Recursive allocations and wealth distribution with multiple goods: Existence, survivorship, and dynamics
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Colacito, R.; Croce, Mariano M.; Liu, Zhao Article Recursive allocations and wealth distribution with multiple goods: Existence, survivorship, and dynamics Quantitative Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Colacito, R.; Croce, Mariano M.; Liu, Zhao (2019) : Recursive allocations and wealth distribution with multiple goods: Existence, survivorship, and dynamics, Quantitative Economics, ISSN 1759-7331, The Econometric Society, New Haven, CT, Vol. 10, Iss. 1, pp. 311-351, https://doi.org/10.3982/QE457 This Version is available at: https://hdl.handle.net/10419/217144 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/
Quantitative Economics 10 (2019), 311–351 1759-7331/20190311 Recursive allocations and wealth distribution with multiple goods: Existence, survivorship, and dynamics R. Colacito Kenan-Flagler Business School, University of North Carolina at Chapel Hill and NBER M. M. Croce Finance Department, Bocconi University, CEPR, and NBER Zhao Liu Department of Economics, Duke University We characterize the equilibrium of a complete markets economy with multiple agents featuring a preference for the timing of the resolution of uncertainty. Utilities are defined over an aggregate of two goods. We provide conditions under which the solution of the planner’s problem exists, and it features a nondegenerate invariant distribution of Pareto weights. We also show that perturbation methods replicate the salient features of our recursive risk-sharing scheme, provided that higher-order terms are included. Keywords. Recursive preferences, multiple agents, equilibrium. JEL classification. C62, F37. 1. Introduction In the context of single-agent economies, recursive preferences have become increasingly relevant for the analysis of issues at the forefront of the macro-finance agenda (see, among others, Hansen and Sargent (1995), Tallarini (2000), Bansal and Yaron (2004), and Backus, Routledge, and Zin (2005)). In models populated by multiple agents, in contrast, the adoption of recursive preferences is less common, as it produces a key challenge in the characterization of the risk-sharing dynamics. With recursive preferences, in fact, optimal allocations are a function not only of aggregate endowment, but also of a possibly time-varying distribution of wealth. As documented by Anderson (2005), in a one-good economy in which agents have risk-sensitive preferences there is typically a tension between ensuring that a nondegenerate distribution of wealth exists and having interesting heterogeneity across agents. The same R. Colacito: [email protected] M. M. Croce: [email protected] Zhao Liu: [email protected] We thank seminar participants at the 2009 Meeting of the Society for Economic Dynamics in Istanbul and at the 2010 Meeting of American Economic Association in Atlanta. We are grateful to Lars Hansen, Thomas Philippon, Tom Sargent, and the participants at the 2010 PhD mini-course on Asset Pricing and Risk Sharing with Recursive Preferences at NYU, which was partly based on this paper. All errors remain our own. ©2019 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at http://qeconomics.org.https://doi.org/10.3982/QE457
312 Colacito, Croce, and Liu Quantitative Economics 10 (2019) paper documents that this tension can be relaxed if multiple preference parameters are changed simultaneously and agents have power-reward functions with risk aversion between zero and one. In this paper, we overcome these challenges by focusing on an economy with multiple goods. We show that rich dynamics of Pareto optimal allocations are obtained even in the case in which all agents share the same risk-sensitivity parameter and have logarithmic period reward functions, provided that they feature a different degree of preference for one of the two goods. Furthermore, we provide conditions under which a nondegenerate ergodic distribution of Pareto weights exists. This is the case in which every agent in the economy has a strictly positive wealth in the long run. An agent with recursive preferences is willing to trade off expected utility for higher conditional moments of future utility. In a world with a Cobb–Douglas aggregate over multiple goods, the intensity of this trade-off is stronger for agents that consume a large share of aggregate resources, that is, agents with high Pareto weights. For example, as shown in a simple two-period model, when agent 1has a high initial share of resources, she will have a strong incentive to buy insurance from agent 2to mitigate future utility uncertainty. In equilibrium, under a preference for early resolution of uncertainty, agent 1accepts a reduction in her expected average share of resources (i.e., her Pareto weight is expected to decline) in exchange for a reduction of future utility variance. This tradeoff between expected utility and conditional volatility of future utility results in a welldefined invariant distribution of Pareto weights. Several authors have documented the theoretical properties of one-good versions of the economy analyzed in this paper (Lucas and Stokey (1984), Ma (1993), and Kan (1995)). In particular, Anderson (2005) shows that in an economy with heterogenous agents and recursive preferences it is difficult to ensure the existence of a nondegenerate ergodic distribution of wealth, unless very extreme forms of heterogeneity are considered (see, e.g., Backus, Routledge, and Zin (2008)). Our focus on the case of a consumption aggregate of multiple goods resolves these problems, and it is important in many economic applications. In a closed economy, we may think of agents featuring different propensities across commodities produced by different firms or sectors. In an open economy, it is common to assume that consumers located in different countries are biased toward the consumption of the domestically produced good (see, e.g., Tretvoll (2018)). The economy analyzed in this paper provides the foundations for the international macro-finance model in Colacito and Croce (2013). In related work, Backus, Coleman, Ferriere, and Lyon (2016) showed that the endogenous variation in Pareto weights in the type of economies that we consider can be interpreted as wedges from the perspective of a frictionless model with additive preferences. Colacito and Croce (2012) applied the results in this manuscript to the heterogeneous-beliefs literature (among others, see Kubler and Schmedders (2012)and Tsyrennikov (2012)).1They show that consumption home bias is isomorphic to endogenous disagreement about the fundamentals of the economy. Under the conditions explored in our paper, the ergodic distribution of wealth is nondegenerate, despite the 1Specifically, Colacito and Croce (2012) interpreted the preferences used in this manuscript as describing a concern for model misspecification, according to the definition of Hansen and Sargent (2008). This results
Quantitative Economics 10 (2019) Recursive allocations and wealth distribution 313 existence of endogenous heterogenous beliefs. For a detailed study of both, the survivorship and risk sharing in economies populated by recursive agents with exogenous heterogeneous beliefs see Boroviˇ cka (2018). From a computational point of view, the characterization of the risk-sharing arrangement with recursive preferences poses additional challenges, as the state space includes the relative wealth of the agents, which in turn, depends on the continuation utilities. We compare a global method that uses value-function iterations and a perturbation-based approach and document that a first-order Taylor expansion about the stochastic steady state of the economy is not appropriate for capturing the dynamics of the economy. This approximation severely deteriorates in regions distant from the steady state, and it produces a counterfactual limiting wealth distribution in which either agent may find herself with zero wealth with probability one. Higher-order approximations are necessary not only to provide a better period-by-period characterization of the dynamics of the model, but also to capture the stationarity of the model. These findings are consistent with the analysis of Anderson, Hansen, and Sargent (2012)and Pohl, Schmedders, and Wilms (2017). Additionally, we show that our setting produces endogenous time variation in higher-order conditional moments of consumption, and hence it offers general equilibrium foundations for the analyses of Bansal, Kiku, Shaliastovich, and Yaron (2014), Kuehn and Boguth (2013), Colacito, Ghysels, Meng, and Siwarasit (2016), and Segal, Shaliastovich, and Yaron (2015). We also study important extensions of our benchmark model by considering the case in which agents have intertemporal elasticities of substitution different from 1and endowment shocks that are persistent. This means that our analysis can be informative for the growing body of the literature that has explored the macro-finance implications of Epstein and Zin (1989) preferences (see, e.g., Bansal et al. (2014)). Furthermore, we show that the introduction of a moderate degree of heterogeneity in the calibration of the two countries may still result in a well-defined ergodic distribution of wealth in equilibrium. This is relevant for the application of our study to economies in which, for example, investors in different countries face a heterogeneous degree of fundamental risk in their endowments or productivities. Baker and Routledge (2017) studied an economy similar to the one analyzed in this paper. Like us, they consider the risk-sharing arrangement between two agents with recursive preferences defined over a Cobb–Douglas aggregate of two goods: oil and a general consumption good. Since the main focus of their paper is matching the price of oil and related futures contracts, they rely on asymmetric calibrations of the two agents. This choice typically results in the survivorship of only one of the two agents in the economy. In this respect, the results that we provide in Section 5, in which we relax the symmetry of the calibration, are informative for the general class of model that they consider. in agent-specific distorted conditional distributions of the endowment processes. Since each probability depends on the utility of a specific agent, when preferences feature heterogenous bias across goods, agents disagree on the transition probabilities across states of the world.
314 Colacito, Croce, and Liu Quantitative Economics 10 (2019) Our paper is organized as follows. In Section 2,weprovidethesetupofourbenchmark economy, featuring unit intertemporal elasticity of substitution and i.i.d. shocks. In Section 3, we discuss the main intuitions of our framework in the context of a simple two periods model and provide the set of conditions under which a nondegenerate limiting distribution of Pareto weights exists in the infinite horizon setting. In Section 4,we compare a numerical solution of the model obtained via value function iteration with firstand higher-order approximations. In Section 5, we present the results of several generalized versions of our baseline setup. Section 6concludes the paper. 2. Setup of the economy In this section, we describe the assumptions that we use in the benchmark version of our model. For the purpose of simplifying the analytical proofs and the intuitions of the model, in our benchmark we assume that the intertemporal elasticity of substitution is equal to one and that the two countries share a symmetrical calibration. In Section 5,we use simulations to show that our results apply to more general settings. The following three assumptions about preferences, consumption, and endowments will be retained throughout the rest of the paper. Assumption 1 (Preferences). Let there be two agents,indexed by 1and 2,whose preferences are recursively defined as Ui(ciqi)=(1−δi)logci+δiθilog z πzexpqiz θi∀i∈{12}(1) where qi(z)gives the utility remaining from the next period on when next-period’s state is z.For each agent i,θi<0. This class of preferences can be interpreted in several ways. First, they correspond to the case of risk-sensitive preferences studied by, among others, Hansen and Sargent (1995), Tallarini (2000), and Anderson (2005). Second, they coincide with a logtransformation of Epstein and Zin (1989) preferences in the case in which the intertemporal elasticity of substitution is equal to one. In this case, the risk-sensitive parameter, θ, is related to risk aversion, γ, by imposing θ=1 1−γ In this paper, we focus on a discrete time setting as opposed to the continuous time approach of Epstein (1987), Duffie, Geoffard, and Skiadas (1994), Geoffard (1996), and Dumas, Uppal, and Wang (2000). Since these preferences depart from the expected utility case, higher moments of continuation utilities matter for the determination of optimal risk sharing. As as example, if continuation utilities qi(z)are normally distributed, the functional form in (1) can be written as Ui(ciqi)=(1−δi)logci+δiEi(qi)+δi 2θi Vi(qi) ∀i∈{12}(2)
Quantitative Economics 10 (2019) Recursive allocations and wealth distribution 315 where Ei(qi)= z πzqiz Vi(qi)= z πzqiz− z πzqiz2 are the conditional mean and variance of the continuation utility, respectively. Although we will work with the general specification in (1), equation (2) is instructive, since it intuitively shows that when the parameters θi’s are less than zero, the risksharing scheme must account for an efficient endogenous trade-off between utility level and utility variance. As the dynamics of second-order moments are crucial for characterizing the equilibrium of the model, in Section 4we also assess the accuracy of several approximations based on how well they can account for the dynamics of volatilities. Assumption 2 (Consumption Bundles). Let consumption cibe an aggregate of two goods,xiand yi.Specifically,let ci=(xi)λi(yi)1−λi(3) be the consumption bundle,with λ1>1/2and λ2<1/2so that the two agents have a bias for different goods. This assumption generalizes the one-good framework studied by Anderson (2005), which obtains as the special case in which λi=1/2,∀i, that is, the case in which there is no preference bias across goods, and hence we are effectively in a one-good economy. The next assumption pertains to the endowment process and is common to Anderson (2005). Assumption 3(Endowments). The endowment of the two goods follows a first-order time-homogenous Markov process (z0z1) which takes values in a finite set N= {1n}.The aggregate supply of the two goods at time tis such that 0<X t=X(zt)<∞, and 0<Y t=Y(zt)<∞. Finally, we need to make sure that the preference parameters are chosen so that the utility recursion converges: Assumption 4 (Contraction). The parameters {λiγiδi}are such that the right-hand side of equation (1)has a modulus smaller than one,∀i={12}. Recursive planner’s problem Let logW∗ i(z ci{qiz}z)be the right-hand side of equation (1). Given the conditions specified by Ma (1993)andMa (1996), the social planner’s value function, denoted as
316 Colacito, Croce, and Liu Quantitative Economics 10 (2019) Qp(z μ1):N×[01]→R, satisfies the following functional equation proposed by Lucas and Stokey (1984)andKan (1995): Qp(z μ1)=max {xiyiqiz}i∈{12}z∈N 2 i=1 μilogW∗ izci{qiz}z(4) subject to μ2=1−μ1 0≤x1≤X(z) 0≤x2≤X(z)−x1 0≤y1≤Y(z) 0≤y2≤Y(z)−y1(5) ci=(xi)λi(yi)1−λi∀i={12} 0≤min μ1(z)∈[01]Qpzμ1z−μ1zq1z−1−μ1zq2z∀z∈N Differentiability and first-order conditions Let the ratio of the Pareto weights be defined as S=μ1 μ2 =μ1 1−μ1 Let Ui(z S),i=12, denote the utility function of agent ievaluated at the optimum when the exogenous state is z,andS∈(0∞). On a consumption path that is bounded away from zero, Ui(zS) is differentiable (see Kan (1995)andAnderson (2005)) and dUi dμi >0μ 1∈(01)μ2=1−μ1 On a consumption path that is bounded away from zero for both agents, Qp(sμ1)is also differentiable with respect to μ1∈(01). The optimality condition in equation (5) implies that dQp dμ1 (z μ1)=U1(z S) −U2(zS) μ1∈(01) (6) d2Qp dμ2 1 (z μ1)=dU1 dμ1 (z S) +dU2 dμ2 (z S) > 0(7) Therefore, Qp(sμ1)is strictly convex with respect to μ1,asinLucas and Stokey (1984). This is relevant because it implies that the unique optimal policy of the planner can be characterized using first-order conditions. According to the first-order conditions, for a given S, the optimal allocation of goods satisfies the following system of equations common to all static Pareto problems with two goods and two agents: (1−δ1)∂logc1 ∂x1 ·S=(1−δ2)∂logc2 ∂x2
Quantitative Economics 10 (2019) Recursive allocations and wealth distribution 317 (1−δ1)∂logc1 ∂y1 ·S=(1−δ2)∂logc2 ∂y2 (8) X=x1+x2Y=y1+y2 The optimal dynamic adjustment of the ratio of the pseudo-Pareto weights is then given by S=S·MzS(9) where MzS≡δ1expU1zS/θ1 z πzexpU1zS/θ1· z πzexpU2zS/θ2 δ2expU2zS/θ2 Equation (9) determines the dynamics of the ratio of the Pareto weights and implicitly generates a continuous function that we denote by fS(z·):[0+∞)→[0+∞): S=fSzS(10) Characterizing the planner’s problem through first-order conditions is useful because it allows us to represent the planner’s problem in (4) as a simple system of firstorder stochastic difference equations, namely (1), (3), (8), and (9). In the next section, we use perturbation methods to solve our dynamic system of equations. Relativepriceofthetwogoods The relative price of the two goods, p, is the equilibrium marginal rate of substitution across goods p=(1−λ1) λ1 ·x1 y1 Given the optimal allocations of x1and y1, we can write the relative price as p=˜ p·X Y where ˜ p≡(1−λ1)/λ1·1+S·(1−λ1) (1−λ2)1+S·λ1 λ2 When the supply of good Xrelative to good Yis high, the price of good Yincreases for two reasons. First, the last term (X/Y) directly affects the relative price. This channel wouldbeatworkevenforλ1=1/2, in which case ˜ p=1,andp=X/Y. Second, since λ1>λ 2it follows that ∂˜ p ∂S =−1−λ1 λ1 ·λ2 1−λ2 ·(λ1−λ2) (λ2+λ1·S)2<0
318 Colacito, Croce, and Liu Quantitative Economics 10 (2019) This means that the price will further increase as long as the ratio of Pareto weights (S) declines when X/Y is large (we prove this statement formally in Proposition 1). Equivalently, the price of good Yis large whenever its supply is low. This effect is magnified in the context of our model since λ1>1/2,λ2<1/2, and the ratio of Pareto weights can move away from a symmetric wealth distribution. This enhanced price adjustment allows agent 2to purchase a larger share of resources whenever the supply of its most preferred good is low. Share of world consumption (SWC) We note that under our Cobb–Douglas aggregator across goods, the relative share of world consumption of agent 1,SW C, evolves as follows: SW C =x1+py1 X+pY =S 1+S=μ1(11) According to equations (8)and(11), the Pareto weight of agent 1has a simple economic interpretation, namely, the relative size of consumption allocated to agent 1. Symmetry So far, we have not imposed any specific assumptions on the conditional probability of the Markov chain governing the supply of the two goods, nor have we imposed any special restrictions on the preference parameters of our agents. In order to have a wellspecified problem, all we need is that Assumptions 1–4hold. In what follows, however, we list further restrictions that are necessary to analytically characterize the main properties of the optimal risk-sharing policy of the planner. These conditions impose symmetry and are sufficient, but in many cases not necessary, for the existence of a stationary distribution. In the next section, we relax many of these assumptions. Assumption 5 (Symmetrical Preferences). Let the preference parameters δiand θibe identical ∀i∈{12}.Let the consumption-bundle’s parameters be symmetrical across agents,that is,λ1=1−λ2>1/2. Assumption 6 (Balanced Endowment Space). Let the support of the endowment of good Xbe given by the vector H=[h1h2hN].Let the support of the endowment of good Ybe Has well.The endowments of the two goods take values in the finite set N,given by all possible pairwise permutations of H.We refer to Nas a balanced endowment space. Definition 1 (Symmetric States). Let the states ziz−i∈Nbe such that zi={Xi= X(i)Yi=Y(i)}and z−i={X−i=Y(i)Y−i=X(i)}.Thenziand z−iare symmetric states. Finally, just to simplify our proof, we focus on a setting with i.i.d. shocks and on symmetric probability distributions.
Quantitative Economics 10 (2019) Recursive allocations and wealth distribution 325 Figure 3. Comparison of the actual dynamics of Pareto weights and that obtained through a first-order Taylor expansion. For the same sequence of shocks, the black line shows the actual path of the agent 1Pareto weight, μ1, while the red line shows the path obtained using a first-order approximation about the unconditional mean of μ1,thatis,μ1=1/2. The curse of the linear approximation A first-order Taylor approximation about the unconditional mean of the ratio of Pareto weights fails to reproduce at least two crucial aspects of the economy. First, it provides a highly inaccurate description of the period-by-period dynamics of the model. Second, and most importantly, it does not capture the mean-reverting property of the model. This results in the possibility that one of the two agents eventually dies and is assigned a steady-state Pareto weight of zero. In order to show these two facts, we proceed as follows. First, we solve the model numerically by value-function iteration (see Appendix Cin the Supplemental Material, available in a supplementary file on the journal website, http://qeconomics.org/supp/ 457/code_and_data.zip) and obtain what we denote as the “actual” solution. Second, we compare the “actual” dynamics of Pareto weights with the dynamics computed using a first-order Taylor expansion about μ1=μ2=05. Figure 3reports this comparison for a simulation of 400,000 periods. For the first part of the simulation, the Pareto weights are in the relatively small neighborhood of 05. In this region, the first-order Taylor expansion does a good job of approximating the actual dynamics of the economy. The approximation, however, starts deteriorating significantly as the economy departs from μ1=05.Onthishistory, according to the first-order Taylor expansion, the Pareto weight of agent 1should level off at zero, even though this is in sharp contrast to the actual dynamics of the model and the survivorship results explained in the previous sections. As a consequence, the long-run implications of the first-order Taylor expansion are unreliable. In this clear-cut example, what may at first look like a small error results in an irreversible misrepresentation of the actual dynamics of the economy and its long-run moments. In this economy, higher-order approximations are needed not only to provide a more accurate description of the period-by-period dynamics, but also to preserve the existence of a well-defined ergodic distribution.
326 Colacito, Croce, and Liu Quantitative Economics 10 (2019) Figure 4. Comparison of first and second conditional moments across several solution methods. We denote the log of the ratio of the Pareto weights by st≡log(St)and its growth by st+1=st+1−st, respectively. The left panel reports the conditional mean E[st+1|st]as a function of st. The right panel reports the conditional variance of st+1with respect to st.Ineach panel, the curve label “Actual” refers to the solution obtained through value-function iterations. The other three curves are based on Taylor approximations of higher order. In the left panel, the probability density function (PDF) is computed using the actual policy. The case for higher-order approximations The linear approximation does not accurately describe the dynamics of the Pareto weights because it impels a first-order integrated process. This is clearly depicted in the left panel of Figure 4, in which we compare the expected growth of the ratio of Pareto weights as a function of the current ratio across different solution methods. The flat line for the first-order approximation suggests that the conditional expected change of the ratio of Pareto weights is identically zero, implying the lack of any kind of mean reversion. The second-order approximation does capture some of the mean reversion, although not enough to be comparable to the actual solution of the model. Furthermore, by looking at the right panel of Figure 4we notice that the second-order approximation does not feature any time variation in the conditional variance of the ratio of Pareto weights. This is in stark contrast to the actual dynamics of the second moments of the actual solution. Equivalently, the second-order approximation completely misses the time variationinthelasttermofequation(2), that is, the key determinant of the risk-sharing motive of our agents. In order to capture time-varying volatilities of both consumption and continuation utilities, we implement a third-order approximation. In contrast to the second-order perturbation, the third-order approximation provides an extremely accurate representation of the dynamics of both the first and second conditional moments. This is certainly the case in a 99% confidence interval of the actual long-run distribution of the ratio of Pareto weights. As expected, the quality of the approximation deteriorates toward the tails of the distribution.
Quantitative Economics 10 (2019) Recursive allocations and wealth distribution 327 To summarize, this class of models produces rich dynamics for both the first and second conditional moments of the Pareto weights and, therefore, consumption shares across agents. To appropriately capture these dynamics, an approximation of at least the third order is required. In the next section, we use third-order approximations to study more general settings. 5. More general environments In this section, we generalize our setting in two respects. First of all, we consider preferences defined as in Epstein and Zin (1989), Uit =(1−δ) ·(Cit )1−1/ψ +δEt(Uit+1)1−γ1−1/ψ 1−γ1 1−1/ψ ∀i∈{12}(14) where ψdenotes the IES and γrepresents RRA. Second, we consider the following endowment process that allows persistence: logXt=μ+ρlogXt−1−τ[logXt−1−logYt−1]+εX t (15) logYt=μ+ρlogYt−1+τ[logXt−1−logYt−1]+εY t εX t εY t∼iidN0 0σX2ρXY σXσY ρXY σXσYσY2(16) where ρ∈[01]and τ∈(01)determine the extent of cointegration when ρ=1. Cointegration is required to have a well-defined ergodic distribution of the relative supply of the two goods, but it plays a minor quantitative role in our analysis as we set it to a very small number. Solving the planner’s problem with global methods and multiple exogenous state variables goes beyond the scope of this manuscript. The reason is that properly capturing the mean reversion of the pseudo-Pareto weights requires a very thin grid, and it exposes us to the curse of dimensionality even with one extra state. Hence in this section we explore the generality of our results through simulations based on a third-order perturbation of our dynamic model, which we detail in Appendix Din the Supplemental Material (available in a supplementary file on the journal website, http://qeconomics. org/supp/457/code_and_data.zip). Reference calibration Our reference calibration features μ=2%,ρ=090,σX= σY=187%,ρXY =035,τ=50E−04,γ=5,ψ=1,δ=096,andλ1=λ2=097.Theparameters for the endowment processes are set in the spirit of Colacito and Croce (2013). In what follows, we first consider different endowment processes and different levels of the IES and RRA while preserving symmetry across agents and goods. We then explore the implications for a small degree of heterogeneity in preference for the two goods (λi) and in fundamental volatility across goods (σXand σY). 5.1 Symmetric environments Theroleofpersistence We vary the persistence of our endowment shocks from zero to one. When ρ=0,wehavei.i.d. level shocks, as in the previous section. When ρ=1, level
328 Colacito, Croce, and Liu Quantitative Economics 10 (2019) Figure 5. The role of persistence and preferences. Our model features Epstein and Zin (1989) preferences as specified in equation (14) and the endowments reported in equation (15). Our reference calibration is detailed in Section 5. We vary one parameter at a time for both countries, leaving the other parameters unchanged. shocks are permanent. We depict key features of the distribution of the log-ratio of the Pareto weights, st, in Figure 5and simulated moments in Table 1. We make several observations. First, as we increase ρ, the endowment shocks become more long-lasting and volatile. As a result, the endogenous process stbecomes more volatile, as documented by its fatter tails (rightmost plot of Figure 5, panel A) and the higher conditional volatility of st(center plot of Figure 5).
Quantitative Economics 10 (2019) Recursive allocations and wealth distribution 329 Table 1. The role of persistence and preferences. Moments of SWC Percentiles of Approx. Errors (%) Parameter Value Cumul. ERR (%) Mean Std. Dev. 50 75 90 95 The role of persistence (ρ) 005018 20E−14 40E−14 70E−14 90E−14 80E−01 0905019 80E−15 20E−14 20E−14 40E−14 70E−04 1050220E−14 40E−14 60E−14 70E−14 50E−05 The role of RRA (γ) 305021 20E−14 30E−14 60E−14 90E−14 10E−02 5050280E−15 20E−14 20E−14 40E−14 70E−04 705016 30E−14 60E−14 80E−14 90E−14 50E−04 The role of IES (ψ) 2/305019 40E−11 40E−11 40E−14 40E−14 10E−01 1050220E−14 40E−14 50E−14 70E−14 50E−05 1505021 40E−14 90E−14 20E−13 20E−13 50E−05 Note:OurmodelfeaturesEpstein and Zin (1989) preferences as specified in equation (14) and the endowments reported in equation (15). Our reference calibration is detailed in Section 5. We vary one parameter at a time, leaving the others unchanged. Both the approximation errors and the cumulative approximation errors (Cumul. ERR) are defined in Appendix Dand are multiplied by 100. All numbers are based on repetitions of long-sample simulations (at least 2million periods) with different starting points for log ratio of the Pareto weights (st). The unconditional volatility of st, however, increases just slightly, as documented in Table 1. This is because as shocks become more long lasting the precautionary motives of these agents become more pronounced and more sensitive to size. As the exogenous endowment persistence (ρ) increases, the endogenous persistence of stdeclines, as captured by the more negative slope of Et[st+1]with respect to st(leftmost plot of Figure 5, panel A). Second, given the lower persistence of st, our cumulative error measure declines to very small numbers with higher values of ρ. This result is reassuring because in many realistic applications the endowment shocks are calibrated to be very persistent. The role of preferences When we vary the subjective discount factor, we do not find significant changes in the dynamics of the log-ratio of the Pareto weights. For this reason, we focus only on the role of risk aversion and IES. Increasing risk aversion amplifies the sensitivity of continuation utility to shocks, and hence it makes the redistribution channel stronger. This intuition is confirmed in panel (b) of Figure 5, where we show that the conditional variation of st+1increases with higher values of γand, at the same time, the mean reversion of the Pareto weights speeds up. Since the endogenous change in mean reversion dominates quantitatively, as γincreases more mass is concentrated in the center of the probability distribution function of st, implying that the unconditional volatility of this process declines. Together, the faster speed of mean reversion and the lower unconditional volatility of stimply lower levels of approximation errors. We conclude this analysis by examining the case in which we vary the IES. The effect of this parameter on the conditional volatility of the ratio of Pareto weights is almost
330 Colacito, Croce, and Liu Quantitative Economics 10 (2019) negligible (center plot of panel (c), Figure 5). The impact on the endogenous persistence of stis a bit more pronounced, but still moderate compared to the case in which we change risk aversion. Qualitatively, agents with a higher IES are more willing to accept fluctuations of consumption over time and hence are more willing to accept very long-lasting reallocations, that is, slower mean reversion in st(leftmost plot in panel (c), Figure 5). Most importantly, we note that in the long-run risk literature the IES is set to values larger than or equal to one. For these values, the approximation errors are small, meaning that when the curvature of the utility function with respect to intertemporal aggregation is moderate (ψ≥1), the quality of our approximation is good. 5.2 Asymmetric environments Asymmetry preferences for the two goods We lower the degree of preference for good Y of agent 2,λ2,from097 to 090, thus increasing the ability of agent 2to smooth fluctuations in her consumption bundle by trading the two goods. All other parameters are unchanged. We depict key features of the distribution of the log-ratio of the Pareto weights, st, in Figure 6and simulated moments in Table 2. Since agent 2faces less consumption uncertainty than agent 1, her demand of insurance is moderate compared to that of agent 1. As a result, the average share of resources allocated to agent 2increases and the curves depicted in the top-left panel of Figure 6 shift to the left. In the Appendix in the Supplementary Material, we show that this intuition is also present in the simple two-period model (see Figure AF-1). We note that the distribution of the log-ratio of the Pareto weights does not shift to the left in a parallel way. As documented in the top portion of Table 2, under the optimal risk-sharing scheme, agent 1accepts a lower average level of resources in exchange for both a reduction in future utility uncertainty and positive skewness of its share of world consumption. That is, agent 1 benefits from a sizeable positive redistribution of resources along histories with a severe downside of the relative supply of good X. Rabitsch, Stepanchuk, and Tsyrennikov (2015) point out that a global approximation is required when countries are subject to asymmetric constraints, such as a borrowing limit, and when their wealth distribution is nonstationary. Since we have a frictionless model with complete markets and a well-defined ergodic distribution of wealth, a perturbation approach provides a good approximation of the equilibrium. Consistent with the findings in Rabitsch, Stepanchuk, and Tsyrennikov (2015), our cumulative errors increase as we make the two agents more asymmetric, but our errors remain as low as 22%. CRRA and heterogeneous degree of preference for the two goods It is useful to study this asymmetric scenario under time-additive CRRA preferences. We choose the intermediate case λ1=097,λ2=095,andsetψ−1=γ=5.3We fix the initial ratio of the Pareto weights to a value that delivers an average SWC of 047, as in the case with recursive preferences. Panel (b) of Figure 6and Table 2confirm that under recursive preferences, the 3Note that since γ>1, the ratio of Pareto weights is no longer constant over time despite the adoption of time additivity of preferences (Cole and Obstfeld (1991)). The implied variation in st, however, is very limited as highlighted in panel (b) of Figure 6.
Quantitative Economics 10 (2019) Recursive allocations and wealth distribution 331 Figure 6. Asymmetric calibrations. Our model features Epstein and Zin (1989) preferences as specified in equation (14) and the endowments reported in equation (15). Our reference calibration is detailed in Section 5. We vary one parameter at a time, leaving the others unchanged. reallocation channel is very pronounced and long-lasting, and it prescribes a significant amount of positive skewness for agent 1, as she is facing more consumption risk because of a higher degree of preference for good X. Heterogeneous volatility In many applications, the properties of the goods traded are asymmetric. In international finance, for example, different countries may be subject to productivity shocks with different volatilities. In the presence of the same degree of preference for the two goods, the agent which prefers the more volatile good faces more consumption, and hence future utility uncertainty. As suggested by our two-period model
332 Colacito, Croce, and Liu Quantitative Economics 10 (2019) Table 2. Asymmetric cases. Moments of SWC Parameter Value Cumul ERR (%) Mean Std Skew Agent 2Bias (λ2)−EZ Case 0.97 05019 0 70E−04 095 047 018 022 26E−03 090 046 017 033 22E+00 Agent 2Bias (λ2)−CRRA Case (γ=ψ−1=5) 095 047 002 003 50E−06 Y-Good Volatility (σY) 100 ·σX05019 0 70E−04 105 ·σX058 019 −027 90E−04 110 ·σX056 022 −027 13E−01 Note:OurmodelfeaturesEpstein and Zin (1989) preferences as specified in equation (14) and the endowments reported in equation (15). Our reference calibration is detailed in Section 5. We vary one parameter at the time, leaving the others unchanged. The cumulative approximation errors (Cumul ERR) are defined in Appendix Dand are multiplied by 100.Allnumbers are based on repetitions of long sample simulations (at least 2million periods) with different starting points for log ratio of the Pareto weights (st). (see Figure AF-1), this agent should be willing to accept a low average SWC in exchange for insurance. Panel (c) of Figure 6and Table 2confirm this finding as we increase σY. In this case, the average share of resources of agent 1increases. The volatility of the SWC increases as well, as a direct result of the higher standard deviation of good Y. Agent 1is willing to insure agent 2against downside risk, and agent 1accepts negative skewness in her own share of resources. Across all cases, the associated cumulative approximation errors are smaller than 014%. Heterogenous RRA A well-known result with multiple agents with risk-sensitive preferences in a one-good economy with growth is that only the agent with the lowest risk aversion remains wealthy in the long run, ceteris paribus (see Anderson (2005)). We confirm this finding in our setting by depicting in Figure 7the expected growth rate of the log-ratio of Pareto weights for the case λ1=λ2and γ2>γ 1. Since the expected growth is positive for all values of st, all resources are allocated to agent 1in the limit. Our risk-sharing mechanism, however, suggests that as we make the degree of preference for the two goods more asymmetric, size matters progressively more for the intensity of the reallocation channel. Hence as the high-risk-aversion agent receives a smaller share of world resources, her willingness to buy further insurance should decline, which in turn results in survivorship. Our simulations confirm this intuition and suggest that there exist regions of the parameter space in which survivorship is also possible with asymmetric risk aversion, provided that multiple parameters are simultaneously adjusted, in the spirit of Anderson (2005). 6. Concluding remarks We have characterized the solution of a planner’s problem with multiple agents, multiple goods, and recursive preferences. The introduction of multiple goods substantially
Quantitative Economics 10 (2019) Recursive allocations and wealth distribution 333 Figure 7. Asymmetric risk aversion. Our model features Epstein and Zin (1989) preferences as specified in equation (14) and the endowments reported in equation (15). Our reference calibration is detailed in Section 5.Wesetγ2=7and γ1=5and vary simultaneously λ1and λ2. changes the dynamics of Pareto optimal allocations. Future research should extend our theoretical results to continuous time and continuous shocks. Asset pricing implications may be particularly appealing, due to the ability of this class of models to endogenously produce time-varying second moments without requiring market frictions. Since the main features of the risk-sharing scheme can be accurately captured through a thirdorder approximation, this class of models could be easily extended to an international business cycle setting as well. Appendix A: Two-period model Environment In this section, we present a simplified two-period version of our model in order to provide intuition on the reallocation motives induced by recursive preferences. Specifically, at time t=1agents receive news ξabout their time-1endowment of goods. At time t=0, that is, before the arrival of the shock, agents i∈{12}exchange a complete set of ξ-contingent securities to maximize their time-0utility, given their initial wealth (reflected in their time-0Pareto weights). Utility and technology In what follows, we take advantage of lognormality wherever possible. Up to a log linearization of the allocation shares, this modeling strategy enables us to obtain a simple closed-form solution. In this spirit, we start by assuming that agents have an IES equal
334 Colacito, Croce, and Liu Quantitative Economics 10 (2019) to 1, that is, their preferences can be expressed as follows: ui 0=⎧ ⎪ ⎨ ⎪ ⎩ δθilogE0expui 1 θi0>θ i>−∞ δE0ui 1θ i→−∞ (A.1) where ui 1=logCi 1 and C1 1=X1 1λ1Y1 1(1−λ1)C 2 1=X2 1(1−λ2)Y2 1λ2(A.2) The resource constraints are specified as follows: X1 1+X2 1=eξY 1 1+Y2 1=e−lξξ∼N0σ2(A.3) where the parameter ldetermines whether agent 2is more (l>1)orless(0<l<1) exposed to the shock ξthan agent 1. We assume that ξaffects both goods to preserve symmetry in our equations. In what follows, we show that the results are driven only by the relative supply of the goods (1+l)ξ. Pareto problem Under complete markets, the allocation can be recovered by solving the following Pareto problem: max {X1 1X2 1Y1 1Y2 1} μ0·u1 0+(1−μ0)·u2 0 subject to the constraints specified in (A.2)–(A.3). Let S0≡μ0/(1−μ0); after simplifying common coefficients, the optimality condition for the allocation of good X1is S1(ξ)∂log C1 1 ∂X1 =∂logC2 1 ∂X1 (A.4) with S1(ξ) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ S0θ i→−∞ S0· expu1 1 θ1 E0expu1 1 θ1expu2 1 θ2 E0expu2 1 θ20>θ i>−∞(A.5) Equation (A.4) establishes that the optimal allocation can be found as in a regular static problem, for a given value of S1.Equation(A.5) pins down S1and yields two important results. First, in the time-additive case, the share of resources is time invariant, that is, it is not affected by the actual realization of ξ. This is consistent with the special log case considered by Cole and Obstfeld (1991).
Quantitative Economics 10 (2019) Recursive allocations and wealth distribution 341 Figure AF-2. Difference of continuation utilities in symmetric states. This figure depicts the difference of the continuation utilities in the states of high supply of good X—low supply of good Y,U iHL, and low supply of good X—high supply of good Y,U iLH ,fori=12.Theredline refers to agent 1,theblacklinetoagent2. Corollary 1. For any two symmetric states ziand z−i,such that X(zi)>Y(z i),there exists a finite * Si 2>1such that U2(zifS(zi* Si 2)) =U2(z−ifS(z−i* Si 2)),where fS(··)is defined in (10), and U2(zifS(ziSi 2)) > U2(z−ifS(z−iSi 2)),∀Si 2>* Si 2. We illustrate the content of the preceding proposition and corollary in Figure AF-2, which depicts the differences of the continuation utilities in the two states of unequal supply of the two goods for the example discussed in Section 3.2 of the main text. When the Pareto weight attached to agent 1(agent 2)isapproaching1, the continuation utility for the state of abundant supply of good X(good Y) is higher than the continuation utility for the state of scarce supply of good X. However, as suggested by Proposition A5,thereexistsa*μ1<1/2(1−*μ1<1/2) past which the ranking of the continuation utilities is reversed. We are now ready to characterize the sign of the covariance term in (B.17). The following definition of conditional covariance in symmetric states is useful in establishing an upper bound on the last term of equation (B.17). Definition 2 (Covariance of Symmetric States). Let ziz−i∈Nbe symmetric states. The conditional covariance between two random variables hand gvalued on {zi,z−i}is covi−i[h g|S]= l={i−i} p(zl)h(zl)g(zl)− l={i−i} p(zl)h(zl) l={i−i} p(zl)g(l) where p(zl)≡π(zl)/(π(zi)+π(z−i)).
342 Colacito, Croce, and Liu Quantitative Economics 10 (2019) Proposition A6. Let cov be the sum of the conditional covariances between exp{U 2/θ} and Sacross all symmetric states: cov = i covi−iexpU 2/θS|S If cov <0,then covexpU 2/θS|S<0(B.23) Proof. By the definition of conditional covariance, covexpU 2/θS|S= n j=1 π(j)expU 2(j)/θS(j) −n j=1 π(j)expU 2(j)/θn j=1 π(j)S(j) =π i covi−iexpU 2/θS|S+coveqexpU 2/θS|S − i*π(i)expU 2/θ i* *π(i)S (B.24) where πis a nonnegative scalar smaller than one; ,icovi−i[exp{U 2/θ}S|S]is the conditional covariance across all symmetric states; coveq[exp{U 2/θ}S|S]is the conditional covariance across all states of equal supply; and {*π(i)}n i=1and {* *π(i)}n i=1are nonnegative sequences of scalars. We know from Proposition 1that the optimal choice of Pareto weights is identical across all states in which the supply of the two goods is the same. This implies that coveqexpU 2/θS|S=0 We can also conclude that the last term in equation (B.24) is nonnegative, being the product of the sums of nonnegative terms. Therefore, we can state that covexpU 2/θS|S≤π i covi−iexpU 2/θS|S which concludes the proof. The sign of cov is key. Suppose that the current ratio of Pareto weights is in the region between zero and mini{* Si 2}as defined in Corollary 1. For any two symmetric states, the utility of agent 2is going to be larger when the supply of good Yis relatively more abundant. Since the parameter θis negative, the ranking of U2/θ is reversed. Proposition 1 suggests that the ratio of Pareto weights is larger when the supply of good Yis larger.
Quantitative Economics 10 (2019) Recursive allocations and wealth distribution 343 Figure AF-3. Upper bound on the conditional covariance between exp{U 2/θ}and Sas a function of the current Pareto weight. Therefore, we can conclude that, in this region, cov, and hence the covariance term in (B.17), is negative. This argument is summarized in Figure AF-3. The next proposition plays a crucial role in our proof for the existence of a welldefined ergodic distribution of the Pareto weights. We focus on the properties of the left tail of the stochastic process S=μ 1 1−μ 1 =fS(zS) and base our proof on all the other propositions developed so far. Proposition A7. The stochastic process - S(S) := min(S mini{* Si 2}),where * Si 2is defined as in Corollary 1is a submartingale. Proof. Follows directly from all the propositions in this Appendix. We conclude this section with the proof of Proposition 2. Proof of Proposition 2. We know from Proposition A7 that the stochastic process - S(S) =min.Smin i* Si 2/ is a submartingale. Since a bounded submartingale cannot converge almost surely to its lower bound (see Sciubba (2005)), then - S(S) cannot converge almost surely to 0. We prove the rest of the proposition by contradiction. Denote by Ω=×∞ t=1ztthe set of sample paths of endowments, with representative element ω=(z1z2zt). Given the initial Pareto weight ratio S0=1and the transition dynamics S=fS(zS), the ratio of Pareto weights at time tcan be written as a function of the sample path: St=St(ω). Suppose Proposition 2does not hold. Then there must exist a set of ωsuch that Pr0ω++lim t→∞ St(ω) =01>0(B.25)
344 Colacito, Croce, and Liu Quantitative Economics 10 (2019) Denote by Q={ω|limt→∞ St(ω) =0}and consider the natural filtration on Q:Q1⊆Q2⊆ ···⊆Qt⊆···⊆Q. Recall that {zt}∞ t=1is i.i.d. As in Sciubba (2005), Qt−1does not restrict the original probability distribution of zt, and hence we have that Pr{zt=z(i)|Qt−1}= π(z(i)). As a consequence, all the results that we have proved so far in this section apply with respect to the conditioning set Qt−1. This means that Stis a bounded submartingale with respect to the conditioning information set Qt−1. Since a bounded submartingale cannot converge almost surely to its lower bound, then Stcannot converge almost surely to 0on Q. This contradicts the definition of Q.Hence,Stcannot converge to 0 with probability 1. Equivalently, the probability that Stconverges to 0is null. Note that μ1(- S) =- S 1+- S,thatis,μ1(- S) is a nonnegative, monotonically increasing function of - S. It follows that μ1cannot also converge to 0almost surely. By repeating all the proofs in the paper for μ2=1−μ1, it follows by symmetry that μ2cannot converge to 0almost surely. Equivalently, μ1cannot converge to 1almost surely either. B.3 Proof of Proposition 3 This section reports the proof of the proposition on the mean-reversion property of the Pareto weight. Proof of Proposition 3. We prove the three statements in Proposition 3in the same order in which they appear in the proposition’s claim. 1. We show that only μ1=1/2is such that E[μ 1|S]=μ1. After imposing a symmetrical calibration, the recursive definition of the ratio of Pareto weights derived in the paper becomes μ 1 1−μ 1 =μ1 1−μ1 ·expU1zS/θ z πzexpU1zS/θ z πzexpU2zS/θ expU2zS/θ By rearranging and taking the conditional expectation operator, we obtain Eμ 1|S=μ1−μ1·covμ 1expU 1/θ|S EexpU 1/θ|S −(1−μ1)·covμ 1expU 2/θ|S EexpU 2/θ|S (B.26) For each μ1=μ∗ 1such that E[μ 1|S]=μ∗ 1,equation(B.26) implies that μ∗ 1=covμ 1expU 2/θ|S EexpU 2/θ|S covμ 1expU 2/θ|S EexpU 2/θ|S−covμ 1expU 1/θ|S EexpU 1/θ|S (B.27)
Quantitative Economics 10 (2019) Recursive allocations and wealth distribution 345 By symmetry, for each μ∗ 1<1/2,theremustexistaμ∗∗ 1=1−μ∗ 1such that E[μ 1|S]=μ∗∗ 1. At μ∗∗ 1=1−μ∗ 1,equation(B.26) implies that μ∗ 1=covμ 1expU 1/θ|S EexpU 1/θ|S covμ 1expU 1/θ|S EexpU 1/θ|S−covμ 1expU 2/θ|S EexpU 2/θ|S (B.28) Combining equations (B.27)and(B.28), we obtain that at μ∗ 1the following condition must hold: covμ 1expU 1/θ|S EexpU 1/θ|S=−covμ 1expU 2/θ|S EexpU 2/θ|S(B.29) By plugging condition (B.29) into equation (B.27)or(B.28), we conclude that the only μ∗ 1∈(01)such that E[μ 1|S]=μ∗ 1is μ∗ 1=1/2. 2. We show that if μ1>1/2, then E[μ 1|S]<μ 1. We know from Proposition A7 that the stochastic process - S(S) =min(S mini{* Si 2}), where mini{* S2}>1is a submartingale. Since, by definition, - S(S) =μ1(- S) 1−μ1(- S),andthe stochastic process μ1cannot converge to 1(see Proposition 2), there must exist a μ+ 1>mini* Si 2 1+mini* Si 2 >1/2such that E[S|S+]=S+,whereS+=μ+ 1 1−μ+ 1 . Since S=μ1 1−μ1is a convex function in μ1, μ+ 1 1−μ+ 1 =ES|S+=Eμ 1 1−μ 1+++S+ =covμ 11 1−μ 1+++S++Eμ 1|S+E1 1−μ 1+++S+ ≥covμ 11 1−μ 1+++S++Eμ 1|S+1 1−Eμ 1|S+ Since μ 1and 1 1−μ 1 are both monotonically increasing functions of μ 1,cov[μ 11 1−μ 1 |S+]> 0. This implies that μ+ 1 1−μ+ 1 ≥covμ 11 1−μ 1+++S++Eμ 1|S+1 1−Eμ 1|S+ >E μ 1|S+1 1−Eμ 1|S+ from which we obtain Eμ 1|S+<μ + 1 Since we know that there exists only one μ∗ 1=1/2such that E[μ 1|μ∗ 1 1−μ∗ 1 ]=μ∗ 1,bythe continuity of fSwith respect to S,itmustbethecasethatE[μ 1|S]<μ 1,∀μ1∈(1/21).
346 Colacito, Croce, and Liu Quantitative Economics 10 (2019) 3. We show that if μ1<1/2, then E[μ 1|S]>μ 1. This proof mirrors the one that we just provided for μ1>1/2, and thus for parsimony we omit it here. Appendix C: Recursive method and global solution Given the conditions in Ma (1993)andMa (1996), the social planner’s value function, denoted as Qp(zμ1):N×[01]→R, satisfies the following functional equation proposed by Lucas and Stokey (1984), and Kan (1995): Qp(z μ1)=max {xiyiqiz}i∈{12}z∈N 2 i=1 μilogW∗ izci{qiz}z(C.30) subject to μ2=1−μ1 0≤x1≤X(z) 0≤x2≤X(z)−x1 0≤y1≤Y(z) 0≤y2≤Y(z)−y1(C.31) ci=(xi)λi(yi)1−λi∀i={12} 0≤min μ1(z)∈[01]Qpzμ1z−μ1zq1z−1−μ1zq2z∀z∈N Remark Appendix C.1. Let Assumptions 1–4hold. When μi=0, interpret μilog W∗ i(z ci{qiz}z)=0. Since 0<X(z)<∞and 0<Y(z)<∞,∀z∈N, it can be proved that there exists a unique bounded and continuous solution to (C.30)–(C.31). A feasible allocation is Pareto optimal if and only if it is generated recursively from the solution to (C.30)–(C.31). Assumptions 1–4are sufficient to apply the contraction mapping theorem to the planner’s problem described by the system of equations (4)–(5). This result is parallel to those proposed by Lucas and Stokey (1984), Kan (1995), and Anderson (2005). IES =1: A two-step approach It is convenient to use the definition of risk sensitive preferences in equation (1)tonotice that the recursive planner’s problem can be decomposed in two parts: Qp(sμ1)=QA p(s μ1)+QB p(s μ1) where QA p(s μ1)is the static social planning problem QA p(s μ1)≡max {xi≥0yi≥0}i∈{12} μ1(1−δ1)logc1+(1−μ1)(1−δ2)logc2(C.32)
Quantitative Economics 10 (2019) Recursive allocations and wealth distribution 347 subject to: X(z) ≥x1+x2Y(z)≥y1+y2 ci=(xi)λi(yi)1−λii=12; and QB p(s μ1)is related to the allocation of future utilities QB p(z μ1)≡max Dz μ1δ1log z exp(1−γ1)(q2z+Dz)πs +(1−μ1)δ2log z exp(1−γ2)q2zπs(C.33) subject to q2z=QpzdQp dμ1 −1zDz−Dz dQp dμ1 −1zDz∀z∈N(C.34) where Dzis defined as the difference of the future utilities in state z:Dz≡q1z−q2z. Equation (C.34) defines the Pareto frontier and is obtained using equations (5)–(7). By equation (7), the first derivative of Qpis strictly increasing, hence, invertible. Equation (6) implies that: μ1(z) =dQp dμ1 −1 (z q1z −q2z) (C.35) Appendix D: Allocation as a function of Pareto weights Let Wi t=W(C i tUi t+1)be the right-hand side of equation (14). If we denote the partial derivatives of the aggregator Wias Wi 1t := ∂W i t ∂Ci t W i 2t := ∂W i t ∂Ui t+1 then the stochastic discount factor is equal to Mi t+1=Wi 2tWi 1t+1 Wi 1t ∀i={12}(D.36) The optimality condition for the allocation of good Xtfor t=12 in each possible state is μh 0·t−1 2 j=0 W1 2j·W1 1tC1 t λ1 x1 t =(1−λ2) x2 t C2 tW2 1t ·t−1 2 j=0 W2 2j·μ2 0(D.37)
348 Colacito, Croce, and Liu Quantitative Economics 10 (2019) Define the date tPareto weights as μi t=μi 0·t−1 2 j=0 Wi 2j·Wi 1tCi t =μi t−1·Wi 2t−1·Wi 1t Wi 1t−1 ·Ci t Ci t−1 =μi t−1·Mi t·expci t∀i∈{hf} It follows that equation (D.37) can be rewritten as μ1 t·λ1 x1 t =(1−λ2) x2 t ·μ2 t(D.38) Let St:= μ1 t/μ2 t. Then the optimality condition in equation (D.38) can be represented by the following system of recursive equations: St λ1 x1 t =(1−λ2) x2 t St=St−1 M1 tec1 t M2 tec2 t (D.39) where Mit+1=δCit+1 Cit −1 ψU1−γ it+1 EtU1−γ it+11/ψ−γ 1−γ (D.40) A similar first-order condition applies with respect to good Y. Approximation methods and errors We use perturbation methods to solve our system of equations and compute our policy functions using the dynare++4.2.1 package. All variables are expressed in log-units. In Tables 1and 2, we report statistics regarding the maximum cumulative approximation error (Cumul. ERR) for our relative pseudo-Pareto weights, that is, the key drivers of both consumption shares. The maximum is taken across agents, and we account for the fact that the relative Pareto weights are persistent and errors can accumulate over longer simulations. More specifically, we construct recursively the following processes: ˜ Sjt =˜ Sjt−1· ˆ Mjt ˆ M1t ·ˆ Cjt /ˆ Cjt−1 ˆ C1t/ˆ C1t−1∀t≥1(D.41) whereweadoptthe“ˆ·” notation to indicate approximated variables. We initialize the recursion at different starting points, ˜ Sj0and we then define the error for agent jas follows: errj t=1 t t τ=1++++ ˜ Sjτ ˆ Sjτ −1++++·100
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