maeSome Implications of Belief in the Afterlife and the Allocation of Time to Spirituality∗
Constantino Hevia† University of Chicago September 2004

Abstract An otherwise standard model of intertemporal consumer choice is extended to incorporate the allocation of time to spiritual activities along the lines of the human capital literature. Several testable implications are analyzed. We study exogenous and endogenous changes in life expectancy, and we argue that the traditional value of life or willingness to pay formulas for reductions in risks to life have to be modified when we account for afterlife utility. The model is then extended to rationalize the existence of suicide bombings and to discuss the complementarity between religiosity and patience. Jews, Christians, and Muslims all profess belief in immortality, but the veneration paid to the first century of life is proof that they truly believe only in those hundred years, for they destine all the rest, throughout eternity, to rewarding or punishing what one did when alive. [J.L. BORGES, ”The immortal”.]
COMMENTS WELCOME!. I’d like to thank Gary Becker for his encouragement and the very helpful comments. I also thank the participants of the workshop of Applications of Economics at the University of Chicago for their comments. † E-mail: chevia@uchicago.edu.
∗

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1

Introduction

Since the seminal paper of Azzi and Ehrenberg [1975] (AE hereafter) there has been an increasing interest in the economics of religion. This literature is growing substantially, specially since the work of Iannaccone ( e.g. [1984], [1990], [1992]). The main idea of this field is to use the tools provided by economic theory to analyze questions about the allocation of time and resources to religious activities, the choice of religious denomination, competition among religious groups, etc. Even though the main object of study of sociology (and economics in particular) is group behavior, it is necessary first to understand individual choice to derive conclusions about the aggregate. AE’s paper first and Iannaccone’s papers next ([1984], [1990]), provided a rationale for some robust empirical facts, for example, that older people are those more involved with religious activities. The first paper introduced afterlife consumption as the main motive behind religious participation while the latter introduce the concept of religious human capital. Agents are subject to religious habit formation and the model resembles those of the "rational addiction" literature (Becker and Murphy [1988]). There are some reasons why the afterlife motive was de-emphasized in favor of the human capital approach. For example, Ehrenberg’s [1977] empirical analysis use a database of American Jews to test their model, but Iannaccone [1998] argues that during the period 1972-1990 only 30 percent of them believe in an afterlife. Besides the last observation, several religions (e.g. Catholic, Muslim) assert that an afterlife exists and more importantly, that our fate there depends on what we did while alive. Notwithstanding the thirty years since the original publication, very few implications of the afterlife assumption on individual behavior were analyzed. This paper tries to fill, at least partially, this gap. We do not only study the life cycle implications in terms of religious participation but several other questions as well: the basic model is extended in several dimensions to discuss how religious activities are related to changes in health (endogenous and exogenous) and patience. For example, we analyze under what circumstances our model predicts a positive correlation between religiosity and health, a robust empirical finding. Moreover, we argue that the afterlife provides a sound rationale for the existence of suicide bombers. Specifically, an otherwise standard model of intertemporal consumer choice is extended to incorporate the allocation of time to spiritual activities. We follows the human capital literature (e.g. Becker [1993]) by assuming that 2

agents accumulate a stock of spiritual capital that will be useful in their afterlife. In particular, the higher the amount of spiritual capital accumulated, the higher the expected afterlife utility. Our model is consistent with the existence of an afterlife consumption (as in AE’s paper) or with a heaven-hell interpretation of the afterlife. Contrary to Azzi and Ehrenberg’s model, we work in a stochastic environment. Uncertainty enters the model through random life-spans. This assumption allows us to study changes in health, diseases and willingness to pay for increases in longevity, for example. Since the model is quite different from AE’s, we also show that our framework is consistent with the life cycle pattern of religious activities. Section 2 develops the basic framework, solves the agent’s problem and derives the life cycle implications of the model. In section 3 we analyze the relation between health, the value of life (defined as the willingness to pay for reductions in risks to life) and religion. Section 4 extends the model to allow for suicidal attacks and shows that this framework provides a sound rationale for this type of activities. We next study the complementarity between patience and religion and section 6 concludes.1

2

The Model

An otherwise standard model of intertemporal choice is extended with the introduction of a new state variable interpreted as a stock of spiritual capital. Time is discrete and at the beginning of each period an agent can be either dead or alive. The agent enters period i with two state variables, assets Ai and a stock of spiritual capital Ki . The key assumption of the model is that agents believe that a higher stock of spiritual capital increases afterlife utility. The two state variables play different roles: assets are only valuable if the agent is alive while spiritual capital is only valuable if the agent is dead. This is an extreme assumption since we could also assume that spiritual capital affects utility if alive (this is the approach followed by Iannaccone [1984], for example). Nevertheless we focus on this case to simplify the analysis and to highlight the afterlife motive on decision making. There exists an investment technology through which it is possible to modify the stock of spiritual capital. This technology can be interpreted as
A full mathematical appendix with detailed proofs is available upon request from the author.
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being determined by the particular religion this person belongs to, which we take as given.2 Since spiritual activities are time intensive we take the extreme position of assuming that the only input in the investment technology is time. The law of motion for spiritual capital is given by Ki+1 = (1 − δ) Ki + f (ti ) (1)

where δ is a depreciation rate between zero and one, f (ti ) represents the investment technology (which we assume to have decreasing returns to scale), ti is time devoted to the accumulation of spiritual capital in period i, and K0 is given. For the time being, the depreciation rate δ will be taken as given. It is reasonable to assume, however, that the stock of spiritual capital depreciates endogenously by particular actions taken by the agent. This can be easily accounted for in our model by letting δ be a function of some good x, which can be interpreted as a sin-good. While alive, agents derive utility from two nondurable goods: a consumption good and leisure denoted by c and respectively. Both goods are normal and the period utility function, denoted by u (c, ), is increasing in both arguments and concave. If the agent dies at the beginning of period i + 1, he enjoys a perpetual flow of utility of (1 − β) J (Ki+1 )3 . The function J (·) is strictly increasing and twice differentiable. This function captures the intuition that dying with a higher stock of spiritual capital increases afterlife utility. For most of the time we will assume that J (·) is concave (this assumption is needed to make sure that the first order conditions are also sufficient) but with enough concavity in the investment technology, J (·) can be assumed convex and the first order conditions being still sufficient. This latter case generates increasing returns in K. There are two possible interpretations for J (Ki+1 ): since Ki+1 potentially takes any positive value, the first interpretation is that Ki+1 represents the level of afterlife consumption (or the input to an afterlife production function that provides afterlife consumption). In the second interpretation, there is a heaven and a hell, which generates the flow of utilities (1 − β) V and (1 − β) V respectively. The probability of going to heaven is function of the stock of spiritual capital, P (K), thus J (Ki+1 ) = P (Ki+1 ) V + (1 − P (Ki+1 )) V .
We are not studying the choice of religion but the economic implications once this choice has been made and there is no possibility of changing religious denomination. 3 We are assuming that after dying the depreciation rate δ is identically zero.
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The utility of living exactly from period 0 to i is i X j=0

β j u (cj , j ) + β i+1 J (Ki+1 )

where 0 < β < 1 is the subjective discount factor. Denote by gi+1 the probability of dying at the beginning of period i + 1, or equivalently, the probability of living exactly until period i. Then the expected utility function is " i # ∞ X X gi+1 β j u (cj , j ) + β i+1 J (Ki+1 ) . i=0 j=0

Changing the order of summation and denoting by Si the probability of being P alive at period i (i.e. Si = ∞ gj+1 ), we can rewrite the last equation as j=i V0 =
∞ X i=0

β i Si [u (xi , i ) + βhi+1 J (Ki+1 )]

(2)

where hi+1 ≡ (Si − Si+1 ) /Si is the death rate at period i + 1. (hi+1 is the probability of dying at the beginning of period i + 1 conditional on being alive at period i.) The last equation has a natural interpretation: if the agent is alive at period i, he enjoys an instantaneous utility of u (ci , i ) and faces a probability of being dead the next period of hi+1 . If he dies, he enjoys a utility of J (Ki+1 ), and discounted back to period i it is valued as βJ (Ki+1 ). But since preferences are measured at period zero, that flow of utility has to be weighted by the unconditional probability of being alive at period i, Si , and discounted at the rate β i . Note that given a sequence {cj , j , Kj+1 }∞ we can rewrite the above j=0 preferences in a recursive form. If Vi is used to denote the utility of an alive agent at the beginning of period i, it must satisfy the following recursive equation Vi = u (ci , i ) + β [hi+1 J (Ki+1 ) + (1 − hi+1 ) Vi+1 ] . This equation is the basis of our dynamic programming approach. Using recursive methods has the advantage of focusing on policy functions instead of sequences, and thus we will be able to analyze how consumption, leisure, time spent on religiosity and savings depend on Ki and Ai . 5

We assume that there exists an insurance institution where all agents share their death risks among identical persons at each period (this is the one period equivalent to an annuity market, Rosen [1988]). The institution separates groups of identical agents and manages their savings. Each person contributes Ai dollars in period i − 1 and the institution invests the total amount of money at the risk-free interest rate ri . A proportion hi die and their savings are shared equally among the survivors; this amounts to Ai (1 + ri ) hi / (1 − hi ) for each survivor. Thus conditional on being alive the next period, the consumer receives Ai (1 + ri ) + Ai (1 + ri ) hi / (1 − hi ) = Ai (1 + ri ) / (1 − hi ) . This implies that the flow budget constraint at period i given initial assets Ai is 1+r + wi . Ai+1 + ci + wi (ti + i ) ≤ Ai 1 − hi Full income (the right hand side) is spent on consumption, leisure, time spent on religiosity and savings.4 An alive agent enters each period with a state vector (K, A) and its problem is represented by the following Bellman equation Vi (K, A) = subject to 1 + ri + wi − c − wi ( + t) 1 − hi K 0 = (1 − δ) K + f (t) c, , t ≥ 0; +t≤1 A0 ≤ A where we used the standard tradition of denoting next period values with a prime. Since the death rates in general differ by age, the value functions will be time dependent. Standard arguments can be used to prove the following results, which are very useful to characterize the optimal policies: 1. Monotonicity: The value function Vi (K, A) is strictly increasing in both arguments.
4

{c, ,t,A

max 0

,K 0 }

hu (c, ) + β [hi+1 J (K 0 ) + (1 − hi+1 ) Vi+1 (K 0 , A0 )]i

Total time per period has been normalized to 1.

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2. Concavity: (a) If u (·) and J (·) are both strictly concave, the value function Vi (K, A) is strictly concave. (b) If u (·) is strictly concave and J (K) = αK, the value function takes the form Vi (K, A) = φi K + Ui (A), where Ui is strictly concave. Case 2.b is an interesting example to analyze. It has the advantage that the policy functions will depend only on A, and this simplifies considerably the analysis. From now on we will make the following natural assumption: death rates are increasing through time (i.e. hi ≤ hi+1 for all i) and starting from ¯ some period I ≥ 0 they become constant at the level hi = h for all i ≥ I. Increasing death rates can be interpreted as aging and the constancy of death ¯ rates after some period I is consistent with h = 1, so that everyone eventually ¯ dies. Everything we derive does not depend on h = 1, however. The optimality conditions of our problem are u (ci , i ) = wi uc (ci , i ) ¡ 0 ¢ ¡ 0 ¢¤ £ β hi+1 J 0 Ki+1 + (1 − hi+1 ) VKi+1 Ki+1 , A0i+1 f 0 (ti ) = u (ci , i ) ¡ 0 ¢ β (1 − hi+1 ) VAi+1 Ki+1 , A0i+1 = uc (ci , i ) (3) (4) (5)

The first equation is the usual intratemporal condition between consumption and leisure. The second equates the marginal value of time spent on religiosity with the marginal value of time spent in leisure, while the last condition equates the marginal value of savings with the marginal value of consumption. Given the normality assumption, we can solve for consumption as an increasing function of leisure, as c = λ ( ) with λ0 > 0. Moreover, from now on, we will assume that uc (λ ( ) , ) is a decreasing function of . The envelope conditions can be written as VKi (K, A) = uc (λi ( i (K, A)) , i (K, A)) wi (1 − δ) f 0 (ti (K, A)) i (6) (7)

VAi (K, A) = uc (λi ( i (K, A)) , 7

(K, A))

1 + ri 1 − hi

In what follows, we focus on the case where J (·) is strictly concave. In the appendix we show that the policy functions have the following properties: • ci (K, A), • ci (K, A), i i

(K, A), ti (K, A) and A0i+1 (K, A) are increasing in A. (K, A) is increasing in K

• ti (K, A) ais decreasing in K and A0i+1 (K, A) can be either increasing or decreasing in K. The dependence of the policy functions on A is standard: afterlife consumption is a good, hence a higher level of assets imply higher consumption, leisure, savings and time spent on religiosity. The novel thing is the dependence of the policy functions on K. A higher stock of spiritual capital implies that everything else equal, afterlife consumption is higher. Hence agents will find it optimal to increase consumption and leisure and given the decreasing returns to K, to decrease ti . Remark: ti being decreasing in K depends crucially on the strict concavity of J (·). If this function were convex, a higher stock of spiritual capital increases the marginal utility of additional spiritual capital K. In this case it could be optimal to have ti increasing in K and consumption and leisure decreasing in K. Unfortunately, since this depends on the relative concavities of J (·), f (·) and u (·), we don’t have a general result. Convexity of J (·) is consistent with an increasing specialization on spiritual activities in detriment of consumption and leisure through time. As will be seen below, increasing returns in J (·) is crucial to generate a positive correlation between health and religiosity. Example: u (c, ) strictly concave and J (K) = αK The value function takes the form Vi (K, A) = φi K + Ui (A) where U (·) is strictly concave. In the appendix we show that the constants {φi } solve the following difference equation £ ¤ (8) φi = β hi+1 α + (1 − hi+1 ) φi+1 and φi ≤ φi+1 with strict inequality if and only if hi < hi+1 . The last result implies that the value of an additional unit of spiritual capital is increasing through time. Moreover, if the death rate in period i + 1 is strictly higher that the death rate in period i, then the value of capital is 8

strictly higher in period i + 1. This can be seen by considering the envelope condition of the Bellman’s equation, VKi (K, A) = φi . A higher death rate means that there is a higher probability that the person will die and since his utility only depends on K in that state, each additional unit of capital is more valuable. In this example, the optimality conditions (4) and (5) become φi f 0 (ti ) = uc (λi ( i ) , i ) w (9)

0 (10) β (1 − hi+1 ) Ui+1 (Ai+1 ) = uc (λ ( i ) , i ) ¡ 0 ¢ This immediately implies that the policy functions Ai+1 , ci , i , ti only depend on A. Moreover, by the strict concavity of u, the right side of (9) and (10) are both strictly decreasing in . The strict concavity of f (·) and Ui+1 (·) plus the budget constraint imply that the policy functions ci (A) , i (A) , ti (A) 0 and A0i+1 (A) are strictly increasing in A. Furthermore, Ki+1 (K, A) = (1 − δ) K+ ti (A) is strictly increasing in both arguments. In this example consumption, leisure, savings and time spent on religiosity are independent of the stock of spiritual capital accumulated so far and are strictly increasing in the stock of assets.

Remark: Our model, as the last example shows, imply that the stock of spiritual capital {Ki } could be either increasing or decreasing through time. There is nothing strange with a decreasing sequence of spiritual capital. It could well be the case that people start with a high stock of spiritual capital and it depreciates through time by evil actions (δ in our case). A positive amount of time spent on religiosity would be, in this case, to avoid even larger decreases in the stock of spiritual capital.5

2.1

Life Cycle Implications

A robust empirical fact is that time spent on religiosity over the life cycle has either a ∪-shape (with younger and older agents devoting more time to religiosity than middle aged people) or it is always increasing. We will show
5

I owe this insight to Steve Levitt.

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that this model is consistent with this fact. To that end we will focus on the example J (·) = αK. The optimality condition (9) between periods i and i + 1 implies φi f 0 (ti ) uc (λi ( i ) , i ) wi = . 0 (t φi+1 f i+1 ) uc (λi+1 ( i+1 ) , i+1 ) wi+1 Using the intertemporal Euler equation uc (λi ( i ) , i ) = β (1 + ri+1 ) uc (λi+1 ( (obtained using (5) and (7)) we can rewrite the last condition as φ wi f 0 (ti ) (1 + ri+1 ) . = β i+1 0 (t f i+1 ) φi wi+1 (11) i+1 ) , i+1 )

There are two effects that determine the slope of the path of time spent on spirituality: first the intertemporal price of leisure between periods i and i+1, given by wwi (1 + ri+1 ) , and second, the ratio βφi+1 /φi (i.e. the marginal i+1 rate of substitution between spiritual capital in period i+1 and period i). The first effect captures the intertemporal substitution effect driven by changes in the relative price of time, while the second captures the fact that as the agent becomes older the marginal value of an additional unit of spiritual capital increases. (Remember that φi+1 ≥ φi with strict inequality if and only if hi+1 > hi .) If the right side of (11) is higher (smaller) than one, the strict concavity of f (·) implies that ti+1 > ti (ti+1 < ti ). To interpret the last equation, focus on the case β (1 + ri+1 ) ' 1 and assume φi+1 = φi for all i (so that hi+1 = hi for all i). This implies that ti will mirror the movements in wi . Since wages have an inverted ∪-shape through the life cycle, this implies that time spent on religiosity will have a ∪−shape through the life cycle. However since death rates increase through time, we have φi+1 ≥ φi for all i. This force points toward an increase in time spent on religiosity over the life cycle. Thus, this example is consistent with the fact that the sequence {ti } is either increasing, or has a ∪-shape over the life cycle. Furthermore, under standard specifications of the period utility function u (x, ), the ratio of leisure between adjacent periods i+1 / i follows a similar pattern as ti+1 /ti does. This is a well known fact and a thorough exposition can be found in Becker [1993]. The proportional increase in leisure i+1 / i will be negatively related to the growth rate of wages and positively related to the interest rate. This implies that leisure and religious activities tend to move together over the life cycle, at least as induced by changes in wages and interest rates. 10

3

Health, the Value of Life and Spiritual Capital

How do advances in health affect religiosity? Does the existence of an afterlife modify the willingness to pay for reductions in risks to life? If so, in what direction? Do more religious people place a higher or lower value to reductions in risks to life? This are some of the questions that we will address in this section. Since our model is a straightforward generalization of standard models used to analyze changes in risks to life and changes in survival probabilities (e.g. Rosen [1988], Murphy and Topel [1999]), the results are easily comparable. Figure 1 shows a typical life-cycle pattern of survival rates, defined as {1 − hi }∞ . Our first experiment will be to analyze an exogenous improvei=0 ment in survival rates as shown in the same figure, that is, the fast decline in survival rates is delayed. Changes in death rates do not come isolated: the typical individual retires later and the life cycle decline in wages is also delayed. This considerations and the fact that death rates also enter the budget constraint imply that there are considerable wealth effects. To simplify the analysis we assume that those effects are netted out. Furthermore, we will focus on the case J (K) = αK. In that example we saw that older agents devote the highest amounts of time to the accumulation of spiritual capital. Under this conditions, (8) and (11) imply that the increase in time allocated to the accumulation of spiritual capital will also be delayed. Intuitively, the higher wages together with the greater longevity decrease the incentives to allocate time to spiritual activities because the relative price of time is higher and because the marginal valuation of spiritual capital is lower - the latter is given by φi , which is decreasing in hi . A more general analysis of the relation between health and religiosity would allow for arbitrary changes in the survival probabilities. In this section we will analyze the willingness to pay for arbitrary (differentiable) changes in the survivor function, defined as the sequence {Si }∞ . Assume that Si is i=0 a (differentiable) function of a health index variable z.6 The change in the objective function given a perturbation in the survivor function is obtained
6

Since the dependence of Si on z is arbitrary, this is without loss of generality

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Survival rates

1

{ −hi} 1

{ −h'i} 1

0

Age

Figure 1: Exogenous improvement in survival rates through the Envelope Theorem,
∞ X i=0

"∞ X ¡ 0 ¢ ª © 0 0 β Si (z) u (ci , i ) + Si (z) − Si+1 (z) βJ (Ki+1 ) −µ Ri Si0 (z) [ci − wi (1 − i i=0

i

− ti )]

#

¸ X ∞ ¢ βJ (Ki+1 ) ¡ 0 Ri Si0 (z) − Si+1 (z) V= i − ti ) + uc (ci , i ) i=0 i=0 (12) Since the change in the survivor function is arbitrary, the last equation is useful to analyze different scenarios. The first term in the last formula is what Rosen [1988] calls the expected consumer surplus along the optimal path {c, , t}∞ , while the second term captures the influence of afterlife i=0 utility. (The division by the marginal utility of consumption is to value afterlife utility in terms of consumption or wealth at period zero.) In fact, denoting by εi the elasticity of the period utility function with respect to ci , d log u/d log ci , the first term in square brackets becomes µ ¶ 1 − εi ci + wi (1 − i − ti ) εi
∞ X

where µ is the Lagrange multiplier on the intertemporal budget constraint. Using the optimality condition β i uc (ci , i ) = µRi , dividing by µ (the marginal utility of wealth at period 0) and rearranging we obtain the willingness to pay for a change in the survivor function from {Si (z)}∞ to {Si (z) + Si0 (z)}∞ : i=0 i=0 Ri Si0 u (ci , i ) − ci + wi (1 − (z) uc (ci , i ) ·

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and this object is interpreted as the consumer surplus at period i (see the discussion in Rosen [1988]). If the perturbation of the health index z is feasible under current medical knowledge and the marginal cost of acquiring an additional unit of z is α, optimality requires V = α. If the perturbation in z is not feasible, equation (12) can still be used to assign a monetary value to the proposed change. The last is the basis of the Value of Life, or reductions in risks to life computations of Arthur [1981], Rosen [1988] and Murphy and Topel [1999]. Since the effects on the willingness to pay derived from expected consumer surplus are well understood (e.g. Arthur [1981], Rosen [1988]), we will focus on the last term of (12). Given the optimal path {Ki+1 , ci , i , ti } 0 and a perturbation in z, Si0 (z) − Si+1 (z) is the change in the unconditional probability of dying between periods i and i+1.7 A health improvement typically increases longevity by reducing the unconditional probability of dying in some age interval [a, b] and increasing that probability in some future age interval [c, d]. In other words, the mass of the unconditional probability of 0 death is moved to later periods. The term Si0 (z) − Si+1 (z) will be negative in the interval [a, b] and positive in [c, d]. The net effect on total utility is uncertain: it will be positive, for example, if the stock of spiritual capital increases sufficiently and the discounted afterlife utility gained during the years [c, d] is higher than the loss caused by the negative weight on afterlife utility at ages [a, b]. Equivalently, a higher longevity gives the agent more periods to accumulate spiritual capital if he desires to do so, but the discount on those utilities will be higher. Moreover, the decline in the probability of dying younger (say, between 40 and 50 years old) gives less weight to the expected afterlife utility in that age interval as measured at time zero. The standard way of estimating the value of life is by comparing wage risk-premia in occupations with varying risks of death. Since the traditional value of life literature normalizes afterlife utility to zero, the obtained number is assigned completely to expected consumer surplus. However, equation (12) shows that when we endogeneize afterlife utility, it is incorrect to assign the whole monetary value to consumer surplus. A part of it is due to the fact that a perturbation in the survivor function changes, on average, the lifespan. Hence, as long as the sequence of spiritual capital is not constant, a
7 Remember that we are talking about unconditional probabilities. The unconditional probability of dying at age 110 is very small, but of course, the death rate at age 100 is very close to 1

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different life expectancy changes the expected afterlife utility as of period zero. If this effect is positive (negative), ignoring afterlife utility introduces an upward (downward) bias in the estimated expected consumer surplus.

3.1

The correlation between Health and Religiosity

Since the late eighties increasing attention was devoted to the relation between health and religiosity in the medical science. The positive correlation between several indicators of religiosity and physical and mental health is a robust empirical fact (Levin [1994] and Koenig [1997] to name just a few). Higher levels of spirituality are negatively correlated with morbidity and mortality rates, adjusting for a variety of factors. There are several explanations for this evidence. It is argued, for example, that religiosity promotes healthy behavior and lifestyles, that belonging to a religious group provides support and care from the community fellows, that it reduces distress, anxiety and depression, etc. The direction of causality between health and religion is far from clear, however (Levin [1994]). Under some circumstances, another explanation can be added to the above list. As the next discussion will make it clear, increasing returns in J (K) is crucial to generate a positive correlation between health and religiosity, measured as time spent on spiritual activities. A simple example illustrates this point. Assume A0 = K0 = 0, w = 1, r = 0, δ = 0 and u (c, ) = log c + log . Furthermore, assume that without investment in health, life ends for sure in period one. In this case the agent’s problem becomes max hlog c + log + J (f (t))i (A) subject to c+ +t=1 On the other hand, if the agent invest 2q dollars in health, the life-span is extended one period. In this case it is optimal to have c0 = c1 , 0 = 1 and t0 = t1 . Thus the problem reduces to max h2 [log c + log ] + J (2f (t))i subject to 2 (c + + t) + 2q = 2 (B)

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In both cases it is optimal to have c = . The optimality conditions for (A) and (B) are, respectively J 0 (f (t)) f 0 (t) = 1/c 2c + t − 1 = 0 and J 0 (2f (t∗ )) f 0 (t∗ ) = 1/c 2c∗ + t∗ − 1 + q = 0 where a star denotes the optimal choice if the agent invests in health. Thus if q is sufficiently small, the agent will find it optimal to invest in health. The question is, how does t relates in the two cases?. It is clear that the answer depends on the concavity or convexity of J (·). In the limit, if q = 0 and J 0 (K) = α, then t = t∗ and c = c∗ . If q = 0 and J (·) is concave, then t∗ < t and c∗ > c. Finally, if q = 0 and J (K) is convex then t∗ > t and c∗ < c.8 By continuity, for q sufficiently small and J (K) convex, we obtain t∗ > t and c∗ < c, that is, the agent finds it optimal to invest in health and at the same time to increase time spent on religiosity. In other words, the last example shows that to obtain a positive correlation between health and religiosity it is crucial the assumption of increasing returns to scale in spiritual capital. Intuitively, the longer life-span together with the increasing returns in K makes specialization in religiosity more appealing.

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4.1

Some Extensions
Rationalizing suicide bombings

Why do some people decide to engage in suicide bombings giving up their lives? This is a deep and difficult topic. However, I believe that this simple model can shed some light into this question. A key element in our explanation lies in the role of indoctrination and a promised reward to be enjoyed in the afterlife. To simplify the exposition
Remember that with enough concavity in f (t), the first order condition are still sufficient.
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we will focus on the time invariant model and we will assume that a suicide bombing is always successful. At the beginning of each period the agent has the choice of committing a suicide bombing or not (we do not allow for randomized policies). It is reasonable to assume that the reward of the attack, denoted by Φ (K), is (weakly) increasing in the stock of spiritual capital but does not depend on the level of assets. An agent deciding whether to commit an attack or not solves the following problem V (K, A) = max {U (K, A) ; Φ (K)}
{not attack; attack}

where U (K, A) =
{c, ,t,K 0 ,A0 }

max

hu (c, ) + β [hJ (K 0 ) + (1 − h) V (K 0 , A0 )]i

subject to the constraints 1+r + w − c − w ( + t) 1−h K 0 = (1 − δ) K + f (t) c, , t ≥ 0; +t≤1 A0 ≤ A The term U (K, A) is the utility if the agents does not commit suicide the current period, while the second term is the utility if he does. The function Φ (K) is assumed to be increasing or constant. Standard arguments can be used to show that U (K, A) and V (K, A) are increasing in both arguments but concavity does not longer hold. The form of the function Φ (K) becomes crucial. Figure 2 presents two interesting examples. Panel I shows the case ¯ where Φ (K) is sufficiently flat. Given A, there exist a unique K (A) such ¯ (A) it is optimal to commit the attack. This example has that if K < K a problem, however: suicide bombers are generally associated with religious fanaticism, but in the last example the agents with low stock of spiritual capital are precisely those who become suicide bombers. In panel II we have an example where Φ (K) is sufficiently steep. In this case it is optimal to ¯ commit the attack if K > K (A). Intuitively, the gains of becoming a suicide bomber has increasing returns. Note that this example is consistent with an agent accumulating spiritual capital for a while and then becoming a suicide ¯ bomber when K is sufficiently high. In both cases, as A increases, K (A) decreases: the reason is that as wealth increases the agent is able to generate 16

more utility living, thus the stock of spiritual capital has to be lower the wealthier is the agent to engage in a suicidal attack. Figure 3 shows two examples in the assets dimension. In both cases there ¯ ¯ is a cut-off level of assets A (K) such that if A < A (K) the agent becomes a suicide bomber. Intuitively, given the promised reward Φ of dying in a bombing, the poorer is the agent, the lower the level of utility attainable if alive. The key difference between panel I and II lies in the form of Φ (K). If it is sufficiently flat (panel I), as the stock of spiritual capital increases, the ¯ cut-off point A (K) decreases. The reason is that the gains of committing an attack hardly moves with K, and a higher K implies higher utility in the afterlife if the agent dies of any other cause but a suicide bombing, thus an agent with a higher stock of spiritual capital will be less likely of becoming a suicide bomber. Panel II shows a completely different scenario, with Φ (K) ¯ sufficiently steep in K. In this case the cut-off point A (K) is increasing in K. As K increases, the gains of becoming a suicide bomber are higher than the increase in the utility of living, U (K, A). Take for example K2 > K1 , ¯ ¯ agents with assets between A1 and A2 would become suicide bombers for K2 but not for K1 .
I
U (K, A2 )

II
U (K, A1 ) Θ(K )

Θ(K )

U (K, A2 )
U (K, A1 )

K2

K1

K

K1 K 2

K

A1 < A2

Figure 2: Decision rules in the K’s dimension Throughout our explanation the function Φ (K) was arbitrary. Where does it come from? Here is where the role of indoctrination and persuasion becomes crucial. An organization that for some reason derives utility from terrorist acts will try to persuade their followers that becoming a suicide 17

I

U(K2, A) U(K1, A)

II

U(K2, A)

U(K1, A)

Θ(K2 )

Θ(K1 ) = Θ(K2 )

Θ(K1 )

A2

A1

A

A1
K1 < K2

A2

A

Figure 3: Decision rules in the A’s dimension bombers gives right to an afterlife full of joy and happiness. All this is captured in the function Φ (K). Krueger et. al. [2002] present some evidence suggesting that Hezbollah’s suicide bombers tend to be above the poverty line and with higher education that the average Lebanese. We can construct examples consistent with this observation. If the reward of a bombing, Φ (K), is sufficiently low and flat for low levels of spiritual capital and then it increases at increasing rates (as in panel II of figure 2) our model is consistent with an agent accumulating spiritual capital, assets, increasing consumption and leisure for a while and eventually becoming a suicide bomber.

4.2

Patience and religion

What is the relation between patience and religiosity? In our model, religiosity determines the level of expected afterlife utility; since by definition afterlife utility is enjoyed in the future, agents with high expected afterlife utility have more incentives to be patient. In this section we build on Becker and Mulligan [1997] (BM hereafter) to analyze this connection explicitly. BM study the relation between patience and future utilities by constructing a model where the determination of the time preference parameter β is endogeneized. If the consumer foresees that future utilities will be high, there 18

are incentives to reduce the discount on those utilities. BM assume that this objective can be accomplished by investing in ‘future-oriented’ capital that, roughly speaking, consists of resources spent on imagining future circumstances and utilities so that they seem less remote. They argue that among a long list of variables that could affect the time and goods spent to reduce the discount rate is religion. We will assume, as BM do, that each agent is able to modify at period zero the discount factor through investment in future-oriented capital, denoted by q. To rule out unbounded returns we will assume that for all q ≥ 0, β (q) < 1. The implication of this assumption is that no matter how much effort they put in reducing the discount rate, it is not feasible to give as much weight to future utilities as to present utilities. We will also assume that β 0 (q) > 0 and β 00 (q) ≤ 0. However, as discussed in BM, concavity of β, U and J is not sufficient to guarantee global concavity of the objective function (so that the first order conditions are sufficient). The interested reader is referred to their paper. Given the optimal sequence {ci , i , ti Ki+1 }∞ , the agent’s problem is i=0 max subject to
∞ X i=0

β (q)i Si [u (ci , i ) + β (q) hi+1 J (Ki+1 )]

where ψ is the price of q. The first order condition with respect to future oriented capital can be written as (∞ ) X i−1 β 0 (q) Si β (q) [iu (ci , i ) + (i + 1) β (q) hi+1 J (Ki+1 )] = ψuc (c0 , 0 ) i=0 ∞ X i=0

Ri Si [ci − wi (1 −

i

− ti )] + ψq = A0

This is the equation, slightly modified, that BM obtain. As we can see, and given the concavity of the β (·) function, agent’s that expect higher levels of utility are those who will devote more resources to reduce the discount rate. Therefore anything that raises future utilities without changing the marginal utility of consumption at period zero reduces the discount on future utilities. BM argue that there is a complementarity between patience and future utilities. The above equation, of course, shows the same complementarity, but 19

now future utilities are not only derived from consumption and leisure but also from expected afterlife utility. Afterlife utility is determined by the stock of spiritual capital accumulated while alive, thus people with a higher stock of spiritual capital have higher willingness to pay to decrease the discount rate. Equivalently, future utilities determine the choice of the discount factor, but the discount factor determines the accumulation of spiritual capital and future consumption (through savings) which in turn, determines future utility. In other words, agents with good perspectives in the afterlife will be prone to invest in reducing the discount rate. Furthermore, higher wealth implies higher consumption and higher amounts of time spent on spiritual activities, which in turn increase the incentives to be more patient.

5

Concluding remarks

A simple model of the allocation of time to religious activities was developed. We obtained several implications. For example, time allocated to spiritual activities tend to have either a ∪-shape or it is always increasing over the life cycle. Thus, a robust implication is that older agents are those who devote more time to spirituality. The two forces behind this result are, a substitution effect driven by changes in the relative price of wages over the life cycle and the fact that aging increases the value of spiritual capital. Among a group of identical agents (in terms of preferences) the model predicts a positive correlation between wealth (i.e. the level of assets) and religiosity. In section 3 we discussed the relation between health and religiosity. Exogenous improvements in longevity tend to delay the allocation of time to spirituality. We next showed that the endogeneity of afterlife utility modifies the willingness to pay formulae used in the Value of Life literature. We argued that the current valuation of expected consumer surplus derived from changes in life expectancy is biased, since a part of it has to be assigned to the change in expected afterlife utility. A robust empirical fact is the positive correlation between religiosity and health. We showed that a crucial assumption for our model to be consistent with this observation is the existence of increasing returns of the afterlife utility with respect to stock of spiritual capital. This does not require time spent on religiosity to have increasing returns. Section 4 extends the model in two directions. First we argued that the 20

model can be used to rationalize the existence of suicidal attacks. We analyzed several examples and examined the choice of becoming a martyr or not. The incentives to become a suicide bomber are given by a promised reward to be enjoyed in the afterlife. Contrary to the traditional wisdom, the typical suicide bomber does not differ significantly from the average person in terms of education and wealth. Our model is consistent with that observation. The second extension of the model in section 4 analyzes the relation between patience and religiosity. We built on Becker and Mulligan [1997] and endogeneized the discount factor. We discussed the complementarity between patience and future utilities concluding that in general, the model predicts that religiosity and patience are complementary.

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[14] Rosen, Sherwin. ” The Value of Changes in Life Expectancy.” Journal of Risk and Uncertainty 1988, pp. 285-304.

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