ECONOMICS SERIES SWP 2012/1. Is It Really Good to Annuitize? James Feigenbaum. and Emin Gahramanov. Faculty of Business and Law

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1 Faculty of Business and Law School of Accounting, Economics and Finance ECONOMICS SERIES SWP 212/1 Is It Really Good to Annuitize? James Feigenbaum and Emin Gahramanov The working papers are a series of manuscripts in their draft form. Please do not quote without obtaining the author s consent as these works are in their draft form. The views expressed in this paper are those of the author and not necessarily endorsed by the School or IBISWorld Pty Ltd.

2 Is It Really Good to Annuitize? James Feigenbaum Utah State University Emin Gahramanov Deakin University March 6, 212 Abstract Although rational consumers without bequest motives are better off investing exclusively with annuitized instruments in partial equilibrium, we demonstrate the welfare effect of annuitization is ambiguous in general equilibrium on account of the pecuniary externality. Accidental bequests improve consumption allocations by transferring capital mostly to young people rather than to the old, for whom the present value of the transfer is much less. If households are not borrowing constrained in the rational competitive equilibrium where they annuitize, there will exist a consumption/investment rule involving nonannuitized investments that confers higher utility in general equilibrium while maintaining the same equilibrium capital stock. Thus it may be that households eschew annuitization because society has learned it is suboptimal. Regardless of the explanation for this behavior, policymakers should not take steps to encourage more annuitization by the public. JEL Classification: C61, D11, E21 Keywords: consumption, saving, coordination, learning, general equilibrium, pecuniary externality, annuities puzzle, bequests, mortality risk, overlapping generations, optimal irrational behavior, Golden Rule The authors thank Frank Caliendo, Tom Davidoff, Jim Davies, Scott Findley, Hui He, Ian King, Val Lambson, Geng Li, Jochen Mierau, Moshe Milevsky, Svetlana Pashchenko, Eduardo Saucedo, Eytan Sheshinksi, and participants at seminars at the Australasian Economic Theory Workshop; BYU; CEF Conference; Deakin University; the University of Hawai i; the IFID Workshop; University of Pittsburgh; SABE Conference; Utah State; University of Sydney; and the Workshop on Optimal Control, Dynamic Games, and Nonlinear Dynamics for their views and comments. We also appreciate financial support provided by the TAER program at Deakin. Corresponding author J.Feigen@aggi .usu.edu. Website: 1

3 Annuities, i.e. investment instruments that pay an income stream that terminates upon the owner s death, present a puzzle to economists. In their optimal format, annuities perfectly insure against longevity risk by giving surviving investors, on top of the ordinary return to capital, a premium that increases with the probability of dying. 1 the premiums of surviving investors. disposal of their assets after death. Deceased investors surrender their investment, and these assets are used to pay Without bequest motives, rational households are indifferent to the Since annuities earn a higher return, such households should invest all their wealth in annuities (Yaari (1965)). Even households with bequest motives ought to annuitize the wealth intended to finance their own consumption. Nevertheless, private annuity investments account for only 1% of total household wealth for households over age 65 in the United States, 2 and only 6% of households in this age range participate in private annuities (Pashchenko (211)). Although many researchers believe the near total rejection of annuities by the public can be explained if all relevant frictions are properly accounted for, this stylized fact remains a diffi cult challenge for the rational-expectations paradigm. 3 Moreover, even if one finds that annuities products available to the public today are not attractive to rational agents, this does not explain why the annuities market is so thin to begin with. We consider the issue in a different light. the presumption that people hurt themselves when they fail to annuitize. Most of the literature on the annuities puzzle is motivated by In the basic partial-equilibrium model of Yaari (1965), this is certainly true from the perspective of a single individual, but is it true for society as a whole in general equilibrium? If agents behave rationally, the answer is ambiguous, depending on the parameterization of the model, though for our baseline calibration welfare will be higher if households do not have access to annuities. 4 If we expand our focus from individually rational behavioral rules to any behavioral rule consistent with market-clearing constraints, the answer is more straightforward. 5 If households are not borrowing constrained in a rational competitive equilibrium where households have access to annuities, there will exist a market-feasible consumption rule employing nonannuitized investments that confers higher lifetime utility while maintaining the same capital stock. In our baseline calibration, the optimal consumption and saving rule can nearly achieve the maximal welfare of the Golden Rule without extramarket transfers of consumption These unintuitive results are a consequence of the pecuniary externality (McKean (1958), Prest and Turvey (1965)), the property intrinsic to markets that people s actions affect prices, which in turn affect people s behavior. Although this two-way causal relationship is the centerpiece of modern economics, 1 In practice, a financial intermediary will do the actual work of maintaining assets that finance the recipient s consumption stream. Frictions in the annuities market will reduce the effective return on annuities and the consequent consumption stream. See Sheshinski (28) for a review of the general theory of annuities. 2 This is according to the 2 Health and Retirement Study (Johnson, Burman, and Kobes (24)). Social Security and defined benefit pension plans are not accounted for here, though they are effectively annuities, albeit suboptimal annuities that do not provide the intracohort risk-sharing benefits of financial annuities (Guo, Caliendo, and Hosseini (212)). Transaction costs and borrowing constraints may deter poor households from optimizing their portfolios, but this cannot explain why so few wealthy households do so. 3 Davidoff, Brown, and Diamond (25) and Leung (21) show that the management costs of annuitization would have to be huge to prevent households from annuitizing a substantial portion of their wealth. See Pang and Warshawsky (29), Lockwood (29), and Pashchenko (211) for estimates of how much annuitization will occur in lifecycle models with frictions. 4 Heijdra, Mierau, and Reijnders (21) have also found this in a two-period model. 5 There is no annuities puzzle if households are not required to be fully rational. See Brown (27), Hu and Scott (27), and Milevsky and Young (27). 2

4 economists generally assume that households take prices as given and ignore the effect their actions have on prices. The Welfare Theorems prove this is an innocuous assumption in infinite-horizon, representativeagent models for which Pareto optimality is equivalent to welfare maximization. However, this equivalence does not hold for overlapping-generations models, even when competitive equilibria are Pareto optimal, because there are multiple agents. Recent work (Feigenbaum and Caliendo (21), Feigenbaum, Caliendo, and Gahramanov (FCG) (211)) has shown that different cohorts can coordinate their consumption and saving behavior across generations to exploit the pecuniary externality and improve welfare for everyone in the steady state without extramarket transfers of consumption. 6 In the present context, we see this result extends to portfolio allocation rules that apportion savings between annuities and nonannuitized investments. Holding the received bequest fixed in a partial-equilibrium fashion, households do better investing only in annuities. But if everyone follows this strategy, there will be no bequest to inherit. increase their welfare if they coordinate on a strategy of not annuitizing their wealth. Thus households can This behavior is irrational in that individual consumers could increase their welfare further by purchasing annuities while enjoying the bequests left behind by their less selfish peers. Thus FCG (211) term this coordination optimal irrational behavior. In terms of game theory, markets in an overlapping-generations setting induce a Prisoners Dilemma game for which the socially optimal outcome is not a Nash equilibrium. What is remarkable about the application of optimal irrational behavior to annuities is that, whereas preceding papers have hypothesized that households might engage in socially optimal, yet irrational behavior just as experimenters observe that people sometimes do cooperate in one-shot Prisoners Dilemma games, here we actually have a real-world example where almost everyone behaves irrationally in a manner that improves their welfare via macroeconomic mechanisms. Bequests improve welfare by allowing better intertemporal consumption allocations. Both annuities and bequests preserve capital by transferring the assets of deceased agents to living agents. However, annuities primarily transfer this wealth to the elderly, who receive the largest insurance premiums. Bequests transfer this capital across the whole population. 7 budget constraint, so it purchases more consumption. A transfer received when young is discounted less in the household The benefits of bequests could be further increased if larger bequests are given to younger agents (Feigenbaum and Gahramanov (211)). Thus the mystery of why people do not annuitize can be accounted for by the fact that they are better off as a whole, though not individually, if they abstain from annuitization. error for policymakers to correct here. There is no friction or cognitive To the extent that real-world annuities are complicated financial products, the widespread aversion to expending the time and effort necessary to understand them actually protects us from a suboptimal world in which fewer people inherit bequests. While we do not observe people following the best market-feasible consumption rule, which would bring us much closer to the Golden Rule, people deviate from this rule primarily along the dimension of choosing how much to save, not in terms of 6 Early generations will be hurt when this coordination is implemented, so the Pareto optimality of dynamically effi cient competitive equilibria is not violated. 7 Here we focus on the case where bequests are spread uniformly across the whole population, but this is not essential as long as some assets are inherited by the young. Feigenbaum and Findley (211) study the benefits of bequests in a model where bequests are distributed across the surviving population more realistically with a greater portion going to the middle-aged children of deceased agents than goes to their grandchildren. 3

5 how they save. With the retirement of the Baby Boom generation, there is great concern about how this surge in retirees will convert their savings into consumption. Since annuities make this process easier than other investments, many economists are now advocating for legislation that would facilitate the rollover of savings into annuities, for example by making this the default option for the termination of 41(k) plans (Poterba, Venti, and Wise (211)). Our findings here demonstrate that such policies would be shortsighted. We should encourage people to save more, but we should not interfere with their natural inclination to abstain from annuitization. The paper is organized as follows. Section 1 describes the basic model that will be used throughout the paper. Section 2 compares what happens in rational competitive equilibria if markets are complete so households have access to annuities or if markets are incomplete so households cannot invest in annuities. Section 3 then establishes the result that a market-feasible consumption/investment rule that uses uninsured investments will exist that gives higher utility in general equilibrium than the rational competitive equilibrium with annuities. In Section 4, we derive the optimal consumption-saving rule both for a regime where only annuities are used and a regime where only uninsured investments are used. Section 5 concludes. 1 The General Model We consider a continuous-time overlapping generations model that generalizes Feigenbaum and Caliendo (21) to allow for an uncertain lifetime. This embeds Regime D of Yaari (1965) within a general-equilibrium context as in Hansen and Imrohoroglu (28). At each instant, a continuum of agents of unit measure is born. Their lifespan is stochastic with a maximum value of T. 8 Let Q(t) denote the probability of surviving until age t, which is a strictly positive, strictly decreasing, C 1 function. Denote the flow of consumption at age t by c(t). An agent values allocations of consumption over his lifetime by the utility function U = Q(t) exp( ρt)u(c(t)), (1) where u(c) is the period utility function and ρ is the discount rate. relative risk aversion (CRRA) family We restrict attention to the constant u(c; γ) = { 1 1 γ c1 γ γ 1 ln c γ = 1, (2) where γ > is the risk aversion coeffi cient and, more relevantly, the reciprocal of the elasticity of intertemporal substitution. For all γ >, marginal utility is u (c; γ) = c γ. (3) 8 For our calculations in Section 4 the assumption of a maximum lifespan is not innocuous. 4

6 Note that this is the standard preference model for a finitely lived household. We could include nonstandard features such as hyperbolic discounting, but many such features are already known to overturn the Yaari (1965) result and discourage agents from annuitizing even in partial equilibrium. Here we have a preference model in which rational households definitely would annuitize, and we can analyze whether annuitization increases or decreases welfare in general equilibrium. At age t, the agent is endowed with e(t) effi ciency units of labor, which he supplies inelastically to the market in return for the real wage w per effi ciency unit. Thus labor income at t is we(t). The consumer allocates this income between consumption and saving. He has a choice of two saving instruments: annuities and risk-free bonds. The flow of bonds b(t) earns a fixed net return r. The flow of annuities a(t) earns the return r plus an insurance premium h(t) equal to the hazard rate of dying d ln Q(t) h(t) = >. (4) We assume for now that deceased agents bequeath their unannuitized wealth assets to everyone currently alive. 9 Let B denote this constant bequest. Then the consumer must choose the consumption path c(t) subject to the budget constraint c(t) + db(t) + da(t) = we(t) + rb(t) + (r + h(t))a(t) + B, (5) the boundary conditions b() = b(t ) = (6) a() = a(t ) =, (7) and borrowing constraints a(t), b(t) t [, T ]. (8) The latter are necessary to prevent households from accruing infinite wealth by exploiting the arbitrage opportunity that arises because of the higher return paid to annuities. For much of the paper we will consider cases where households either invest exclusively in annuities or exclusively in uninsured bonds, the latter situation being closer to empirical reality. In either of these two restrictive cases, we dispense with the borrowing constraints. To complete the model, we give the economy a Cobb-Douglas production technology F (K, N) = K α N 1 α (9) of labor N and capital K. The latter depreciates at the rate δ >. Labor is supplied inelastically by households, so the aggregate supply is N = 9 Feigenbaum and Gahramanov (211) explore what happens if we loosen this assumption. Q(t)e(t). (1) 5

7 The supply of capital equals the aggregate of household investments K = Q(t)(b(t) + a(t)). (11) Firms behave competitively, so factor prices must satisfy the profit-maximizing conditions w = w(k) (1 α) r = r(k) α ( ) α K (12) N ( ) α 1 K δ. (13) N Finally, in equilibrium we must also have the bequest satisfy the inheritance-flow balance equation B Q(t) = Q(t)h(t)b(t), (14) where the righthand side is the value of nonannuitized wealth belonging to recently deceased agents. In the following, we are going to explore different consumption rules and compare their welfare in equilibrium. Since we do not restrict our attention to consumption rules that maximize individual utility, we need to generalize the usual definition of a market equilibrium to encompass a larger class of consumption rules. Following FCG (211), we define a generalized (steady-state) market equilibrium as a consumption rule c(t), a demand for bonds b(t), a demand for annuities a(t), a capital stock K >, a bequest B, a real wage w, and an interest rate r such that (i) the consumption rule c(t) and asset demands b(t) and a(t) satisfy the budget constraint (5), boundary conditions (6)-(7), and borrowing constraints (8) given B, r, and w; (ii) the capital stock K is the aggregate of the consumers asset demands, satisfying (11); (iii) the factor prices w and r satisfy the profit-maximizing conditions (12) and (13) given K; and (iv) the bequest B satisfies the inheritance-flow balance equation (14) given b(t). 1 of a generalized market equilibrium is called market-feasible. A consumption/investment rule that is part A generalized market equilibrium differs from the usual notion of a competitive equilibrium in that we only require c(t), b(t), and a(t) be affordable and do not confine attention to consumption allocations that maximize the utility (1) given B, r, and w, subject to the budget and borrowing constraints. A generalized steady-state market equilibrium that satisfies this additional optimization condition constitutes a rational competitive (steady-state) equilibrium (RCE). 2 Rational Competitive Equilibria First let us review what happens under the standard lifecycle approach. We consider separately what happens if markets are complete and consumers can sell claims contingent on their survival, and what 1 Here we abstract from population or technological growth, though it is straightforward to generalize this concept to allow for a balanced-growth market equilibrium. 6

8 happens if consumers have no access to annuities markets. 2.1 Complete Markets A rational household proceeds by maximizing (1) subject to the constraints (5)-(8) for a given set of factor prices r and w and the bequest B. This problem has the Lagrangian density L d = Q(t) exp( ρt)u(c(t)) + ρ a (t)a(t) + ρ b (t)b(t) (15) [ +λ(t) we(t) + rb(t) + (r + h(t))a(t) + B c(t) da(t) db(t) ]. The Euler-Lagrange equations for this problem are Together the last two conditions imply L d c(t) = Q(t) exp( ρt)u (c(t)) λ(t) = (16) L d b(t) d L d (db(t)/) = λ(t)r + ρ b(t) + dλ(t) = (17) L d a(t) d L d (da(t)/) = λ(t)(r + h(t)) + ρ a(t) + dλ(t) =. (18) λ(t)h(t) + ρ a (t) = ρ b (t). Since we have assumed h(t) > for all t, the Kuhn-Tucker conditions for (8) imply the corresponding Lagrange multipliers must satisfy ρ b (t) > ρ a (t) for all t. in bonds (Yaari (1965)) in this environment without intrinsic bequest motives. Thus a rational household will never invest This is purely an arbitrage result. The household does not actually care whether it is insuring against mortality risk or not. It invests in annuities solely because annuities pay a higher return, which the household earns by selling off the rights to its annuitized wealth in the event of its death. Note that we introduce the borrowing constraints in Section 1 to eliminate arbitrage opportunities and not because we were particularly concerned about the debt of the dead (Feigenbaum (28), Yaari (1965)). Here and in the following, when only one of the two assets is held in positive quantities in this case the annuities we ignore the borrowing constraint on that asset, which tremendously simplifies the computation. After this simplification, Eq. (18) reduces to d ln λ(t) = (r + h(t)). (19) This has the well-known solution λ(t) = λ Q(t) Q() exp( rt), 7

9 where λ is an integration constant. Meanwhile, (16) implies c(t) = ( ) 1/γ λ(t) exp ( ργ ) Q(t) t. (2) If we define the lifecycle consumption profile will be c = ( ) 1/γ λ, (21) Q() ( ) r(k) ρ c(t) = c exp t. (22) γ After setting b(t) = for all t and B =, we can rewrite the budget constraint (5) as [ d (Q(t) exp( rt)a(t)) = Q(t) exp( rt) (r + h(t))a(t) + da(t) ] = Q(t) exp( rt) [we(t) c(t)]. (23) Upon integrating (23), the borrowing conditions (7) determine c : c = w(k) Q(t)e(t) exp( r(k)t) ( ). (24) Q(t) exp (1 γ)r(k) ρ γ t By investing in annuities, the household eliminates any effect of mortality risk on the shape of its consumption profile. The survivor function only enters (22) to the extent that the expected present value of the household s labor endowment and consumption stream are weighted by Q(t) in (24). that we say that annuities allow the household to insure against mortality risk. to consume, the path of consumption is independent of the realization of the time of death. The demand for annuities is also determined by integrating (23): a(t) = The equilibrium capital stock K ann rce t It is in this sense Conditional on being able Q(s) exp(r(k)(t s))[w(k)e(s) c(s)]ds. (25) Q(t) is then determined by the market-clearing condition K = Q(t)a(t). (26) Note that in the special case where r(k) = the lifecycle consumption profile (22) corresponds to the profile of the Golden Rule allocation described in Appendix A. of the exogenous parameters. However, this will only happen for special choices 8

10 2.2 Incomplete Markets In the absence of annuities markets, the budget constraint becomes db(t) and the capital stock is simply the aggregate of nonannuitized bonds: = we(t) + rb(t) + B c(t), (27) K = Q(t)b(t). (28) Now a rational household maximizes (1) for a given B, r, and w subject to the constraint (27) and boundary conditions (6). In this case, the Lagrangian density for the household s problem is L inc d [ = Q(t) exp( ρt)u(c(t)) + λ(t) we(t) + rb(t) + B c(t) db(t) ]. (29) The Euler-Lagrange equations for this problem are (16) and L inc d L inc d b(t) d (db(t)/) = λ(t)r + dλ(t) =. (3) The latter has the solution λ(t) = λ exp( rt). (31) Substituting this into (2), we now get the lifecycle consumption profile ( ) ( ) 1/γ r(k) ρ Q(t) c(t) = c exp t, (32) γ Q() where again c is defined by (21). In this case, the budget constraint (27) can be rewritten d (exp( rt)b(t)) = exp( rt) [we(t) + B c(t)]. (33) The boundary conditions (6) then imply the lifetime budget constraint exp( r(k)t) [w(k)e(t) + B c(t)] = (34) and thereby determine c : c = exp( r(k)t) [w(k)e(t) + B] ( ) 1/γ ( ). (35) Q(t) Q() exp (1 γ)r(k) ρ γ t Notice the difference between the lifecycle consumption profile with complete markets given by (22) and (24) 9

11 and the consumption profile with incomplete markets given by (32) and (35). Under complete markets, the consumption profile only depends on the survivor function through c. shape of the consumption profile, only on the level of consumption. Mortality risk has no effect on the In contrast, under incomplete markets, consumption is proportional to Q(t) 1/γ, but the present value of expected income, i.e. the numerator in (35) is not directly dependent on the survivor function. 11 With incomplete markets mortality risk does affect the shape of the consumption profile (Bullard and Feigenbaum (27), Feigenbaum (28), Hansen and Imrohoroglu (28)). Integrating Eq. (33) gives us the demand for bonds b(t) = t exp(r(k)(t s))[w(k)e(s) + B c(s)]ds. (36) An equilibrium is computed by solving the two remaining conditions (28) and (14) for the bequest B beq rce and the capital stock Krce. beq The possibility for ineffi ciency stemming from the absence of life-insurance markets can be demonstrated by the fact that when r(krce) beq = the lifecycle consumption profile (32) cannot equal the consumption profile of the Golden Rule allocation in Appendix A, for the latter does not depend on Q(t). Nevertheless, there exist calibrations of the model in which expected utility is higher in the equilibrium with incomplete markets than it is in the equilibrium with complete markets, though both expected utilities must then be less than the Golden Rule expected utility. 2.3 Calibration and Numerical Results For our baseline calibration, we set the share of capital to α =.3375, the consumption to output ratio C/Y =.75, and the capital to output ratio K/Y = 3., all common values from the literature. These targets determine α and δ =.83. This leaves two parameters: the elasticity of intertemporal substitution γ 1 and the discount rate ρ. Since these preference parameters are not separately identified by steady-state behavior, we consider three common values of γ from the literature:.5, 1, and 3. For each choice of γ, we set ρ so the rational competitive equilibrium with incomplete markets satisfies K/Y = 3. We use the model with incomplete markets rather than complete markets to set the calibration since most people do not annuitize. Thus the incomplete-markets model is more relevant empirically. We assume households live for T = 75 years for a lifespan that corresponds to real ages of 25 to 1. The survivor function Q(t) is taken from Feigenbaum (28). For t < T R = 4, the endowment profile e(t) is proportional to the income profile of Gourinchas and Parker (22). For t T R, we assume consumers retire and e(t) =. Representative macroeconomic variables describing the rational competitive equilibria, both with complete (COM) markets, so all investments are annuitized, and with incomplete (INC) markets, so no investments are annuitized, are presented in Table 1 for the three choices of γ. The corresponding lifecycle consumption profiles for the incomplete markets equilibria are plotted in Fig. 1. To see the effect of annuitization without changing macroeconomic observables (as opposed to keeping 11 In equilibrium, the numerator will depend on the survivor function since the equilibrium bequest depends on Q(t). 1

12 γ ρ K/Y U EV f B COM INC COM INC % 5.24% % 6.89% % 9.2% Table 1: Macroeconomic observables for rational competitive equilibria, both with complete markets, where only annuities are employed, and with incomplete markets, where only uninsured investments are used. Three calibrations are considered with representative values of γ. the structural parameters constant), we also plot the lifecycle consumption profile under complete markets for a separate calibration in which K/Y = 3. This consumption profile is the same for all choices of γ since (r ρ)/γ is the same for each of these equilibria, and γ and ρ do not appear elsewhere in (22). 12 For γ =.5, our a priori intuition that consumers should be better off with complete markets is borne out. However, for γ = 1 and γ = 3, that is not the case. In these calibrations, utility is higher when consumers are forced to leave bequests. Since it is diffi cult to interpret the economic significance of differences in utility, Table 1 also reports the equivalent variation EV Q(t) exp( ρt)u(c INC (t)) = such that Q(t) exp( ρt)u((1 + EV )c COM (t)), (37) i.e. the fraction by which consumption would have to be augmented at all instants in the complete markets equilibrium to achieve the same lifetime utility as in the incomplete markets equilibrium. For γ = 1, the change in utility from complete markets to incomplete markets is only slightly positive. However, for γ = 3, it is huge. Consumption would have to be augmented by 6% in the complete markets equilibrium to achieve the same utility as would arise without annuities. of eliminating idiosyncratic risk to be only 1-2%. Why is annuitization welfare-improving only for low values of γ? effects on utility. For comparison, Vidangos (28) estimates the benefit Annuitization has two competing First, as can be seen in Fig. 1, annuitization allows for smoother consumption streams. The household will never run out of assets no matter how long it survives. Thus consumption drops by only 35% from the beginning of life to the maximal lifespan for the complete markets profile in Fig. 1. In contrast, those households that survive to age 1 see their consumption drop by 85% or more with incomplete markets. This is the advantage of annuitization identified by Yaari (1965). The second effect is that annuitization eliminates the possibility of receiving a bequest. lifetime wealth accounted for by accidental bequests in the incomplete markets equilibrium f B = The fraction of B exp( rt) exp( rt)[we(t) + B] (38) is reported in the last column of Table For all other exercises in this paper, we hold the structural parameters constant when comparing between complete and incomplete markets, rather than the macroeconomic observables. 11

13 Figure 1: Rational competitive equilibrium lifecycle consumption profiles for the baseline calibration with γ =.5, 1, 3 for both complete and incomplete markets. ρ is chosen so K/Y = 3 in all graphs. 12

14 Between Fig. 1 and Table 1, we see that as γ increases, two things happen. First, as the elasticity of intertemporal substitution decreases, the household does a better job of smoothing its consumption on its own with incomplete markets, so the smoothing benefits of annuitization become less important. Second, because households run down their assets more slowly, they leave a larger estate when they die. Thus the fraction of lifetime wealth contributed by the bequest in the incomplete markets equilibrium increases. Together these two effects imply that utility will be higher under incomplete markets for suffi ciently high γ. Since most estimates of intertemporal elasticity put γ somewhere between 1 and 3, this suggests that households are actually better off because most do not annuitize. The bequest effect is a general-equilibrium effect that is a consequence of the pecuniary externality. The bequest B that households receive in the incomplete-markets equilibrium is accidental and determined by the equilibrium conditions. The capital stock K that arises under the two equilibria is also determined by equilibrium conditions. However, they are not determined separately. There is an interaction between the capital stock and the bequest since a bequest allows the household to save and possibly increase K. The bequest adds to income directly, but it can also affect income indirectly through its effect on the capital stock. 13 Does utility increase when we shut down annuitization because of the direct effect of bequests or the indirect effect via the capital stock? To assess this question we report equivalent variations relative to the complete markets general equilibrium in various partial equilibria for γ = 1 and γ = 3. These are reported in Table 2. The first and second columns report the equivalent variation from the complete and incomplete markets general equilibria respectively. The third and fourth columns show the equivalent variation for partial equilibria respectively under complete and incomplete markets but with the factor prices from the general equilibrium of the other market regime. For the incomplete markets partial equilibrium, the bequest is the same as in the incomplete markets general equilibrium. Finally, in the fifth column we consider what happens in a quasi-general equilibrium of the incomplete markets model with factor prices from the complete markets general equilibrium but with the equilibrating bequest that satisfies the balance equation (14). All equivalent variations are computed starting from the complete-markets general equilibrium. For the γ = 1 case, the capital stock is actually higher in the complete markets general equilibrium than it is in the incomplete markets general equilibrium. Thus factor prices are more favorable with complete markets since, holding the market structure constant, utility increases as we switch from the incompletemarkets prices to the complete-markets prices. In this case the higher utility under incomplete markets must be due to the direct effect of the bequest, and this is supported by the numbers in Table 2. If we go partway from the complete markets general equilibrium to the incomplete markets general equilibrium by changing the factor prices to their incomplete market values while maintaining annuitization, utility decreases by the equivalent of one percent of consumption. If, instead, we change from annuitization to uninsured investments while maintaining the complete markets factor prices, utility increases by the equivalent of 1.5% of consumption. Thus the direct effect of having accidental bequests has a larger effect on lifetime utility than the change in factor prices. 13 Heijdra, Mierau, and Reijnders (21) focus on the capital channel. 13

15 γ COM GE INC GE Annuities PE (INC GE Prices) Bequest PE (COM GE Prices) Bequest Quasi GE 1.%.11%.89% 1.5%.4% 3.% 6.15% 2.45% 4.69% 5.16% Table 2: Equivalent variations relative to the complete markets general equilibrium for various partial equilibria under the γ = 1 and γ = 3 calibrations. The size of the bequest is also important. In comparing the quasi-general equilibrium to the complete markets equilibrium, we are keeping factor prices the same and looking at what happens with the smoothing effects of annuitization vs what happens where there is no smoothing and a smaller bequest. The bequest constitutes 5.6% of lifetime wealth in the quasi-general equilibrium as opposed to 6.9% in the full general equilibrium. The smoothing benefits of annuitization slightly outweigh the benefit of this smaller bequest. For the γ = 3 case, the capital stock is higher under incomplete markets so this is more consistent with the story that the bequest improves utility by increasing the capital stock. Nevertheless we still see that the direct effect of the bequest is larger than the indirect effect via the capital stock. When we change factor prices to their incomplete market prices while preserving the use of annuities, utility increases by the equivalent of 2.5% of complete-markets consumption. If, on the other hand, we change from annuity investments to uninsured investments while preserving the complete markets factor prices, utility increases by the equivalent of 4.7% of complete-markets consumption. Thus the larger increase comes through the switch from annuities to bequests than from more favorable factor prices. To further establish that accidental bequests improve welfare because they deliver the income that comes from redistributing the assets of the dead at earlier ages than the mortality premium on annuities would, we can see that a positive time value of money is important to get this improvement with rational agents. Let us consider another set of partial equilibria where factor prices are the same both in the annuities regime and in the bequest regime. Specifically, we consider the case where r =, so income received at different points in the lifecycle is valued the same. In the bequest regime we set B to solve the bequest-balance equation (14) as in the quasi-general equilibrium of Table For the γ =.5, γ = 1, and γ = 3 calibrations we find that utility is always higher in the annuities regime. The equivalent variation of the bequest partial equilibrium relative to the annuities partial equilibrium is between 1% and 2% for all three calibrations. 3 Market-Feasible Consumption Rules The results of the previous section demonstrate that annuitization will not be good for the economy as a whole if the elasticity of intertemporal substitution is suffi ciently low. However, this finding contributes nothing to the annuities puzzle since fully rational households still ought to completely annuitize their savings if they can. For our next exercise we take a more evolutionary perspective. Instead of assuming 14 Ideally we would like to compare between partial equilibria while keeping the present value of income the same. However, this is problematic since present value is computed differently in the two regimes. By equilibrating B in the bequest regime, we ensure that the assets of the dead are fully recouped in both regimes. 14

16 that households individually maximize their utility, we follow FCG (211) and assume that households may adopt any market-feasible consumption rule, i.e. any rule that satisfies the instantaneous budget constraint and clears markets in the aggregate. Suppose there exists a market-feasible consumption rule that confers higher lifetime utility than the individually rational consumption rule. If individuals adopt consumption rules taught to them by society rather than through individually rational cognition, then society may learn to inculcate this higher-utility consumption rule. To characterize the optimal market-feasible consumption rule, it is convenient to introduce a social planner. This social planner maximizes (1) subject to the constraints (5), (8), (11), and (14) and the boundary conditions (6)-(7). This imposes more constraints than a rational household would, as described in Section 2, since the market-clearing conditions are imposed during the optimization instead of as an equilibrium condition that must be satisfied afterwards. This new problem also differs from the problem faced by a standard Pareto social planner who allocates goods to maximize (1) subject only to the constraint that the economy is capable of producing all the goods allocated. The solution to that problem is what we refer to as the Golden Rule allocation and is described in Appendix A. 15 The Lagrangian for our social planner is L s = {Q(t) [exp( ρt)u(c(t)) + µ(a(t) + b(t)) + ξ(h(t)b(t) B)] +ρ a (t)a(t) + ρ b (t)b(t) + λ(t)[w(k)e(t) + r(k)b(t) + (r(k) + h(t))a(t) + B c(t)} [ da(t) λ(t) + db(t) ] µk, (39) and the corresponding Lagrangian density is The Euler-Lagrange equations are L s = Q(t)[exp( ρt)u(c(t)) + µ(a(t) + b(t)) + ξ(h(t)b(t) B)] +ρ a (t)a(t) + ρ b (t)b(t) µk + λ(t)[w(k)e(t) + r(k)b(t)] [ T +λ(t) (r(k) + h(t))a(t) + B c(t) da(t) db(t) ]. (4) L s c(t) = Q(t) exp( ρt)u (c(t)) λ(t) = (41) L s b(t) d L s (db(t)/) = Q(t)[µ + ξh (t)] + ρ b(t) + λ(t)r(k) + dλ(t) = (42) 15 Here we assume the social planner has a zero generational discount rate, as Ramsey (1928) argued it should be, so we can focus on steady-state equilibria. FCG (211) show in a dynamic model that similar results would accrue with a positive generational discount rate. 15

17 L s a(t) d L s (da(t)/) = Q(t)µ + ρ a(t) + λ(t)(r(k) + h(t)) + dλ(t) = (43) L T s K = µ + λ(t)[w (K)e(t) + r (K)(a(t) + b(t))] =. (44) Assuming the bequest must be nonnegative, the first-order condition for the bequest is where inequality holds only if B =. L s B = [λ(t) ξq(t)], (45) As in Feigenbaum and Caliendo (21), we next focus our attention on finding the optimal market-feasible consumption rule consistent with a given choice of (B, K) R + R ++. (B, K), which is governed by the Bellman equation subject to the constraints c(t) + db(t) V (B, K) = + da(t) We refer to this as Subproblem max Q(t) exp( ρt)u(c(t)) (46) c(t),b(t),a(t) = w(k)e(t) + r(k)b(t) + (r(k) + h(t))a(t) + B b(t), a(t) t [, T ] K = Q(t)(b(t) + a(t)) (47) and the boundary conditions B Q(t) = Q(t)h(t)b(t), (48) b() = b(t ) = a() = a(t ) =. In the event that there is no market-feasible choice of c(t), b(t), and a(t) that satisfies these constraints, we define V (B, K) =. Note that if K rce is the equilibrium capital stock in the general model of Section 1, including the borrowing constraints, then the rational consumption rule with full annuitization will be the solution to Subproblem (, K rce ). The problem of a rational household imposes fewer constraints than Subproblem (, K rce ) while maximizing the same objective so it must earn utility U rce V (, K rce ). Yet since the rational consumption rule satisfies the market-clearing conditions (47) with K = K rce and (48) with B =, it meets all the constraints of Subproblem (B, K), so V (, K rce ) U rce. Thus the rational consumption rule with full 16

18 annuitization is the optimal market-feasible consumption rule consistent with (, K rce ). The Lagrangian for Subproblem (B, K) remains (39), only now we view B and K as exogenous. Envelope Theorem and (44)-(45) then give us the derivatives of the value function: The V (B, K) K = λ(t)[w (K)e(t) + r (K)(a(t) + b(t))] µ (49) V (B, K) B = [λ(t) ξq(t)]. (5) Note that the first-order condition for c(t), (41), is unchanged from the first-order condition (16) for the rational competitive equilibrium. On the other hand, if we combine the Euler-Lagrange equations (42)-(43) for b(t) and a(t), we now get ρ b (t) ρ a (t) = h(t)[λ(t) ξq(t)]. (51) For the rational competitive equilibrium, we only had h(t)λ(t) on the righthand side. This is the additional return, measured in utility, that can be earned immediately at t by investing an extra unit in annuities as opposed to bonds. function. This must be positive on account of (41) and our assumptions about the survivor Consequently, a rational household would never invest in uninsured bonds. Our social planner, in contrast, must also consider the countervailing term ξh(t)q(t). bequest produced by investing an extra unit in bonds as opposed to annuities. The factor h(t)q(t) is the additional This is weighted by ξ, the shadow price of bequests, to determine the return in utility that is obtained by this investment. the social planner would advise households to invest in bonds rather than annuities if ξq(t) > λ(t). This situation, of course, can only arise if ξ is positive, but from (45) we see that at the optimal (B, K) the shadow price must satisfy ξ The first inequality can only be strict if B =. λ(t) Thus Q(t) >. (52) For an interior solution, the shadow price at the optimal (B, K) is the sum of marginal utilities across the entire lifespan, since a uniform bequest provides a constant income at each t, divided by the population the bequest is spread over. One might naively conclude that (45) automatically implies that a rational competitive equilibrium must be suboptimal since ξ presumably ought to be zero for the rational competitive equilibrium whereas the fraction in (52) must be strictly positive. However, ξ does not appear in the equations of motion (41) and (43) that carry over from the rational competitive model with just annuities. These equations revert to their rational competitive equilibrium counterparts (16) and (18) when µ =, but it is not necessary to set ξ =. Rather, when B =, ξ is determined so that ρ b (t) for all t. Indeed, the shadow price of bequests ought not to be zero when B = precisely because there is a benefit to having a bequest. As an example, when it is possible to obtain the Golden Rule as a rational competitive equilibrium, λ(t) is proportional to Q(t) for all t, and ξ > is the constant of proportionality. or annuities is the same for all t. In this special case the effective return from bequests 17

19 We can establish the following theoretical result that generalizes the main proposition from FCG (211). Proposition 1 Assuming households have access to annuities, in a rational competitive equilibrium that does not conform to the Golden Rule and in which households are only borrowing constrained on a subset of [, T ] of measure zero, there will exist a market-feasible consumption rule involving nonannuitized investments that produces higher lifetime utility U while maintaining the same capital stock K rce. Proof. The rational consumption rule will be the solution to Subproblem (, K rce ), where K rce is the equilibrium capital stock with full annuitization. Let a(t) be the lifecycle profile of annuities under this rule. Define S a = {t [, T ] : a(t) > } and S b = {t S a : ρ b (t) > }. have λ(t) ξq(t) for all t S a. λ(t) = ξq(t). Since ρ a (t) = ρ b (t) for all t S a, we Furthermore, for t S b we have λ(t) > ξq(t), and for t S a S b we have By assumption [, T ] S a has measure zero, so (5) implies V (, K rce ) B = [λ(t) ξq(t)] = [λ(t) ξq(t)] = S a [λ(t) ξq(t)]. S b If S b has positive measure, V (, K rce )/ B >. In that case, for some ε >, V (ε, K rce ) > V (, K rce ). Thus a market-feasible consumption rule with B = ε must exist that produces higher utility while maintaining the same capital stock. The only exception occurs when S b has measure zero, in which case V (, K rce )/ B =. Then (41) implies that c(t) = ξ 1/γ exp ( ργ ) t for all but a set of measure zero of t. Since we are only considering smooth consumption functions, (53) must hold for all t [, T ], and this is the Golden Rule consumption profile. The caveat in this proposition that households are never borrowing constrained must be emphasized here. In all three calibrations of the model in 2.3, households hold negative amounts of annuities or bonds for small t, so households would be borrowing constrained if we enforced the borrowing constraints in these examples. While it is true that a rational competitive equilibrium will necessarily involve only annuities, Proposition 1 does not necessarily imply that a social planner can improve upon any non-golden-rule rational competitive equilibrium. This is because it does not apply to such equilibria if both borrowing constraints bind on a set of positive measure. The proposition would permit V (, K)/ B for some K > if ρ a (t) > ρ b (t) > for all t in the subset of positive measure that satisfy λ(t) < ξq(t), though we have not found an example where this occurs. (53) 4 Optimal Irrational Behavior A natural question to ask at this point is what is the best market-feasible (BMF) consumption rule. This is what FCG (211) refer to as the optimal irrational behavior. 16 We obtain this by finding (B, K) 16 This behavior is irrational in the sense that it requires actions contrary to the solution of an individual household s optimization problem. Households could get higher utility by deviating from the optimal irrational rule and following the fully 18

20 R + R ++ that maximizes V (B, K). Computationally, this problem is facilitated by our knowledge of the first-order partial derivatives. Note that, unless the Golden Rule allocation coincides with the complete-markets rational competitive equilibrium, there will be infinitely many market-feasible consumption rules that confer higher utility than the rational consumption rule. The best market-feasible rule is of interest primarily because it puts an upper bound on how much utility could be obtained within markets. As we will see below, the best rule is no more consistent with empirical behavior than the rational rule that requires full annuitization. There may be nonmaterial costs that prevent adoption of the best market-feasible rule. For example, sticking to the best market-feasible rule will often require tremendous self control since it may defer a great deal of consumption till the end of life. Computing equilibria in which both borrowing constraints may possibly bind is extremely diffi cult and beyond the scope of the present paper. Instead, we consider what happens if we shut down either the annuities market or the uninsured bond market, in which case we can relax the two borrowing constraints without creating arbitrage opportunities. 4.1 With Annuities Analogous to our procedure in 3, we define as Subproblem K the problem of choosing, for a given K >, c(t) and a(t) to maximize subject to V ann (K) = c(t) + da(t) (26), and (7). The Lagrangian for this problem is max Q(t) exp( ρt)u(c(t)) c(t),a(t) = w(k)e(t) + (r(k) + h(t))a(t), (54) L ann s = + Q(t) [exp( ρt)u(c(t)) + µa(t)] µk (55) [ λ(t) w(k)e(t) + (r(k) + h(t))a(t) c(t) da(t) ], and the associated Lagrange density is L ann s = Q(t)[exp( ρt)u(c(t)) + µa(t)] (56) [ +λ(t) w(k)e(t) + (r(k) + h(t))a(t) c(t) da(t) ] µk T. rational rule. However, if all households do this, they will all end up worse off. Some have suggested that optimal irrational behavior might better be characterized as hyperrational behavior. 19

21 The Euler-Lagrange equations for Subproblem K are (41) for c(t), which is unchanged from Section 2, and L ann s L ann s a(t) d (da(t)/) Meanwhile the Envelope Theorem gives us ann(k) = Lann s K = µ + V = µq(t) + λ(t)[r(k) + h(t)] + dλ(t) The differential equation for λ, (57), has the solution =. (57) λ(t)[w (K)e(t) + r (K)a(t)] =. (58) ( ) λ Q() λ(t) = exp( r(k)t) µ r(k) [1 exp( r(k)t)] Q(t) r(k) ( ) λ Q() µt Q(t) r(k) =, (59) where λ is an integration constant. Let us define recalling our definition of c, (21). lifecycle consumption profile c exp c(t) = χ = µq() λ = µc γ, (6) Converting marginal utilities to consumption via (2), we obtain the ( ) ( r(k) ρ γ t c exp r(k) [exp(r(k)t) 1] ) 1 γ 1 χ r(k) ) ( ργ t (1 χt) 1 γ r(k) =. (61) Note that this is exactly the same consumption profile as was obtained by Feigenbaum and Caliendo (21) without any mortality risk, although c is determined by the instantaneous budget constraint (23) that we used in Section 2.1 and so does depend on the survivor function: c = ( exp (1 γ)r(k) ρ γ t exp( r(k)t)q(t)e(t) ) Q(t) ( ) 1 1 χ r(k) [exp(r(k)t) 1] γ w(k) (62) for r(k) and c = Q(t)e(t) T exp( ρ γ t)q(t) (1 w(k) (63) 1 χt) γ for r(k) =. Likewise, the asset demand profile is computed using (25). The only remaining variable that needs to be determined is χ. constraint (26). Suppose r(k). Then we must have This is obtained by solving the capital χ [exp(r(k)t) 1] 1 (64) r(k) 2

22 for all t [, T ] in order for c(t) to be defined for all t. 17 χ and decreasing otherwise. condition for the consumption profile to be defined is The lefthand side of (64) is increasing in t if Since the condition is always satisfied for t =, a necessary and suffi cient χ r(k) exp(r(k)t ) 1. (65) This imposes an upper bound on the set of χ we must search over to solve Eq. (26). V ann (K). We find the best market-feasible capital stock K ann by maximizing V ann (K). We can then compute If we set χ =, (61) and (62) revert to the solutions, (22) and (24), for a rational household with annuities given factor prices r(k) and w(k). Thus χ can be interpreted as a measure of how much the consumption profile that solves Subproblem K deviates from the a rational consumption profile. Since the rational competitive equilibrium solves Subproblem K ann rce, it must be the case that V ann (K ann) V ann (K ann rce ). We refer the reader to Feigenbaum and Caliendo (21) for a characterization of the properties of the best market-feasible consumption profile with annuities, which aside from the dependence of c on the survivor function are the same as in a model without mortality risk. 4.2 Without Annuities In the absence of annuities markets, we define Subproblem (B, K) for (B, K) R + R ++ to be V beq (B, K) = max Q(t) exp( ρt)u(c(t)) c(t),b(t) subject to (27), (28), (14), and (6). The Lagrangian for the problem with a time-varying hazard rate 18 is L beq s = + Q(t) [exp( ρt)u(c(t)) + µb(t) + ξ (h(t)b(t) B)] (66) [ λ(t) w(k)e(t) + r(k)b(t) + B c(t) db(t) ] µk. The Lagrange density is L beq s = Q(t) [exp( ρt)u(c(t)) + µb(t) + ξ (h(t)b(t) B)] (67) [ +λ(t) w(k)e(t) + r(k)b(t) + B c(t) db(t) ] µk T. 17 For r(k) =, it is necessary and suffi cient that χt The problem with a constant hazard rate has to be treated differently since the bequest balance equation implies that B is proportional to K, so the constraint qualification does not hold. Since the empirical hazard rate is not constant, we do not consider this special case here. 21

23 The Euler-Lagrange equation (41) for c(t) is unchanged from Section 4. The remaining equation is L beq s L beq s b(t) d (db(t)/) = λ(t)r(k) + [µ + ξh(t)]q(t) + dλ(t) The Envelope Theorem then gives the partial derivatives of V beq : V beq (B, K) K V beq (B, K) B =. (68) = Lbeq s K = λ(t)[w (K)e(t) + r (K)b(t)] µ =. (69) = Lbeq s B For this model, the solution to the λ equation (68) can be written = ξ Q(t) + λ(t) =. (7) λ(t) = exp( r(k)t)(λ µg(t) ξh(t)), (71) where we define G(t) = and H(t) = t t Q(s) exp(r(k)s)ds (72) Q(s)h(s) exp(r(k)s)ds. (73) If we again use the notation c, defined by (21), and we introduce new Lagrange multipliers ν = µ λ (74) ζ = ξ, λ (75) we obtain from (2) the lifecycle consumption profile ( ) ( ) 1/γ r(k) ρ Q(t) c(t) = c exp t [1 νg(t) ζh(t)] 1/γ. (76) γ Q() The intertemporal budget constraint is the same as in Section 2.2, when we considered the rational competitive equilibrium without annuities markets. boundary conditions (6) determine c : Thus the demand for bonds b(t) is given by (36) and the c = exp( r(k)t)[w(k)e(t) + B] ( ) ( 1/γ. (77) exp (1 γ)r(k) ρ γ t Q(t) Q()) [1 νg(t) ζh(t)] 1/γ Not surprisingly, when ν = ζ =, (76) and (77) reduce to the solution (32) and (35) for a rational household in a partial equilibrium with factor prices r(k) and w(k). This special case must then be the 22

24 solution of Subproblem (Brce, beq Krce), beq which is therefore nested within the set of optimal market-feasible rules for each (B, K). Consequently, generalizing the result from Section 4.1, we have V beq (Bbeq, K beq ) V beq (Brce, beq Krce), beq where Bbeq is the best market-feasible bequest and K beq is the best market-feasible capital stock, both with annuities markets closed. Using (7) we can isolate the value of ξ at the optimal B, which, assuming B >, is ξ = λ(t) Q(t) >. (78) Inserting (71) into (78), we obtain Using (74) and (75), this simplifies to ξ Q(t) = exp( r(k)t)(λ µg(t) ξh(t)). ζ Q(t) = exp( r(k)t)(1 νg(t) ζh(t)), which we can solve for ζ: ζ(k, ν) = exp( r(k)t)(1 νg(t)) (79) [Q(t) + exp( r(k)t)h(t)]. Note that ν and K are the only endogenous variables in (79). Having isolated ζ, we can solve for the optimal equilibrium bequest B given K and ν: where Z(K, ν) = B(K, ν) = w(k) These are derived in Appendix B. exp( r(k)t)[z(k, ν) H(t)]e(t) (8) [Q(t) + (H(t) Z(K, ν)) exp( r(k)t)], ( ) ( 1/γ H(t) exp r(k)(1 γ) ρ γ t Q(t) Q()) [1 νg(t) ζ(k, ν)h(t)] 1/γ ( ) ( 1/γ exp (1 γ)r(k) ρ γ t Q(t) Q()) [1 νg(t) ζ(k, ν)h(t)] 1/γ. (81) Define ν(k) to be the ν that satisfies the capital constraint (28) given the capital K and B = B(K, ν). Then we can define V beq (K) = V beq (B(K, ν(k)), K), and we have reduced the problem of finding the best market-feasible behavior to a one dimensional optimization of V beq (K) with respect to K. Lagrange multiplier ν(k). The primary diffi culty of computing V beq (K) is then to find the Analogous to Section 4.1, we can restrict attention to ν that satisfy νg(t) + ζ(k, ν)h(t) 1 t [, T ]. (82) 23

25 Since G and H are both strictly increasing in t, for the case where ν, ζ(k, ν), a necessary and suffi cient condition for (82) to hold is that ν satisfy νg(t ) + ζ(k, ν)h(t ) 1. (83) What does the best market-feasible consumption profile look like? Using (2), we can write the growth rate of consumption as a function of the decay rate of the instantaneous marginal utility λ(t): d ln c(t) = 1 γ From (71), (72), and (73), we obtain the decay rate of λ: Plugging this into (84), we derive d ln c(t) [ ] d ln λ(t) ρ h(t). (84) d ln λ(t) = r(k) + (µ + ξh(t)) Q(t) λ(t). (85) = 1 γ [ r(k) ρ h(t) + (µ + ξh(t)) Q(t) ]. (86) λ(t) In the special case where the Lagrange multipliers µ and ξ vanish, (86) reduces to the growth rate of consumption for an individually rational household. Ignoring the pecuniary externality, a rational household will increase (decrease) its consumption when the interest rate is greater (less) than the effective discount rate ρ + h(t), inclusive of mortality risk. This arises because a unit of consumption deferred at t will earn the household 1 + r units of consumption at t +, but a unit of consumption at t + is only worth 1 (ρ + h(t)) at t +. However, the (B, K) that maximizes V beq must have that the shadow price of capital is µ = from (69) while the shadow price of bequests ξ is given by (78). d ln c(t) = 1 γ [ r(k) ρ h(t) + λ(t)[w (K)e(t) + r (K)b(t)] (87) Thus we can rewrite (86) as Q(t) λ(t ) T λ(t) Q(t ) Q(t )[w (K)e(t ) + r (K)b(t )] + h(t) ] Q(t) λ(t ) λ(t) Q(t ). (88) The second term, involving w (K) and r (K), also appeared in the consumption growth equation without mortality risk in Feigenbaum and Caliendo (21). This term accounts for the fact that, by deferring a unit of consumption from t to, the household increases the capital stock by. t will change by Q(t )[w (K)e(t ) + r (K)b(t )] at every moment t of the lifecycle. Thus expected income at age Since marginal utility is λ(t)/q(t), the factor (λ(t )/Q(t ))/(λ(t)/q(t)) then converts the utility gain at t into a corresponding utility at t. The third term, involving h(t), accounts for the fact that there is a probability h(t) that the 24

26 household will die between t and t +. surviving population. a time-t measure of this utility gain. In that event, the extra saving will be spread across the whole Again the utility gain at t must be weighted by (λ(t )/Q(t ))/(λ(t)/q(t)) to obtain Overall these contributions from the effect of deferring consumption on K and B introduce a term inversely proportional to marginal utility in (86). The best market-feasible consumption profile must deviate from the standard Lifecycle/Permanent-Income Hypothesis profile of a rational household in order for it to be consistent in equilibrium with a capital stock K and a bequest B. It deviates most when the deviation will have the smallest impact on utility, i.e. when marginal utility is lowest. Now let us see how the consumption growth rate evolves over the lifecycle since this will determine the shape of the consumption profile. d 2 ln c(t) 2 = 1 γ In Appendix C we derive [( ξ Q(t) ) dh(t) λ(t) 1 + (µ + ξh(t)) Q(t) ( r(k) h(t) + (µ + ξh(t)) Q(t) )]. (89) λ(t) λ(t) Let us suppose that the hazard rate of dying h(t) is strictly increasing with age as it is empirically for adults. For an individually rational household, µ = ξ = and (89) reduces to ( d 2 ) ln c(t) 2 RCE = 1 γ dh(t) Thus the consumption profile for a rational household will be strictly concave. case is when h(t) = as in Feigenbaum and Caliendo (21). d 2 ln c(t) 2 = µ h(t)= γ <. (9) Then (89) simplifies to ( Q(t) r(k) + µ Q(t) ). λ(t) λ(t) Another interesting special For r(k)µ, which will happen if the economy is dynamically effi cient and the shadow price of capital is positive, the consumption profile will be strictly convex. For the general case, the sign of (89) depends on both the rate of change of h(t), whether the interest rate is bigger or smaller than the hazard rate, and how ξq(t)/λ(t) compares to 1. Conceivably we could obtain best market-feasible consumption profiles with many different shapes depending upon the parameters Numerical Results Now we compute the best market-feasible equilibria, both for the case where households only annuitize and where households never annuitize, for the same calibration as we used in 2.3 and compare them to the corresponding rational competitive equilibrium. Table 3 reports macroeconomic observables for these equilibria for the γ = 1 calibration, and Fig. 2 displays the corresponding lifecycle consumption profiles. Table 4 and Fig. 3 provide the same information for the γ =.5 calibration, and Table 5 and Fig. 4 for the 19 The empirical consumption profile is hump-shaped (Gourinchas and Parker (22)). According to (9), a rational household will also have a hump-shaped consumption profile. However, Feigenbaum (28) has shown it is diffi cult to find parameters for the present model with bequests that give a rational competitive equilibrium consumption profile that matches empirical data. 25

27 Regime K B U EV K/Y C/Y f B RCE Annuity % RCE Bequest % BMF Annuity % BMF Bequest % Golden Rule 388 N/A % N/A Table 3: Macroeconomic observables under the baseline calibration with γ = 1 for rational competitive equilibria and best market-feasible equilibria both under annuities and bequests as well as for the Golden Rule allocation. γ = 3 calibration. We focus on the γ = 1 calibration. We have already seen that the rational competitive equilibrium with accidental bequests confers slightly higher lifetime utility than the rational competitive equilibrium with annuities. The best market-feasible equilibrium with annuities gives higher utility than both of the rational competitive equilibria, providing the equivalent of a 3% increase in consumption in the rational competitive equilibrium with annuities. However, consistent with our finding in Section 3, the best marketfeasible equilibrium with accidental bequests does substantially better, providing the equivalent of a 9% increase in consumption relative to the rational equilibrium with annuities. Indeed, there is not much room for improvement over the best market-feasible equilibrium with bequests. The Golden Rule allocation, which confers the highest utility of any feasible allocation, only gives a.36% bigger increase in equivalent consumption. Both of the best market-feasible equilibria have capital stocks higher than the Golden Rule capital stock, so they are both dynamically ineffi cient. Thus there would be a market failure if households adopt the best market-feasible rule either with annuities or accidental bequests. The government could achieve the Golden Rule allocation by implementing wealth transfers, possibly even with a Pareto optimal transition. However, in the equilibrium with bequests, the possible gain from this effort would be negligible. Looking at Fig. 2, one might be concerned about how the BMF equilibrium with bequests is achieving such high utility. Both with annuities and accidental bequests, consumption shoots up as t T. This spike in terminal consumption was also observed in Feigenbaum and Caliendo (21), though it is blatantly counterfactual. Is all the utility coming from a gigantic consumption binge enjoyed by the small fraction of the population that survives to the maximum lifespan? In fact only.25% of lifetime utility comes from the last year of life, so the answer is no. The spike is an artifact of the assumption that everyone dies at T. 2 Since there is no bequest motive, everyone intends to die broke, yet the household keeps a huge amount of savings that must be liquidated before it dies. We could eliminate the spike if we introduce a small terminal bequest motive so a household that survives to T can pass on a portion of its wealth to the next generation. The unimportance of the spike for utility can also be seen by the fact that the BMF consumption profile with bequests nearly coincides with the Golden Rule allocation up until age 9. Since the BMF profile is 2 Since the behavior of the model at large t depends so crucially on T, it is not clear how to compute the model with an unbounded lifespan. 26

28 Figure 2: Lifecycle consumption profiles for rational competitive equilibria (RCE) and best market-feasible (BMF) equilibria both with annuities and with bequests and for the golden-rule allocation in the baseline calibration with γ = 1. 27