Optimal irrational behavior

Journal of Economic Behavior & Organization 77 (2011) 285–303

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Journal of Economic Behavior & Organization journal homepage: www.elsevier.com/locate/jebo

Optimal irrational behavior James Feigenbaum a,∗ , Frank N. Caliendo a , Emin Gahramanov b a b

Utah State University, United States Deakin University, Australia

a r t i c l e

i n f o

Article history: Received 4 May 2010 Received in revised form 29 October 2010 Accepted 3 November 2010 Available online 19 November 2010 JEL classification: C61 D11 E21 Keywords: Consumption Saving Coordination Overlapping generations Lifecycle/permanent-income hypothesis SMarT Plan General equilibrium Pecuniary externality Transition dynamics Bounded rationality

a b s t r a c t Contrary to the usual presumption that welfare in markets is maximized if consumers behave rationally, we show in a two-period overlapping generations model that there always exists an irrational consumption rule that can weakly improve upon the lifecycle/permanent-income rule in general equilibrium. The market-clearing mechanism introduces a pecuniary externality that individual rational households do not consider when making decisions but a publically shared rule of thumb can exploit. For typical calibrations, the improvement of the welfare of irrational households is robust to the introduction of rational agents. Although transitions to the optimal irrational steady state are not Pareto improving, transitions do exist that will improve a Pareto social welfare function with a sufficiently small generational discount rate. Generalizing to a more realistic lifecycle model, we find that the Save More TomorrowTM (SMarT) Plan, if properly parameterized, can confer higher lifetime utility than the permanent-income rule. © 2010 Elsevier B.V. All rights reserved.

1. Introduction The dominant paradigm in economics for at least the last three decades has assumed that people behave rationally, meaning they individually maximize their utility. Nevertheless, if we test whether people follow the basic lifecycle/permanent-income (LCPI) consumption rule that comes out of the benchmark preference specification, then empirically people often fall short. However, while there is strong evidence that households do not maximize utility, clearly they do not behave completely randomly either. The question addressed by this paper is, if irrational households interacting in a market follow a shared consumption rule, then what is the optimal rule for them to adopt? In particular, is the optimal rule the LCPI rule? If the answer to this question is yes, then it would still be reasonable to suppose that households will adopt a rule close to the LCPI rule, in which case the aggregate effects of deviations from rationality ought to be small. Remarkably, however, the answer is no. We find that, in an overlapping generations (OLG) environment, the optimal irrational (OI) consumption rule actually confers higher lifetime utility than the LCPI rule in the steady state, and the gains

∗ Corresponding author at: Department of Economics and Finance, Utah State University, Old Main Hill, Logan, UT 84322-3565, United States. Tel.: +1 4357972316. E-mail address: [email protected] (J. Feigenbaum). 0167-2681/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jebo.2010.11.002

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from switching to the OI rule can be huge. As a consequence, there is no inclination for households interacting in markets to evolve toward rational behavior in the long run. The main criticism rationalists volley at behavioral economics is the lack of a formal theory. We believe that charting the properties of irrational behaviors in market equilibria is the first step toward developing a formal, systematic theory of irrational behavior. In particular, this will help to judge what irrational behaviors should be favored by natural selection in a market environment. That arrangements exist in OLG models under which households do better than the LCPI rule is not surprising. By definition, the golden-rule allocation, which maximizes both consumption and lifetime utility only under the constraint of feasibility, must produce weakly higher utility than a rational competive equilibrium. The lure of Marxist-Leninism was that a dictator could in principle allocate goods to achieve utopia, but this could not be done in practice without substantial transition costs. Our innovation is to consider only allocations consistent with a market structure, which excludes the golden-rule allocation except for special parameterizations. The agents in our model are irrational, but they still have to finance their consumption by selling their labor and investing in competitive factor markets. From a policy perspective, if households are not following the OI rule – and we make no claim that they currently are – then the protection of markets will greatly reduce the costs of transitioning to a steady state governed by this rule. In partial equilibrium, it would not be possible for irrational agents to do better than rational agents since, by definition, rational behavior maximizes lifetime utility given exogenous prices. But in general equilibrium, markets exhibit a pecuniary externality (McKean, 1958; Prest and Turvey, 1965) in that household behavior affects prices. For example, if households save more, the capital stock will be larger, which lowers the interest rate and raises wages. In competitive markets where no one has any pricing power, rational households maximize their own utility assuming their actions do not affect prices, even though collectively their actions do affect prices. Thus individually rational households will not exploit the pecuniary externality. In an infinite-horizon, representative-agent model, the Welfare Theorems imply that rational households disregard the pecuniary externality without consequence. However, this inference is not valid for overlapping-generations models. The distinction arises because, in a representative-agent model, the decision to save more will affect future factor prices, but it has no effect on current prices. In an OLG model, a mechanism arises by which households can exploit the pecuniary externality. If a household of a given age saves more today, then a household that will be the same age next period will enjoy higher wages. Thus if all cohorts agree to save more at this age, they all receive higher wages in the steady state. A publically shared consumption rule can act as a focal point to coordinate behavior across generations, much as a rule to always do what the woman wants (or what the man wants) can avoid coordination failures in the Battle of the Sexes game. Our contribution is to quantify how much room for improvement there is if agents employ irrational consumption rules. These gains can be quite significant, which calls into question the mainstream view that agents ought to behave rationally on theoretical grounds. At first blush, our result may appear to contradict the Welfare Theorems.1 There is no inconsistency because the OI rule does not generate a Pareto optimal allocation. There is no free lunch here. If the entire population adopts the OI rule, future generations will benefit, but the first generation will lose utility because of an unfavorable change in factor prices.2 For explicative purposes, we begin with a two-period OLG model à la Diamond (1965). In this simple example, the household has only one decision: how much of his initial income should he consume or save. The LCPI rule is nested within this one-parameter family of consumption rules. We show that the LCPI rule coincides with the OI rule only for knife-edge parameterizations where the rational competitive equilibrium also coincides with the golden-rule steady state. If the LCPI rule confers a dynamically efficient market equilibrium then the OI rule is to save more. Conversely, if the LCPI rule produces a dynamically inefficient market equilibrium, then the OI rule saves less. In either case, the OI rule weakly improves upon the LCPI rule. Since no one has any power to force agents to follow the OI consumption rule, one might be concerned that this improvement will disappear if there exist rational agents who can work out the LCPI consumption rule that maximizes their own individual utility. We consider the experiment where a fraction of agents are rational and the remainder follow the OI rule (derived assuming everyone follows the rule). For the baseline calibration, two thirds of the population must be rational to create a situation where the irrational followers do worse than they would if everyone followed the LCPI rule. Thus even if some agents do behave rationally, this need not destroy the result that everyone can be made better off than they would be in the rational market equilibrium. Next, as a concrete application of our idea, we consider a family of rules that have actually received popular interest for practical implementation. In Thaler and Benartzi’s (2004) Save More Tomorrow (SMarT) Plan, employees commit today to save some fraction of future wage increases. The plan is also referred to as escalated saving. Several leading investment institutions in the U.S. have given a pool of workers numbering into the millions the option of participating in such a plan, and the response has been quite positive.3

1 The Welfare Theorems are not strictly applicable to OLG models, which differ from their finite-horizon Arrow–Debreu counterparts (Geanakoplos, 2008; Weil, 2008) because of a lack of market clearing at infinity. This allows for the possibility of dynamic inefficiencies. However, for plausibly calibrated OLG models, the rational competitive equilibrium will be Pareto efficient. 2 This neoclassical embodiment of Keynes’ (1936) paradox of thrift is not just a property of irrational consumption rules. This also happens if the population initially saves at a less than rational rate and then becomes rational. 3 In the 2006 Retirement Confidence Survey, 65 percent of workers said they would like to participate in a SMarT Plan (Helman et al., 2006). As evidence of the widespread political support, the recent Pension Protection Act of 2006 creates incentives for firms with 401(k) plans to make the default setting a

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Extending the two-period model to a continuous-time OLG model, we find under standard calibrations that the SMarT economy dominates the permanent-income economy for a wide range of rates for saving out of income increases. If we begin with a general-equilibrium economy of naive consumers who follow the very simple consumption rule of saving 3.5 percent of all income (just as in the pilot implementations of the SMarT Plan at a U.S. manufacturing firm), then these naive savers would do worse than consumers in the rational competitive equilibrium. However, if these consumers all commit to save between 55 percent and 85 percent of future wage increases on top of the 3.5 percent from their current income, lifetime utility will exceed the level reached in the permanent-income economy.4 The paper is summarized as follows. In Section 2, we show in the two-period model that there is always a consumption rule that can weakly improve upon the LCPI rule in general equilibrium, and we consider experiments where the economy transitions between different consumption rules. In Section 3, we present quantitative results for the implementation of the SMarT Plan in a more realistic lifecycle model. We conclude in Section 4 with some discussion of the larger implications of these results. 2. A simple two-period example 2.1. The basic model The pecuniary externality that allows us to improve upon the LCPI rule can best be understood in terms of a two-period OLG model. In each period, a continuum of agents of unit measure is born and lives with certainty for two periods. Consumers value allocations over the lifecycle according to the preferences U(c0 , c1 ) = u(c0 ) + ˇu(c1 ),

(1)

where the discount factor ˇ > 0, u( · ) is a strictly increasing, strictly concave, twice-differentiable utility function, and ct is consumption at age t = 0, 1. A consumer at age t has an endowment of et efficiency units of labor that he supplies inelastically to the market, for which he is compensated at the real wage w. The consumer can invest in capital K at age 0, for which he is compensated at the (gross) rate of return R. Thus the consumer faces the budget constraints c0 + K = we0

(2)

c1 = we1 + RK.

(3)

A rational consumer will choose c0 , c1 , and K to maximize (1) subject to (2) and (3). This can be reduced to the problem of choosing the saving K to maximize Ld (K|R, w) = u(we0 − K) + ˇu(we1 + RK),

(4)

where d indicates this is the allocation problem for a decentralized economy. However, we do not assume that all the agents in this economy are rational. While all consumers have preferences described by (1), some or all of them may not be capable of solving the problem of maximizing Ld (K|R, w). On the production side of the economy, we assume there is a competitive firm with the production function Y = F(K, N),

(5)

where K is the capital stock and N is the supply of labor as measured in efficiency units. In equilibrium, the aggregate capital stock K must equal the saving of young agents and the labor supply must satisfy N = e0 + e1 .

(6)

We assume that F exhibits constant returns to scale, so the production function can be rewritten F(K, N) = f

K  N

N,

(7)

where f is a strictly increasing, strictly concave, twice-differentiable function. Factor prices are determined by marginal principles under perfect competition. Capital depreciates at the rate ı ∈ [0, 1], so the gross rate of return on capital (net of depreciation) is R(K) = FK (K, N) + 1 − ı = f





K e0 + e1



+1−ı

(8)

SMarT Plan, meaning workers would have to actively opt out to not participate (Moore, 2006). 4 Such saving rates may seem preposterously high, but in actual practice people did commit to such rates in the pilot program (Thaler and Benartzi, 2004).

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while the real wage is w(K) = FN (K, N) = f



K e0 + e1



−

K f e0 + e1



K e0 + e1

 .

(9)

In this simple example we focus on steady-state no-growth equilibria, which we define as follows. A generalized steadystate market equilibrium consists of a consumption rule (c0 , c1 ), a capital stock K, and factor prices (R, w) such that (i) the consumer’s budget constraints (2) and (3) are satisfied and (ii) the firm’s profit-maximizing conditions (8) and (9) are satisfied.5 If, in addition, K maximizes Ld (K|R, w), we will refer to the equilibrium as a rational competitive equilibrium, but in general we forgo this rationality condition. We consider (c0 , c1 , K, R, w) to be a generalized equilibrium as long as these variables are feasible and consistent with market-clearing. Instead of assuming that consumers independently choose to follow a consumption rule that maximizes their utility, we take the view here that the bulk of households follow consumption rules inculcated by parental or societal authorities.6 We can then ask what is the optimal consumption rule for the population to coordinate upon. 2.2. Optimal irrational steady state The generalized steady-state market equilibrium concept allows us to straightforwardly define what we mean by an optimal rule for consumption and saving decisions. We cast this problem as a social planner’s problem. However, this problem differs from the familiar Pareto social planner’s problem, which has Lagrangian Lg (c0 , c1 , K) = u(c0 ) + ˇu(c1 ) + [F(K, e0 + e1 ) − c0 − c1 − ıK],

(10)

since we require more than that allocations be technologically feasible. Because our households continue to interact through markets, a consumption rule must produce an allocation that satisfies (2) and (3) and (8) and (9) in addition to technological feasibility. Note that our social planner should only be viewed as a convenient fiction. In particular, the social planner has no dictatorial power to enforce an allocation that could not be achieved through existing institutions. We could imagine that the social planner advises households what consumption rule to use, but we do not model the process by which households acquire their consumption rule, so we could just as well imagine that it is the result of some process of natural selection. In this simple two-period OLG model, the conditions that define a generalized market equilibrium express c0 , c1 , R, and w as functions of the capital stock K. Thus the OI social planner’s problem reduces to choosing K to maximize Lc (K) = u(w(K)e0 − K) + ˇu(w(K)e1 + R(K)K),

(11)

where the c subscript denotes that there is coordination across generations. The distinction between (4) and (11) is that the market-clearing conditions are imposed after the decentralized household maximizes (4) whereas in (11) the marketclearing conditions are included as constraints on the optimization.7 Let us define Kc to be the capital stock in the OI generalized market equilibrium that maximizes Lc , Kd the capital stock for the (decentralized) rational competitive equilibrium, and Kg the golden-rule capital stock, defined by R(Kg ) = 1. For the case of the two-period model presented above, the following proposition, proved in Appendix A, addresses the question of whether the LCPI consumption rule is optimal in the steady-state market equilibrium. Proposition 1. The optimal irrational consumption rule consistent with a generalized market equilibrium coincides with a rational competitive equilibrium only if Kc = Kd = Kg . If the functional form of f and u is such that Lc is a strictly concave function, then if the rational competitive equilibrium is dynamically efficient (i.e. Kd ≤ Kg ) Kc ≥ Kd , with equality only if Kd = Kg . If the rational competitive equilibrium is dynamically inefficient, the opposite inequalities hold.8 The LCPI consumption rule will only be optimal for the knife-edge set of parameters where the rational competitive equilibrium is the golden-rule steady state obtained by maximizing Lg . A dictatorial social planner who can ignore markets will choose Kg as the capital stock and impose a consumption allocation equivalent to the LCPI rule with R = R(Kg ) = 1. Note that a dictatorial planner with a Pareto social welfare function will also impose a consumption allocation equivalent to the LCPI rule, though with R > 1. Since in general the OI rule will not correspond to an LCPI rule for any R, the OI rule will generally not be Pareto optimal. The intuition behind Proposition 1 can best be understood by comparing the present model to one where there is a single cohort that lives for two generations with a given initial capital stock. In this single-cohort model, we will get the same

5

These conditions are easily generalized to allow for a balanced-growth equilibrium as in Section 3. Allen and Carroll (2001) have argued there simply is not enough time for individual consumers to adopt the LCPI rule (or even a linear approximation to it) within one lifecycle in a model with endogenous learning. 7 Households that follow the consumption rule that maximizes (11) are not individually rational according to traditional terminology, which is why we classify them as irrational, but one could in some sense view them as being hyperrational. 8 In Appendix B, we show the condition for this stronger result holds if the production function is Cobb–Douglas. With this specific functional form, Feigenbaum (2010) has gone on to prove the even stronger result that for a dynamically efficient rational competitive equilibrium Kd ≤ Kc ≤ Kg and for a dynamically inefficient rational competitive equilibrium Kg ≤ Kc ≤ Kd . 6

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consumption allocation in each of three regimes: (i) we assume that agents individually maximize their lifetime utility subject to market-determined budget constraints while assuming prices, which are determined by market-clearing conditions, are fixed; (ii) a social planner maximizes the cohort’s lifetime utility subject to the technological feasibility constraint; (iii) a social planner maximizes the cohort’s lifetime utility subject to market-determined budget constraints while accounting for the dependence of prices on the allocation. Regimes i and ii correspond respectively to the usual decentralized and usual social planner’s problem, and the Welfare Theorems guarantee these problems have the same solution. Regime iii is the variation on the planner’s problem that we introduce here. Its equivalence to the other regimes arises because the initial capital stock is exogenous. Only in Regime iii will the social planner exploit the pecuniary externality (since there are no factor prices in Regime ii), but with a constant-returns-to-scale production function, derivatives of factor prices with respect to the second-period capital stock cancel out. Since the initial capital stock is not a choice variable, any influence it might have on factor prices is irrelevant. Thus the solution to the problem in Regime iii is the same as in Regime ii and thereby Regime i. On the other hand, if we solve for a steady state of the OLG model, maximizing Ld , Lg , and Lc (corresponding to Regimes i, ii, and iii respectively) will, in general, have three different answers as we demonstrate in the following example. The driving distinction between the OLG and single-cohort environments is that in a steady state of an OLG model the capital stock is endogenous. If we require agents to trade through markets, the effect of the capital stock on factor prices matters in an OLG model. 2.3. Calibrating the example To demonstrate that Proposition 1 is not just an abstract result but can actually matter for typical calibrations of the macroeconomy, let us consider a numerical example. If we assume u(c) = lnc and a Cobb–Douglas production function with depreciation, the two-period model has five free parameters: the share of capital ˛, the discount factor ˇ, the depreciation rate ı, and the income endowments e0 and e1 for young and old workers respectively. We set the technology parameters to typical values from the literature: ˛ = 1/3 and ıan = 0.10. Likewise, we set ˇan = 0.96.9 It is easier to obtain equilibria with bizarre properties, such as dynamic inefficiency, if old agents have no income endowment since young consumers will have to save at any interest rate, so we normalize e0 to 1 and set e1 = 1/3, imagining that the first period of life covers ages 25–55 and the second period covers the working years 55–65 as well as a retirement period. In this two-period model, the determination of the capital stock K invested by young workers fully specifies the consumption rule, but it is more natural to characterize the saving rule in terms of the saving rate s(K) =

K w(K)e0

(12)

of young workers out of their labor income. Let sd denote the saving rate of the rational competitive equilibrium, i.e. for the decentralized equilibrium where consumers make their choices individually (and rationally) to maximize the decentralized objective (4). Let sc denote the OI saving rate that produces the capital stock that maximizes the coordinated social planner’s objective (11). Fig. 1 shows lifetime utility U as a function of the saving rate s (i.e. U(s) = Lc (K(s)), where K(s) is the inverse of s(K) as defined by (12)). The dashed horizontal line is the lifetime utility that would be obtained under the decentralized equilibrium, and the double line is utility from the golden-rule steady state. For our baseline parameters, the decentralized saving rate is sd = 16 %. The OI saving rate that maximizes (11) is sc = 39 %. Note that any saving rate between sd = 16% and 65% will give rise to equilibria where the consumer is better off than in the decentralized equilibrium. It is not necessary to increase saving all the way to 40 percent to improve welfare for everyone, if that should prove unsustainable. Fig. 1 demonstrates the optimality of sd and sc for their respective problems. The thick red curve plots lifetime utility as a function of the saving rate in general equilibrium. This is the objective Lc (K(s)) of the social planner and is maximized at sc . The thin blue curve plots lifetime utility as a function of the saving rate in a partial equilibrium where prices are held fixed at their decentralized equilibrium values. This is the objective Ld (sw(K(sd ))e0 ) of a rational household in the decentralized equilibrium and is maximized at sd . (For interpretation of the references to color in text, the reader is referred to the web version of the article.) Since we do not have any growth of population or technology in this simple model, a generalized market equilibrium will be dynamically efficient if the net interest rate r is nonnegative. Fig. 2 shows the equilibrium interest rate (per annum) as a function of the saving rate s and for the decentralized equilibrium. Note that the threshold saving rate above which r ≤ 0 is sr=0 = 69 %, so for this example all saving rates that improve upon the decentralized equilibrium also lead to dynamically efficient equilibria, including the OI saving rate sc . Fig. 2 also demonstrates that the variation in interest rates between these two equilibria is quite large, for r decreases from 4.78 percent per annum to 1.83 percent. We see similar variation in other important macroeconomic observables. Fig. 3 shows the variation of aggregate output Y as a function of the saving rate. Going from the decentralized equilibrium to the

T 9 A period is viewed as lasting for T = 30 years, so the per-period discount factor is ˇ = ˇan and the per-period depreciation rate is ı = 1 − (1 − ıan )T . All time-dimensionable quantities specified in the text will be given in annual terms.

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Fig. 1. Lifetime utility in the baseline two-period model as a function of the saving rate s in general equilibrium and in partial equilibrium, holding the prices fixed at both the decentralized equilibrium prices and the optimal irrational (coordinated) equilibrium prices. The dotted line corresponds to lifetime utility in the decentralized equilibrium while the double line is the utility of the golden-rule steady state.

OI saving rate, output increases by 50 percent. Thus if consumers coordinate on a higher saving rate than is accorded by the LCPI rule, the economy can achieve a remarkable degree of greater prosperity. Even if consumers only deviate from the LCPI rule by a modest amount, say increasing their saving rate to 18 percent, output would increase by 5 percent. It is difficult to interpret directly how much of an improvement the change in lifetime welfare from U = − 2.29 for the decentralized equilibrium to U = − 2.05 for the OI generalized equilibrium actually confers. A more informative measure of how the two equilibria compare is the compensating variation , the fraction by which the consumption in each period of a household in the decentralized equilibrium would have to be increased to equalize the utility of a given generalized equilibrium. For a generalized steady-state equilibrium with consumption ct at age t,  solves the equation u((1 + )c0d ) + ˇu((1 + )c1d ) = u(c0 ) + ˇu(c1 ),

(13)

14.00% saving rule decent. equil. optimal rule

12.00%

interest rate

10.00%

8.00%

6.00%

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0.00%

-2.00% 0

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Fig. 2. Net interest rate r (per annum) in the baseline two-period model as a function of the saving rate s and for the decentralized equilibrium. The interest rate for the OI saving rate sc is represented by a large dot.

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1 0.9 0.8 saving rule decent. equil. optimal rule

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saving rate Fig. 3. Aggregate output Y in the baseline two-period model as a function of the saving rate s and for the decentralized equilibrium. Output for the OI saving rate sc is represented by a large dot.

where ctd is consumption at age t in the decentralized equilibrium. For the case of logarithmic utility, Eq. (13) is easily solved analytically to obtain



 = exp

U − Ud 1+ˇ



− 1,

(14)

where U is lifecycle utility in the generalized equilibrium and Ud is utility for the decentralized equilibrium. For the OI generalized equilibrium, the compensating variation is 20.4 percent. Given that the compensating variation from eliminating business cycles is only half a percentage point (Lucas, 2003), the potential benefits of coordinating across generations are huge. Nevertheless, while the consumers in the OI generalized market equilibrium are better off than they would be in the decentralized equilibrium, as individuals they could increase their lifetime utility even more if they maximize their own utility. The dashed purple curve in Fig. 1 plots lifetime utility as a function of the saving rate in a partial equilibrium where prices are held fixed at their values from the OI generalized equilibrium. This corresponds to Ld (sw(K(sc ))e0 ), which would be the objective function of a rational household if everyone else follows the OI consumption rule. It is maximized at 7.5 percent, the saving rate prescribed by the LCPI rule, for which the rate of consumption growth c1 /c0 equals ßR. If households save sc = 39 %, their consumption will increase by 65 percent from the first period to the second, but ˇR = 0.50 in per-period terms. Because the interest rate is so low, the consumer could increase his welfare from −2.05 to −1.86 if he saves only 7.5 percent, enjoying the bulk of his consumption while young and letting his consumption drop by 50 percent when he is old. If only a couple agents deviate in this way, this should not affect aggregate quantities. But if a measurable fraction of the population behaves rationally, this will have a negative impact on the equilibrium. So next we consider what happens if rational agents invade the population. 2.4. An influx of rational agents Now suppose that a fraction  of the population is rational and chooses its saving kr (R, w) =



w e1 ˇe0 − R 1+ˇ



(15)

to maximize Ld (kr |R, w) given the market-determined R and w. The remaining fraction 1 −  follows the OI saving rate sc that maximizes Lc (K(s)) and chooses its saving according to ki (w) = sc we0 given

w.10

(16)

The aggregate capital stock will be

K = kr + (1 − )ki .

10 We could also consider the case of a smarter social planner who takes into account the fact that some agents will deviate from the rule of thumb. This fine tuning would improve welfare even more for the irrational agents, but it would require information that may be difficult to obtain.

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Fig. 4. Lifetime utility for both rational and irrational agents in the baseline two-period model as a function of the fraction of rational agents . Lifetime utility for the decentralized equilibrium is the dashed line. 0.14 irrational rational

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Fig. 5. Saving (in absolute terms) by a rational agent kr and by an irrational agent ki as functions of the fraction of rational agents  for the two-period baseline model.

A generalized market equilibrium will then be determined by the capital stock K that solves the market-clearing condition for capital K = kr (R(K), w(K)) + (1 − )ki (w(K)). As a concrete example, let us consider what happens if we inject rational agents into the model under the calibration of Section 2.3. Fig. 4 graphs the lifetime utility of both rational and OI agents as a function of the fraction of rational agents . As we discussed above, rational agents always do better than OI agents and, moreover, always do better than in the decentralized equilibrium. Indeed, the rational agents are essentially taking advantage of the OI agents, who through their high saving allow the economy to enjoy high wages. However, unlike in most models where rational and irrational agents coexist, while the rational agents end up with higher lifetime utility they do not end up with higher wealth. Fig. 5 shows how the individual saving of the two types of agents varies with . In all cases, an OI agent saves substantially more than a rational agent.11

11 Of course, in this model there is no advantage to having more wealth. If we added wealth to the utility function as in Carroll (2000), this would reinforce the gains from coordination across generations that would accrue from higher saving since this would increase utility directly (instead of just through more advantageous factor prices).

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1

Fig. 6. Aggregate output Y in the baseline two-period model as a function of the fraction of rational agents .

The lifetime utility of both rational and OI agents decline with , and they both do so nearly linearly. Consequently, a small influx of rational agents does not destroy the result that the OI rule can improve upon the decentralized equilibrium. Indeed, the utility of OI agents only goes below the decentralized utility in Fig. 4 for  > 2/3. As long as less than two thirds of the population is rational, both the rational and the OI agents continue to enjoy greater lifetime utility than they could attain in the decentralized equilibrium where everyone is rational.12 Fig. 6 shows how aggregate output varies with the fraction of rational agents . For the baseline parameters, output is a monotonic, nearly linear function of . Thus while rational agents diminish the material prosperity that the social planner could achieve if everyone followed his advice, they only do so in proportion to . Macroeconomic quantities behave continuously in the limit as  → 0 and are not especially sensitive to small changes in . 2.5. Transition dynamics with irrational start Because we did not solve a standard Pareto problem to arrive at the OI rule, there is no reason to expect that we can arrive at the OI steady state in a Pareto-improving fashion, and, indeed, we cannot. Since a common criticism of steadystate optimality criteria (Arrow, 1999) is that they lead the social planner to immiserate early cohorts in order to build up the optimal capital stock, we conclude our discussion of the two-period model by considering what happens in various transitional experiments. As in the Solow (1956) model, if consumers follow a given consumption rule and if there is a unique steady-state equilibrium consistent with this rule, the economy will converge to this steady state from most initial capital stocks. Given an initial capital stock K0 , we define a generalized dynamic market equilibrium as a sequence of consumption rules {c0,t , c1,t }∞ ,a t=0 sequence of factor prices {Rt , wt }∞ t=0 , and a saving rule Kt+1 (wt ) such that (i) the budget constraints c0,t + Kt+1 = wt c1,t = Rt Kt are satisfied for all t, and (ii) the profit-maximizing conditions wt = w(Kt ) Rt = R(Kt ) are satisfied for all t. We consider experiments where the economy starts in either the rational competitive equilibrium steady state or a steady state market equilibrium with a saving rate even less than the LCPI rate, and at t = 0 either households follow the OI saving rule or begin to behave rationally. Since most evidence suggests present-day households save less than is dictated by the LCPI rule, let us first consider what happens if the economy begins in a steady-state market equilibrium with a very low saving rate of s = 3.5 %. For this particular experiment, we assume there is no income in old age, so we obtain an analytic expression for the rational dynamics. Thus we set e1 = 0 and normalize e0 = 1, leaving other parameters the same as in Section 2.3. Lifetime utility in the initial state is −3.15, and the compensating variation relative to the rational steady state is −53 %.

12

Interestingly, the cross-sectional mean of lifetime utility across the population is higher than the decentralized utility for all values of .

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Fig. 7. The compensating variation relative to the rational competitive equilibrium steady state as a function of cohort birth date in two transitional experiments starting from a steady state with a 3.5 percent saving rule: going to the LCPI rule and the optimal irrational saving rule.

We consider two types of generalized dynamic market equilibria. Given factor prices Rt+1 (which turns out to be irrelevant under our assumptions) and wt faced by an individual born at time t, the saving rule that maximizes (1) is the rational, LCPI saving rule r = Kt+1

ˇ wt , 1+ˇ

(17)

which also happens to be a constant saving rule.13 Meanwhile, if we have irrational consumers who follow the consumption rule that is optimal in the steady state, the saving rule will be i = sc wt . Kt+1

(18)

After setting e1 = 0, the LCPI saving rate is sd = ˇ/(1 + ˇ) = 23 % and confers steady-state lifetime utility of −2.18. For this calibration, the OI saving rate remains sc = 39 %, though now the steady-state lifetime utility is −2.06, for which the compensating variation relative to the rational steady state is 9 percent. If households adopt the OI saving rule at t = 0, the first generation has to reduce its lifetime utility to −3.38 (or a compensating variation of −60 %), but everyone is better off than in the initial steady state from there on out. In Fig. 7, we plot the compensating variation (relative to the rational steady state) of the lifetime utility for each cohort as a function of the cohort’s birth date. The fact that the first generation suffers a utility loss is not a property of following an irrational consumption rule. If households become rational at t = 0, the first generation will still reduce its lifetime utility from −3.15 to −3.19, again because of the resulting decrease in interest rates.14 While it is true that adopting the OI saving rule will involve some pain for early adopters, the same would also be true if economists convinced more households to behave rationally. Moreover, it is only the first generation that fares worse during the transition to the OI steady state than they would if they transitioned to the rational steady state. Starting with the second generation, all future generations are better off. As the next experiment demonstrates, if households only gradually ramp up their saving rate to the OI saving rate, they can spread the pain over several generations. This is not possible if households suddenly become rational. 2.6. Transition dynamics with rational start As an alternative experiment in transition dynamics, let us now consider what happens if the economy begins in the rational competitive equilibrium steady state. Beginning at t = 0, households subscribe to the advice of a social planner about

13 If e1 > 0, we would have to account for the present value of future income w(Kt+1 )e1 /R(Kt+1 ), and we would not obtain an analytic expression relating Kt+1 to Kt . 14 If the initial saving rate is unrealistically low, it is possible that implementing the LCPI rule can be done without any utility loss by the first generation.

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25.00%

20.00%

compensating variation

15.00%

10.00% ν = 3.33 ν = 0.019

5.00%

ν = 0.008 ν = 0.004 ν = 0.002

0.00% 0

10

20

30

40

50

60

70

80

90

ν = 0.0001

100

ν = 0.00002

-5.00%

-10.00%

-15.00%

cohort birth date -20.00%

Fig. 8. The compensating variation relative to the rational competitive equilibrium steady state as a function of cohort birth date starting from a rational competitive equilibrium steady state and transitioning to the OI steady state for various choices of the convergence rate ␯.

how much to save. Since this experiment does not require us to solve for rational behavior during the transition, we return to the baseline calibration of Section 2.3, where e1 = 1/3. First suppose the social planner advises households to immediately follow the OI saving rate sc = 0.39. The compensating variation relative to the rational steady state during this transition is the thick black line in Fig. 8. Households reach the OI steady state within 15 generations. However, just as in the transition experiments of Section 2.5, the first generation will suffer a utility loss because of the decrease in interest rates. In the case of this immediate transition, the initial utility loss is quite significant, equivalent to a 16 percent decrease in consumption in both periods. However, a social planner need not instruct households to jump directly from the rational saving rate sd = 0.16 to the OI saving rate of 0.39. Alternatively, he could introduce a time-varying saving rule, slowly ramping up the saving rate so it converges to sc at the rate v: s(t) = sc + (sd − sc ) exp(−t).

(19)

Fig. 8 also displays the transition path of the compensating variation relative to the rational steady state for various choices of the convergence rate  (measured in annual terms in the legend). Clearly, we cannot transition from the rational competitive equilibrium to the OI steady state in a Pareto-improving fashion. Nevertheless, we can ask whether Pareto social planners exist who might prefer an irrational allocation to the rational steady state. Consider a social planner with an intergenerational discount rate  > 0. The social welfare function maximized by this planner will be W = exp()ˇu(c1,0 ) +

∞ 

exp(−t)[u(c0,t ) + ˇu(c1,t+1 )].

(20)

t=0

Assume the initial capital stock is the steady state capital Kd of the rational competitive equilibrium. The allocation of consumption obtained in this equilibrium is Pareto optimal because it maximizes W for  = lnR(Kd ). Thus if the planner’s generational discount rate equals the rational competitive interest rate, no consumption allocation can confer a higher value of W than the rational competitive allocation. For this social welfare function, there is no transition path to the OI steady state that can outperform the rational competitive equilibrium. But what happens if the social planner discounts future generations at a lower rate than lnR(Kd )? We consider transition paths parameterized by  as in Eq. (19). Table 1 shows the  that maximizes W for various choices of  < lnR(Kd ). These are also the  for which the corresponding transition path is plotted in Fig. 8. In each of these six examples, where  ranges Table 1 Optimal convergence rate  and compensating variation w for representative generational discount rates . Both  and  are measured annually. 

4.77%

4.7%

4%

3%

2%

1%

 w

1.6 × 10−5 2.09 × 10−8

0.00015 1.94 × 10−6

0.0016 0.000287

0.0043 0.00276

0.0080 0.0131

0.019 0.0526

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between 1 percent and just under the equilibrium interest rate of 4.78 percent (measured annually), the optimal  confers higher social welfare than what would be obtained if the economy remained in the rational competitive equilibrium steady state. To see this, the compensating variation of W for the optimal transition relative to the rational steady state is also given in the table. To be more precise, the compensating variation w of the consumption allocation cs,t relative to the allocation d solves the equation cs,t d )+ exp()ˇu(c1,0

∞ 

d d exp(−t)[u((1 + w )c0,t ) + ˇu((1 + w )c1,t+1 )]

t=0

= exp()ˇu(c1,0 ) +

∞ 

exp(−t)[u(c0,t ) + ˇu(c1,t+1 )],

t=0 d (leaving consumption unchanged for the initial old agents). setting W equal for the allocations cs,t and (1 + w )cs,t If  = 4.77 %, the transition to the OI steady state must be done extremely slowly to achieve higher welfare than the rational steady state, and even then the equivalent gain in consumption is only two parts in one hundred million. As we decrease , the rate of convergence increases, and the compensating variation becomes more substantial. For a generational discount rate of 1 percent, the social planner can achieve the equivalent of increasing consumption by 5 percent if he commands his charges to transition to the OI steady state. As  → 0, the convergence rate  → ∞, and the compensating gain will converge to the 20 percent gain enjoyed by all but the very first generations after an instantaneous transition to the OI saving rate. For any generational discount rate between zero and the rational interest rate, OI behavior is preferable to rational behavior, even after accounting for the transition. Although a generational discount rate of 4.78 percent or more may seem reasonable in the abstract, it would mean the well-being of each succeeding generation should be valued at 24 percent of its parents’. The U.S. Constitution did not even discount slaves by that much, so it would be difficult for a policymaker to argue that he ought to prefer the rational steady state over every feasible transition to the OI steady state. Weitzman (1999), for example, suggests that a generational discount rate close to the market rate should only be used in the short term. At horizons of 50 years, corresponding to 1–2 periods in our model, he would set the rate to 2 percent per annum. For horizons on the order of hundreds of years he would reduce it to 1 percent per annum, and in the limit of very large horizons he would set it close to 0.

3. SMarT plan in general equilibrium To show that the results of the previous section do not depend on the special assumption that agents live for two periods, we now consider the opposite extreme of a continuous-time OLG model. This greatly enlarges the space of possible consumption rules since consumers now face multiple saving decisions. We focus on the one-dimensional family of rules that fall under the heading of Thaler and Benartzi’s (2004) Save More Tomorrow or SMarT Plan, which policymakers actually have encouraged people to adopt.15

3.1. The continuous-time lifecycle model First, we describe the environment faced by an individual agent “born” at time  ∈ R, which is a generalization of the model in Bullard and Feigenbaum (2007). The agent starts work at , retires at time  + T, and passes away at time  + T¯ , where 0 < T < T¯ . During his working life, the agent earns the real wage w(t), which grows at the rate x that corresponds to the rate of technological growth for the economy. The agent can borrow or save at the rate of return r.16 Let c(t, ) denote the flow of consumption at t of an agent born at  and k(t, ) denote the corresponding stock of savings. Then the agent faces the differential budget constraint dk(t, ) = dt



rk(t, ) + w(t) − c(t, ) rk(t, ) − c(t, )

t ∈ [,  + T ] t ∈ [ + T,  + T¯ ]

(21)

along with the boundary conditions k(, ) = k( + T¯ , ) = 0.

(22)

15 The SMarT Plan consumption rule, as presented in Section 3.1, is actually characterized by two parameters: the SMarT Plan saving rate and the base saving rate. However, we would view the base saving rate as an exogenous parameter since it describes how consumers would behave if left to their own devices. Only the SMarT Plan saving rate is a policy instrument under the control of our fictional social planner. 16 Both w(t) and r are market-determined as we describe in Section 3.2.

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Analogous to the two-period model, we assume that the agent values consumption allocations according to the utility function



T¯

e−t u(c( + t, ); )dt,

U() ≡

(23)

0

where



1 c 1− 1− ln c

u(c; ) =

= / 1 =1

for > 0. However, the consumer is not able to compute the consumption rule that maximizes (23) subject to (21) and (22). Instead, during the working life the agent would, absent any direction, follow the consumption rule of saving a constant fraction of income. After retirement, we assume the agent’s savings account pays out the annuity value of the account,

+T 

a() =

(w(t) − c(t, ))er(+T −t) dt

+T¯ +T

,

(24)

er(+T −t) dt

that confers a perfectly smooth flow of consumption while running savings down to zero at the time of death  + T¯ . Thus the basic consumption rule is



cb (t, |s) =

(1 − s)w(t) a()

t ∈ [,  + T ] . t ∈ [ + T,  + T¯ ]

Now suppose the social planner implements a SMarT Plan that requires agents to save at a higher rate ≥ s out of any wage increases that accrue at t ≥ . Thus under the SMarT Plan, the consumption rule is



cS (t, |s, ) =

w(t) − (sw() + [w(t) − w()]) a()

t ∈ [,  + T ] . t ∈ [ + T,  + T¯ ]

Note that this nests the basic consumption rule with = s. The path of the individual’s savings account is



t

{sw() + [w(u) − w()]}er(t−u) du

k(t, ) =

t ∈ [,  + T ]

(25)





+T¯

a()er(t−u) du

k(t, ) =

t ∈ [ + T,  + T¯ ].

(26)

t

3.2. The continuous-time OLG model In general equilibrium, we must consider all agents alive at a given time. At each instant t a new cohort of population N(t) is born and the oldest cohort, born at t − T¯ dies. The size of each successive cohort grows at rate n (and hence the total population also grows at rate n), so the size of a cohort born at time t is N(t) = ent , where we normalize the cohort population at t = 0 to one. We endogenize factor prices by incorporating a production sector that employs capital and labor as factors. In equilibrium, the capital stock K(t) is equated to the aggregate sum of savings of consumers



t

K(t) =

N()k(t, )d.

(27)

t−T¯

Capital also depreciates at the rate ı. Unretired consumers inelastically supply one unit of labor, so the labor supply is



t

N()d = L(0)ent .

L(t) =

(28)

t−T

Firms behave competitively and have a constant returns to scale production function Y (t) = K(t)˛ [A(t)L(t)]1−˛ , where A(t) = ext is a labor-augmenting technology factor that grows at the constant rate x. Factor prices are determined competitively:

 K(t) ˛−1

r(t) = ˛

A(t)L(t)

−ı

(29)

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-24.4

-24.6

Utility

-24.8

-25

-25.2

-25.4

-25.6

-25.8

-26 0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

SMarT Rate Fig. 9. Lifetime utility for three types of economies with = 2 and  = 0.02. Thin black line is the permanent-income rule. Blue dashed and red solid lines are rule-of-thumb economies respectively with high (7 percent) and low (3.5 percent) base saving rates. (For interpretation of the references to color in text, the reader is referred to the web version of the article.)

 K(t) ˛

w(t) = (1 − ˛)

A(t)L(t)

A(t).

(30)

For this model, a generalized steady-state market equilibrium is characterized by a consumption rule c(t, ) = c0 (t − )ext in which consumption depends only on the household’s age t −  (after accounting for technological growth), a saving rule k(t, ) = k0 (t − )ext , a time-independent interest rate r, and a wage function w(t) = w0 ext such that (i) c(t, ) and k(t, ) satisfy (21) and (22) given r and w(t) and (ii) r and w(t) satisfy (29) and (30) given k(t, ). For comparison with the LCPI rule, we will also consider rational competitive equilibria, which satisfy the additional condition that c(t, ) and k(t, ) maximize (23) under the constraints (21) and (22). 3.3. The optimal SMarT plan saving rate Let us consider the optimal steady-state SMarT Plan saving rate if we calibrate the model using parameter values typical of the literature on general-equilibrium lifecycle models. The rate of population growth n is set to 1 percent to match the U.S. experience over the last few decades. Following Bullard and Feigenbaum (2007), the rate of technology growth x is set to 1.56 percent. The technology parameters are set to common values of ˛ = 0.35 for the share of capital and ı = 8 % for the depreciation rate. The worklife T is 40 years and the lifespan T¯ is 55 years, which if we assume that the working life begins at age 25 is consistent with an average lifespan of roughly 80 years. Finally, we set the preference parameters to = 2 for the inverse elasticity of intertemporal substitution and  = 2 % for the discount rate, which are both near the middle of the broad range of values that researchers have considered for these difficult-to-pin-down parameters. Fig. 9 plots three curves. The first is a flat line (black) set equal to the level of lifetime utility in the rational competitive equilibrium for our calibration of and . The other two curves depict lifetime utility as a function of the SMarT saving rate

in steady-state market equilibria where consumers participate in a SMarT Plan. For the dashed, blue curve, we assume consumers have a default saving rate of s = 7 % whereas for the solid, red curve s = 3.5 %. We see that the optimal SMarT rate is near 60 percent when s = 7 % and is near 70 percent when s = 3.5 %. Keep in mind these are saving rates out of raises, and not with respect to total income. It is clear that the result proved in Section 2 is not academic. It can be generalized to more realistic and empirically relevant models with consumption rules that are straightforward enough that consumers could practically implement them. Economy-wide participation in a program such as the SMarT Plan can push an economy in general equilibrium from a level of utility that is initially below the permanent-income utility (i.e. when = s) to a new level that is even higher than the permanent-income utility. Fig. 10 is exactly the same as Fig. 9 except that the discount rate  has been lowered from 2 percent to 0 percent. Comparing Figs. 9 and 10, we find that the optimal SMarT rate is not terribly sensitive to the discount rate. However, in this case we find that the LCPI rule achieves superior lifetime utility than the SMarT Plan for any value of the SMarT saving rate

. This does not, however, imply that there is no consumption rule that can outperform the LCPI rule in general equilibrium

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-38

-38.5

Utility

-39

-39.5

-40

-40.5 0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

SMarT Rate Fig. 10. Lifetime utility for three types of economies with = 2 and  = 0. Thin black line is the permanent-income rule. Blue dashed and red solid lines are rule-of-thumb economies respectively with high (7 percent) and low (3.5 percent) base saving rates. (For interpretation of the references to color in text, the reader is referred to the web version of the article.)

since we have only considered a one-dimensional subspace of the entire space of feasible consumption rules. Indeed, for the high-saving economy, the optimal SMarT Plan comes very close to matching the utility of the LCPI rule, so it is likely that a small perturbation of the SMarT Plan rule could outperform the permanent-income economy.17 One criticism of the SMarT Plan is that if consumers opt to choose a SMarT saving rate that is too high then they may actually do worse than if they stick to the default consumption rule of saving a small fraction of their income. In partial equilibrium, Findley and Caliendo (2010) have shown that this criticism may be warranted. Here though we see that if all consumers adopt the same SMarT saving rate then in general equilibrium the SMarT Plan will outperform the default rule for any > s. To show the importance of the pecuniary externality for arriving at these results, we repeat the  = 0.02 experiment in partial equilibrium, keeping factor prices constant at their general-equilibrium values for the permanent-income economy. The results are depicted in Fig. 11. The dashed blue line shows lifetime utility for a SMarT Plan partial-equilibrium economy with a base saving rate s = 3.5 %, plotted as a function of the SMarT saving rate . That is, we fix factor prices at their values from the permanent-income economy and we do not allow these factor prices to change as we adjust the SMarT rate. For comparison, the straight black line and solid red line are the same as in Fig. 9. The former is lifetime utility for the permanent-income economy, which, by construction, has not changed. The solid red line is the level of lifetime utility in a SMarT Plan economy with general-equilibrium effects included. It is clear that in partial equilibrium the rule-of-thumb economy can never dominate the permanent-income one, but it is also interesting that even in partial equilibrium SMarT Plan participation can take lifetime utility almost to the permanent-income level. One concern that needs to be addressed is whether these steady-state market equilibria that dominate their permanentincome counterparts might be dynamically inefficient. In Fig. 12 we plot the equilibrium rates of return for our baseline economies. The solid black line is the equilibrium rate of return from the permanent-income economy. The dotted line is the rate of growth of aggregate income n + x. An economy will be dynamically inefficient if the rate of return is less than this growth rate. Clearly, the permanent-income economy is not dynamically inefficient. The solid red and dashed blue lines give the equilibrium interest rates for SMarT Plan economies as a function of the SMarT saving rate with base rates s = 3.5 % and s = 7 % respectively. Large dots mark where the SMarT saving rate is optimal. Both these points are above the dotted line, indicating these equilibria are also dynamically efficient. At even higher SMarT Plan saving rates, we do obtain dynamically inefficient equilibria. Notice, however, that some of these dynamically inefficient equilibria still deliver higher utility than the permanent-income economy.18

17

Feigenbaum and Caliendo (2010) solve for the optimal continuous consumption path. They also conduct a more extensive sensitivity analysis. It is possible in this model for the equilibrium with the optimal SMarT rate to be dynamically inefficient while also delivering a level of lifetime utility that dominates the permanent-income rule. For example, if the base saving rate is 15 percent, the optimal SMarT rate is near 40 percent, and the level of utility exceeds the permanent-income level but the rate of return in the rule-of-thumb economy is below the growth rate of the economy. Thus, some dynamically inefficient parameterizations can dominate dynamically efficient ones. 18

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Fig. 11. Lifetime utility for = 2,  = 0.02, and base saving rate s = 3.5 %. Thin black line is for permanent-income consumer. Blue dashed and red solid lines are rule-of-thumb economies respectively in partial and general equilibrium. (For interpretation of the references to color in text, the reader is referred to the web version of the article.)

While we have established that a social planner can improve upon a permanent-income economy with respect to lifetime utility if he convinces households to adopt a SMarT Plan with a sufficiently high saving rate, utility is not observable, so to close this section let us consider how sensitive macroeconomic observables are to the SMarT Plan saving rate. Fig. 13 plots steady state values of equilibrium output Y as a function of the saving rate both with a base saving rate s of 3.5 percent (red solid line) and of 7 percent (blue dashed line). For comparison, the rational competitive equilibrium value is represented by the thin black line. The effect on output of going from the LCPI rule to the optimal SMarT Plan is not as pronounced as in the corresponding experiment for the two-period model, pictured in Fig. 3. Here, output only increases from the permanent0.09

0.08

0.07

Interest Rate

0.06

0.05

0.04

0.03

0.02

0.01

0 0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

SMarT Rate Fig. 12. Equilibrium interest rate for = 2,  = 0.02. Thin solid black line is for permanent-income economy. Blue dashed and red solid lines are rule-ofthumb economies for high (7 percent) and low (3.5 percent) base saving rates. Big circles correspond to optimal SMarT saving rates. Thin dashed black line is the exogenous growth rate (population plus technology). (For interpretation of the references to color in text, the reader is referred to the web version of the article.)

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70.00

65.00

Output

60.00

55.00

50.00

45.00

40.00 0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

SMarT Rate Fig. 13. Equilibrium aggregate output Y for = 2,  = 0.02. Thin black line is for permanent-income economy. Blue dashed and red solid lines are rule-ofthumb economies for high (7 percent) and low (3.5 percent) base saving rates. Big circles correspond to optimal SMarT saving rates. (For interpretation of the references to color in text, the reader is referred to the web version of the article.)

income economy by 7 percent for the two calibrations. It is very plausible that the two-period model exaggerates the possible gains from exploiting the pecuniary externality, which is why serious investigation of lifecycle models requires finer periods. Of course, the optimal parameterization of the SMarT Plan is not the OI consumption rule. Additional gains in output might be obtained if the social planner varied the consumption rule along other dimensions.

4. Conclusion Contrary to conventional intuition, we have shown it is possible in a general-equilibrium OLG model for consumers interacting in a market to attain higher lifetime utility than they would obtain if they behaved rationally. This is still possible even if a large fraction of agents, in some cases even a majority, do behave rationally. Moreover, this is not just an abstract result that involves peculiar rules of thumb that no one in their right mind would consider. On the contrary, we can improve upon the LCPI rule simply by implementing the Save More TomorrowTM Plan with a sufficiently high rate of saving out of future raises. Our results imply the logical underpinnings of the rational paradigm are quite wobbly. Most arguments for assuming agents are rational start from the notion that humans will evolve toward optimal behavior (Friedman, 1953). However, Feigenbaum (2009) shows that biological natural selection only favors individually rational behavior in the absence of externalities that relate the fertility of one individual to the behaviors of other individuals. This includes the pecuniary externality. Fortunately, the overlapping-generations construct is easily modified to incorporate natural selection, so we have the tools to study what spectrum of behaviors will evolve in the presence of the pecuniary externality. If a large fraction of households do follow rules of thumb, then a policy instrument which political leaders need to be mindful of is their influence over these rules of thumb. Cross-country comparisons reveal that American households often save a smaller fraction of their income than their counterparts abroad. This fact is hard to account for in rational-choice models without invoking ad hoc explanations such as that preferences differ across countries, which is difficult to test since preferences are not observable. In contrast, it is much easier to observe what rules of thumb governments advocate. If we consider the history of the United States, during World War II citizens on the home front were repeatedly exhorted to buy war bonds. In contrast, during the recent Middle-East conflicts, Americans were urged to spend, spend, spend. There has not been a conscious effort by policymakers in the United States to encourage people to coordinate on rules of thumb that involve high saving. While there have been various attempts to enact policies that encourage saving such as the introduction of IRAs, at the same time these are often undermined by other policies such as high taxes on capital. Rather than focusing so much on the positive question of how consumers make consumption-saving decisions, perhaps economists should take a closer look at the issue of how to guide consumers toward increasing their saving. Experiments might be a useful tool for learning what messages from the government and other sources best motivate subjects to save more of their income. One innovative idea that might also be politically feasible, given the clamor to eliminate estate taxes, would be to reduce taxes on estates bequeathed to young adults (but not to older heirs) who put this inheritance in illiquid trusts that prevent early consumption (Feigenbaum and Findley, 2010). This would create new aggregate saving and also exploit another aspect of the pecuniary externality involving bequests.

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Appendix A. Proof of optimal irrational two-period consumption rule Given w and R, Ld (K|R, w) = −u (we0 − K) + ˇRu (we1 + RK), and Ld (K|R, w) = u (we0 − K) + ˇR2 u (we1 + RK) < 0 since we have assumed u is strictly concave. Thus Ld is strictly concave, and Kd = arg max Ld (K|R, w)

(31)

is unique, satisfying u (we0 − Kd ) = ˇRu (we1 + RKd ).

(32)

In particular, Kd must satisfy (32) in a generalized market equilibrium where w = w(Kd ) and R = R(Kd ). Since we have assumed u and f are twice-differentiable, Lc is differentiable. Thus a maximum of the social planner’s objective (11) must satisfy Lc (K) = 0, where Lc (K) = (w (K)e0 − 1)u (w(K)e0 − K) + ˇ(w (K)e1 + R (K)K + R(K))u (w(K)e1 + R(K)K). Substituting (32) into (33),

(33)



Lc (Kd ) = ˇ R(Kd )(w (Kd )e0 − 1) + w (Kd )e1 + R (Kd )Kd + R(Kd ) × u (w(Kd )e1 + R(Kd )Kd ) = ˇ[w (Kd )(R(Kd )e0 + e1 ) + R (Kd )Kd ]u (w(Kd )e1 + R(Kd )Kd ). Using (8) and (9), R (K) =

1 f  e0 + e1

w (K) =

1 f e0 + e1

and







K e0 + e1

K e0 + e1



−

1 f e0 + e1

Thus (e0 + e1 )w (K) + R (K)K =

K f  e0 + e1





K e0 + e1

K e0 + e1



−



−

K (e0 + e1 )

K f  e0 + e1



f  2



K e0 + e1

K e0 + e1





=−

K (e0 + e1 )

f  2



K e0 + e1

 > 0.

= 0.

Therefore Lc (Kd ) = ˇ(R(Kd ) − 1)w (Kd )e0 u (w(Kd )e1 + R(Kd )Kd ). Since u and w are strictly increasing, Lc (Kd ) has the same sign as R(Kd ) − 1. Kd can only be the OI capital stock if R(Kd ) = 1. Since there is no population or technological growth in this simple model, a net interest rate of zero characterizes the golden-rule steady state. Thus the OI steady state is a rational competitive equilibrium only if the rational competitive equilibrium coincides with the golden-rule steady state. More generally, if we assume that Lc is strictly concave, then there is a unique solution to the social planner’s problem. If the rational competitive equilibrium is dynamically efficient, R(Kd ) ≥ 1, so Lc (Kd ) ≥ 0. Thus if the rational competitive equilibrium is not the golden-rule steady state, the OI capital stock will be greater than Kd . Conversely, if the rational competitive equilibrium is dynamically inefficient, the OI capital stock will be less than Kd . Appendix B. Concavity of social planner’s objective The second derivative of the social planner’s objective function is Lc (K) = w (K)e0 u (w(K)e0 − K) + (w (K)e0 − 1) u (w(K)e0 − K) 2

+ ˇ[w (K)e1 + R (K)K + R(K)] u (w(K)e1 + R(K)K) + ˇ[w (K)e1 + R (K)K + 2R (K)]u (w(K)e1 + R(K)K). 2

Suppose that the production function is Cobb–Douglas with share of capital ˛ ∈ (0, 1), so f(k) = k˛ . Then

 K ˛

w(K) = (1 − ˛)

N

J. Feigenbaum et al. / Journal of Economic Behavior & Organization 77 (2011) 285–303

 K ˛−1

R(K) = ˛

N

+ 1 − ı.

Since w (K) = −

303

˛(1 − ˛)2 N2

 K ˛−2 N

and R (K)K + 2R (K) =

˛2 (˛ − 1) N

<0

 K ˛−2 N

< 0,

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