M = ( P µ , µ , b µ , σ µ , δ µ , ϑ µ , r µ ) as ab ov e, with the giv en consisten t iso elastic preference structure. Deﬁne  ξ µ ( x ) , c µ ( x ) , π µ ( x )  the form ulas of Theorem 2 satisfying the explicit formulas (36) – (37) (and the corre- sp onding p ortfolio representation), with initial net wealth y µ ( x ) = y ( x ), and initial ﬁnancial wealth ξ µ 0 ( x ) = y ( x ) + L µ 0 ( x ), and lab or income Q µ t ( x ), so − L µ t ( x ) = I µ t ∂ t η t Z ∞ t Ψ µ t s ( x ) ds = ( H µ t ) − 1 E  Z ∞ t H µ u Q µ u ( x ) du    F t  , is the current v alue of future lab or at time t in the constructed market. It follows b y Corollary 2, that ξ µ t = L µ t + ( H µ t ) − 1 η t , and therefore by Equation (34) , P µ t = ξ µ t . Also, by Itˆ o’s rule, and aggregation, it follows from Equation (23) H µ t L µ t − Z t 0 H µ u Q µ u du = − E h Z ∞ 0 H µ u Q µ u du i + Z t 0 H µ u  ( π µ Q ) u ( σ µ u ) ⊺ − L µ u ( ϑ µ u ) ⊺  dW ( u ) , (46) where π µ Q is the aggregate p ortfolio (that hedges L µ t ) implied b y the represen tation as lo cal martingale of the term on the left of the last equation. W e observe that H µ t P µ t + Z t 0 H µ u D µ u du = E h Z ∞ t ( − ∂ u η u ) du − Z ∞ t Q µ u I µ u ( − ∂ u η u ) du  | F t i + Z t 0  I µ u − Q µ u  ( − ∂ u η u ) I µ u du = E h Z ∞ 0 ( − ∂ u η u ) du − Z ∞ 0 Q µ u I µ u ( − ∂ u η u ) du  | F t i 24 JAIME A. LONDO ˜ NO is a martingale. On the other hand, Itˆ o’s Lemma implies H µ t P µ t + Z t 0 H µ u D µ u du = P µ 0 + Z t 0 H µ u P µ u  b µ u + δ µ u − r µ u − ( σ µ u ) ⊺ ϑ µ u  du + Z t 0 H µ u P µ u (( σ µ u ) ⊺ − ( ϑ µ u ) ⊺ ) dW ( u ) (47) Since the left-hand side of Equation (47) is a martingale, it follows that the b ounded v ariation part is 0, implying that b µ u + δ µ u − r µ u = ( σ µ u ) ⊺ ϑ µ u and then the market, as deﬁned ab ov e, is free of state-tame arbitrage opp ortunities. F rom Theorem 2, and Corollary 2 and Equation (33), it follows that c µ t = − ∂ t η t H µ t = I µ t (48) Moreo v er from Theorem 2, and Corollary 2, and Equation (48) it follows that H µ t ξ µ t + Z t 0 H µ u ( c µ u − Q µ u ) du = ξ 0 + Z t 0 H µ u  π µ u ( σ µ u ) ⊺ − ξ µ u ( ϑ µ u ) ⊺  dW ( u ) (49) T aking diﬀerences of equations (49) and (47) , and since ξ µ t = P µ t , and since D µ t = I µ t − Q µ t = c µ u − Q µ u b y equation (48) it follows that P µ u = π µ u , proving the clearing of the money market, and the clearing of the sto ck market. This completes the existence part and therefore the pro of. □ Corollary 3. Assume the c onditions of The or em 3. L et η t := Z D exp  − Z t 0 γ ( φ u ( x )) du  y ( x ) dµ ( x ) , (50) b e the p opulation-weighte d eﬀe ctive aggr e gate pr o c ess. Then the state pric e pr o c ess and the c onsumption–we alth r atio satisfy H t = − ∂ t η t c t , c t P W t = − ∂ t η t η t = − ∂ t log( η t ) , (51) wher e c t = c µ t is aggr e gate c onsumption and P L t := ( H t ) − 1 E  Z ∞ t H u Q µ u du     F t  = − L µ t , P W t := P t + P L t = ( H t ) − 1 η t is the total we alth (physic al we alth plus lab or we alth) owne d by the so ciety at time t . Mor e over, sinc e η is a.s. absolutely c ontinuous in t , the volatility σ W t of P W t is e qual to the market pric e of risk ϑ µ t . R emark 7 . The right-hand side of Equation (51) implies that the consumption– w ealth ratio is the instantane ous pr op ortional de c ay r ate of η t , where η t represen ts the impatience-adjusted so cial wealth: it is a cross–sectional a v erage of initial w ealth y ( x ), weigh ted by the cumulativ e impatience factor along eac h tra jectory through the eﬀectiv e impatience ﬁeld γ ( φ u ( x )). A consequence of the latter is that the only factor that accounts for changes in the consumption–wealth ratio is derived from c hanges in the preference structure of the p opulation, and not from factors such as pro ductivit y , p opulation growth, or inﬂation, at least directly . See Lustig et al. [49] for empirical analysis of the wealth–consumption ratio, and Lettau and Ludvigson [39] for the closely related c ay v ariable, which has b een shown to forecast sto ck SHOR T-HORIZON DUESENBERR Y EQUILIBRIUM 25 returns and whose ﬂuctuations, in the present framework, are driven entirely b y shifts in the cross-sectional impatience structure η t . V olatilit y decomp osition. If σ c t denotes the v olatilit y of aggregate consumption (relativ e diﬀusion) and σ − ∂ η t = 1 − ∂ t η t Z D  ∇ γ ( φ t ( x )) ϱ ( φ t ( x ))  exp  − Z t 0 γ  φ u ( x )  du  y ( x ) dµ ( x ) is the volatilit y (relative diﬀusion) of − ∂ t η t , then the volatilit y σ W t of total wealth P W t (relativ e diﬀusion) satisﬁes σ W t = ϑ t = σ c t − σ − ∂ η t , where ϑ t is the market price of the risk pro cess. Hence, asset-price risk premia and mark et volatilit y (which are gov erned by the diﬀusion of the state-price pro cess) need not b e tied to consumption volatilit y: even if aggregate consumption is smo oth (small ∥ σ c t ∥ ), aggregate wealth (or the state-price) can b e highly volatile whenev er impatience risk generates large ﬂuctuations in − ∂ t η t . This provides a direct v aluation channel through which time-v arying eﬀective impatience can reconcile low consumption volatilit y with volatile markets. W e note that this mec hanism is distinct from the long-run risk channel of Bansal and Y aron [6] , where p ersisten t sho c ks to consumption growth drive asset-price volatilit y , and from the external habit mechanism of Campb ell and Co chrane [11] , where a time-v arying surplus ratio ampliﬁes risk premia. In our framework, the additional source of mark et v olatilit y arises from heterogeneous and time-v arying impatience across the p opulation, enco ded in the term σ − ∂ η t . R emark 8 . Equity premium decomp osition. The volatilit y decomp osition ϑ t = σ c t − σ − ∂ η t of Remark 7, together with the no-arbitrage relation (8) , yields an explicit decomp osition of the equity premium of any traded asset (or p ortfolio) with price pro cess P Σ t , rate of return b Σ t , dividend yield δ Σ t , and volatilit y vector σ Σ t : b Σ t + δ Σ t − r t = ( σ c t ) ⊺ σ Σ t | {z } consumption risk premium − ( σ − ∂ η t ) ⊺ σ Σ t | {z } impatience risk premium (52) The ﬁrst term is the classical Consumption-CAPM premium (see Breeden [9] ): the co v ariance of aggregate consumption growth with asset returns. The second term is the imp atienc e risk pr emium : the cov ariance of the growth of the Duesenberry loading − ∂ t η t with asset returns. F rom market v olatilit y to aggregate w ealth volatilit y . A conv enient wa y to read (52) is that equity premia are gov erned by an inner pr o duct of volatilities . In fact, as a consequence of Equation (51) , the market price of risk ϑ t = σ W t , where σ W t is the volatilit y of total wealth (physical w ealth plus lab or w ealth) P W t , and therefore Equation (52) can b e rewritten as b Σ t + δ Σ t − r t = ( σ W t ) ⊺ σ Σ t , where σ Σ t denotes the mark et (equit y) v olatilit y . In particular, the proxy σ W ≈ σ Σ w ould predict an equity premium of order ( σ Σ ) 2 . T able 1 collects long-run U.S. estimates from three indep enden t sources and compares the observed equity premium with the v alue c EP : = ( σ Σ ) 2 predicted by this proxy . Our framework do es not require aggregate wealth to b e “as smo oth as equity”. The e quilibrium identiﬁes ϑ t with a volatility-typ e obje ct that naturally admits 26 JAIME A. LONDO ˜ NO T able 1. Observ ed versus predicted equity premium under the pro xy σ W ≈ σ Σ : long-run U.S. estimates. Source P erio d σ Σ c EP = ( σ Σ ) 2 EP ϑ Mehra [51] 1889–2000 0 . 186 3 . 5% 6 . 9% 0 . 37 Jorda et al. [31] 1870–2015 0 . 192 3 . 7% 6 . 94% 0 . 36 Jorda et al. [31] 1963–2015 0 . 163 2 . 7% 5 . 69% 0 . 35 Campb ell [10] 1891–1995 0 . 186 3 . 5% 4 . 74% 0 . 25 Campb ell [10] 1947–1996 0 . 155 2 . 4% 6 . 78% 0 . 44 Campb ell [10] 1970–1996 0 . 174 3 . 0% 4 . 54% 0 . 26 Notes. σ Σ is the annualized standard deviation of real equity returns. c EP = ( σ Σ ) 2 is the equity premium predicted by the proxy σ W ≈ σ Σ . EP is the observed mean excess return of equities ov er bills, annualized. ϑ : = EP /σ Σ is the implied market price of risk, which under the equilibrium coincides with σ W . In every sample EP > c EP , equiv alen tly ϑ > σ Σ . Data for Mehra [51] : T able 1 and Sharp e-ratio calculation on p. 60 therein (S&P 500, 1889–2000). Jorda et al. [31] : T ables 3–5 (U.S. equity excess returns ov er bills; full and post-WW2 balanced panels). Campb ell [10] : T able 2 therein (log returns on a broad U.S. stock index ov er short-term governmen t debt). an interpretation as an aggr e gate we alth volatility . As T able 1 conﬁrms, empirical estimates of ϑ range from 0 . 25–0 . 37 in long-run U.S. data (Mehra 51 ; Jorda et al. 31 ; Campb ell 10 ) to roughly 0 . 44 in the p ost-w ar quarterly sample of Campbell 10 , implying an unobserv ed volatilit y of total wealth of the same order; the “puzzle” is shifted from consumption volatilit y to the volatilit y of (partly unobserved) aggregate w ealth. Connection to credit conditions and a forthcoming extension. T able 1 rev eals that ϑ = σ W consisten tly exceeds σ Σ in every sample, implying that the v olatilit y of aggregate wealth is substan tially larger than market v olatilit y alone. A natural explanation arises from the comp osition of total w ealth P W t = P t + P L t : the present v alue of future lab or income P L t is discounted, in the mo del, at the mark et interest rate r t . Moreov er, the eﬀe ctive rates at which households actually discoun t future consumption streams—consumer credit, credit-card, and unsecured b orro wing rates—are b oth signiﬁcantly higher and considerably more volatile than the risk-free rate. Because P L t t ypically accounts for tw o-thirds or more of aggregate w ealth (Lustig et al. [49] ), even mo derate ﬂuctuations in these consumer-relev an t discoun t rates translate into large swings in P W t , generating an aggregate wealth v olatilit y σ W w ell in excess of equity volatilit y σ Σ . The ab o v e remark suggests an extension— Duesenb erry e quilibria with endo genous b orr owing pr emia [ 48 ] —in whic h the relev an t discounting of lab or cash-ﬂo ws is p erformed at a household b orr owing r ate (or, more generally , a state-dep enden t eﬀectiv e rate generated by a hazard/intensit y mec hanism) rather than at r t . Ho w ev er, as shown in our related analysis (see Londo ˜ no [47] ), introducing a state-dep endent adjustmen t to the eﬀective b orro wing rate—in terpretable as a hazard intensit y λ ( φ t ( x )) reﬂecting default and liquidit y risks—does not mo dify the aggregate pricing k ernel H t nor the optimal consumption path (expressed relative to the relev an t SHOR T-HORIZON DUESENBERR Y EQUILIBRIUM 27 presen t v alue). It only re-ev aluates the collateralizable present v alue of lab or income en tering the b orrowing constrain t. Hence, large ﬂuctuations in total wealth arise from state-dep endent discounting of future lab or ﬂo ws, without altering aggregate risk pricing or consumption dynamics. W e develop this extension in a forthcoming w ork (see Londo ˜ no [48]). In terest rate decomp osition. A parallel decomp osition holds for the short rate. Applying Itˆ o’s formula to H t = ( − ∂ t η t ) /c t and writing dc t /c t = µ c t dt + ( σ c t ) ⊺ dW t and d ( − ∂ t η t ) / ( − ∂ t η t ) = µ − ∂ η t dt + ( σ − ∂ η t ) ⊺ dW t for the relative drifts and diﬀusions of aggregate consumption and the Duesenberry loading, resp ectiv ely , one obtains r t =  µ c t − µ − ∂ η t  − ( σ c t ) ⊺ ϑ t . (53) The ﬁrst term captures the net drift of aggregate consumption o v er the Duesenberry loading; the second is a precautionary-sa vings correction that dep ends on the co v ariance of consumption growth with the market price of risk. In the constan t- impatience case, µ − ∂ η t = − γ and (53) reduces to r t = µ c t + γ − ∥ σ c t ∥ 2 , the standard expression. Risk-free rate puzzle. As sho wn in Londo ˜ no [45 , Section 5.1 ] , the constant- impatience formula r = µ c + γ − ∥ σ c ∥ 2 implies a nominal short rate of approxi- mately 6 . 73% on p ost-war U.S. data (using µ c ≈ 5 . 55%, γ ≈ 1 . 2%, ∥ σ c ∥ ≈ 1 . 24%), w ell ab o v e the observed mean of roughly 4 . 73% ov er the perio d 1952.I–2011.IV (Lustig et al. [49] ). The discrepancy p ersists in real terms: with real consumption gro wth µ c ≈ 2 . 31%, the constant-impatience form ula yields a real rate of ab out 3 . 2%, whereas p ost-war real s hort rates hav e av eraged 1–2% (Campb ell [10] ; Jorda et al. [31]). This ov erprediction is the risk-fr e e r ate puzzle of W eil [65]. The source of the puzzle is transparent in (53) : when the p opulation is homo- geneous, ϑ t = σ c t and the precautionary-savings correction reduces to ( σ c t ) ⊺ ϑ t = ( σ c t ) ⊺ σ c t ≈ 0 . 02%, a negligible reduction. In the heterogeneous-impatience economy , ho w ev er, ϑ t = σ c t − σ − ∂ η t is the ful l mark et price of risk (Remark 7), so the correction b ecomes ( σ c t ) ⊺ ϑ t = ∥ σ c t ∥ 2 − ( σ c t ) ⊺ σ − ∂ η t , whic h is substantially ampliﬁed whenever ( σ c t ) ⊺ σ − ∂ η t < 0, that is, when consumption gro wth and the Duesenberry loading co-v ary negatively . Using the empirical con- sumption risk premium estimate ( σ c ) ⊺ ϑ ≈ 2 . 38% of Lustig et al. [49] and retaining µ − ∂ η ≈ − γ as the leading-order drift, Equation (53) gives r ≈ ( µ c + γ ) | {z } ≈ 6 . 75% − ( σ c ) ⊺ ϑ | {z } ≈ 2 . 38% ≈ 4 . 37% , broadly consistent with the observ ed nominal rate of 4 . 73% (our estimation). In real terms, the same calculation yields r real ≈ 2% + 1 . 2% − 2 . 2% = 1 . 0%, within the range of 0 . 8–2% rep orted for the post-war U.S. real short rate (Campb ell [10] ; Jorda et al. [31] ). Thus, the ampliﬁcation of the precautionary c hannel through heterogeneous impatience—the same mechanism that resolves the equity premium puzzle—simultaneously resolves the risk-free rate puzzle without requiring an implausibly low or negative rate of time preference. R emark 9 . Comparison with comp eting approac hes to the equity premium puzzle. The decomp osition (52) and the calibration exercise ab ov e relies exclusiv ely on mark et observ ables: the Sharp e ratio ∥ ϑ ∥ , aggregate consumption v olatilit y ∥ σ c ∥ , and the consumption risk premium ( σ c ) ⊺ ϑ , and the equity v olatility σ Σ . No 28 JAIME A. LONDO ˜ NO preference parameter needs to b e calibrated b ey ond the requirement that the cross- sectional impatience distribution b e non-degenerate. This stands in sharp contrast to the leading approaches in the literature, each of which introduces at least one laten t v ariable or a p oorly identiﬁed parameter to generate the observed equity premium. (i) Representativ e-agent CRRA models (Jorda et al. [31] , Campb ell [10] ) require a co eﬃcient of relative risk av ersion for equities of at least 37–48 (on the p ost- WW2 U.S. sample, dep ending on sample and data source), far ab o v e the range considered economically plausible; our decomp osition nests this case with σ − ∂ η = 0, so the failure of the CRRA mo del is precisely the absence of the impatience risk premium. (ii) The external habit mo del of Campb ell and Co chrane [11] matc hes aggregate moments but relies on a reverse-engineered sensitivity function λ ( S t ), en tailing anomalous welfare prop erties: Ljungqvist and Uhlig [40] demonstrate that destro ying part of the endowmen t can raise welfare under the Campb ell–Co c hrane sp eciﬁcation. (iii) The long-run risk mo del (Bansal and Y aron [6] ; Epstein and Zin [19] ) rests on a latent p ersisten t comp onen t in consumption growth whose implied predictabilit y is not found in the data (Beeler and Campb ell [7] ), a disputed elasticit y of intertemporal substitution (Hall [28] ; Y ogo [66] ), and an implausibly strong preference for early resolution of uncertaint y (Epstein et al. [20] ). (iv) The incomplete-mark ets heterogeneit y of Constantinides and Duﬃe [13] is an existence result whose quantitativ e assessment requires countercyclically heteroscedastic id- iosyncratic sho cks of debated magnitude (Ko c herlak ota [35] ). Brav et al. [8] sho w that the channel can account for the equity premium when household-level con- sumption data are used, but this requires micro-lev el consumption panels that are una v ailable in most countries and do not settle the question at the aggregate level. In each case, the additional state v ariable is either latent, gov erned by p oorly iden tiﬁed parameters, or tied to preference sp eciﬁcations with implausible side eﬀects. The Duesenberry equilibrium requires none of these ingredien ts: σ − ∂ η is not a free parameter, but an identit y implied by the equilibrium conditions (51) , fully determined once ϑ and σ c are measured using market data. R emark 10 . Inﬂation and the nominal short rate. The left-hand side of Equation (51), combined with the dynamics dH t = − r t H t dt − ϑ ⊺ t H t dW t , implies that if inﬂation do es not alter the imp atienc e p ar ameter of the p opulation , then inﬂation is transmitted line arly into the nominal short rate in the long run (since it only enters H t through aggregate consumption c µ t = I µ t ). How ev er, the transmission of inﬂation to the nominal short rate need not b e purely linear, since a pure price-level eﬀect rescales aggregate nominal consumption, and inﬂation might generate short-run distributional wedges b ecause nominal lab or income adjusts sluggishly (relative to prices and aggregate consumption). In the present framework, this sluggishness matters b ecause (51) implies that inﬂation can aﬀect not only the denominator c t but also the numerator − ∂ t η t through c hanges in eﬀective impatience and wealth/income weigh ts. Hence, deviations from a one-for-one mapping can b e in terpreted as equilibrium adjustmen ts in the impatience-adjusted w ealth comp onen t induced by inﬂation-driven income misalignment. See Fisher [22] , Mishkin [53] for the Fisher eﬀect and its empirical evidence. See also Auclert [5] , Do epke and Sc hneider [14] for literature studying the eﬀect of inﬂation on the distribution of SHOR T-HORIZON DUESENBERR Y EQUILIBRIUM 29 w ealth (see also Erosa and V entura [21] for an equilibrium analysis of inﬂation as a regressive consumption tax), and see Mian et al. [52] , Rannenberg [59] , Rachel and Summers [57] , among others, for studies on the eﬀect of wealth distribution on in terest rates. P opulation growth. As a consequence of the left-hand side of Equation (51) , p op- ulation growth aﬀects aggregates through the p opulation structure (hence through c t = I µ t ) in a weighte d manner, reﬂecting income w eigh ts. In particular, assuming an absolutely con tin uous p opulation growth structure as in this pap er, its aggregation w eigh t by wealth contributes linearly to the short-term interest rate and do es not alter the volatilit y of aggregate consumption or the market volatilit y . See Carv alho et al. [12] , Gagnon et al. [26] , Aksoy et al. [2] for references relating demographic c hanges to changes in the short interest rate, and Eggertsson et al. [18] for a quan titativ e equilibrium analysis of how population aging depresses the natural rate. Multiplicativ e in v ariance. Any purely multiplicativ e scaling of the aggregate income pro cess (from inﬂation or p opulation size) rescales c t and P W t in the same prop ortion, while leaving c t /P W t = − ∂ t log( η t ) unchanged. 5. Examples The purp ose of this section is to illustrate tw o complementary (and essentially equiv alent) wa ys to p ar ametrize tractable short-horizon Duesen berry equilibria under smo oth market conditions. Example 2 is most useful when the mo deler wishes to treat the valuation of human we alth (the present v alue of lab or) as the endogenous primitive, whereas w ages are a derived quantit y . Example 3 reverses the construction and is tailored to situations where the lab or inc ome pr o c ess is the endogenous primitive. W e assume the following for this section: Assume D ⊂ R d and let ( φ t ) t ≥ 0 b e a common Brownian ﬂow of C 2 –diﬀeomorphisms generated by dφ t ( x ) = ρ ( φ t ( x )) dt + ϱ ( φ t ( x )) dW t φ 0 ( x ) = x, where W is an n -dimensional Brownian motion and the co eﬃcien ts ensure a non- explosiv e strong solution and Kunita–t yp e regularity . Assume a p opulation structure µ with p opulation gro wth rate h , and assume an iso elastic consistent preference structure for a p opulation U with eﬀective impatience ﬁeld γ ( φ t ( x )( ω )) of U t ( ω )( x ), where γ ( · ) is a function of type C 2 ( D ), satisfying Assumption 2. Assume a p ositive income function I ∈ C 2 ( D ), and an initial total wealth y of type C 2 with y ( x ) > 0 for all x satisfying Equation (30) . Deﬁne the type-dep endent income and aggregate income as I t ( x ) : = Λ t ( x ) I ( φ t ( x )) , I µ t = Z D I t ( x ) dµ ( x ) where the population w eights Λ s,t ( x ) as in Deﬁnition 2 and the (ﬁnite-v ariation) k ernel are given by Λ t ( x ) = Λ 0 ,t ( x ) = exp  Z t 0 h ( φ u ( x )) du  , Z t ( x ) = exp  − Z t 0 γ ( φ u ( x )) du  y ( x ) , and deﬁne the p opulation-w eigh ted eﬀectiv e aggregate pro cess η t and the aggregate Duesen b erry loading − ∂ t η t as deﬁned in Theorem 3. 30 JAIME A. LONDO ˜ NO Example 2 . Assume the conditions at the b eginning of this section (Section 5). Moreo v er, let B t ( x ) b e a b ounded-v ariation, absolutely contin uous pro cess in t with con tin uous deriv ativ e ∂ t B t ( x ) with: dB t ( x ) = ∂ t B t ( x ) dt, B t ( x ) ≥ 0 , ∂ t B t ( x ) ≤ 0 for all x ∈ D , with Z D B 0 ( x ) y ( x ) dµ ( x ) < Z D y ( x ) dµ ( x ) (54) Also, deﬁne the pricing kernel H t > 0, by Equation (42), where H t : = − ∂ t η t I µ t , dH t H t = − r t dt − ϑ ⊺ t dW t , ϑ t ∈ R n . Deﬁne L t ( x ) = η t B t ( x ) I µ t ∂ t η t y ( x ) R D y ( x ) dµ ( x ) , where L t ( x ) ≤ 0 for all x ∈ D , and 0 ≤ t < ∞ . Deﬁne, Q t ( x ) = y ( x ) R D y ( x ) dµ ( x ) I µ t ( ∂ t η t ) ∂ t  η t B t ( x )  = y ( x ) I µ t ( − ∂ t η t ) R D y ( x ) dµ ( x )  B t ( x )( − ∂ t η t ) − η t ∂ t B t ( x )  ≥ 0 . (55) A straightforw ard computation shows under Assumption 2, that for all x ∈ D   H t L t ( x )   ≤ B 0 ( x ) y ( x ) , Z ∞ 0 H t Q t ( x ) dt = y ( x ) B 0 ( x ) implying that L t ( x ) is a subsistence random ﬁeld and Q t ( x ) is a rate of endowmen t structure as in Deﬁnition 5. Moreov er, H t L t ( x ) − R t 0 H s Q s ( x ) ds is constant and therefore a martingale. Also, deﬁne P µ t = L µ t + η t H t = η t I µ t ∂ t η t R D B t ( x ) y ( x ) dµ ( x ) R D y ( x ) dµ ( x ) − η t I µ t ∂ t η t > 0 for all t ≥ 0. If w e deﬁne Σ I t : = Z D Λ t ( x ) ( ∇ I ϱ )( φ t ( x )) dµ ( x ) , Σ γ t : = Z D Z t ( x ) ( ∇ γ ϱ )( φ t ( x )) dµ ( x ) , (56) b y Itˆ o’s form ula it follows that the volatilit y σ t of the price pro cess P µ t and the v olatilit y ϑ t of the state price pro cess H t are identical: σ t = ϑ t = Σ I t I µ t + Σ γ t ∂ t η t . (57) If σ t is non-degenerate for all t , then the market satisﬁes the Equation (35) , and the existence construction of Theorem 3 applies. A simple condition which guarantees the non-degenerate condition for n ≥ 2 is the existence of nonzero vectors u, v ∈ R n suc h that, almost surely with u ⊤ Σ γ t = 0 , v ⊤ Σ I t = 0 , u ⊤ Σ I t + v ⊤ Σ γ t  = 0 , (58) whic h using Equation (56) , which can b e interpreted as requiring that income-risk and impatience-risk enter (at least partially) through distinct Brownian directions, so that the tw o comp onen ts cannot cancel identically in (57). SHOR T-HORIZON DUESENBERR Y EQUILIBRIUM 31 R emark 11 . A particular sub case of Example 2, is an economy with no lab or income, i.e., Q t ( x ) ≡ 0 for all ( t, x ) (i.e., B t ( x ) ≡ 0, which also forces L t ( x ) ≡ 0). Then, the total income is entirely generated by the dividend/endo wmen t stream, so the aggregate income reduces to I µ t . The state-price density and aggregate price simplify to H t = − ∂ t η t I µ t , P µ t = η t H t = I µ t η t ( − ∂ t η t ) . Example 3 . Assume the conditions at the b eginning of this section (Section 5). Moreo v er, assume a non-negativ e contin uous random ﬁeld χ t ( x ) such that for eac h x ∈ D the map t 7→ χ t ( x ) is a.s. absolutely con tin uous, with χ t ( x ) ≥ 0 and ∂ t χ t ( x ) ≤ 0 for t ≥ 0 , x ∈ D . Assume that u t ( x ) := χ t ( x ) + η t ∂ t η t ∂ t χ t ( x ) , Z D y ( x ) u t ( x ) dµ ( x ) < 1 , t ≥ 0 (59) and deﬁne Q t ( x ) := y ( x ) I µ t u t ( x ) , L t ( x ) := η t ∂ t η t I µ t y ( x ) χ t ( x ) . Since ∂ t η t < 0 and ∂ t χ t ( x ) ≤ 0, w e hav e u t ( x ) ≥ χ t ( x ) ≥ 0, hence Q t ( x ) ≥ 0, while L t ( x ) ≤ 0. Also, Assumption 2 implies that for all x ∈ D   H t L t ( x )   ≤ η 0 y ( x ) χ 0 ( x ) , Z ∞ 0 H t Q t ( x ) dt = η 0 y ( x ) χ 0 ( x ) , implying that L t ( x ) is a subsistence random ﬁeld and Q t ( x ) is a rate of endowmen t as in Deﬁnition 5. Moreov er, H t L t ( x ) − R t 0 H s Q s ( x ) ds is a constant and therefore a martingale. Deﬁne the aggregate price pro cess P µ t := L µ t + η t H t = η t I µ t − ∂ t η t  1 − Z D y ( x ) χ t ( x ) dµ ( x )  . Because u t ( x ) ≥ χ t ( x ), (59) implies R D y ( x ) χ t ( x ) dµ ( x ) < 1, hence P µ t > 0. More- o v er, since χ is of ﬁnite v ariation (no Brownian term), the diﬀusion co eﬃcient of P µ coincides with that computed in Example 2; in particular the smo oth market condition (35) holds (one ma y take the scalar multiplier to b e equal to 1). Therefore, b y Theorem 3, the implied economy is a short-horizon Duesenberry equilibrium. Alternativ ely , ﬁx a contin uous function f : D → [0 , ∞ ) and assume a non- negativ e random ﬁeld u t ( x ) such that for each x the map t 7→ u t ( x ) is a.s. absolutely con tin uous and Z D y ( x ) u t ( x ) dµ ( x ) < 1 , t ≥ 0 , together with the tw o inequalities η t u t ( x ) + Z t 0 ( − ∂ s η s ) u s ( x ) ds = η 0 u 0 ( x ) + Z t 0 η s ∂ s u s ( x ) ds ≥ η 0 f ( x ) ≥ Z t 0 ( − ∂ s η s ) u s ( x ) ds, t ≥ 0 , x ∈ D , (60) (F or instance, the left-hand side of Equation (60) holds whenever u t ( x ) is non- decreasing in t and u 0 ( x ) ≥ f ( x ).) Deﬁne χ t ( x ) by the relation χ t ( x ) η t := η 0 f ( x ) + Z t 0 ( ∂ s η s ) u s ( x ) ds. 32 JAIME A. LONDO ˜ NO Then the right-hand side of (60) yields χ t ( x ) ≥ 0, and the left-hand side of Equa- tion (60) is equiv alen t (via integration by parts) to ∂ t χ t ( x ) ≤ 0; moreov er, u t ( x ) = χ t ( x ) + η t ∂ t η t ∂ t χ t ( x ) . Hence, deﬁning Q t ( x ) := y ( x ) I µ t u t ( x ) , L t ( x ) := I µ t ∂ t η t y ( x )  η 0 f ( x ) + Z t 0 ( ∂ s η s ) u s ( x ) ds  w e hav e that H t L t ( x ) − R t 0 H s Q s ( x ) ds is constant, therefore a martingale, and with P µ t := L µ t + η t H t = η t I µ t − ∂ t η t  1 − Z D y ( x ) χ t ( x ) dµ ( x )  > 0 . As in Example 2 the v olatilit y σ t of the price process P µ t and the volatilit y of the state price pro cess ϑ t are giv en b y Equation (57) . It also applies that the mo del satisﬁes the Smo oth Market Condition as long as the condition given by Equation (58) holds. It follows that Theorem 3 yields again, a short-horizon Duesenberry equilibrium. 6. Concluding Remarks In this pap er, we develop a framework for mo deling ﬁnancial markets that can accoun t for the changing behavior of utilities for consumption and inv estment within inﬁnite p opulations. W e establish explicit suﬃcien t conditions (see Theorem 3) on the primitives of the economy to guarantee the existence of a short-horizon Duesen b erry equilibrium without assuming market completeness a priori . The analysis concludes that incorp orating the evolution of consumer preferences regarding relativ e income oﬀers a robust theoretical framework to address b oth the “equit y premium puzzle” and the “risk-free rate puzzle.” F urthermore, this equilibrium framework circumv en ts common theoretical limita- tions of classical equilibrium theories and provides a tractable metho d for pricing assets in incomplete markets driven by Bro wnian ﬂows. Three directions for future research are particularly promising. First, the interest- rate decomp osition (53) and its connection to the consumption–wealth ratio suggest a natural extension to the term structur e of inter est r ates : the Duesenberry load- ing − ∂ t η t enco des cross-sectional impatience information that should propagate in to the yield curve, p otentially explaining b oth the level and the slop e of the term structure through demographic fundamentals rather than latent factors. Sec- ond, the present framework assumes aggregate income and dividends as exogenous primitiv es; embedding the mo del in a pr o duction e c onomy , where these pro cesses arise endogenously from pro duction technology and lab or supply decisions, w ould pro vide a fully general-equilibrium foundation and sharp en the empirical predictions regarding the co-mov ement of consumption, inv estmen t and asset returns. Third, the credit-conditions discussion in Remark 8 p oints tow ard Duesenb erry e quilibria with endo genous b orr owing pr emia , in which the present v alue of lab or incom e is discoun ted at a state-dep enden t household b orro wing rate—reﬂecting default and liquidit y risks—rather than at the risk-free rate. This extension arises naturally from combining the equilibrium framework developed here with the life-insurance completeness and state-dep enden t hazard-rate machinery of Londo ˜ no [47] : the latter sho ws that introducing a state-dep enden t eﬀective b orrowing rate do es not alter the aggregate pricing k ernel, while the former provides the equilibrium structure in SHOR T-HORIZON DUESENBERR Y EQUILIBRIUM 33 whic h such rates generate large ﬂuctuations in total wealth through state-dep enden t discoun ting of future lab or ﬂo ws. In particular, this approach has the p otential to accoun t for the high levels of aggregate wealth volatilit y implied b y the equilibrium (T able 1), where the market price of risk ϑ = σ W consisten tly exceeds equity volatil- it y σ Σ , thereby oﬀering a complete quantitativ e resolution of the equit y premium puzzle. W e develop this program in Londo ˜ no [48]. Appendix A. Aggrega tion and Semimar tingale Proper ties This app endix establishes the semimartingale prop erties of p opulation-w eigh ted aggregates that underpin the equilibrium analysis in the main text. W e ﬁrst present a general result on aggregated Itˆ o processes (Prop osition 1), then develop the cocycle structure of p opulation weigh ts (Lemma 1), and ﬁnally sp ecialize to discrete and con tin uous p opulations (Prop ositions 2 and 3). A.1. Semimartingale prop ert y of aggregated pro cesses. In this subsec- tion we discuss some prop erties of the aggregation of pro cesses of the form R D f ( t, φ s,t ( x )) dµ ( x ) where φ is a consistent semimartingale pro cess, and µ is a probabilit y distribution on D . Such pro cesses arise naturally in the context of mo deling inﬁnite p opulations, where each individual is represented by a p oint x ∈ D , and φ s,t ( x ) represen ts the sto chastic b ehavior of the individual x at time t giv en that at time s the state of this individual w as x . The function f ( t, x ) represents the relationship b etw een the state of the individual at time t and some v ariable of in terest. Finally , the measure µ represen ts the initial distribution of individuals at time s . In this section, we pro vide conditions under whic h suc h aggregated pro cesses are semimartingales and provide their sto chastic representations. Prop osition 1. L et f : [0 , ∞ ) × D → R b e a c ontinuous function with c on- tinuous p artial derivatives ∂ 2 f /∂ x i ∂ x j and ∂ f /∂ t , for i, j = 1 , . . . , d (that is, f ∈ C 1 , 2 ([0 , ∞ ) × D ) ), wher e D ⊂ R d is an op en subset. Supp ose that φ is a c onsistent semimartingale pr o c ess with values in D and sto chastic r epr esentation (2) , wher e ρ and ϱ ar e functions of class C m,χ with m = 0 and χ = 1 or m ≥ 1 , and χ > 0 . Assume that µ is a pr ob ability distribution on D with c ontinuous density g ( x ) . Deﬁne a ij ( t, x ) : = n X k =1 ϱ i,k ( t, x ) ϱ j,k ( t, x ) , i, j = 1 , . . . , d. Assume that for any s < t < ∞ : i f ( t, · ) is µ -inte gr able for any t . ii ∂ f /∂ t , P i ρ i ∂ f /∂ x i , P i ϱ i,k ∂ f /∂ x i for every k = 1 , . . . , n , and P i,j a ij ∂ 2 f /∂ x i ∂ x j ar e Lipschitz c ontinuous. iii E  R D | φ i s,t ( x ) | dµ ( x )  < ∞ for i = 1 , . . . , d . 34 JAIME A. LONDO ˜ NO Then, for e ach ﬁxe d s the pr o c ess R D f ( t, φ s,t ( x )) dµ ( x ) is a c ontinuous ( F T s,t ) t ∈ [ s,T ] semimartingale pr o c ess almost P -a.s, and Z D f ( t, φ s,t ( x )) dµ ( x ) = Z D f ( s, x ) g ( x ) dx + Z t s Z D ∂ ∂ t f ( u, φ s,u ( x )) g ( x ) dx du + Z t s X i Z D ∂ ∂ x i f ( u, φ s,u ( x )) ρ i ( φ s,u ( x )) g ( x ) dx du + 1 2 Z t s X i,j Z D ∂ 2 ∂ x i ∂ x j f ( u, φ s,u ( x )) a ij ( φ s,u ( x )) g ( x ) dx du + X k Z t s X i Z D ∂ f ∂ x i ( u, φ s,u ( x )) ϱ i,k ( φ s,u ( x )) g ( x ) dx dW k s ( u ) . (61) Pr o of. W e in troduce some notation. Let a = ( a 1 , · · · , a d ) < b = ( b 1 , · · · , b d ), and let P i = { t i 0 = a i < t i 1 <, · · · , < t i p i = b i } for i = 1 · · · d b e a partition of the h yp er-rectangle [ a, b ]. F or each t = ( t 1 i 1 , · · · , t d i d ) ∈ P = P 1 × · · · × P d , with t < b w e deﬁne the next corner t + = ( t 1 i 1 +1 , · · · , t d i d +1 ). W e also deﬁne a sample for each t ∈ P to b e a p oin t t ⋆ ∈ [ t, t + ), where for [ t, t + ) = { x ∈ R d : t ≤ x < t + } . T o each partition P , we deﬁne the norm or mesh | P | = max t ∈ P,t 0 suc h that | E [ φ s,t ( x ) − x ] | ≤ K (1 + | x | ( t − s )) for 0 ≤ s ≤ t (see Kunita [37 , Theorem 4.2.5 ] ). It follows that under the assumption of Lipschitz contin uity for ρ and ϱ , assuming R D | x i | g ( x ) dm < ∞ for all i where dµ/dm = g , implies that condition iii of Prop osition 1 holds. The latter follows using the dominated con v ergence theorem for sto chastic in tegrals. A.2. Co cycle prop erties of p opulation weigh ts. Lemma 1 (Co cycle and identit y prop erties) . A dopt Deﬁnition 2. The fol lowing c o cycle and identity pr op erties hold P -a.s.: (i) (Multiplicative weigh t co cycle.) F or al l 0 ≤ r ≤ s ≤ t and x ∈ D , Λ r,t ( x ) = Λ r,s ( x ) Λ s,t ( φ r,s ( x )) . (63) (ii) (Kernel co cycle.) F or al l 0 ≤ r ≤ s ≤ t and Bor el B ⊂ D , ν r,t ( x, B ) = Z D ν s,t ( y , B ) ν r,s ( x, dy ) , i.e. ν r,t = ν r,s ◦ ν s,t . (64) (iii) (Measure co cycle.) F or any (deterministic or F r -me asur able) ﬁnite me asur e µ on D , µ r,t = ( µ r,s ) ν s,t . (65) (iv) (0-based aggregate identit y .) Fix µ at time 0 and set µ s : = µ 0 ,s as in Deﬁni- tion 2. Then for any Bor el function f such that the inte gr als b elow ar e ﬁnite, and any 0 ≤ s ≤ t , Z D f ( φ s,t ( x )) Λ s,t ( x ) µ s ( dx ) = Z D f ( φ 0 ,t ( y )) Λ 0 ,t ( y ) µ ( dy ) . (66) Equivalently, the p opulation-weighte d aggr e gate c ompute d on [ s, t ] fr om the shifte d p opulation µ s c oincides p athwise with the aggr e gate c ompute d on [0 , t ] fr om the initial p opulation µ . Pr o of. Iden tit y (63) follo ws from additivity of the time integral and the ﬂo w property φ r,u = φ s,u ◦ φ r,s for u ≥ s . Then (64) is immediate since ν r,t ( x, B ) = Λ r,t ( x ) 1 { φ r,t ( x ) ∈ B } = Λ r,s ( x )Λ s,t ( φ r,s ( x )) 1 { φ s,t ( φ r,s ( x )) ∈ B } , and ν r,s ( x, · ) = Λ r,s ( x ) δ φ r,s ( x ) ( · ). Iden tit y (65) is obtained b y integrating (64) against µ . 36 JAIME A. LONDO ˜ NO Finally , (66) is just (65) with ( r, s, t ) = (0 , s, t ) written in test-function form: Z f ( z ) µ t ( dz ) = Z f ( φ s,t ( x ))Λ s,t ( x ) µ s ( dx ) , and R f ( z ) µ t ( dz ) = R f ( φ 0 ,t ( y ))Λ 0 ,t ( y ) µ ( dy ) by deﬁnition of µ t . □ A.3. Discrete p opulations. Assumption 3 (Discrete p opulation and p opulation weigh ts) . Assume a diﬀe o- morphic Br ownian ﬂow that satisﬁes Assumption 1. Fix s ≥ 0 . L et the p opulation b e discr ete, µ = X i ∈ I w i δ x i , X i ∈ I | w i | < ∞ , wher e I is ﬁnite or c ountable and x i ∈ D . L et h : D → R b e a c ontinuous C 2 , 0+ function and deﬁne the p opulation weight as in Deﬁnition 2, wher e it is assume d that P φ is the set of ﬁnite me asur es. Assume that for e ach i ∈ I , t 7→ ψ µ t ( x i ) is a c ontinuous semimartingale with de c omp osition ψ µ t ( x i ) = ψ µ 0 ( x i ) + Z t 0 ζ µ u a  φ u ( x i )  du + Z t 0 ( υ µ u ) ⊺ diag  b ( φ u ( x i ))  dW ( u ) , for c ontinuous functions a (r e al-value d) and b ( R n -value d), c onsistent with r esp e ct to the p opulation structur e µ , and c ontinuous semimartingale pr o c esses r e al value d ζ µ u , and R n value d ( υ µ ) u (which is typ e indep endent). Assume either I is ﬁnite, or for every T > 0 , X i ∈ I | w i | Z T 0    Λ u ( x i ) ζ µ u a  φ u ( x i )    +   Λ u ( x i ) h  φ u ( x i )  ψ u ( x i )    du < ∞ , and X i ∈ I | w i |  Z T 0 Λ u ( x i ) 2   ( υ µ u ) ⊺ diag  b ( φ u ( x i ))    2 du  1 / 2 < ∞ . Prop osition 2 (Population-w eigh ted aggregation: discrete p opulation) . Under Assumption 3, deﬁne the p opulation-weighte d aggr e gate (cf. Deﬁnition 2) ψ µ t : = Z D Λ t ( x ) ψ µ t ( x ) dµ ( x ) = X i ∈ I w i Λ t ( x i ) ψ t ( x i ) . Then, t 7→ ψ µ t is a c ontinuous semimartingale for 0 ≤ t < ∞ . Mor e over, ψ µ t = ψ µ 0 + Z t 0 X i ∈ I w i h Λ u ( x i ) ζ µ u a  φ u ( x i )  + Λ u ( x i ) h  φ u ( x i )  ψ u ( x i ) i du + Z t 0 X i ∈ I w i Λ u ( x i ) ( υ µ u ) ⊺ diag  b ( φ u ( x i ))  dW ( u ) . Pr o of. F or eac h i , Λ t ( x i ) has ﬁnite v ariation with d Λ t ( x i ) = Λ t ( x i ) h ( φ t ( x i )) dt . Hence, Λ t ( x i ) ψ t ( x i ) is a contin uous semimartingale by integration by parts, with drift and diﬀusion as stated. The summability conditions justify exchanging the (coun table) sum with the Lebesgue and sto chastic integrals, which yield the aggregate decomp osition. □ SHOR T-HORIZON DUESENBERR Y EQUILIBRIUM 37 A.4. Con tin uous p opulations. Assumption 4 (Contin uous p opulations) . Assume a diﬀe omorphic Br ownian ﬂow that satisﬁes Assumption 1. L et h : D → R b e a c ontinuous C 2 , 0+ function and deﬁne the p opulation weight as in Deﬁnition 2. L et µ b e a ﬁnite Bor el me asur e. Assume that for e ach x ∈ D , ψ µ t ( x ) = ψ 0 ( x ) + Z t 0 ζ µ u a  φ u ( x )  du + Z t 0 ( υ µ u ) ⊺ diag  b ( φ u ( x ))  dW ( u ) is a c ontinuous semimartingale in t , for c ontinuous r e al-value d functions a and R n -value d b , c onsistent with r esp e ct to the p opulation structur e µ , and c ontinu- ous semimartingale pr o c esses r e al value d ζ µ u , and R n value d ( υ µ ) u (which is typ e indep endent). Assume that for every T > 0 , almost everywher e Z D Z T 0    Λ u ( x ) ζ µ u a  φ u ( x )    +   Λ u ( x ) h  φ u ( x )  ψ u ( x )    du dµ ( x ) < ∞ , and Z D  Z T 0 Λ u ( x ) 2   ( υ µ u ) ⊺ diag  b ( φ u ( x ))    2 du  1 / 2 dµ ( x ) < ∞ . Prop osition 3 (Aggregate semimartingale: contin uous p opulation) . Under As- sumption 4, the p opulation-weighte d aggr e gate ψ µ t : = Z D ψ t ( x ) Λ t ( x ) dµ ( x ) is a c ontinuous semimartingale for 0 ≤ t < ∞ . Mor e over, ψ µ t = ψ µ 0 + Z t 0 Z D h Λ u ( x ) ζ µ u a  φ u ( x )  + Λ u ( x ) h  φ u ( x )  ψ u ( x ) i dµ ( x ) du + Z t 0 Z D Λ u ( x ) ( υ µ u ) ⊺ diag  b ( φ u ( x ))  dµ ( x ) dW ( u ) . Pr o of. Apply the pro duct rule to x 7→ ψ t ( x )Λ t ( x ) to identify its drift and diﬀusion from Assumption 4. Then in tegrate in x and justify exc hanging the x -in tegral with the Lebesgue/sto c hastic in tegrals b y the same dominated-con vergence and sto c hastic- F ubini argument used in Prop osition 1, proving the stated decomp osition. □ Corollary 4 (Aggregate semimartingale and restart prop erty) . Assume Assump- tion 3 or Assumption 4, and adopt the p opulation structur e of L emma 1. Fix t ≥ 0 and set µ t : = µ 0 ,t . (R estarte d aggr e gate pr o c ess). 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Londo ˜ no, Dep ar t amento de Ma tem ´ aticas y Est ad ´ ıstica, F acul t ad de Ciencias Exact as y Na turales, Universidad Nacional de Colombia, Sede Manizales, Manizales 170003, Colombia Email address : jaime.a.londono@gmail.com

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