# Optimal consumption and investment strategies with a perishable and an indivisible durable consumption good

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<ul><li><p>Journal of Economic Dynamics & Control 28 (2003) 209253www.elsevier.com/locate/econbase</p><p>Optimal consumption and investment strategieswith a perishable and an indivisible durable</p><p>consumption good</p><p>Anders Damgaarda, Brian Fuglsbjergb, Claus Munkc;aDanske Research, Danske Bank, DK-1092 Copenhagen K, Denmark</p><p>bSimCorp A/S, DK-2100 Copenhagen O, DenmarkcDepartment of Accounting, Finance & Law, University of Southern Denmark-Odense, Campusvej 55,</p><p>DK-5230 Odense M, Denmark</p><p>Abstract</p><p>We study the consumption and investment choice of an agent in a continuous-time economywith a riskless asset, several risky 0nancial assets, and two consumption goods, namely a per-ishable and a durable good with an uncertain price evolution. Assuming lognormal prices and amultiplicatively separable, isoelastic utility function, we provide an explicit Merton-type solutionfor the optimal strategies for the case where the durable (and all other assets) can be tradedwithout transaction costs. For the case where the durable good is indivisible, in the sense thatdurable trades imply transaction costs proportional to the value of the current durable holdings,we show analytically that the optimal durable trading policy is characterized by three constantsz z 3z. As long as the ratio z of the total current wealth to the value of current durableholdings of the investor is in (z; 3z), it is optimal not to trade the durable. At the boundaries ofthis interval it is optimal to trade the durable to attain z = z. The model is used to examinethe optimal substitution between perishable and durable consumption and the importance of thedurable price uncertainty and the correlation between the price of the durable good and 0nancialasset prices.? 2002 Elsevier B.V. All rights reserved.</p><p>JEL classi/cation: G11; D11; D91; C61</p><p>Keywords: Optimal consumption and investment; Durable goods; Transaction costs; Viscosity solutions;Numerical solution</p><p> Corresponding author. Tel.: +45-6550-3257; fax: +45-6593-0726.E-mail address: cmu@sam.sdu.dk (C. Munk).</p><p>0165-1889/02/$ - see front matter ? 2002 Elsevier B.V. All rights reserved.PII: S0165 -1889(02)00135 -5</p></li><li><p>210 A. Damgaard et al. / Journal of Economic Dynamics & Control 28 (2003) 209253</p><p>1. Introduction</p><p>Modern economies oEer an enormous variety of consumption goods. For modelingpurposes each good is typically classi0ed either as a perishable good or a durablegood. A perishable good cannot be stored and provides utility only at the time of pur-chase. A durable good provides utility to its owner over a period of time and can beresold so that it also acts as an investment that transfers wealth over time. Traditionalmodels of optimal consumption and investment problems consider either a single per-ishable consumption good, cf. Merton (1969), or a single durable consumption good,cf. Grossman and Laroque (1990). In this paper, we merge these settings by allowingfor both a perishable and a durable good with a stochastically evolving relative price.This enables us to study optimal behavior in an economically more appealing setupand to address questions that cannot be dealt with in traditional single-good models,such as how optimal perishable and durable consumption policies are related and howthe uncertainty about future relative consumption prices aEects optimal consumptionand investment decisions.More speci0cally, we examine the optimal consumption and investment choice of an</p><p>agent in a continuous-time economy with one riskless and several risky 0nancial assetsand both a perishable and a durable consumption good. The durable consumption goodis indivisible in the sense that in order to change the stock of the good (beyond theassumed depreciation), the agent must sell his entire current holdings of the good andthen buy the desired new stock, which is the case for houses and cars for example. Weassume that in doing so the agent must pay transaction costs proportional to the valueof the current stock of the durable. The perishable good and the 0nancial assets aretraded without transaction costs. The agent extracts utility from the rate of consumptionof the perishable good and the stock of the durable good. We study the case where theagent has an in0nite time horizon and a utility function of the multiplicatively separable,isoelastic form U (c; k) = (ck1)1=(1 ), where c and k are the current perish-able consumption rate and the current stock of the durable, respectively. Furthermore,measured in terms of perishable consumption units, the price of the durable good andthe prices of the risky 0nancial assets follow correlated geometric Brownian motions.Our 0rst contribution is to derive an explicit solution to the consumptioninvestment</p><p>problem for the case of no transaction costs. The optimal strategy is to keep boththe perishable consumption rate, the value of the durable holdings, and the amountinvested in each of the risky 0nancial assets as 0xed fractions of wealth. This resultgeneralizes the solution to the single perishable consumption good problem of Merton(1969). The set of risky 0nancial assets exhibits two-fund separation in that the optimalinvestment strategy combines only the mean-variance tangency portfolio and a durablehedge portfolio, which is the portfolio with the highest possible absolute correlationwith the price of the durable consumption good.The optimal strategy for the no transaction costs problem involves continuous re-</p><p>balancing of the stock of the durable due to Huctuations in 0nancial asset prices andthe price of the durable and also due to the physical depreciation of the stock ofthe durable. With transaction costs, such a strategy is clearly not optimal. Our secondcontribution is to characterize the optimal consumption and investment policies with</p></li><li><p>A. Damgaard et al. / Journal of Economic Dynamics & Control 28 (2003) 209253 211</p><p>transaction costs. We show that the optimal behavior is completely determined by theratio z = x=(kp) of the total current wealth, x, to the product of the current stock ofdurable, k, and the current price of durable, p. In addition, using the notion of vis-cosity solutions, we demonstrate that there are two critical values z 3z such that it isoptimal to refrain from trading the durable good as long as z (z; 3z). At the boundariesof this interval, it is optimal to trade the durable. When z = z, it is optimal to shift toa lower stock of the durable, and when z= 3z, it is optimal to shift to a higher stock ofthe durable. In both cases, the optimal transaction is such that, immediately after thetransaction, the new value of z, z, is in the open interval (z; 3z). If the initial valuesare such that z (z; 3z), the optimal policy involves an initial transaction to z (z; 3z).Since there is a rather small loss in utility from deviating a little from the optimaldurable consumption level, adjustments of the durable holdings will be infrequent evenwith small transaction costs. Concerning the optimal investment strategies, we showthat the presence of transaction costs on the durable good does not change the fundseparating structure of the set of risky 0nancial assets.Our third contribution is to provide numerical results illustrating several important</p><p>economic eEects of transaction costs in our setting. We 0nd that the relative riskaversion associated with the indirect utility of wealth varies over the interval (z; 3z)such that the relative risk aversion is low close to the boundaries (and, in particular,immediately before a durable trade) and high near the target value z (immediatelyafter a durable trade). We show that this variation in risk aversion has a signi0cantimpact on the weights of the two separating risky funds in the optimal portfolio. Wedemonstrate that the no-trade interval (z; 3z) widens as transaction costs increase and thattransaction costs generally have smaller quantitative eEects than in a model with only adurable consumption good (the Grossman and Laroque model discussed below), sincethe agent in our model will substitute perishable consumption for durable consumptionto reduce transaction costs. For example, the expected period of time between durabletrades is signi0cantly lower than found by Grossman and Laroque. We 0nd that theperishable consumption propensity, i.e. the optimal rate of perishable consumption asa fraction of total wealth, can vary substantially over the interval (z; 3z), in contrast tothe no transaction costs case where it is kept constant. The precise relation is highlydependent on the exogenous preference parameters and and the endogenouslydetermined relative risk aversion associated with the indirect utility of wealth. Forsome parameter values, the perishable consumption propensity is a decreasing functionof z and, hence, increasing in the price of the durable and decreasing in 0nancialwealth. For other parameter values, the relation is non-monotonic. Finally, we showthat the optimal behavior can be highly sensitive to the correlation between the price ofthe durable good and 0nancial asset prices and, especially, the risk-return relationshipof the durable good.The analysis of the paper generalizes that of Grossman and Laroque (1990) who</p><p>consider the simpler problem with only a durable consumption good that acts asthe numeraire good. In their simpler setup, they give a similar characterizationof the optimal durable trading strategy and have similar results on risk aversion andthe dependence of the no-trade region on transaction costs. Our analysis shows thattheir approach and conclusions carry over to the more general and economically more</p></li><li><p>212 A. Damgaard et al. / Journal of Economic Dynamics & Control 28 (2003) 209253</p><p>appealing framework with two types of consumption goods with an uncertain relativeprice, but that the quantitative eEects of transaction costs are smaller in the two-goodeconomy. Moreover, our general setting allows us to study the relation between per-ishable and durable consumption and the impact of the uncertainty of the durable goodprice and its correlation with 0nancial asset prices on optimal behavior.A few other papers study the implications of durable consumption goods on optimal</p><p>behavior. Cuoco and Liu (2000) examine a model with a single consumption good,namely a divisible durable good, e.g. furniture, where adjustment of the stock of thedurable requires the payment of transaction costs proportional to the change in the stockof durable, not the current stock. Under this assumption it is optimal to keep the ratioz in a closed interval [zl; zu], but at the boundaries the optimal transaction is the mini-mal needed to keep z in this interval, an eEect also found in models with proportionalcosts of transacting 0nancial assets, cf., e.g., Davis and Norman (1990). In mathe-matical terms, the optimal durable trading strategy is of a local time nature. Hence,the adjustments to the stock of the durable are small and frequent (continuous at theboundaries), contrary to our setup where the changes are infrequent, non-in0nitesimaljumps into the interior of the interval. The reason for this diEerence is that in ourmodel the transaction costs are similar to a 0xed cost in an optimal stopping problem.Hindy and Huang (1993) discuss a model with utility derived from the stock of</p><p>a durable good that cannot be resold once bought. With power utility and lognormal0nancial asset prices, it is optimal to keep the ratio of wealth to the stock of durablebelow a critical level and purchase durable good only at this critical level.Detemple and Giannikos (1996) study a model with both a perishable and a durable</p><p>good where the durable good provides a utility both through current purchases of thegood (status) and through the current stock of the good (services). The investoris not allowed to sell out of his stock of durable and the price of the durable good (interms of units of the perishable good) is spanned by the 0nancial asset prices, whereaswe allow for a durable-speci0c price risk. Using martingale methods for this completemarkets optimization problem, they characterize the optimal consumption processes interms of the state-price density.The rest of the paper is organized as follows. Section 2 describes the details of the</p><p>model we use. In Section 3, we provide an explicit solution to the utility maximizationproblem for the special case where the durable consumption good can be traded withouttransaction costs. For the problem with transaction costs, we derive analytically someimportant properties of the value function and the optimal strategies in Section 4.Since it seems impossible to derive explicit expressions in the latter case, we turn tonumerical solution methods. Section 5 presents and discusses numerical results. Finally,Section 6 concludes the paper with a summary and a discussion of possible extensions.All proofs are in the appendices.</p><p>2. The model</p><p>We consider an economy with two consumption goods, namely a perishable goodand a durable good, and n+1 0nancial securities. The price processes are de0ned below</p></li><li><p>A. Damgaard et al. / Journal of Economic Dynamics & Control 28 (2003) 209253 213</p><p>using the perishable good as the numeraire, i.e. security prices and the price of thedurable good are stated in terms of perishable consumption units. One of the 0nancialassets is a riskfree security paying a constant continuously compounded interest rater R+. The remaining n 0nancial assets are risky. We model the uncertain evolutionof the risky asset prices and the durable good price in terms of an n-dimensionalWiener process w1 and a one-dimensional Wiener process w2 uncorrelated with w1,where w1 and w2 are de0ned on a given probability space (;F;P). The informationstructure of the economy is the 0ltration {Ft}t0 de0ned as the augmentation of thenatural 0ltration generated by the Wiener process w = (w1; w2). We assume that then-dimensional price vector St of the risky 0nancial assets evolves according to thestochastic diEerential equation</p><p>dSt = diag(St)[ dt + dw1t]; t 0; S0 = s; (1)</p><p>where is a constant, non-singular n n matrix, is a constant vector in Rn, andthe term diag(St) denotes the square matrix with the vector St along the diagonal andzero in all other entries. Hence, the prices of the risky 0nancial assets are assumed tofollow correlated geometric Brownian motions.The unit price of the durable good, Pt , is also assumed to follow a geometric Brow-</p><p>nian motion</p><p>dPt = Pt[P dt + P1 dw1t + P2 dw2t]; t 0; P0 = p; (2)</p><p>where P and P2 = 0 are constant scalars, and P1 is a constant vector in Rn.For future reference we de0ne the (n + 1)-dimensional vector P = (P1; P2)</p><p>. Byconstruction, the price of the durable consumption good is imperfectly correlated withthe 0nancial asset prices, so that it is impossible to hedge away all the risk associatedwith the durable good price by appropriate investments in the 0nancial market.We assume that the stock of the durable good depreciates at a physical depreciation</p><p>rate over time. More precisely, letting Kt be the number of uni...</p></li></ul>

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