How does the no-Ponzi game condition work in an optimal consumption problem?

Tamegawa, Kenichi Article How does the no-Ponzi game condition work in an optimal consumption problem? Economics & Finance Research Provided in Cooperation with: Taylor & Francis Group Suggested Citation: Tamegawa, Kenichi (2013) : How does the no-Ponzi game condition work in an optimal consumption problem?, Economics & Finance Research, ISSN 2164-9499, Taylor & Francis, Abingdon, Vol. 1, Iss. 1, pp. 42-44, https://doi.org/10.1080/21649480.2013.810553 This Version is available at: https://hdl.handle.net/10419/147690 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/3.0/ Economics & Finance Research, 2013 Vol. 1, 42–44, http://dx.doi.org/10.1080/21649480.2013.810553 How does the no-Ponzi game condition work in an optimal consumption problem? Kenichi Tamegawa Meiji University, School of Commerce, 1-1 Kanda-Surugadai, Chiyoda-ku, Tokyo 101-8301, Japan In an optimal consumption choice problem, in which households have assets yielding interest rates, it is difficult to guarantee the existence of a solution without some restrictions for the consumption space, if the assumed utility function is unbounded. In this article, we formally state how the no-Ponzi game condition is used to guarantee an existence of optimal solutions. Furthermore,weprovidetheconditioninwhichasolutionattainsthefiniteintertemporalutility. I. Introduction Economists often assume that infinitely lived households behave such that they maximize their lifetime utility subject to an intertemporal budget constraint. However, in solving this maximization problem,there arecases whenone needsto usecertain techniquesto guaranteetheexistenceofasolutionpathiftheassumedutilityfunction is unbounded. In particular, this is the case when households have financial assets yielding an interest rate. This is because the domain of an objective function is not compact in the usual norm. Stokey and Lucas (1989) describe how we confirm the existence of an optimal path in a capital accumulation problem in which the utility function is unbounded. In this case, the domain of an objective function can be bounded. However, in asset choice problems, this approach is difficult to apply. Boyd (1990) copes with the unboundedness using the specific norm for a domain. Later, this approach is made applicable to a dynamic programming techniquebyDuran(2000).Forotherapproachesofboundedreturns,see Alvarez and Stokey (1998), Streufert (1990), Le Van and Morhaim (2002),Rincon-ZapateroandRodriguez-Palmero(2003)andLeVan and Vailakis (2005). These works tend towards generality, but we limit our attention to financial asset choice problems. By doing this, we can show that the no-Ponzi game condition is sufficient for showing the existence of an optimal path under the usual assumptions imposed on a temporal utility function. In this article, we formally state how the no-Ponzi game condition is used to guarantee an existence of optimal solutions. The rest of this article is organized as follows. In Section 2, we describe the model setting and state the main theorem. In Section 3, we apply our theorem to a problem with the Constant relative risk aversion (hereafter, CRRA) utility function. Finally, in Section 4, we conclude the article. ∗E-mail: tamega[email protected] II. The Existence Theorem Consider a sequence of consumption, 0c=(c0,c1,...)∈R∞ +, where Rdenotes the set of real numbers plus infinity and R+represents non-negative part of R. We define a real-valued intertemporal utility function U:R∞ +→Ras follows: U(0c)= ∞  t=0 βtu(ct) whereurepresentsareal-valuedtemporalutilityfunctionu:R+→ Rand βis a discount factor. The maximization problem we are interested in is as follows: supU(0c),(P) s.t. At+1≤RtAt+wt−ct, lim t→∞ At t i=1Ri=0, given {Rt},{wt}, and A0 where Atrepresents a financial asset yielding interest rate denoted by Rtand wtdenotes income. The third of the constraints is the no-Ponzi game condition. We assume that the sequences {Rt}and {wt}are bounded. Inamaximizationproblemlike Pgivenabove,itisoftendifficult toguaranteetheexistenceofasolution. However, with the no-Ponzi game condition and a suitable norm, this becomes an easy task. Our approach is to first construct infeasible consumption sequences that include the feasible sequences. To do so, we define the following notation: R=inf{Rt};¯ R=sup{Rt};¯w=sup{wt}. The no-Ponzi game condition, limt→∞ At/(t i=1Ri)=0, and the © 2013 Kenichi Tamegawa How does the no-Ponzi game condition work in an optimal consumption problem? 43 budget constraint yield1 c0+c1 R1+···≤w0+w1 R1+w2 R1R2+···+R0A0, ≤¯w+¯w R+¯w (R)2+...+R0A0 This inequality implies that the maximum consumption at the period 0 is constant and finite: c0≤M, where M=¯w+¯w/R+ ...+R0A0. Next, we consider the maximum consumption level at period 1. It can be obtained when the consumption at period 0 is 0. Therefore, we have c1≤¯ RM. The maximum consumption level at period 2 can be constructed in a similar manner: c2≤ (¯ R)2M. Continuing this procedure, we can confirm that for any t,ct≤(¯ R)tM.2Thus, the no-Ponzi game condition yields the maximum consumption plan denoted by 0¯c, which is infeasible: 0¯c=(M,¯ RM,...,(¯ R)tM,...). To facilitate our analysis, we employ the following sup–norm for a consumption sequence t 0c=(c0,c1,...,ct), as in Boyd (1990): t 0c=sup s∈{0,1,...,t} cs (¯ R)s Note that 0¯cand therefore, 0cis finite for any plan of feasible consumption plans. Consider the set of the feasible consumption plans denoted by C.We denote by ld ∞the consumption space that is finite for any 0cin the norm of limt→∞ t 0c. Since Cis not compact in the sup–norm,3we consider the weak topology that is generated by the set of all bounded linear functionals on ld ∞. Since (¯ R)tM→∞, we have the possibility of U(0c)=∞.To cope with this in the following main theorem, we define by U(t 0c) the partial sums of the discounted temporal utility function denoted as follows: U(t 0c)= t  s=0 βsu(cs) Inthefollowing,aconsumptionplan0cisoptimalifU(0c)≥U(0c) for all 0c∈C. Here, we state the main theorem. Theorem Supposethattheno-Ponzigameconditionholds.Ifu(ct) isanon-decreasingweakupper-semicontinuousfunction,thenthere exists an optimal consumption plan for the problem P. Proof If U(0c)=−∞for every 0c∈C, then those plans are trivially optimal. Therefore, consider that we have elements such that U(0c+)>−∞ for some 0c+∈C. Denote by C∗the set of feasible consumption plans that exclude a plan attaining U(0c)=−∞. Furthermore, to facilitate the proof, we define asymptotic cones as follows: for any 0c∈C∗ U∞(0c)=d∈R+|lim t→∞  U(t 0c) at =dfor some |at|→+∞  If one considers the case of a with γ>1, then limt→∞ |U(t 0c)/a|for any 0c∈C∗. Therefore, U∞(0c)is non-empty for 1Note that limt→∞ At/(t i=1Ri)= 0 is allowed if it is finite. 2We assume R>1. If R≤1, then we can say ct≤Mfor any t. 3See Majumdar (1975). any 0c∈C∗. Note that we have two cases: (i) U∞(0c)={0}for any 0c∈C∗; (ii) U∞(0c)={0}for some 0c∈C∗, and U∞(0c)={0}for any 0c∈C∗/{0c}.4 In case (i), since U(0c)is bounded, limt→∞ βt|u(ct)|=0 for any ct∈C∗. Since |U(0c)−U(t 0c)|=∞ s=t+1βs|u(cs)|→0 for any ct∈C∗,U(t 0c)uniformly converges to U(0c). Therefore, U(0c)is weakupper-semicontinuous.Withthisupper-semicontinuity, there exists a neighbourhood {0cε}of 0c+such that {0cε}={ 0c∈C|U(0c+)+ε≥U(0c)}for any ε>0. Here, define the following set Cε=0c∈R∞ +|U(0cε)≤U(0c)and 0c∈C ThecompactnessofCandtheupper-semicontinuityofUimplythat Cεiscompact.Theextreme-valuetheoremguarantees the existence of an optimal consumption plan. In case (ii), there exists consumption plans {0c}such that limt→∞ U(t 0c)/|at|becomes a convergent sequence since uis nondecreasing; for example, a t=βtu(c t). Since U(0c)is infinite, 0c is trivially the optimal plan in the problem P. If a solution attains infinite utility, it would be meaningless in terms of a policy analysis. Hence, our interest is in case (i) in the above proof. Whether the optimization problem is case (i) or case (ii) can be checked by confirming βtu(( ¯ R)tM)→0. If it is satisfied, then we are in case (i). Now we provide the following useful proposition.  Proposition Suppose that the no-Ponzi game condition holds. If u(ct)is a non-decreasing weak upper-semi continuous function and βtu(( ¯ R)tM)→0,then there existsan optimalconsumptionplanfor the problem P and the solution attains finite utility. III. Examples In this section, we introduce an example in which the assumption of the above proposition is satisfied. Suppose that a temporal utility function is the CRRA type, u(ct)=(ct)1−δ/(1−δ). In this case, we can easily check when the assumption of the above proposition is satisfied: Case (I): δ=1 (that is, u(ct)=logct) In this case, since βtlog(¯ R)tM→0, we have U(0¯c)=logM 1−β+β 1−β2log ¯ R<∞ Case (II): δ= 1 Ifβ(¯ R)1−δ<1,thenβt([(¯ R)tM]1−δ/(1−δ)) →0. In this case, U(0¯c)=M1−δ 1−δ 1 1−β(¯ R)1−δ<∞ Note that if β<1 and ¯ R>1, then β(¯ R)1−δ<1 for any δ>0. 4If U∞(0c)={0}, then U(0c)=∞. 44 K. Tamegawa IV. Conclusion In this article, we show the sufficiency of the no-Ponzi game condition for guaranteeing the existence of optimal solutions for asset choiceproblems,usingthespecific norm. Further,we provideasufficient condition in which a solution attains the finite intertemporal utility. Acknowledgements I am grateful to anonymous referees and Shin Fukuda for their help-ful comments. This work was supported by a Grant-in-Aid for ScientificResearch(No.22330090)fromtheMinistryofEducation, Culture, Sport, Science and Technology, Japan. 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