WILEY Economics Department of the University of Pennsylvania Institute of Social and Economic Research--Osaka University Optimal Investment and Consumption Strategies Under Risk, an Uncertain Lifetime, and Author(s): Nils H. Hakansson Source: International Economic Review, Vol 10, No. 3(Oct, 1969), pp. 443-466 Published by: Wiley for the Economics Department of the University of Pennsylvania and Institute of Social and Economic Research --Osaka University StableUrl:http://www.jstor.org/stable/2525655 Accessed:11/09/20130234 Your use of the JSTOR archive indicates your acceptance of the Terms Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp JStOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support( @jstor. org Wiley, Economics Department of the University of pennsylvania, Institute of social and Economic Research Osaka University are collaborating with JSTOR to digitize, preserve and extend access to International Economic review 的d http://www.jstororg This content downloaded from 202. 115.118.13 on Wed, I I Sep 2013 02: 34: 55 AM All use subject to STOR Terms and Conditions
Economics Department of the University of Pennsylvania Institute of Social and Economic Research -- Osaka University Optimal Investment and Consumption Strategies Under Risk, an Uncertain Lifetime, and Insurance Author(s): Nils H. Hakansson Source: International Economic Review, Vol. 10, No. 3 (Oct., 1969), pp. 443-466 Published by: Wiley for the Economics Department of the University of Pennsylvania and Institute of Social and Economic Research -- Osaka University Stable URL: http://www.jstor.org/stable/2525655 . Accessed: 11/09/2013 02:34 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. . Wiley, Economics Department of the University of Pennsylvania, Institute of Social and Economic Research -- Osaka University are collaborating with JSTOR to digitize, preserve and extend access to International Economic Review. http://www.jstor.org This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:34:55 AM All use subject to JSTOR Terms and Conditions
INTERNATIONAL ECONOMIC REVIEW Vol. 10, No 3, October, 1969 OPTIMAL INVESTMENT AND CONSUMPTION STRATEGIES UNDER RISK, AN UNCERTAIN LIFETIME, AND INSURANCE* BY NILS H. HAKANSSON 1. INTRODUCTION AND SUMMARY IN A PREVIOUS ARTICLE [8, a normative model of the individuals economic decision problem under risk was presented. In addition, certain implications of the model with respect to individual behavior were deduced for the class of utility functions, 2i=rai-lu(ej),0<a<l, where c; is the amount of con sumption in period 3, such that either the risk aversion index -u(e)/u'(a), or the risk aversion index -ou'(e)lu'(e), is a positive constant for all a20. In a second paper [6], it was further shown that this model, developed with the individual in mind, also gives rise to an induced theory of the firm under risk for the same class of utility functions. In the foregoing model, it was assumed that the individual' s horizon was infinite(or known with certainty ) In this paper, we consider the same basic model with three modifications. First, we postulate that the individuals lifetime is a random variable with a known probability distribution. Second we introduce a utility function intended to represent the individuals bequest motive. Third, we offer the individual the opportunity to purchase insurance on his life. It is found that when some or all of these modifications are made all of the more important properties possessed by the optimal consumption and investment strategies under a certain horizon are preserved, albeit only d In Section 2, the various components of the decision process are constructe In the earlier model, the individuals objective was assumed to be the maxi- mization of expected utility from consumption over time. Here, we postulate more generally, that his objective is to maximize expected utility from con sumption as long as he lives and from the bequest left upon his death. As before, the individual's resources are assumed to consist of an initial capital position(which may be negative)and a non-capital income stream. The latter, which may possess any time-shape, is assumed to be known with certainty and to terminate upon his death. In addition to insurance available at a"fair rate, the individual faces both financial opportunities(borrowing and lending) and an arbitrary number of productive investment opportunities. The interest rate is presumed to be known but may have any time shape. The returns from the productive opportunities are assumed to be random variables, whose probability distributions may differ from period to period but are assumed to satisfy the"no-easy-money"condition. While no limit is placed on borrow- ing the individual is required to be solvent at the time of his death with probability 1, that is, all debt must be fully secured at all times Manuscript received November 22, 1967, revised June 3, 1968. This content downloaded from 202.115.118.13 on Wed, II Sep 2013 02: 34: 55 AM
INTERNATIONAL ECONOMIC REVIEW Vol. 10, No. 3, October, 1969 OPTIMAL INVESTMENT AND CONSUMPTION STRATEGIES UNDER RISK, AN UNCERTAIN LIFETIME, AND INSURANCE* BY NILs H. HAKANSSON 1. INTRODUCTION AND SUMMARY IN A PREVIOUS ARTICLE [8], a normative model of the individual's economic decision problem under risk was presented. In addition, certain implications of the model with respect to individual behavior were deduced for the class of utility functions, Z' , ai-lu(ci), 0 0. In a second paper [6], it was further shown that this model, developed with the individual in mind, also gives rise to an induced theory of the firm under risk for the same class of utility functions. In the foregoing model, it was assumed that the individual's horizon was infinite (or known with certainty). In this paper, we consider the same basic model with three modifications. First, we postulate that the individual's, lifetime is a random variable with a known probability distribution. Second, we introduce a utility function intended to represent the individual's bequest motive. Third, we offer the individual the opportunity to purchase insurance on his life. It is found that when some or all of these modifications are made, all of the more important properties possessed by the optimal consumption and investment strategies under a certain horizon are preserved, albeit only under special conditions. In Section 2, the various components of the decision process are constructed. In the earlier model, the individual's objective was assumed to be the maximization of expected utility from consumption over time. Here, we postulate, more generally, that his objective is to maximize expected utility from con-- sumption as long as he lives and from the bequest left upon his death. As before, the individual's resources are assumed to consist of an initial capital position (which may be negative) and a non-capital income stream. The latter, which may possess any time-shape, is assumed to be known with certainty and to terminate upon his death. In addition to insurance available at a "fair'" rate, the individual faces both financial opportunities (borrowing and lending) and an arbitrary number of productive investment opportunities. The interest rate is presumed to be known but may have any time-shape. The returns from the productive opportunities are assumed to be random variables, whose probability distributions may differ from period to period but are assumed to satisfy the "no-easy-money" condition. While no limit is placed on borrowing, the individual is required to be solvent at the time of his death with probability 1, that is, all debt must be fully secured at all times. * Manuscript received November 22, 1967, revised June 3, 1968. 443 This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:34:55 AM All use subject to JSTOR Terms and Conditions
444 NILS H. HAKANSSON The components de in Section 2 are assembled into formal model in Sections 3, 4.6. 7.8 The fundamental approach taken is that the portfolio composition the financing decision, the consumption decision and, where applicable the insurance decision, are all analyzed in one model The vehicle of analysis is discrete-time dynamic programming Sections 4, 6, 7, and 8 consider the four possible combinations of no bequest motive/ bequest motive and no insurance/insurance Explicit solutions are derived, where possible, for that class of one-period utility functions whose proportional risk aversion indices are positive constants and are found have the same form as when the horizon is known. A review of the prop erties and implications of these solutions is given in Section 5: it is noted that due to the solvency constraint, the solution does not always exist in this form for all functions in the class In Section 9, the amount of insurance to be purchased in each period is included among the decision variables. when this is done the solution found to be of the indicated form only under highly specialized conditions the optimal insurance strategy is found to be linear increasing in the future installments of the non-capital income stream In Sectio is shown that the models developed in this paper give rise o an induced theory of the firm under risk, which may be viewed as ar extension of the theory developed for the case in which the horizon is certai is shown in Section 11 that when the premium charged is fair", an individual can in most instances increase his expected utility by selling insurance to others. Thus, any given individual may be able to make himself better off both by the purchase of insurance on his own life and the sale of insurance on the lives of others. Furthermore, both a supply of and a demand for insurance will exist in an economy of individuals whose utility functions belong to the class examined ASSUMPTIONS AND NOTATION In this section, the postulates concerning the individuals preferences,re- sources, and opportunities will be specified. As the various building blocks are introduced, we also give the notation to be used in the following sections 2. 1. Resources and opportunities. We assume that the individual has the opportunity to make decisions at diserete points, called decision points, which are equally spaced in time. The space of time intervening between the two adjacent decision points 3 and j+1 will be denoted period Let pi>0 be the individuals probability of dying in the j-th period, j= 1,.,n, where 2j=1pi=l; thus n is the last period in which death may occur, We now observe that (1) pm≡p∑p m,y=1,…,n(m≤j expresses the probability that the individual will pass away in period j given that he is alive at the beginning of period m. e denote the amount of the individuals monetary(capital) resources at This content downloaded from 202.115.118.13 on Wed, II Sep 2013 02: 34: 55 AM
444 NILS H. HAKANSSON The components developed in Section 2 are assembled into formal models in Sections 3, 4, 6, 7, 8, and 9. The fundamental approach taken is that the portfolio composition decision, the financing decision, the consumption decision, and, where applicable, the insurance decision, are all analyzed in one model. The vehicle of analysis is discrete-time dynamic programming. Sections 4, 6, 7, and 8 consider the four possible combinations of no bequest motive/bequest motive and no insurance/insurance. Explicit solutions are derived, where possible, for that class of one-period utility functions whose proportional risk aversion indices are positive constants, and are found to have the same form as when the horizon is known. A review of the properties and implications of these solutions is given in Section 5; it is noted that due to the solvency constraint, the solution does not always exist in this form for all functions in the class. In Section 9, the amount of insurance to be purchased in each period is included among the decision variables. When this is done, the solution is found to be of the indicated form only under highly specialized conditions; the optimal insurance strategy is found to be linear increasing in the future installments of the non-capital income stream. In Section 10, it is shown that the models developed in this paper give rise to an induced theory of the firm under risk, which may be viewed as an extension of the theory developed for the case in which the horizon is certain [6]. Finally, it is shown in Section 11 that when the premium charged is "fair", an individual can in most instances increase his expected utility by selling insurance to others. Thus, any given individual may be able to make himself better off both by the purchase of insurance on his own life and the sale of insurance on the lives of others. Furthermore, both a supply of and a demand for insurance will exist in an economy of individuals whose utility functions belong to the class examined. 2. ASSUMPTIONS AND NOTATION In this section, the postulates concerning the individual's preferences, resources, and opportunities will be specified. As the various building blocks are introduced, we also give the notation to be used in the following sections. 2.1. Resources and opportunities. We assume that the individual has the opportunity to make decisions at discrete points, called decision points, which are equally spaced in time. The space of time intervening between the two adjacent decision points j and j + 1 will be denoted period j. Let ]5j > 0 be the individual's probability of dying in the j-th period, j = 1, *.., n, where E =fpj = 1; thus n is the last period in which death may occur. We now observe that (1) Pmj-Pi l2Pac m,=, i n(,m< j) k-m expresses the probability that the individual will pass away in period j given that he is alive at the beginning of period m. We denote the amount of the individual's monetary (capital) resources at This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:34:55 AM All use subject to JSTOR Terms and Conditions
INVESTMENT AND CONSUMPTION 445 the j-th decision point, given that he is alive at that point, by j. In the event the individual passes away in period j-1, the amount of his resources at the end of that period will be termed his estate and will be designated wj We assume that the individual may also be the recipient of a non-capital income stream during all or part of his life-time. If the individual is alive at decision point j, he will be paid the(finite)installment pertaining to period 3, 1320, at the end of that period; if he is not alive, he will receive nothing. In this paper, we make the fairly strong assumption that the individuals tential non-capital income stream is exogenously determined and is known in advance. It may be thought of as consisting of the income from labor, pensions, unemployment compensation, ete We postulate that the individual faces both financial and productive oppor- tunities in each period. The first of these is the opportunity to borrow or lend arbitrary amounts of money in each period at the riskless(finite)rate r-1>0 on the condition that any borrowings (including interest)must be fully secured. The amount saved at decision point j will be denoted z1i; negative aii will then indicate borrowing For cont ce, we shall define (2) Y rirj+i where Y, may be interpreted as the present value of the individual's potential non-capital income stream at the i-th decision point. The productive opportunities faced by the individual consist of the possi bility of making risky investments. Let the total number of different risky (productive) opportunities available to the individual at decision point j be Mi-l, of which the first S,-1 s M-l may be sold short. A short sale will be defined as the opposite of a long investment, that is, if the individual ells opportunity i short in the amount e, he will receive a immediately (te do with as he pleases)in return for the obligation to pay the transformed value of e at the end of the period. The net proceeds realized at the end of of that period will be denoted Bij. Thus, returns to scale are assumed to period j from each unit of capital invested in opportunity i at the beginni stochastically constant, all investments are assumed to be realized in cash at the end of each period, and taxes and conversion costs, if any, are assumed to be proportional to the amount invested The amount invested in opportunity ,讠=2,……,M, at the j-th decision point will be denoted zij, and is,as indicated earlier, a decision variable along with z1i It will be assumed that the joint distribution functions F, given by (3)Fa2,m,…,mM)≡Pr{a≤m,月3s3,…,M;≤},j=1,…… are known and independent In addition, we shall postulate that the [Bish 1 In real world situations, the individual would, of course, be forced to derive h own subjective probability distributions. Numerous descriptions of how this may be ccomplished, on the basis of postulates presupposing certain consistencies in behavior, are available in the literature; see, for example the accounts of Savage [ 14]an Marschak [1 This content downloaded from 202.115.118.13 on Wed, II Sep 2013 02: 34: 55 AM
INVESTMENT AND CONSUMPTION 445 the j-th decision point, given that he is alive at that point, by x;. In the event the individual passes away in period j - 1, the amount of his resources at the end of that period will be termed his estate and will be designated x. We assume that the individual may also be the recipient of a non-capital income stream during all or part of his life-time. If the individual is alive at decision point j, he will be paid the (finite) installment pertaining to period j, yj ? 0, at the end of that period; if he is not alive, he will receive nothing. In this paper, we make the fairly strong assumption that the individual's potential non-capital income stream is exogenously determined and is known in advance. It may be thought of as consisting of the income from labor, pensions, unemployment compensation, etc. We postulate that the individual faces both financial and productive opportunities in each period. The first of these is the opportunity to borrow or lend arbitrary amounts of money in each period at the riskless (finite) rate rj- 1 > 0 on the condition that any borrowings (including interest) must be fully secured. The amount saved at decision point j will be denoted z,j; negative zlj will then indicate borrowing. For convenience, we shall define (2) yj - yj Yi+1 ... + Yn j1* ,n r3 rjrj+i ri ... rn where Yj may be interpreted as the present value of the individual's potential non-capital income stream at the j-th decision point. The productive opportunities faced by the individual consist of the possibility of making risky investments. Let the total number of different risky (productive) opportunities available to the individual at decision point j be Mj- 1, of which the first Sj - 1 < Mj -- 1 may be sold short. A short sale will be defined as the opposite of a long investment, that is, if the individual sells opportunity i short in the amount 0, he will receive a immediately (to do with as he pleases) in return for the obligation to pay the transformed value of 0 at the end of the period. The net proceeds realized at the end of period j from each unit of capital invested in opportunity i at the beginning of that period will be denoted ,Bj. Thus, returns to scale are assumed to be stochastically constant, all investments are assumed to be realized in cash at the end of each period, and taxes and conversion costs, if any, are assumed to be proportional to the amount invested. The amount invested in opportunity i, i = 2, ..., Mi, at the j-th decision point will be denoted zij, and is, as indicated earlier, a decision variable along with zlj. It will be assumed that the joint distribution functions Fj given by (3) Fi(x2, X3, * * *, XMj) - Pr{j92j < X2, l3j < X3* X8, ', imjj x XMj} 'j = 1, **, n are known and independent'. In addition, we shall postulate that the {J9ij} 1 In real world situations, the individual would, of course, be forced to derive his own subjective probability distributions. Numerous descriptions of how this may be accomplished, on the basis of postulates presupposing certain consistencies in behavior, are available in the literature; see, for example, the accounts of Savage [14] and Marschak [11]. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:34:55 AM All use subject to JSTOR Terms and Conditions
446 NILS HAKANSSON satisfy the following conditions for all j and all finite Bi; such that 0 i20 for all i>s, and 0,j+0 for at least one i.( 5) is known as the"no-easy-money condition for the case when the lending rate equals the borrowing rate [8]. This condition states that no combination of productive investment opportunities exists in any period which provides, with probability 1, a return at least as high as the(borrowing)rate of interest; no combination of short sales is available in which the probability is zero that a loss will exceed the (lending)rate of interest; and no combination of productive investments made from the proceeds of any short sale can guarantee against loss In some variants of the basic model the individual has the opportunity to purchase term insurance on his own life and to sell (purchase)term insurance on the lives of others in each period. Let t20 denote the premium paid by the individual at the j-th decision point for life insurance on his own life during period 3. If the individual dies during this period, which by(1)has probability pii of happening, we assume that his estate will receive ti/pi; at the end of period 3; if he is alive at decision point 3+1, he will receive nothing Since in this contract the mathematical expectation of the"return equals the cost we shall say that the insurance is available at a"fair" rate We assume that insurance is issued only when pjj<l, i.e. at decision points We shall allow the possibility of contracting in advance for purchases of insurance on the individuals own life. Such an arrangement will be called an insurance contract. The unexpired portion of such a contract at decision point 3 will be denoted (ti, ti+i,., tw-i, where tu/pek is the amount of insurance the individual will keep in force in period k given that he is alive at the ke-th decision point (when the premium tk is paid) For convenience we define 7=2+ We also assume that t≤x+B, where B, denotes the maximum an individual may borrow at the j-th decision point on the security of his non-capital income stream and his insurance contract. Since no insurance can be issued at the nth decision point, it is clear that (8) Bw=yn/r. and that t计+1+B This content downloaded from 202.115.118.13 on Wed, II Sep 2013 02: 34: 55 AM
446 NILS H. HAKANSSON satisfy the following conditions: (4) O? jij 0 for all j and all finite Oij such that Oij > 0 for all i > Sj and Oiji 0 for at least one i. (5) is known as the "no-easy-money" condition for the case when the lending rate equals the borrowing rate [8]. This condition states that no combination of productive investment opportunities exists in any period which provides, with probability 1, a return at least as high as the (borrowing) rate of interest; no combination of short sales is available in which the probabilityis zero that a loss will exceed the (lending) rate of interest; and no combination of productive investments made from the proceeds of any short sale can guarantee against loss. In some variants of the basic model the individual has the opportunity to purchase term insurance on his own life and to sell (purchase) term insurance on the lives of others in each period. Let tj ? 0 denote the premium paid by the individual at the j-th decision point for life insurance on his own life during period j. If the individual dies during this period, which by (1) has probability pjj of happening, we assume that his estate will receive tj/pjj at the end of period j; if he is alive at decision point j + 1, he will receive nothing. Since in this contract the mathematical expectation of the "return"' equals the cost, we shall say that the insurance is available at a "fair" rate. We assume that insurance is issued only when pjj < 1, i.e., at decision points. I, * **, n-1. We shall allow the possibility of contracting in advance for purchases of insurance on the individual's own life. Such an arrangement will be called an insurance contract. The unexpired portion of such a contract at decision point j will be denoted (tj, tj+?, * *, tn-1), where tklPkk is the amount of insurance the individual will keep in force in period k given that he is alive. at the k-th decision point (when the premium tk is paid). For convenience we define (6) Tj-tj+ ti+ ... + tn-i j-1,...,n-1. rj ri ... rn-2 We also assume that (7) tj < xj +Bj, j-1, * l ,nwhere Bj denotes the maximum an individual may borrow at the j-th decision point on the security of his non-capital income stream and his insurance contract. Since no insurance can be issued at the n-th decision point, it is, clear that (8) Bn = yn/rn, and that (9) Bj = min { r__+ ' _ jtj B, } = X-1 ri ri~~~I This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:34:55 AM All use subject to JSTOR Terms and Conditions
INVESTMENT AND CONSUMPTION 0 The purchase and sale of insurance on the lives of others will be viewed s a subset of the productive opportunities. Each person will be assumed to give rise to a separate investment opportunity, the return of which is ependent of the returns of all other opportunities 2.2. The Utility Function. The amount spent on consumption in period j will be designated c. As indicated, c, is a decision variable; in order to give it economic meaning, we require it to be nonnegative. We now postulate that the individuals preference ordering at the beginning of period m, conditioned on the event that death occurs in period k 2 m, is representable by a numerical utility function Umk. This utility function is de fined on the Cartesian product of all possible consumption programs(cm, ... ck and the amount of his estate ak+ at the end of period k; thus, the utility function is independent of the opportunities faced by the individual. We assume in this paper that the conditional utility function Umk has the form (11)Um&cm c,xk+1)=-1 ∑(量)00+…m-a m,k=1 (m8≤k) Implicit in this form is the assumption that preferences are independent over time We shall call u(c) the one-period utility function of consumption and g(')the utility function of bequests. The constant aj>0(ao= 1)is the patience factor linking the (one-period) utility functions of periods j and j+1 given that the individual will be alive at decision point j+l.When ai<1(a,2 1)we shall say that impatience (patience) prevails in period j with respect to period 3+1. Similarly, the constant &; expresses the relative weight attached to bequests by the individual at decision point 3, given that death will occur in period 3. Since ar, and a; are constants, we note that the rate of patience, while dependent on time, is independent of the overall level of utility(see [10]). We also postulate that the individual obeys the von Neumann-Morgenstern postulates [15]; accordingly, the individuals objective is to maximize the expected utility attainable from consumption over his life-time and the estate remaining and bequeathed upon his death. We also assume that the individual always prefers more consumption to less in any period, i.e., that u(c)is monotone increasing, and that the bequest function g(ar)is non-decreasing Finally, we assume that the individual is risk averse with respect to con- plies that u(c) is strictly concave, and that u(c) and g(a') are twice differentiable The notation developed in the previous section is summarized below before e proceed to construct our basic model z We assume, however, that the continuity postulate has been modified in such a way as to permit unbounded utility functions In congruence with this premise, we assume that the functions(11)are cardinal This content downloaded from 202.115.118.13 on Wed, II Sep 2013 02: 34: 55 AM
INVESTMENT AND CO.NSUMPTION 447 where, by assumption, (10) tn = 0. The purchase and sale of insurance on the lives of others will be viewed as a subset of the productive opportunities. Each person will be assumed to give rise to a separate investment opportunity, the return of which is independent of the returns of all other opportunities. 2.2. The Utility Function. The amount spent on consumption in period j will be designated Cj. As indicated, cj is a decision variable; in order to give it economic meaning, we require it to be nonnegative. We now postulate that the individual's preference ordering at the beginning of period m, conditioned on the event that death occurs in period k ? m, is representable by a numerical utility function Umk. This utility function is defined on the Cartesian product of all possible consumption programs (cm, ... , Ck) and the amount of his estate xk+1 at the end of period k; thus, the utility function is independent of the opportunities faced by the individual. We assume in this paper that the conditional utility function Umk has the form (11) Umk(Cm, * * Ck, Xk+?) = aE au)(cj) + am a ak-lOkg(Xk+1) am-1 j=m \irm-1 m, k-1, *,n(m 0 (ao 1) is the patience factor linking the (one-period) utility functions of periods j and j + 1 given that the individual will be alive at decision point j + 1. When a. < 1 (aj ? 1) we shall say that impatience (patience) prevails in period j with respect to period j + 1. Similarly, the constant sj expresses the relative weight attached to bequests by the individual at decision point j, given that death will occur in period j. Since aj and dj are constants, we note that the rate of patience, while dependent on time, is independent of the overall level of utility (see [10]). We also postulate that the individual obeys the von Neumann-Morgenstern postulates [15];2 accordingly, the individual's objective is to maximize the expected utility attainable from consumption over his life-time and the estate remaining and bequeathed upon his death.3 We also assume that the individual always prefers more consumption to less in any period, i.e., that u(c) is monotone increasing, and that the bequest function g(x') is non-decreasing. Finally, we assume that the individual is risk averse with respect to consumption, which implies that u(c) is strictly concave, and that u(c) and g(x') are twice differentiable. The notation developed in the previous section is summarized below before we proceed to construct our basic model: 2 We assume, however, that the continuity postulate has been modified in such a way as to permit unbounded utility functions. 3 In congruence with this premise, we assume that the functions (11) are cardinal. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:34:55 AM All use subject to JSTOR Terms and Conditions
448 NILS H. HAKANSSON Puy probability of death in period j(j s n), pj probability of death in period 3(2m), given that the individual is alive at the beginning of period m capital position at decision point estate at the end of period j-l, given that death occurs in period y, non-capital income received at the end of period 3 if the individual is alive at the beginning of period 3 Y present value at decision point j of the potential non-capital income tream rs-1 interest rate in period j, s amount lent at decision point 3 M; number of investment opportunities in period 3, number of investment opportunities which may be sold short in period 3 net proceeds realized at the end of period j from each unit invested in opportunity i,i=2,., M,, at the beginning of period j, Fs joint distribution function of B2; ,.. AM it zi amount invested in opportunity i, i=l,., Mi, at the beginning of t insurance premium paid at the beginning of period 3 for insurance in period t contractual insurance premium payable at the beginning of period 3 if individual is alive at that point, T present value at decision point j of potential premiums t;, tj+l,., tn-1 amount of consumption in period 3 one-period utility function of consumption utility function of bequests a, patience factor linking periods j and j+I if the individual remains alive at the end of period j patience factor linking periods j and 3+l if the individual passes 3. DERIVATION OF THE BASIC MODEL Te shall now identify the relation which determines the amount of capital (debt)on hand at each decision point in terms of the amount on hand at the previous decision point. This leads to the pair of difference equations: =∑2计+721+y x+=∑1+T1+y+tP √=1 where This content downloaded from 202.115.118.13 on Wed, II Sep 2013 02: 34: 55 AM
448 NILS H. HAKANSSON pj5 probability of death in period j(j ? n), Pmi probability of death in period j(2 m), given that the individual is alive at the beginning of period m, Xj capital position at decision point j, Xi estate at the end of period j - 1, given that death occurs in period j-l, yi non-capital income received at the end of period j if the individual is alive at the beginning of period j, Y3 present value at decision point j of the potential non-capital income stream, rj- 1 interest rate in period j, Z1j amount lent at decision point j, M6 number of investment opportunities in period j, Si number of investment opportunities which may be sold short in period j, 13ij net proceeds realized at the end of period j from each unit invested in opportunity i, i = 2, ***, Mj, at the beginning of period j, Fj joint distribution function of 2,j, *, 19M6j, zi6 amount invested in opportunity i, i = 1, ***, Mj, at the beginning of period j, tj insurance premium paid at the beginning of period j for insurance in period j, ti contractual insurance premium payable at the beginning of period j if individual is alive at that point, T3 present value at decision point j of potential premiums t6, t6+1, *.* *, cj amount of consumption in period j, Umk utility function at the beginning of period mn of consumption and bequests given that the individual passes away in period k ? m, u one-period utility function of consumption, g utility function of bequests, Ja, patience factor linking periods j and j + 1 if the individual remains alive at the end of period j, ai patience factor linking periods j and j + 1 if the individual passes away in period j. 3. DERIVATION OF THE BASIC MODEL We shall now identify the relation which determines the amount of capital (debt) on hand at each decision point in terms of the amount on hand at the previous decision point. This leads to the pair of difference equations: Mj (12) xj+l = E fiijzij + rjzlj + yj, -,*** n -1 i=2 and M j (13) xj+1 = X Iijzij + rjzlj + yj + tjlpjj, j- = * *, n i=2 where This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:34:55 AM All use subject to JSTOR Terms and Conditions
INVESTMENT AND CONSUMPTION (14) by direct application of the definitions given in Section 2. 1. The first terms of(12)and (13)represent the proceeds from productive investments the e secon terms the pay ment of the debt or the proceeds from savings, the third terms the non-capital income received, and the fourth term in (13)the proceeds from life insurance Inserting (14)into(12)and (13) we obtain (5)2+=(56=)十(--)+,=1,…n-1 (16)+1=∑(1-r)2;+;m--t)++tp,了=1,……, The restriction that only the first S, opportunities may be sold short in period (17) 0,=S+1,…,M must hold while the assumption that all borrowing must be fully secured implies that a must satisfy the condition Pr{m≥0}=1 j=2, y()it follows that there is an upper limit on consumption in period jj=1, 3+Bi-tj which, since c:20, must be non-negative in order that a feasible solution exist in period 3. We shall now define f(ai as the maximum expected utility attainable b the individual over his remaining life-time, as of the beginning of period j on the condition that he is alive at that point and that his capital is w Utilizing(1)and(11), we may write this definition formally ∫x)≡ max elpis(e,x)+p,+1U,;+(e,,x +pin(c max Elu(ci)+pj dig(3+1)+.2 pika jiu(cj+1) +p,+100;+g(x3+2 ……+pn By the principle of optimality,(20) may be written, using (1) The principle of optimality states that an optimal strategy has the property that whatever the initial state and the initial decision are the remaining decisions must onstitute an optimal strategy with regard to the state resulting from the first decision. See [2,(83)1 This content downloaded from 202.115.118.13 on Wed, II Sep 2013 02: 34: 55 AM
INVESTMENT AND CONSUMPTION 449 Mj (14) Zl; = xj-c;-tj- Zij, j=1,***, n i=2 by direct application of the definitions given in Section 2.1. The first terms of (12) and (13) represent the proceeds from productive investments, the second terms the payment of the debt or the proceeds from savings, the third terms the non-capital income received, and the fourth term in (13) the proceeds from life insurance. Inserting (14) into (12) and (13) we obtain Maj (15) xj+1 = , (Aii -rj)zij + ri(x - cj- ti) + yj, j=, n*** n-1 i=2 and Mj (16) x3 +l= E (~ij- rj)zij + ri(x - cj- tj) + yj + tjlpjj, j=1***, n. i=2 The restriction that only the first Sj opportunities may be sold short in period j implies that (17) Zij 2 0 , i = Sj + 1, * ,Mj, j = 1, *** must hold while the assumption that all borrowing must be fully secured implies that x; must satisfy the condition (18) Pr{x >2o}=1, j=2, *- ,n+1. By (5) it follows that there is an upper limit on consumption in period j,j=1, j * ,n, given by (19) xj + Bj -t, jz=1, n which, since Cj ? 0, must be non-negative in order that a feasible solution exist in period j. We shall now define fi(xx) as the maximum expected utility attainable by the individual over his remaining life-time, as of the beginning of period j, on the condition that he is alive at that point and that his capital is xj. Utilizing (1) and (11), we may write this definition formally: fj(xi) max E[pjj Ujj(cj, x>+1) + Pj,j+lUj,j+I(Ci, Cj+l, X42) + * + P.Un(C, ** C, xn+l)] j = 1, ***, n - max E[u(cj) + pjj3jg(xj+i) + E Pikaju(cj+i) (20) k=j+l + pj,j+jayjj+jg(x4+2) + > Pikajaj+IU(Cj+2) k=j+2 + + Pinaj *.. an-1Jng(X$n+1)] j = 1, n By the principle of optimality, (20) may be written, using (1):4 4 The principle of optimality states that an optimal strategy has the property that whatever the initial state and the initial decision are, the remaining decisions must constitute an optimal strategy with regard to the state resulting from the first decision. See [2, (83)]. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:34:55 AM All use subject to JSTOR Terms and Conditions
NILS H. HAKANSSON 22)(x1)=max{t(c)+Epg(x3+)+a(1-p)f+12x+ 1,……,n8 Letting a≡P b;≡a(1-p;), (22) may be written more concisely as f,(ai)= max u(ci)+ Elaig(a;+1)+ b fi+(ecj+iJI We shall now attempt to obtain the solutions to(25) for certain classes of the functions u(c)under different sets of assumptions concerning the bequest function g(a)and the availability of insurance More specifically, we shall consider the class of functions u(c) such that u(e) satisfies one (or more)of the functional equations u(ey)= v(ew(y), u(ay)=v(e)+ wly for c20. The functional equations(26) and(27)in three unknowns the set of equations usually referred to as the generalized Cauchy or Pexider's equations. That subset of their solutions, which is increasing and strictly concave in u, is given (leaving out v and (28) 00 Model il ule= le Model III Note that since u(c)is a cardinal utility function, the solutions(28)-(30)also include every solution 21+ 22u(c) to(26)and(27) where a, and 22>0 are In [9, it was also noted that (28)-(30)is the solution to the differential cv"(c)+r(c)=07>0 Thus,(28)-( 80) are also the only monotone increasing and strictly concave utility functions for which the proportional risk aversion index q*(c)≡-cw"c)/u'(c is a positive constant 4. NO BEQUEST MOTIVE, NO INSURANCE We shall first consider the simplest case, namely that in which there is no bequest motive and no insurance is available. The absence of a bequest motive implies that 6;g(x3+)=0, This content downloaded from 202.115.118.13 on Wed, II Sep 2013 02: 34: 55 AM
450 NILS H. HAKANSSON (22) fj(xj) = max {u(cj) + E[6jpjig(x'+1) + aj(1 -pjj)fj+,(xj+,)]l ji=,**, n. Letting (23) aj3 5jp and (24) b aj(l a - pjj) (22) may be written more concisely as (25) fj(xj) = max {u(cj) + E[aig(xj'+) + bjfj_1(xj+1)]}, j 1, ***, n We shall now attempt to obtain the solutions to (25) for certain classes of the functions u(c) under different sets of assumptions concerning the bequest function g(x') and the availability of insurance. More specifically, we shall consider the class of functions u(c) such that u(c) satisfies one (or more) of the functional equations (26) u(xy) = v(x)w(y), (27) u(xy) = v(x) + w(y), for c 2 0. The functional equations (26) and (27) in three unknowns belong to the set of equations usually referred to as the generalized Cauchy equations or Pexider's equations. That subset of their solutions, which is monotone increasing and strictly concave in u, is given (leaving out v and w) by [9]: (28) u(c) = cy 0 0 Model II (30) u(c) = log e Model III . Note that since u(c) is a cardinal utility function, the solutions (28)-(30) also include every solution 21 + 22u(c) to (26) and (27) where 21 and 22 > 0 are constants, if simultaneously, g(x') is represented by ;,2g(X'). In [9], it was also noted that (28)-(30) is the solution to the differential equation (31) cu"(c) + yu'(c) = 0 r > 0 . Thus, (28)-(30) are also the only monotone increasing and strictly concave utility functions for which the proportional risk aversion index (32) q*(c) -cu"(c)/u'(c) is a positive constant. 4. NO BEQUEST MOTIVE, NO INSURANCE We shall first consider the simplest case, namely that in which there is no bequest motive and no insurance is available. The absence of a bequest motive implies that (33) aig(xi+)=, j1,* ** ,n . This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:34:55 AM All use subject to JSTOR Terms and Conditions
INVESTMENT AND CONSUMPTION For (88)to hold, it is clearly necessary and sufficient either that 0=0 for all j or that g(a) is identically zero. A utility function(11) for which(33) holds has been called a Fisher utility function [17]. The unavailability of insurance of course implies that 4 t;=t;=0 B=yi/r j=1, Utilizing (23), (388),(15), (84),(18), and(16), and letting Z;=(z PROBLEM A ∫(x)=max{ac)+b2f+1(∑(月;-T);+r42;-c)+v where fn+1(xn+)≡0 c;≥ 2≥0,讠=S;+1,…,M (39) Pr∑(;-T)z;+rx-c)+y≥0}=1 Before attempting to obtain the solution to this problem, we shall state two preliminary results LEMMA: Let uc), Bi, and r, be defined ag in Section 2. Then the functions (40) (S(问-r)n+7 8ubject to (41) Pr{(A-)m+r≥0}=1 ≥0 have(finite) maxima and the maximizing vi(= v)are bounded and unique for all i and COROLLARY: Let u(c), Bij, and r, be defined as in Section 2. Moreover, let (c)be such that it has no lower bound. Then the vector uj=(v% which ma.roma 8ubject to(41)and (42)is interior with res pect to(41) that is This content downloaded from 202.115.118.13 on Wed, II Sep 2013 02: 34: 55 AM
INVESTMENT AND CONSUMPTION 451 For (33) to hold, it is clearly necessary and sufficient either that aj = 0 for all j or that g(x') is identically zero. A utility function (11) for which (33) holds has been called a Fisher utility function [17]. The unavailability of insurance of course implies that (34) tj=tj=0, n-1 which in turn, by (9) and (10), implies that (35) B = yj/rj, j=1,**,n. Utilizing (23), (33), (15), (34), (18), and (16), and letting (zj2j, ** , z ij), (25) now becomes PROBLEM A. (36) fj(xj) = max {u(cj) + biE j+ , (3ij - rj)zij + ri(xj - cj) + yj cj, ij i=2 where (37) f.+1(x.+?) 0 (38) c; 2 0, j = 1, n (17) zij 2 0, i = Sj + l, *-- *,Mj j = 1, *,n, and (39) Pr { (,iji- rj)zij + r(xj -cj) + yj ? 0} = 1 j = 1,* **,n. i$=2J Before attempting to obtain the solution to this problem, we shall state two preliminary results. LEMMA: Let u(c), 3ij, and rj be defined as in Section 2. Then the functions (40) hj(v2i, *, vXjj)= ELu(E (3ij - rj)vii + ri)] subject to (41) Pr (j3i. - rj)vij + rj ? 0} = 1 and (42) v?ji0, i=Sj+1, ,M have (finite) maxima and the maximizing vij(= v;) are bounded and unique for all i and j. COROLLARY: Let u(c), ~ii, and rj be defined as in Section 2. Moreover, let u(c) be such that it has no lower bound. Then the vector iY (v*, * *, which maximizes (40) subject to (41) and (42) is interior with respect to (41), that is This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:34:55 AM All use subject to JSTOR Terms and Conditions
