optimal consumption and investment strategies with a perishable and an indivisible durable...

Journal of Economic Dynamics & Control 28 (2003) 209–253www.elsevier.com/locate/econbase

Optimal consumption and investment strategieswith a perishable and an indivisible durable

consumption good

Anders Damgaarda, Brian Fuglsbjergb, Claus Munkc;∗aDanske Research, Danske Bank, DK-1092 Copenhagen K, Denmark

bSimCorp A/S, DK-2100 Copenhagen O, DenmarkcDepartment of Accounting, Finance & Law, University of Southern Denmark-Odense, Campusvej 55,

DK-5230 Odense M, Denmark

Abstract

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.

JEL classi/cation: G11; D11; D91; C61

Keywords: Optimal consumption and investment; Durable goods; Transaction costs; Viscosity solutions;Numerical solution

∗ Corresponding author. Tel.: +45-6550-3257; fax: +45-6593-0726.E-mail address: [email protected] (C. Munk).

0165-1889/02/$ - see front matter ? 2002 Elsevier B.V. All rights reserved.PII: S0165 -1889(02)00135 -5

mailto:[email protected]

210 A. Damgaard et al. / Journal of Economic Dynamics & Control 28 (2003) 209–253

1. Introduction

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 anagent 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) = (c�k1−�)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 consumption–investmentproblem 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-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

A. Damgaard et al. / Journal of Economic Dynamics & Control 28 (2003) 209–253 211

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 importanteconomic 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) whoconsider 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


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 optimalbehavior. 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 ofa 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 durablegood 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 themodel 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.

2. The model

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


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}t¿0 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

dSt = diag(St)[� dt + � dw1t]; t¿ 0; S0 = s; (1)

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-nian motion

dPt = Pt[�P dt + ��P1 dw1t + �P2 dw2t]; t¿ 0; P0 = p; (2)

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)�. 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 depreciationrate � over time. More precisely, letting Kt be the number of units of the durable goodheld by an agent at time t, then, in time intervals where the agent does not trade thedurable, Kt evolves according to

dKt = −�Kt dt: (3)

Moreover, we assume that it is costly to trade the durable good, i.e. to change the stockof the durable good beyond the depreciation. More precisely, we shall assume, follow-ing Grossman and Laroque (1990), that the transaction costs are a fraction �∈ [0; 1]of the value KtPt of the current stock of the durable. This is a reasonable assumptionfor indivisible durable goods, such as a house, where the only way one can change theposition in the durable is to sell the current house and buy another house of the desiredsize and quality. 1 Cuoco and Liu (2000) consider divisible durable goods, where thetransaction costs are proportional to the change in the durable stock, which is morereasonable for furniture, clothing, and similar durable consumption goods.

1 Our assumption models proportional transactions costs of selling and no transactions costs of buyingdurables, such that the price Pt is best interpreted as the unit purchase price of the durable. The modelcould easily be accommodated to proportional buying costs, but that would only complicate the notation andnot change the qualitative results of the paper.


The agent must choose a strategy for consuming the perishable good and trading thedurable good and the 0nancial assets. Denote by Ct the agent’s consumption rate ofthe perishable good at time t. A consumption strategy is a non-negative progressivelymeasurable process, C = {Ct}t¿0 ∈L1, where

Lq ={{Ft}-adapted processes X :

∫ T

0‖Xu(!)‖q du¡∞;

for P-a:e: !∈�; ∀T ¿ 0} ; q = 1; 2:

The set of consumption processes is denoted by C. Again using the perishable consump-tion good as numeraire, we denote by "0t the amount held inthe riskfree security at time t. Similarly, for each i∈{1; : : : ; n}, we let "it denotethe amount held in the ith risky security at time t, and de0ne "t = ("1t ; : : : ; "nt)�. Asabove, Kt denotes the holdings of the durable at time t, and the process K = {Kt}t¿0

indirectly represents the trading strategy for the durable good. The set $ of tradingstrategies consists of the (n + 2)-dimensional progressively measurable stochastic pro-cesses ("0; "; K) valued in R×Rn ×R+ that satisfy "0 ∈L1; "∈L2, and K ∈L2.

We de0ne the wealth of the agent as the sum of his investments in the 0nancialassets and the value of his current stock of the durable measured as units times thecurrent unit purchase price. His time t wealth Xt is therefore given as

Xt = "0t + "�t 1 + KtPt: (4)

If the agent follows a perishable consumption strategy C and a self-0nancing tradingstrategy {"0; "; K}, his wealth Xt = X";K;C

t evolves as

dX";K;Ct = [r(X";K;C

t − KtPt) + "�t (� − r1) + (�p − �)KtPt − Ct] dt

+ ["�t � + KtPt��

P1] dw1t + KtPt�P2 dw2t (5)

in time intervals where the durable is not traded. At a time % where the investor choosesto sell his durable good, there is a jump in his wealth due to the transaction costs:

X";K;C% = X";K;C

%− − �K%−P%: (6)

We require that the agent chooses consumption and trading strategies satisfying thesolvency condition that his total wealth is always greater than the transaction costsinvolved in an immediate sale of his durable holdings. To be more precise, assumingthat the agent is endowed with k = K0− units of the durable good and a total wealthof x=X0−, we say that a policy (";K; C) is admissible if ("0; "; K)∈$; C ∈C, and

(X";K;Ct ; Kt ; Pt)∈ 3S; P-a:s: for all t¿ 0; (7)

where the solvency region 3S is the closure of the set

S = {(x; k; p)∈R3+: x¿�kp}:

We let A(x; k; p) denote the set of admissible policies. We shall assume that (x; k; p)∈3S, which implies that A(x; k; p) is a non-empty set.


We consider an investor with an in0nite time horizon and an additively time-separableutility function∫ ∞

0e−&tU (Ct; Kt) dt;

where &¿ 0 is a parameter reHecting the investor’s time preferences. In the followingwe assume that the instantaneous utility function is of the multiplicatively separable,isoelastic form

U (c; k) =1

1 − �(c�k1−�)1−�; (8)

where �; �∈ (0; 1). A similar utility function was considered in a discrete-time settingwith no transaction costs on the durable good by Dunn and Singleton (1986) and Lax(1999). 2

If the investor pursues the admissible policy (";K; C)∈A(x; k; p), his time zeroexpected utility is

J";K;C(x; k; p) = E[ ∫ ∞

0e−&tU (Ct; Kt) dt

]:

Given an initial endowment such that (X0−; K0−; P0) = (x; k; p)∈ 3S, the investor’sobjective is to 0nd the (";K; C)∈A(x; k; p) that maximizes his expected utility. There-fore, the investor’s value function is de0ned as

V (x; k; p) = sup(";K;C)∈A(x;k;p)

J";K;C(x; k; p): (9)

3. An explicit solution to the no transaction costs problem

In this section we provide explicit expressions for the optimal consumption andinvestment strategies and the value function of the investor for the case where theposition in the durable good can be changed costlessly corresponding to � = 0. Ourresult generalizes the solution to the consumption–investment problem in the standardMerton setting, since we allow for both a perishable and a durable consumption goodwith a stochastically evolving relative price. The solution to the no transaction costscase also provides an important benchmark against which the optimal strategies withtransaction costs will be compared and gives an upper bound on the value functionwith transaction costs.

When the stock of durable can be changed without costs, the consumption andinvestment policies can be chosen without regarding how total wealth is decomposedinto 0nancial wealth and the value of the durable stock. Therefore, the optimal policies

2 It seems more reasonable that the instantaneous utility the agent gets from his durable holdings is relatedto the services he can extract from his stock of durable rather than the stock itself as we have assumed in (8).But if the services are proportional to the holdings of the durable and the utility of perishable consumptionand durable services have the same form as in (8), our representation of preferences lead to the same optimalstrategies.


and the value function will depend only on total current wealth x and the current priceof durable p. We will denote the value function for the no transaction costs case by3V (x; p) and the corresponding optimal policies by 3C; 3K , and 3". In Appendix A we

derive explicit expressions for both the value function and the optimal policies. We dothat by solving the Hamilton–Jacobi–Bellman (HJB) equation associated with 3V (x; p)and verifying the transversality condition appropriate for this in0nite horizon stochasticcontrol problem.

Before stating the solution, we introduce some auxiliary parameters. De0ne theconstants

)0 = −&�

+1 − ��

{r − (1 − �)�P +

12

(1 − �)[1 + (1 − �)(1 − �)]‖�P‖2}

+1 − �2�2 (� − r1− (1 − �)(1 − �)��P1)�(��)−1(� − r1− (1 − �)

× (1 − �)��P1);

)1 = (1 − �)�2P2 +

11 − �

(r − �P + � + (� − r1)�(��)−1�P1);

)2 =(

�1 − �

+1 − �

2

)�2P2:

If )0 ¡ 0, the quadratic equation

)0 + )1*k + )2*2k = 0 (10)

will have a single strictly positive solution, which we will denote *k in the following.If we furthermore assume that

)0 ¡− 12 (1 − �)�2

P2*2k ; (11)

then the constant

*c = −�)0 − 12 �(1 − �)�2

P2*2k

will be positive, as will the constant

*v = �*�(1−�)−1c *(1−�)(1−�)

k : (12)

Finally, de0ne the constant vector in Rn

*, =1�

(��)−1(� − r1) −(

1 − �(1 − �)�

+ *k − 1)

(��)−1�P1: (13)

The solution for the no transaction costs problem can now be stated as follows.

Theorem 3.1. Suppose )0 ¡ 0 and let *k denote the unique positive solution to (10).If (11) is satis/ed, then the value function for the no transaction costs problem is


given by

3V (x; p) =1

1 − �*vp−(1−�)(1−�)x1−� (14)

and the optimal controls are given in feedback form as

3Ct = *c 3X t; (15)

3Kt = *k 3X t=Pt ; (16)

3"t = *, 3X t; (17)

where 3X t is the wealth process generated by these controls.

The optimal strategy is to use a constant fraction *c of wealth for consumption ofthe perishable good, a constant fraction *k of wealth for consumption of the durablegood (at a unit price of Pt), and to keep a constant fraction of wealth invested ineach of the 0nancial assets. The optimal controls for the no transaction costs case arehence a natural extension of the optimal controls in the standard Merton setting withpower utility from a single perishable consumption good and lognormal asset prices,cf. Merton (1969). Due to substitution eEects between the two consumption goods,the fraction of wealth devoted to perishable consumption is generally diEerent fromthat in Merton’s single good setting. The condition (11) ensures that the solution iseconomically meaningful and satis0es the transversality condition. The condition canbe rewritten as

&¿ (1 − �){

12��2

P2*2k + r − (1 − �)�P +

12

(1 − �)[1 + (1 − �)(1 − �)]‖�P‖2

+12�

(� − r1− (1 − �)(1 − �)��P1)�(��)−1(� − r1− (1 − �)

(1 − �)��P1)}

; (18)

which reduces to the condition on & in Merton’s problem where �=1, and �P; �P1; �P2,and *k are all zero.

We see from (13) and (17) that the vector of fractions of wealth invested in therisky 0nancial assets can be interpreted as a combination of two portfolios, namely themean-variance tangency portfolio corresponding to (��)−1(� − r1) and a correctionor hedge portfolio corresponding to (��)−1�P1, i.e. the portfolio having the highestpossible absolute correlation between changes in portfolio value and changes in thedurable price. The set of risky 0nancial assets therefore exhibits two-fund separation inthe sense that any agent is equally well oE with access to trade in the two portfoliosdescribed above as with access to the complete set of risky 0nancial assets. 3 In additionto these two funds, the agent will also trade in the riskless 0nancial asset and the twoconsumption goods.

3 This two-fund separation result is valid for more general speci0cations of the instantaneous utility functionthan the one we assumed in (8), but the precise weighting of the two funds in (13) is due to that assumption.


The correction portfolio appears for two reasons. Firstly, the durable good is arisky investment asset just as the risky 0nancial assets and will, hence, enter themean-variance portfolio of all risky assets. In (13) we have separated out the mean-variance portfolio of the risky 0nancial assets, which is, therefore, to be adjusted forthe presence of the remaining risky investment asset, i.e. the durable. Secondly, sincethe agent extracts utility from the holdings of the durable good, she will seek to hedgechanges in the price of the durable by adjusting her 0nancial investment strategy. 4

To isolate the two components of the correction portfolio term, consider the casewhere � = 1, corresponding to the situation where the agent has no direct utility ofdurable holdings, but still use the durable as an investment asset. Letting -t = KtPt

denote the value of the durable holdings, we can rewrite the wealth dynamics (5) as

dX";-;Ct =

[rX";-;C

t + ("�t ; -t)

(� − r1

�P − �− r

)− Ct

]dt

+("t;-t)�(

� 0

��P1 �P2

)dwt

and apply the well-known solution to Merton’s problem. Assuming �P − � − r¿(�− r1)�(��)−1�P1, the optimal control policies are of the same form as in Theorem3.1 but with diEerent multipliers *c; *k , and *,. In particular, 5

*k =1

��2P2

(�P − �− r − (� − r1)�(��)−1�P1);

*, =1�

(��)−1(� − r1) − (��)−1�P1*k :

Comparing with (13), we see that the presence of the durable good in the instanta-neous utility function aEects the optimal investment in the 0nancial assets both throughthe extra term −(��)−1�P1(1 − �)(1 − �)=�, which reHects the risk aversion and thecorrelation between the durable price and 0nancial asset prices, and through a changein the optimal investment in the durable asset, i.e. the multiplier *k .

4. Analytical results for the transaction costs problem

After having solved in closed form the consumption and investment problem forthe no transaction costs case, we will next consider the problem with transaction costson durable trading, i.e. �¿ 0. The presence of transaction costs changes the natureof the problem and it seems impossible to 0nd an explicit solution. Nevertheless, wewill in this section derive some important properties of the value function and the

4 If the durable asset only served as an investment asset, the agent would not hedge price changes, sincethe investment opportunity set is non-stochastic, cf. Merton (1973).

5 We can obtain the same expressions by multiplying (10) through by 1 − � and letting � → 1 in theresulting expression and in (13).


optimal strategies by analytic arguments before we turn to the numerical solution of theproblem. With the assumed transaction costs on the durable good, the optimal durabletrading strategy will be to trade the durable at most countably many times at strictlyseparated points in time, in contrast to the optimal trading strategy if the transactioncosts are proportional to the change in the durable holdings, cf. Cuoco and Liu (2000)and the discussion in the introduction. 6 The main result of this section is a completeand simple characterization of the states in which the durable should be traded.

Throughout the section we assume that (11) holds and that the initial endowments aresuch that (X0−; K0−; P0) = (x; k; p)∈ 3S. We will also make the following assumption:

Assumption 4.1. For all (x; k; p)∈ 3S, an optimal policy exists and the value functionis 0nite and satis0es the dynamic programming principle so that for all {Ft}-stoppingtimes %

V (x; k; p) = sup(";K;C)∈A(x;k;p)

E[ ∫ %

0e−&tU (Ct; Kt) dt + e−&%V (X%; K%; P%)

]: (19)

In particular, let % correspond to the 0rst time the durable good is traded, so thatKt = ke−�t for t ¡ %, and at time % there is a decrease in total wealth given by (6).The controls (";C) must be chosen such that the solvency condition (7) for t ¡ % issatis0ed with Kt=e−�tk. We let A′(x; k; p) denote the set of such controls. Furthermore,the new level of durable holdings, K%, must be chosen such that the solvency conditioncontinues to hold, i.e. K%6X%−=(�P%) − ke−�%. It follows from (19) that

V (x; k; p) = sup(";C)∈A′(x;k;p);K%6(X%−=�P%)−ke−�%

E[ ∫ %

0e−&tU (Ct; ke−�t) dt

+e−&%V (X%− − �ke−�%P%; K%; P%)]: (20)

We shall apply this equation in the reduction of the dimensionality of the problembelow.

4.1. Properties of the value function

The following theorem gives some important basic properties of the value function.The proof is given in Appendix B.

Theorem 4.1. The value function V (x; k; p) satis/es:

(a) For all (x; k; p)∈ 3S, we have that

11 − �

*p−(1−�)(1−�)(x − �kp)1−�6V (x; k; p)61

1 − �*vp−(1−�)(1−�)x1−�;

6 In mathematical terms, our optimization problem with transaction costs is a combined stochastic controland impulse control problem, cf. Chancelier et al. (2000) and Korn (1997, Chapter 5).


where

* =��(1−�)(1 − �)(1−�)(1−�)r�(1−�)

& + �(1 − �)(1 − �)

and where *v is given by (12).(b) For each (k; p)∈R2

++; V (x; k; p) is strictly increasing and concave in x on{x: x¿ �kp}.

(c) For each (k; p)∈R2++; V (x; k; p) is continuous in x on {x: x¿�kp}.

(d) V (x; k; p) is homogeneous of degree 1 − � in (x; k) and of degree �(1 − �) in(x; p), i.e. for all (x; k; p)∈ 3S and all .¿ 0

V (.x; .k; p) = .1−�V (x; k; p); (21)

V (.x; k; .p) = .�(1−�)V (x; k; p): (22)

As discussed immediately after Theorem 4.2, V (x; k; p) is in fact continuous on thelarger set {(x; k; p)∈R3

++: x¿ �kp}.Because of the homogeneity properties (21) and (22) of the value function, we can

reduce the problem of 0nding V on the set R3++∩S from one with three state variables

(x; k; p) to one with a single state variable, namely the ratio z = x=(kp) of total wealthto durable wealth, since

V (x; k; p) = k1−�p�(1−�)v(x=(kp)); (x; k; p)∈R3++ ∩S; (23)

where v is de0ned as

v(z) = V (z; 1; 1); z ∈ 3S1 (24)

and 3S1 is the closure of the set S1 = (�;∞). In particular, v(�) =V (�; 1; 1) = 0, sincethe only feasible strategy with the initial conditions (x; k; p) = (�; 1; 1) is to sell theunit of durable, which leaves the agent with zero wealth and zero consumption in thefuture. From Theorem 4.1, we know that v is strictly increasing and concave on [�;∞),continuous on (�;∞), and that

11 − �

*(z − �)1−�6 v(z)61

1 − �*vz1−�: (25)

De0ne the process Zt = Xt=(KtPt) with initial value Z0 = z = x=(kp) and introducethe scaled controls Ct = Ct=(PtKt) and "t = "t=(PtKt). Note that Ct=Zt = Ct=Xt and"t=Zt = "t=Xt . Substituting (23) into (20), we get

p�(1−�)v(z) = sup(";C)∈A(z);%∈T

E[

11 − �

∫ %

0e− 3&tC�(1−�)

t P�(1−�)t dt

+e− 3&%P�(1−�)% (Z%− − �)1−� M

1 − �

]; (26)


where 3& = & + �(1 − �),

M = supK%6ke−�%(Z%−−�)=�

(1 − �)(ke−�%(Z%− − �)

K%

)�−1

v(ke−�%(Z%− − �)

K%

)

= (1 − �) supz¿�

z�−1v(z)

and A(z) is the set of ("; C)∈L2 ×C with Zt =Z"; Ct ¿ � for all t¿ 0. The problem

can now be seen as a combined stochastic control and optimal stopping problem withthe stopping value v(z)=f(z), where f(z)=(z−�)1−�M=(1−�) is the stopping rewardfunction. 7 It follows from (25) that *6M6 *v. Also note that

z∗ = arg maxz¿�

z�−1v(z) (27)

is the value of the transformed state variable after the optimal change in the durableholdings at time %, since

Z% =X%

K%P%=

X%− − �K%−P%

K%P%=

K%−K%

(Z%− − �) = z∗:

The new level of durable holdings can be expressed in terms of z∗ as K% = K%−(Z%− − �)=z∗.

The HJB equation associated with the reduced problem can be written as 8

0 = max{H (z; v; v′; v′′); (z − �)1−� M

1 − �− v(z)

}; ∀z ∈S1; (28)

where

H (z; v; v′; v′′) = sup,∈Rn; c∈R++

{L,; cv(z) +

11 − �

c�(1−�) − 3&v(z)}

(29)

with the operator L given by

L,; cv(z) = v(z)�(1 − �)[�P − 12 [1 − �(1 − �)]‖�P‖2]

+ v′(z)[(z − 1)(r − �P + � + [1 − �(1 − �)]‖�P‖2) − c

+ , �(� − r1− [1 − �(1 − �)]��P1)]

+ 12 v′′(z)[, ��,− 2(z − 1), ��P1 + (z − 1)2‖�P‖2]: (30)

7 Note, however, that M itself depends on the value function v(z), and, therefore, the problem is not astandard optimal stopping problem. In this case the function f(z) is elsewhere referred to as the interventionvalue. See footnote 6 and the references given there.

8 The HJB equation can be derived by substituting (23) into the HJB equation for the original problem(derived under the assumption that V (x; k; p) is twice continuously diEerentiable). Alternatively, we can use(26) and derive the dynamics of the process P�(1−�)

t v(Zt) through an application of Ito’s Lemma.


The reduced value function v(z) is not necessarily smooth and, therefore, it cannotbe expected to satisfy the HJB equation (28) in the classical sense. We show be-low that v is a solution in a generalized sense, namely a so-called viscosity solution.While the precise de0nition of viscosity solutions of (28) is given in Appendix B, wenote here that a viscosity solution of (28) is continuous on S1, but not necessarilydiEerentiable. 9 For other applications of viscosity theory to optimal consumption andinvestment problems, see, e.g., Fleming and Zariphopoulou (1991), Davis et al. (1993),DuSe et al. (1997), Hindy et al. (1997), and Constantinides and Zariphopoulou (1999).The next theorem shows that the reduced value function v is a viscosity solution of(28) and that if the parameter constraint

&¿ (1 − �)(��P − 12 �[1 − �(1 − �)]‖�P‖2 − �) (31)

holds, then v is the only viscosity solution of (28) satisfying some desired properties.The proof is relegated to Appendix B.

Theorem 4.2. The function v de/ned by (24) has the following properties:

(a) v is a viscosity solution of the HJB equation (28).(b) v is continuous on [�;∞).(c) if (31) holds, then v is the only strictly increasing and concave viscosity solution

of (28) which is continuous on [�;∞) and satis/es v(�) = 0.

Since V (x; k; p) = k1−�p�(1−�)v(x=(kp)) for all (x; k; p)∈R3++ with x¿ �kp, (b)

implies that V (x; k; p) is continuous on the set {(x; k; p)∈R3++: x¿ �kp}.

4.2. Properties of the optimal consumption and investment strategies

The solvency region S can be divided into a region where it is optimal not to tradethe durable, a region where it is optimal to reduce durable holdings, and a regionwhere it is optimal to increase durable holdings. In terms of the state variable z of thereduced problem, the no-trade region (continuation region) is the set

N = {z¿�: v(z)¿f(z)};

where f(z) = (z − �)1−�M=(1 − �) is the stopping reward function. The next theoremshows that the no-trade region N is an open interval (z; 3z) and that the value ofz immediately after a durable trade, z∗, is in the open interval (z; 3z). The proof,which applies that v is a viscosity solution of the HJB equation (28), is given inAppendix B.

9 The viscosity solution concept can even be further generalized to discontinuous functions, cf. Chancelieret al. (2000).


Fig. 1. Illustration of the partitioning of the state space.

Theorem 4.3. If (31) holds and H (z; f; f′; f′′)¿ 0 for some z ∈ (�;∞), then numbersz and 3z exist satisfying �¡z¡ 3z such that N = (z; 3z). Also, z∗ de/ned by (27)satis/es z∗ ∈ (z; 3z).

The no-trade interval N corresponds to the cone

N ={

(kp; x): z¡xkp

¡ 3z}

in the (kp; x)-plane as illustrated in Fig. 1. At and below the lower boundary @N1 ofN, the value of the durable holdings is so high relative to the total wealth that it isoptimal to decrease the durable holdings. At and above the upper boundary @N2 of N,the value of the durable holdings is so low relative to the total wealth that it is optimalto increase the durable holdings. In both cases, the optimal position immediately afterthe durable trade is a point on the dashed line where x = z∗kp. As depicted in Fig. 1,this point can be found as the crossing point between the dashed line and a straightline through the point corresponding to the position just before the durable trade. For apoint on the lower boundary @N1 the straight line has a slope of �z∗=(z∗− z + �)¿ 0and for a point on the upper boundary @N2 the slope is �z∗=(z∗ − 3z + �)¡ 0. Notethat from (16) the optimal relation between total wealth and durable wealth withouttransaction costs is given by Xt=(KtPt) = 1=*k , which will generally be diEerent fromz∗ and may fall outside the no-trade interval (z; 3z) for some parameter combinations(see the second line of Part C of Table 1 in Section 5).

We can now conclude that if the initial values are such that (kp; x) �∈ N, the optimaldurable trading strategy will involve an initial trade to the line x=z∗kp. Thereafter, thedurable is only traded when the evolution of uncertain prices is such that (Kt−Pt; Xt−)hits one of the boundaries of N in which case the durable good is traded back tothe relation Xt = z∗KtPt . The strategy therefore consists of infrequent, non-in0nitesimaldurable transactions contrary to the 0ndings in models with proportional transactioncosts to 0nancial asset transactions, cf., e.g., Davis and Norman (1990), and in themodel of divisible durable goods studied by Cuoco and Liu (2000). In those models,


the optimal transactions are of a local time nature and involves frequent, in0nitesimaltrades.

We close this section with a brief discussion of the optimal investment strategies inthe 0nancial assets. Assuming v is twice diEerentiable, the 0rst order conditions forthe scaled control values in (29) are 10

, = − v′(z)v′′(z)

(��)−1(� − r1− [1 − �(1 − �)]��P;1) + (��)−1�P1(z − 1);

(32)

c = (�−1v′(z))1=(�(1−�)−1): (33)

In particular, we can write the vector of fractions of wealth optimally invested in therisky 0nancial assets as

,x

=,z

= − v′(z)zv′′(z)

(��)−1(� − r1)

−(− v′(z)

zv′′(z)(1 − �[1 − �]) +

1z− 1)

(��)−1�P1:

Comparing this expression with the constant vector *, in (13), which by Theorem 3.1gives the optimal fractions invested in the risky 0nancial assets without transactioncosts, we see that the optimal investment strategy again combines the mean-varianceportfolio, represented by (��)−1(� − r1), and the portfolio hedging changes in theprice of the durable good, represented by (��)−1�P1. The weights of the funds arevery similar to the no costs case. Firstly, 1=� in (13) is replaced by −v′(z)=(zv′′(z)), but(by Theorem 3.1) 1=� is the relative risk tolerance of the agent for the no costs caseand −v′(z)=(zv′′(z)) = −@V=@x(x; k; p)=(x@2V=@x2(x; k; p)) is the relative risk tolerancewith transaction costs, which varies with z as we will see from the numerical resultsin Section 5. Secondly, the constant *k in (13), denoting the optimal fraction of wealthheld in the durable without transaction costs, is replaced by 1=z = kp=x, which is thecurrent fraction of wealth held in the durable good.

5. Numerical results for the transaction costs problem

To further investigate optimal behavior in our setting we resort to numerical exam-ples. We solve the HJB equation (28) numerically with the iterative scheme outlinedin Appendix C. While a complete proof of convergence of this method is beyond thescope of this paper, we note that the central requirement for convergence is the

10 If v is not twice diEerentiable, it does not solve the HJB equation in the classical sense, but in theviscosity sense, as shown in Theorem 4.2. Intuitively this means that v can be closely approximated bytwice diEerentiable sub- and supersolutions of the HJB equation. By substituting such approximate solutionsinstead of v into the expressions for the , and c we obtain approximations to the optimal controls. Thenumerically computed optimal controls discussed in the next section are based on these expressions with0nite diEerences replacing the derivatives of the value function.


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 0.5 1 1.5 2 2.5 3z

v(z) − 11−γ M (z − λ )1−γ

•

Fig. 2. The diEerence between the value function and the value of changing durable level is plotted againstz∈ (z; 3z) = (0:17; 2:75). The • represents the location just after durable purchase, where z∗ = 0:85. Withouttransaction costs, the optimal durable holdings are such that x=(kp) = 0:51.

existence of a unique viscosity solution to the HJB equation, which is provided byTheorem 4.2. Also, our approach is closely related to the method proposed by Chancelieret al. (2000) for a similar problem for which convergence to the viscosity solution ofthe HJB equation is guaranteed. In our numerical examples we 0x the number of risky0nancial securities to two, which is identical to the number of separating risky funds.We assume the following parameter values throughout the numerical part:

r = 0:05; � =

(0:1

0:15

); & = 0:02; � = 0:04:

Furthermore, the individual stock volatilities are 0xed to 0.25 and 0.3, respectively.

5.1. Optimal behavior as a function of the state

In this subsection, we consider how consumption and portfolio choice change as theratio z of total wealth to wealth in durable changes. For this investigation we assumethat the two stocks are uncorrelated and that each stock has a positive correlation withthe durable of 0.5. Furthermore, we assume that

�P = 0:07; ‖�P‖ = 0:12; � = 0:5; � = 0:5; � = 0:05:

Fig. 2 shows the diEerence between the value function, v(z), and the value if a trans-action of the durable is performed, i.e. the stopping reward function M (z−�)1−�=(1−�).The shape of the curve is very similar to that of the corresponding diEerencein Grossman and Laroque (1990). Close to the boundaries of the no-trade regionthe value function is less concave than M (z − �)1−�=(1 − �), while it is more con-cave between the two boundaries. This implies that the agent is more risk averse in


0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0 0.5 1 1.5 2 2.5 3z

Relative Risk Aversion

•

Fig. 3. The relative risk aversion is plotted against z∈ (z; 3z). The • indicates the point just after durablepurchase, where relative risk aversion is 0.65. The relative risk aversion without transaction costs is 0.5.

the interior of the no-trade region than close to the boundaries. This is illustrated inFig. 3.

In Fig. 3, we plot the relative risk aversion associated with the indirect utility oftotal wealth x, i.e. −x@2V=@x2(x; k; p)=@V=@x(x; k; p)=−zv′′(z)=v′(z). The 0gure con0rmsthat risk aversion changes over the no-trade region. Near z or 3z the agent is less riskaverse than if transaction costs were absent, whereas the agent is more risk averse inthe middle of the region. When the state is close to z∗, the loss in utility associatedwith transacting the durable is large, cf. Fig. 2, and the agent attempts to avoid newdurable purchases and, therefore, behaves in a more risk averse manner. Closer tothe boundaries of the no-trade region, the monetary loss associated with the potentialtransaction costs is compensated for by a change to the optimal ratio of total wealthto durable wealth, and the agent is therefore much less risk averse.

This change in risk aversion has a considerable impact on consumption and portfoliochoice. In Fig. 4, we plot the fraction of wealth optimally invested in the riskfreesecurity and the two separating risky funds as a function of z. Fund 1 is the tangencyportfolio (��)−1(� − r1), the portfolio with highest possible Sharpe ratio, and Fund2 is the portfolio (��)−1�P1 that optimally hedges changes in the price of the durable.The optimal portfolio choices are quite diEerent from the no transaction costs caseof Section 3, where the fraction of wealth invested in each fund is constant. Notsurprisingly, the lower risk aversion close to the boundaries leads to higher fractionsof the wealth placed in the tangency portfolio when the state variable is close to theboundary of the no-trade region than when the state variable is in the center of theno-trade region. The stocks are positively correlated with the durable price, so durableprice changes are hedged if the fraction of wealth in the hedge portfolio is negative,hence the fraction of wealth placed in Fund 2 is negative. Close to the lower boundz, durable wealth is very high compared to total wealth, and the wealth placed inthe portfolio hedging durable price changes is therefore also high, in absolute terms,


-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.5 1 1.5 2 2.5 3z

Ratio of Wealth in Share funds

Fund 1Fund 2•

••

R f

Fig. 4. Fractions of total wealth placed in the funds. Fund 1 is the tangency portfolio and Fund 2 is theportfolio that hedges changes in the price on the durable good. The • represents the optimal position justafter transacting the durable. Finally, the dotted curve labeled Rf represents the ratio of wealth in the riskfreesecurity.

compared to total wealth. For completeness, the fraction of wealth placed in the risklesssecurity is plotted. The agent will always borrow for the preferences and risk premiain this numerical example. Moreover, with the parameter values assumed, the targetwealth ratio z∗ is smaller than one corresponding to a negative 0nancial wealth, i.e.the agent 0nances the durable holdings with a (net) short position in the 0nancialmarket.

In Fig. 5, the optimal propensity to consume, c=x= c=z, is plotted against z. Withouttransaction costs, a constant fraction of wealth is used for consumption of the per-ishable consumption good, and a constant fraction of wealth is used for consumptionof the durable good. These results are implied by the 0rst order conditions from theconsumption–investment problem, where the marginal utility of perishable consumptionequals both the marginal utility of wealth and the marginal utility of durable consump-tion. With transaction costs, it is in0nitely expensive, and hence suboptimal, to keepthese ratios aligned. Assuming for a moment that v is twice continuously diEeren-tiable, it can be shown that the propensity to consume is increasing in z if and only if−zv′′(z)=v′(z)¿− c@2U=@c2(c; k)=@U=@c(c; k) = 1− �(1− �), i.e. whenever the relativerisk aversion with respect to wealth changes is higher than the “relative risk aversion”with respect to changes in the perishable consumption How. For � = 0:5 the relativerisk aversion with respect to wealth changes is everywhere below 1− �(1− �) = 0:75,cf. Fig. 3, so the propensity to consume is in this case decreasing in z, consistentwith the solid line in Fig. 5. To illustrate a case where the propensity to consume isnon-decreasing, we have included the dotted line, where � = 0:7. The bullets indicateconsumption rates just after a durable transaction.

Next, we will consider how the expected time to next durable transaction varieswith the state. If we let u(z) denote the expected time it takes for Zt to reach either z


0.07

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

0.16

0 0.5 1 1.5 2 2.5 3z

Consumption Rate

γ = 0:5γ = 0:7

••

Fig. 5. The ratio of perishable consumption to total wealth plotted against z. The solid line is for thebenchmark case, � = 0:5. Furthermore, the dotted line represents the case where � is changed to 0.7. The•’s represent choices immediately after transacting the durable.

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.5 1 1.5 2 2.5 3z

Expected Waiting Time

•

Fig. 6. The expected time (in years) to the next durable transaction as a function of the current state. The• represents the expected waiting time immediately after a durable transaction, i.e. at z∗.

or 3z given that Z0 = z, it follows from Karlin and Taylor (1981, Section 15.3) that usatis0es the ordinary diEerential equation (ODE)

12 �Z(z)2u′′(z) + �Z(z)u′(z) = −1 for z¡ z¡ 3z; u(z) = u( 3z) = 0;

where �Z and �Z are the drift and diEusion terms, respectively, of the optimally con-trolled state process Zt = Xt=(KtPt). This ODE is solved numerically by a standardmethod. Fig. 6 shows that the expected waiting time function has a similar shape asthe utility gain from not trading the durable depicted in Fig. 2. Even with the moderatetransaction costs level assumed in this example (� = 0:05), the expected time between


two durable transactions is 3.58 years. Nevertheless, the expected waiting times we 0ndare generally signi0cantly lower than those found by Grossman and Laroque (1990),because to avoid transaction costs the agent can in our model substitute perishable con-sumption for durable consumption. Consequently, the optimal fraction of wealth placedin the durable consumption good is lower than if there were no perishable good. Thisdecrease in the stock of the durable good eEectively decreases the impact of the trans-action costs, leading to more frequent trade in the durable than in the single-goodmodel of Grossman and Laroque.

5.2. Parameter sensitivity

In this subsection, we study how optimal behavior changes as the parameters of themarket change. We emphasize the eEects that cannot be studied in the simpler modelof Grossman and Laroque (1990).

As discussed above, the agent can substitute perishable consumption for durableconsumption to reduce the transaction costs implied by a durable trade. Shifting theimportance of durable consumption relative to perishable consumption (shifting �) inthe direct utility function, as we do in Part A of Table 1, demonstrates the importanceof the alternative good. For low values of �, the durable good has a higher weightin the utility function, making the agent more vulnerable to changes in durable level.Moreover, increasing � increases the perishable consumption rate and decreases thedurable wealth to total wealth ratio, thereby decreasing the costs associated with trans-acting the durable, resulting in more frequent trading of the durable and increasing theprobability of buying more of the durable good at next trade. 11

Turning to the importance of the durable price process, we 0rst note that in thebenchmark case the durable consumption good is not attractive as an investment object.Both risky 0nancial securities have a higher expected return to volatility ratio; there aretransaction costs associated with selling the durable; the durable depreciates physically;and the durable is positively correlated with the 0nancial securities, so although thedurable provides further diversi0cation, hedging unwanted changes in the durable levelis associated with a decrease in expected return on 0nancial securities.

Fig. 7 and Part B of Table 1 illustrate how changes in the correlation between the0nancial asset prices and the durable price aEect the optimal behavior. The optimallevel of the state variable, z∗, is an increasing function of the durable-stock corre-lation, corresponding to lower durable holdings for higher correlation. Similarly, thelower bound on the continuation region is increasing in the durable-stock correlation,which is also the case for the upper bound up to a certain, positive correlation afterwhich the upper bound decreases with the correlation. To understand these results notethat there are several consequences of varying the correlation coeScient. Firstly, as thecorrelation increases, the durable becomes less eEective as an instrument to diversifyaway 0nancial risk, which should lead to lower durable holdings. Secondly, varying

11 The probability q(z) that Zt reaches 3z before z given that the current state is Z0 = z can be found as thesolution to the ODE 1

2�Z (z)2q′′(z)+�Z (z)q′(z)=0; z∈ (z; 3z), with the boundary conditions q(z)=0; q( 3z)=1,cf. Karlin and Taylor (1981, Section 15.3).


0

0.5

1

1.5

2

2.5

3

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8Durable Stock correlation

Continuation Region

z

zz*

Fig. 7. The change in the no-trade region as the correlation between the durable consumption good and eachof the risky securities changes.

the correlation from zero towards either +1 or −1 makes it easier to hedge the pricerisk of the durable, which, other things equal, lead to lower durable holdings for zerocorrelation than for either positive or negative correlation. Thirdly, if the correlationis positive (negative), the hedge portfolio consists of negative (positive) positions inthe stocks, so hedging the durable is associated with a wealth loss (gain), because thereturn on the hedge portfolio with the assumed parameters is higher than the returnon the durable. All three eEects support a decrease in optimal durable holdings (anincrease in z∗) as the durable–stock correlation is increased from negative values to-wards zero. As the correlation is further increased to positive values, the second eEecttends to increase durable holdings. In our example it does not dominate the two othereEects at the optimal level, but it does so at the upper boundary of the no-trade re-gion which decreases from a certain value of the correlation coeScient. The size ofthe no-trade region is decreasing in the absolute size of correlation since higher ab-solute correlation increases the possibility of hedging durable price changes, and it istherefore much easier to keep the state variable close to the optimal level. Decreasingthe durable/stock correlations increases the optimal level of durable, thereby increas-ing the costs associated with trading the durable. It is therefore optimal to trade thedurable less frequently so that the expected waiting time to the next durable trade in-creases. Furthermore, it becomes more likely that the next trade of durable is to a lowerlevel.

In Part C of Table 1, we vary the expected return to volatility ratio of the durablegood. With a smaller price volatility the optimal holdings of the durable are substan-tially higher, the no-trade region is narrower, and the expected period of time betweendurable trades is signi0cantly longer. Also note that the impact the assumed transactioncosts has on the optimal durable wealth to total wealth ratio depends heavily on therisk/return relationship of the durable good. In the last case in Part C, where the durablegood is quite attractive for investment purposes, transaction costs have the dramatic


Table 1Columns 1 and 2 are the lower and upper bound on the no-trade region. Column 3 is the optimal durablewealth/total wealth ratio immediately after a durable trade. Column 4 is the corresponding ratio withouttransaction costs. Columns 5 and 6 are the relative risk aversion and optimal consumption rate just afterdurable trade, respectively, and Column 7 represents the optimal consumption rate without transaction costs.Columns 8 and 9 are the expected waiting time until the next durable trade and the probability of tradingto a higher durable level at next durable trade, estimated just after a durable trade

Benchmark z 3z 1z∗ *k RRA c∗ *c E(%) P(z% = 3z)

0.17 2.75 1.16 1.96 0.64 0.096 0.12 3.58 0.94

� Part A: changes in �0.3 0.13 1.99 1.61 2.71 0.71 0.043 0.07 4.13 0.890.7 0.26 4.53 0.69 1.17 0.58 0.218 0.16 3.04 0.96

Corr. Part B: changes in durable/stock correlation−0:5 0.07 1.98 2.41 6.13 0.67 0.035 0.07 4.96 0.440 0.13 2.69 1.35 2.49 0.63 0.074 0.10 4.42 0.72

�P‖�P‖ Part C: changes in durable risk/return relationship0.1/0.05 0.06 1.36 3.38 10.65 0.76 0.031 0.11 6.52 0.870.09/0.01 0.05 1.15 4.91 33.03 0.78 0.021 0.12 11.21 0.93

� Part D: changes in transaction costs0.002 0.27 1.05 1.73 1.96 0.53 0.068 0.12 0.96 0.750.2 0.26 5.58 0.78 1.96 0.74 0.138 0.12 5.00 0.99

The 0rst row in the table represents the benchmark case, described previously in this section. The fourparts illustrate deviations from the benchmark in alternative directions. In Part A, the relative importanceof the durable vs. the perishable good is changed. In Part B, changes in the durable/stock correlation areconsidered. Part C considers changing the volatility and expected return on the price of the durable good,and, 0nally in Part D, we change the transaction costs associated with trading the durable.

eEect of lowering the optimal durable wealth to total wealth ratio by approximately85% (from 33.03 to 4.91).

Finally, we consider the sensitivity of our results to changes in the transaction costsparameter �. In Fig. 8, the no-trade region is plotted for varying transaction coststogether with z∗, the value of the state variable just after trading the durable. Even verysmall transaction costs lead to a relatively wide no-trade region. Without transactioncosts z∗ is 0.51, whereas with � = 0:002 the no-trade region is (0:27; 1:05). As �increases, the size of the continuation region increases in accordance with the intuitionthat higher transaction costs lead the agent to trade the durable more infrequently.Part D of Table 1 shows that with small costs of � = 0:002 the expected waitingtime to the next durable trade is almost 1 year. Fig. 9 provides a close-up of thelower bound of the no-trade region. For low levels of transaction costs, an increasein transaction costs makes the investor more reluctant to trade in the durable, causingz to decrease. As the transaction costs increase, the solvency region becomes smaller,and the potential threat of bankruptcy naturally drives z up. For very high transactioncosts, e.g. � = 1, the durable holdings are reduced because an adverse move in thedurable price may otherwise lead to bankruptcy.


1

2

3

4

5

0 0.05 0.1 0.15 0.2

transactions cost, λ

Continuation Region

zz*

z

Fig. 8. The change in the no-trade region as transaction costs change.

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.05 0.1 0.15 0.2 0.25λ

Lower bound on continuation region

Fig. 9. The solid line is a plot of the lower bound for no-trade region against the transaction costs parameter�. The dotted line is the boundary of the solvency region.

The optimal wealth ratio with transaction costs, z∗, varies from 0.57 for �=0:002 to1.37 for �=0:25, and it is therefore everywhere higher than z∗ without transaction costs,i.e. the optimal durable holdings are lower with transaction costs. The choice of z∗ withtransaction costs is inHuenced by several factors in addition to those without transactioncosts. To compensate for the physical depreciation of the durable and the expectedwealth increase until the agent can trade the durable again, the consumer/investor tendsto purchase a higher level of durable than if there were no transaction costs. On theother hand, the transaction costs when the durable is traded are increasing in thedurable position, and this tends to decrease the durable holdings of the agent. In ourexample, the latter eEect dominates and the durable purchases are lower than in theno transaction costs case. Furthermore, as can be seen from Part D of Table 1, as the


transaction costs increase, durable consumption is shifted to perishable consumptionand the agent will act in a more risk averse manner.

6. Concluding remarks

We have studied an optimal consumption and investment problem in a setting merg-ing the classical models of Merton (1969) and Grossman and Laroque (1990) so thatthe agent extracts utility from consumption of both a perishable and a durable con-sumption good. For the case where the durable good can be traded without transactioncosts, we have derived an explicit Merton-style solution to the problem. With transac-tion costs, we used the notion of viscosity solutions to show that optimal behavior isdetermined by the ratio z of total wealth to wealth in the durable and that as long asz is in the interval (z; 3z), the durable is not transacted. At the boundaries, the durablestock is adjusted such that the new value of the ratio z equals a z∗ ∈ (z; 3z). Furthermore,we have illustrated by numerical examples several important economic eEects that donot appear in the simpler one-good models. Our results demonstrate that the optimalperishable consumption rate as a fraction of total wealth can vary substantially overthe interval (z; 3z) and that optimal behavior can be highly dependent on the risk–returnrelationship of the durable good and its correlation with the 0nancial market prices.

Despite the relatively elaborate setup, several extensions are worth considering tomake the model more realistic, e.g. introducing a 0nite time horizon, stochastic inter-est rates, and a stochastic labor income stream. Until now, such features have only beendiscussed in models with a perishable consumption good; see, e.g., Bodie et al. (1992),Cuoco (1997), and Cocco et al. (1999) for models with stochastic income and Merton(1973) and Campbell and Viceira (2001) for models with utility from perishable con-sumption in a stochastic interest rate environment. The dynamics of both interest ratesand labor income seem to be important factors for individuals’ transactions in housesand other major durable goods. Of course, such extensions will also increase the com-plexity of both the analytical and the numerical analysis. Another direction for futureresearch would be to study the general equilibrium eEects of the single agent behaviorderived in this paper. In simpler settings, equilibrium eEects of durable consumptiongoods have already been addressed by, e.g., Grossman and Laroque (1990), Caballero(1993), Marshall and Parekh (1999), and Lax (1999).

Acknowledgements

We appreciate comments from anonymous referees and from seminar participants atthe Bachelier Finance Society Congress, Paris, June 2000, at the Danske Bank Sym-posium on Asset Allocation and Value at Risk, Middelfart, Denmark, January 2000,and at Copenhagen Business School. In particular, we thank Bjarne Astrup Jensen,Ken Bechmann, and Carsten Sorensen. This paper was written while all three authorswere at the University of Southern Denmark. Damgaard and Munk appreciate 0nancialsupport from the Danish Research Council for Social Sciences.


Appendix A. Solving the no transaction costs problem

The value function of the no transaction costs problem is de0ned by

3V (x; p) = sup(";K;C)∈ 3A(x;p)

E[ ∫ ∞

0U (Ct; Kt) dt

];

where U (C; K) is given by (8), and 3A(x; p) is as the set A(x; k; p) de0ned in Section 2except that the solvency condition now is X";K;C

t ¿ 0 for all t¿ 0. The Hamilton–Jacobi–Bellman (HJB) equation corresponding to this problem can be written as

& 3V (x; p) = sup,∈Rn;c; k∈R+

{1

1 − �(c�k1−�)1−� + (r(x − pk) + , �(� − r1)

+ (�P − �)kp− c)@ 3V@x

(x; p) +12

(, ��, + k2p2‖�P‖2

+ 2, ��P1kp)@2 3V@x2 (x; p) + �Pp

@ 3V@p

(x; p) +12‖�P‖2

×p2 @2 3V@p2 (x; p) + (, ��P1 + ‖�P‖2kp)p

@2 3V@x@p

(x; p)}

: (A.1)

We will show below that under some parameter conditions this highly non-linear partialdiEerential equation has the closed-form solution

3V (x; p) =1

1 − �*vp−(1−�)(1−�)x1−�;

where *v is a constant. Associated with the solution to the HJB equation are maximizingcontrol values of ,; c, and k, which lead to candidate control policies 3"t; 3Ct andKt . We also verify that the appropriate transversality condition is satis0ed. From theveri0cation theorem for controlled diEusions, see, e.g., Theorem IV.5.1 in Fleming andSoner (1993), it follows that 3V (x; p) above is the value function and the candidatecontrol policies are the optimal policies for the no transaction costs case.

A.1. Reducing the dimensionality of the problem

For all .¿ 0, the strategy (";K; C) is admissible with initial wealth x and initialdurable price p if and only if the strategy (.";K; .C) is admissible with initial wealth.x and initial durable price .p. Since U (.C; K) = .�(1−�)U (C; K), it follows that3V (.x; .p) = .�(1−�) 3V (x; p). In particular, 3V (x; p) = p�(1−�) 3V (x=p; 1) ≡ p�(1−�) 3v(x=p).

Substituting this into (A.1) and simplifying yield the highly non-linear ordinary diEer-ential equation

(&− �(1 − �)[�P − 12 ‖�P‖2[1 − �(1 − �)]]) 3v(y)

= sup,∈Rn;c; k∈R+

{1

1 − �(c�k1−�)1−� +

12

(, ��, + ‖�P‖2(y − k)2


− 2, ��P1(y − k)) 3v′′(y) + ((r + [1 − �(1 − �)]‖�P‖2 − �P)(y − k)

−�k + , �(� − r1− [1 − �(1 − �)]��P1) − c) 3v′(y)

}(A.2)

where we have introduced the scaled control values c = c=p and , = ,=p.

A.2. Solving the reduced HJB equation

We claim that the diEerential equation (A.2) has the concavely increasing solution

3v(y) =1

1 − �*vy1−� (A.3)

with the maximizing control values c = *cy; , = *,y, and k = *ky. Substituting (A.3)into (A.2), we get that

(&− �(1 − �)[�P − 12 [1 − �(1 − �)]‖�P‖2])

11 − �

*vy1−�

= sup,∈Rn;c; k∈R+

{1

1 − �(c�k1−�)1−� − �

2(, ��, + ‖�P‖2(y − k)2

− 2, ��P1(y − k))*vy−�−1 + ((r + [1 − �(1 − �)]‖�P‖2 − �P)

×(y − k) − �k + , �(� − r1− [1 − �(1 − �)]��P1) − c)*vy−�

}: (A.4)

Ignoring for now the positivity constraint on c and k, the 0rst order conditions for themaximizing control values are

− *vy−� + �c�(1−�)−1k(1−�)(1−�) = 0; (A.5)

− �(��,− ��P1(y − k)) + (� − r1− [1 − �(1 − �)]��P1)y = 0 (A.6)

and

−�*vy−�−1(, ��P1−‖�P‖2(y − k))−*vy−�(r−�P +�+[1−�(1−�)]‖�P‖2)

+ (1 − �)c�(1−�)k−�−�(1−�) = 0: (A.7)

Inserting control values of the form c = *cy; , = *,y and k = *ky, we 0rst get from(A.5) that

*v = �*�(1−�)−1c *(1−�)(1−�)

k (A.8)

and from (A.6) that

*, =1�

(��)−1(� − r1− (1 − �)(1 − �)��P1) − (��)−1�P1*k : (A.9)


Substituting (A.9) into (A.7) and applying (A.8) yield

*c =�

1 − �(r − �P + � + (1 − �)(1 − �)�2

P2

+ (� − r1)�(��)−1�P1)*k +��

1 − ��2P2*

2k : (A.10)

Substituting the candidate control values back into Eq. (A.4) and simplifying yield

&1 − �

− �(�P − 1

2[1 − �(1 − �)]‖�P‖2

)

=*�(1−�)c *(1−�)(1−�)

k

(1 − �)*v− �

2(*�, ��*, + ‖�P‖2(1 − *k)2 − 2(1 − *k)*�, ��P1)

+(r + [1 − �(1 − �)]‖�P‖2 − �P)(1 − *k)

−�*k + *�, (� − r1− [1 − �(1 − �)]��P1) − *c: (A.11)

Applying (A.8)–(A.10), Eq. (A.11) reduces after tedious manipulations to the quadraticequation

)0 + )1*k + )2*2k = 0; (A.12)

where

)0 = −&�

+1 − ��

{r − (1 − �)�P +12

(1 − �)[1 + (1 − �)(1 − �)]‖�P‖2}

+1 − �2�2 (� − r1− (1 − �)(1 − �)��P1)�(��)−1

×(� − r1− (1 − �)(1 − �)��P1);

)1 = (1 − �)�2P2 +

11 − �

(r − �P + � + (� − r1)�(��)−1�P1);

)2 =(

�1 − �

+1 − �

2

)�2P2:

For the solution above to satisfy the positivity constraints on c and k, the constants*c and *k must be positive, in which case the constant *v will also be positive. Wewill apply that *c can be rewritten as

*c = �)1*k + �()2 − 12 (1 − �)�2

P2)*2k (A.13)

and, using (A.12), as

*c = −�)0 − 12�(1 − �)�2

P2*2k : (A.14)


Consider the quadratic equation (A.12). To determine the number of positive roots,we apply Descartes’ rule of sign. Note that )2 is positive (since �¿ 0 and �∈ (0; 1))and consider the following cases:

1. If )0 and )1 are also positive, there are no positive roots in (A.12).2. Suppose )0 is positive and )1 is negative. Then there are either zero or two positive

roots. In any case, we see from (A.14) that *c will be negative.3. If )0 ¡ 0, there will be a unique positive solution *k to the quadratic equation

(A.12). To ensure a positive *c, we see from (A.14) that a necessary condition isthat )0 ¡− 1

2 (1−�)�2P2*

2k . From (A.13) we see that a suScient condition for *c ¿ 0

is that )1 ¿ 0.

A.3. The transversality condition

From the 0rst two subsections of this appendix, we can conclude that

3V (x; p) =1

1 − �*vp−(1−�)(1−�)x1−�

is a solution to the unreduced HJB equation (A.1) if )0 ¡− 12 (1− �)�2

P2*2k . To ensure

that this is indeed the value function, we must verify the transversality condition. Thedynamics of the wealth of the investor for any given control strategies (";K; C) isgiven by (5). Substituting in the candidate control policies 3Ct = *c 3X t; 3Kt = *k 3X t=Pt ,and 3"t = *, 3X t , the dynamics of wealth 3X t = X

3"; 3K; 3Ct becomes a geometric Brownian

motion

d 3X t = 3X t[r(1 − *k) + *�, (� − r1) + (�P − �)*k − *c] dt

+ 3X t(*�, � + *k��P1) dw1t + 3X t*k�P2 dw2t : (A.15)

In particular, wealth stays strictly positive with probability one. The transversality con-dition to be shown, cf. Theorem IV.5.1 in Fleming and Soner (1993), is thereforethat

limT→∞

e−&TE[ 3V ( 3X T ; PT )] = 0;

i.e. thatlim

T→∞e−&TE[ 3X 1−�

T P−(1−�)(1−�)T ] = 0: (A.16)

From (A.15) and (2) we get that

E[ 3X 1−�T P−(1−�)(1−�)

T ] = x1−�p−(1−�)(1−�)e6T ;

where

6 = (1 − �)

{r(1 − *k) + *�, (� − r1) + (�P − �)*k − *c − �

2*�, ��*,

− �2‖�P‖2*2

k − �*�, ��P1*k +12

(1 − �)[1 + (1 − �)(1 − �)]‖�P‖2

−(1 − �)�P − (1 − �)(1 − �)*�, ��P1 − (1 − �)(1 − �)*k‖�P‖2

}:


Applying (A.9) and (A.10), this reduces after tedious manipulations to

6 = &− (1 − �)()0 + )1*k + )2*2k) + )0 + 1

2 (1 − �)�2P2*

2k

= & + )0 + 12 (1 − �)�2

P2*2k ;

so that the transversality condition (A.16) is satis0ed if and only if

)0 ¡− 12 (1 − �)�2

P2*2k ;

which was already assumed to be the case.

Appendix B. Proofs of analytical results with transaction costs

B.1. Proof of Theorem 4.1

(a) The upper bound follows from the fact that the maximum expected utility isdecreasing in the transaction costs rate �. The lower bound can be obtained as follows.If x = �kp, the only feasible strategy is to liquidate and consequently V (x; k; p) = 0 on@S. If (x; k; p)∈S, a feasible strategy for the agent is to invest 7(x − �kp), where7∈ (0; 1), in the durable good and never trade the durable in the future. In this case,Kt = (7(x− �kp)=p)e−�t for all t¿ 0. If he, furthermore, invests the remainder in theriskless asset and continuously extracts exactly the added interest rate payments forperishable consumption, i.e. Ct = (1−7)(x−�kp)r for all t¿ 0, his life-time expectedutility will be∫ ∞

0

11 − �

e−&t

([(1 − 7)(x − �kp)r]�

[7(x − �kp)

pe−�t

]1−�)1−�

dt:

This expression is maximized by 7= 1− �, for which the expression gives the allegedlower bound.

(b) Fix (k; p) and consider x1 and x2 such that (xi; k; p)∈ 3S; i = 1; 2. Denoteby ("1; K1; C1) and ("2; K2; C2) the optimal strategies for V (x1; k; p) and V (x2; k; p),respectively. De0ne for 7∈ [0; 1]; x7 = 7x1 + (1 − 7)x2, and "7; K7, and C7 similarly.Because of the linearity of the wealth dynamics in (5), we have that

X"7;K7;C7t = 7X"1 ;K1 ;C1

t + (1 − 7)X"2 ;K2 ;C2t ¿ 7�K1tPt + (1 − 7)�K2tPt = �K7tPt

and thus ("7; K7; C7)∈A(x7; k; p). Concavity of the value function now follows fromthe concavity of the utility function, since

V (x7; k; p)¿ J"7;K7;C7(x7; k; p)

= E[ ∫ ∞

0U (C7t; K7t) dt

]

¿ E[ ∫ ∞

0{7U (C1t ; K1t) + (1 − 7)U (C2t ; K2t)} dt

]


= 7J"1 ;K1 ;C1 (x1; k; p) + (1 − 7)J"2 ;K2 ;C2 (x2; k; p)

= 7V (x1; k; p) + (1 − 7)V (x2; k; p):

To show that V is increasing in x, 0x (k; p) and pick x1 and x2 such that (xi; k; p)∈ 3Sand x1 ¡x2. Denote by ("1; K1; C1) the optimal strategy corresponding to V (x1; k; p).Again due to the linearity of the wealth dynamics,

X"1 ;K1 ;C1+(x2−x1)rt = x2 − x1 + X"1 ;K1 ;C1

t

and hence the strategy ("t; Kt; Ct) = ("1t ; K1t ; C1t + (x2 − x1)r) will belong toA(x2; k; p). Therefore,

V (x2; k; p)¿ J"t ;Kt ;Ct (x2; k; p)

= E[ ∫ ∞

0U (C1t + (x2 − x1)r; K1t) dt

]

¿ E[ ∫ ∞

0U (C1t ; K1t) dt

]

= V (x1; k; p):

(c) By concavity the value function is continuous in the interior of the set {x: x¿�kp}.

(d) For all .¿ 0; (";K; C)∈A(x; k; p) if and only (."; .K; .C)∈A(.x; .k; p).Also, with the assumed utility function U (.C; .K)=.1−�U (C; K). Therefore, V (x; k; p)is homogeneous of degree 1 − � in (x; k). Similarly, (";K; C)∈A(x; k; p) if andonly (.";K; .C)∈A(.x; k; .p), and U (.C; K) = .�(1−�)U (C; K), which implies thatV (x; k; p) is homogeneous of degree �(1 − �) in (x; p).

B.2. De/nition of viscosity solutions and proof of Theorem 4.2

We de0ne the viscosity solution concept for the HJB equation (28) as follows:

De(nition B.1. An upper semi-continuous function h :S1 → R is a viscosity subsolu-tion of (28) on the domain S1 if for every ’∈C2(S1) and any local maximum pointz0 ∈S1 of the function h− ’,

max{H (z0; h; ’′; ’′′); (z0 − �)1−� M

1 − �− h(z0)

}¿ 0: (B.1)

A lower semi-continuous function h :S1 → R is a viscosity supersolution of (28) onthe domain S1 if for every ’∈C2(S1) and any local minimum point z0 ∈S1 of thefunction h− ’,

max{H (z0; h; ’′; ’′′); (z0 − �)1−� M

1 − �− h(z0)

}6 0: (B.2)

A continuous function h :S1 → R is a viscosity solution of (28) if it is both asubsolution and a supersolution.


For a general overview of viscosity theory, we refer the reader to Crandall et al.(1992) and Fleming and Soner (1993).

Proof of part (a) of Theorem 4.2. We 0rst prove that v is a viscosity subsolution of(28). Let ’∈C2[�;∞) and suppose z0 is a local maximum point of the function v−’.We can without loss of generality assume that v(z0) =’(z0), and hence v(z)6’(z) insome open ball B(z0; 70) around z0 with radius 70. We assume that there exist strictlypositive numbers ;0 and ;1 such that

H (z; v; ’′; ’′′)¡− ;0 (B.3)

andM

1 − �(z0 − �)1−� − v(z0)¡− ;1 (B.4)

and seek to obtain a contradiction. Since ’∈C2 and v(z0) = ’(z0), (B.3) implies thatthere exists a neighborhood B(z0; 71) around z0 such that

H (z; ’; ’′; ’′′)¡− ;0; ∀z ∈B(z0; 71): (B.5)

Furthermore, because v is continuous, (B.4) holds for all z in some neighborhood of z0,say B(z0; 72). Moreover, by the dynamic programming principle, (B.4) implies that itis optimal to refrain from trading the durable asset for z ∈B(z0; 72) and, consequently,the optimal trajectory (Z∗

t ) is continuous up to the stopping time %2 = inf{t¿ 0: Z∗t �∈

B(z0; 72)}. De0ne the exit time %0 by

%0 = inf{t¿ 0: Z∗t �∈ B(z0; min(70; 71; 72))}:

Then for t ∈ [0; %0] we have by assumption that (B.3)–(B.5) hold, and the optimaltrajectory Z∗

t is continuous.Now applying Ito’s formula to the process {e− 3&tP�(1−�)

t ’(Z∗t )} and taking

expectations yield

E[e− 3&%0P�(1−�)%0

’(Z∗%0

)] = p�(1−�)’(z0) + E

[ ∫ %0

0e− 3&tP�(1−�)

t

×(L’(Z∗t ) − 3&’(Z∗

t )) dt

];

where the operator L is given by (30) with the optimal strategies inserted for , andc. Therefore,

E[e− 3&%0P�(1−�)%0

’(Z∗%0

)]

6p�(1−�)’(z0) + E

[∫ %0

0e− 3&tP�(1−�)

t

(H (Z∗

t ; ’; ’′; ’′′) − C�(1−�)t

1 − �

)dt

]

¡p�(1−�)’(z0) + E

[∫ %0

0e− 3&tP�(1−�)

t

(−;0 − C�(1−�)

t

1 − �

)dt

]:


Here, . ≡ E[∫ %0

0 e− 3&tP�(1−�)t ;0dt]¿ 0, and therefore, using v(z0)=’(z0) and v(Z%0 )6

’(Z%0 ), we get

p�(1−�)v(z0)¿E

[e− 3&%0P�(1−�)

%0v(Z∗

%0) +

∫ %0

0e− 3&tP�(1−�)

tC�(1−�)

t

1 − �dt

]+ .;

which violates the dynamic programming principle, because Z∗ was assumed to be theoptimal trajectory. Hence, v is a viscosity subsolution of (28).

We now prove that v is a viscosity supersolution of (28). Let ’∈C2[�;∞) andsuppose z0 ∈ [�;∞) is a local minimum point of the function v − ’. Again, we as-sume without loss of generality that v(z0) = ’(z0), and note for future reference thatv(z)¿’(z) in some neighborhood of z0, say B(z0; 70). We need to show that

0¿H (z0; v; ’′; ’′′) (B.6)

and

0¿M

1 − �(z0 − �)1−� − v(z0): (B.7)

Since it is always a feasible strategy to rebalance the durable asset, the dynamic pro-gramming principle implies that (B.7) is satis0ed for all z ∈ [�;∞) and thus in particularfor z0. We now prove that (B.6) holds. Let 7∈ (0; 70) be such that B(z0; 7) ⊂ [�;∞).For c¿ 0 and ,¿ 0 consider a policy (C; ")∈ A(z0) satisfying (Ct ; "t) = (c; ,) fort ∈ [0; %7], where %7 = inf{t¿ 0: ZC;"

t �∈ B(z0; 7)}. To simplify notation we shall

in the following write Zt instead of ZC;"t . Applying Ito’s formula to the process

{e− 3&tP�(1−�)t ’(Zt)} and taking expectations yield

p�(1−�)’(z0) = E

[e− 3&%7P�(1−�)

%7 ’(Z%7) −∫ %7

0e− 3&tP�(1−�)

t

×(LC; "’(Zt) − 3&’(Zt)) dt

]: (B.8)

Furthermore, from the dynamic programming principle we have

p�(1−�)v(z0)¿E[ ∫ %7

0e− 3&tP�(1−�)

tc1−�

1 − �dt + e− 3&%7P�(1−�)

%7 v(Z%7)]:

Therefore, using (B.8), v(z0) = ’(z0) and v(Z%7)¿’(Z%7), we get

p�(1−�)’(z0)¿ E[ ∫ %7

0e− 3&tP�(1−�)

tc1−�

1 − �dt + e− 3&%7P�(1−�)

%7 ’(Z%7)]

= p�(1−�)’(z0) + E[ ∫ %7

0e− 3&tP�(1−�)

t (LC; "’(Zt) − 3&’(Zt)) dt

+∫ %7

0e− 3&tP�(1−�)

tc1−�

1 − �dt]:

Thus, for all 7∈ (0; 70),

0¿E[ ∫ %7

0e− 3&tP�(1−�)

t

(c1−�

1 − �+ LC; "’(Zt) − 3&’(Zt)

)dt]


and consequently

0¿ minz∈ 3B(z0 ;7)

{c1−�

1 − �+ Lc;,’(z) − 3&’(z)

}: (B.9)

In particular, (B.9) holds for 7 → 0 and thus, by continuity,

0¿c1−�

1 − �+ Lc;,’(z0) − 3&v(z0): (B.10)

Finally, maximizing the right-hand side of (B.10) over (c; ,) yields (B.6), and hence,v is a viscosity supersolution of (28).

Proof of part (b) of Theorem 4.2. We prove that the original value function, V (x; k; p),is continuous at the boundary @S1 = {(x; k; p)∈R3

++: x − �kp = 0} of the solvencyregion. Since v(z) = V (z; 1; 1), it follows that v is continuous in z = �. We need thefollowing lemma:

Lemma B.1. For any given �¿ 0, there exist constants 7¿ 0, 0¡.6 � + 7, and*¿ 0 such that

V (x; k; p)6 (x; k; p) ≡ *1 − �

(x − �kp)1−�; ∀(x; k; p)∈ I7 ∪ >.;

where

I7 = {(x; k; p)∈R3++: �kp6 x¡ (� + 7)kp};

>. = {(x; k; p)∈R3++: kp6 1; �kp6 x¡.}:

To see that Lemma B.1 implies continuity of the value function on @S1, 0rst notethat if (x; k; p)∈ @S1, the only feasible strategy is to liquidate and, consequently,V = 0 on @S1. Secondly, for any boundary point (x0; k0; p0)∈ @S1 we have that06V (x; k; p)6 (x; k; p) → 0 for (x; k; p) → (x0; k0; p0) through I7 ∪ >..

Proof. To simplify notation put Yt = (Xt; Kt ; Pt) and y = (x; k; p). De0ne the diEer-ential operator F , by

F(y; ; D ; D2 ) = supc;,

{Ac;, (y) + U (c; k) − & (y)};

where D and D2 denotes the vector of partial derivatives and the matrix of secondorder partial derivatives, respectively, and Ac;, is de0ned by

Ac;, (x; k; p) =@ @x

(r(x − kp) + , �(� − r1) + (�P − �)kp− c) − @ @k

�k

+@ @p

�Pp +12@2 @x2 ‖�P‖2p2 +

12@2 @x2 (, ��, + 2, ��P1kp

+ ‖�P‖2k2p2) +@2 @x@p

(, ��P1p + ‖�P‖2kp):


De0ne the auxiliary function ’(z) = *=(1 − �)(z − �)1−�, where * is a so far arbi-trary positive constant. Straightforward computations, utilizing the relation (x; y; k) =k1−�p�(1−�)’(x=(kp)), imply that

F(y; ; D ; D2 ) = k1−�p�(1−�)H(

xkp

; ’; ’′; ’′′)

; (B.11)

where H is de0ned as in (29). Moreover, tedious but straightforward manipulationsshow that there exist constants B;D, and 6¿ 0 such that for all z¿ �

H (z; ’; ’′; ’′′) = *(z − �)1−�(

�1 − �

)0(z − �)2 + B(z − �)

−12�(1 − �)2�2

P2 + D(z − �)6)

;

where )0 is de0ned in Section 3. It follows that for any *, constants 7¿ 0 and.∈ (0; � + 7] exist such that

H (x=(kp); ’7; ’′7; ’

′′7 )6 0; ∀(x; k; p)∈ I7 ∪ >.:

Thus by (B.11)

F(y; ; D ; D2 )6 0; ∀y∈ I7 ∪ >.: (B.12)

De0ne the {Ft}-stopping time %0 = inf{t¿ 0: Yt �∈ I7∪>.}. Ito’s formula for semi-martingales (see, e.g., Protter (1995, Thm. II.33)) applied to the process {e−&t (Yt)}implies

E[e−&%0 (Y%0 )] = (y) + E

[ ∫ %0

0e−&t(AC;" (Yt) − & (Yt)) dt

+∑

06t¡%0

( (Yt) − (Yt−))

]: (B.13)

Furthermore, if Yt jumps at time t, then Xt =Xt−−�Kt−Pt , which implies that (Yt)− (Yt−)6 0 for all t¿ 0. This latter observation together with (B.12) and (B.13) imply

E[e−&%0 (Y%0 )]6 (y) − E[ ∫ %0

0e−&tU (Ct; Kt) dt

]:

It can be shown that

V (X%0 ; K%0 ; P%0 )6 (X%0 ; K%0 ; P%0 )

if * is chosen such that *¿ *v((� + 7)=7)1−�. This together with (B.13) imply

E[ ∫ %0

0e−&tU (Ct; Kt) dt + e−&%0V (Y%0 )

]6 (y):

Maximizing the left-hand side over all admissible controls (C; K;")∈A(y) leads toour claim.


Proof of part (c) of Theorem 4.2. As shown earlier, the function v de0ned by (24)is a viscosity solution of (28) with the stated properties. That v is the only suchsolution follows from the comparison result stated in the following lemma. (Note thatan increasing concave function has sublinear growth.)

Lemma B.2. Suppose that (31) is satis/ed. Let v and 3v be a viscosity subsolutionand a viscosity supersolution, respectively, of (28) on (�;∞). Furthermore, supposethat both v and 3v are continuous and increasing with sublinear growth on [�;∞)and satisfy 0 = v(�) = 3v(�) and v(z)¿M=(1 − �)(z − �)1−� for all z ∈ [�;∞). Thenv(z)6 3v(z) for all z ∈ [�;∞).

Proof. Observe 0rst that the operator H de0ned in (29) can be written as

H (z; v; v′; v′′) = G(z; v′; v′′) − &v;

where G :R3 → R is de0ned by

G(z; a; b) = supc∈R+ ;,∈Rn

{12b[, ��,− 2(z − 1), ��P1 + (z − 1)2‖�P‖2]

+1

1 − �c�(1−�) + a[(z − 1)(r − �P + � + [1 − �(1 − �)]‖�P‖2)

−c + , �(� − r1− [1 − �(1 − �)]��P1)]}

and where

& = &− (1 − �)(��P − 1

2�[1 − �(1 − �)]‖�P‖2 − �

)

by (31) is positive.We assume that

supy∈[�;∞)

{v(y) − 3v(y)}¿ 0 (B.14)

and aim to 0nd a contradiction. By assumption, constants .¿ 0 and F∈ (0; 1) existsuch that v(y)6 .(1+y)F for all y∈ [�;∞). Let F∈ (F; 1). It follows from (B.14) thatfor suSciently small ;¿ 0 we have

supy∈[�;∞)

{v(y) − 3v(y) − ;yF}¿ 0: (B.15)

Moreover, because v and 3v are continuous and increasing and 3v¿ 0, there exists anumber y∗ ∈ [�;∞) such that

supy∈[�;∞)

{v(y) − 3v(y) − ;yF} = v(y∗) − 3v(y∗) − ;(y∗)F:

Note that if y∗ = �, we have a contradiction to (B.15), since we assumed that0 = v(�) = 3v(�).


De0ne the auxiliary functions ’; : [�;∞) × [�;∞) → R by

’(y; z) = (*(z − y) − 47)4 + ;zF;

where *¿ 1; 7¿ 0, and ;¿ 0, and

(y; z) = v(z) − 3v(y) − ’(y; z):

We shall apply the following results which are demonstrated below:

Claim 1. (a) The function is bounded on [�;∞) × [�;∞) and its supremum point(y*; z*) is attained in the compact set [�; 3.] × [�; 3.], where 3.¿ 0 is a /nite constantindependent of *; 7, and ;. (b) The maximum point (y*; z*) has the following properties

lim*↑∞

|z* − y*| = 0; (B.16)

lim*↑∞

(*(z* − y*) − 47)4 = 0; (B.17)

lim;↓0

lim*↑∞

;zF* = 0: (B.18)

Claim 2. For all *¿ 1; 7¿ 0, and ;¿ 0, there exist numbers Z ¡ 0 and Y ¿ 0 withY + Z ¡ 0 such that

&v(z*)6G(z*;

@’@z

; Z)

(B.19)

& 3v(y*)¿G(y*;−@’

@y;−Y

)(B.20)

and 12

(Z 0

0 Y

)6

@2’@z2

@2’@z@y

@2’@z@y

@2’@y2

=

(A + ;B −A

−A A

)(B.21)

with A = 12*2(*(z* − y*) − 47)2¿ 0 and B = F(1 − F)zF−2* 6 0.

Subtracting (B.20) from (B.19) yields

&(v(z*) − 3v(y*))

6G(z*;

@’@z

; Z)− G

(y*;−@’

@y;−Y

)

12 For two symmetric n-dimensional matrices C and D, we write C¿D if C −D is positive de0nite, thatis, if 6�C6¿ 6�D6 for all 6∈Rn.


= supc∈R+ ;,∈Rn

{12Z[, ��,− 2(z* − 1), ��P1 + (z* − 1)2‖�P‖2]

+1

1 − �c�(1−�) +

@’@z

[(z* − 1)(r − �P + � + [1 − �(1 − �)]‖�P‖2)

−c + , �(� − r1− [1 − �(1 − �)]��P1)]}

− sup3c∈R+ ; 3,∈Rn

{−1

2Y [ 3, �� 3,− 2(y* − 1), ��P1

+(y* − 1)2‖�P‖2] +1

1 − �3c�(1−�)

−@’@y

[(y* − 1)(r − �P + � + [1 − �(1 − �)]‖�P‖2)

− 3c + 3, �(� − r1− [1 − �(1 − �)]��P1)]}: (B.22)

Since Y ¿ 0 and Z ¡ 0, the maximizers in (B.22) are

c =(

4*(*(z* − y*) − 47)3 + ;FzF−1*

�

)1=(�(1−�)−1)

;

3c =(

4*(*(z* − y*) − 47)3

�

)1=(�(1−�)−1)

;

, =4*(*(z* − y*) − 47)3 + ;FzF−1

*

−Z(��)−1(� − r1− [1 − �(1 − �)]��P1)

+ (z* − 1)(��)−1�P1;

3, =4*(*(z* − y*) − 47)3

Y(��)−1(� − r1− [1 − �(1 − �)]��P1)

+(y* − 1)(��)−1�P1:

Tedious, but straightforward, computations yield

G(z*;

@’@z

; Z)− G

(y*;−@’

@y;−Y

)

=(

11 − �

− �)(

c�(1−�) − 3c�(1−�))+ (r − �P + � + (� − r1)�(��)−1�P1

+[1 − �(1 − �)�2P2])(4*(z* − y*)(*(z* − y*) − 47)3

+(z* − 1);FzF−1* ) +

12�2P2(Z(z* − 1)2 + Y (y* − 1)2)


−12

(� − r1− [1 − �(1 − �)]��P1)�(��)−1(� − r1− [1 − �(1 − �)]��P1)

×(

(4*(*(z* − y*) − 47)3 + ;FzF−1* )2

Z

+(4*(*(z* − y*) − 47)3)2

Y

):

Claim 3. (a) lim;↓0 lim7↓0 lim*↑∞ (c�(1−�) − 3c�(1−�)) = 0,(b) lim;↓0 lim7↓0 lim*↑∞ (Z(z* − 1)2 + Y (y* − 1)2) = 0,(c) there exist sequences (*n) ↑ ∞; (7n) ↓ 0, and (;n) ↓ 0 such that the term

−12

(� − r1− [1 − �(1 − �)]��P1)�(��)−1(� − r1− [1 − �(1 − �)]��P1)

×(

(4*(*(z* − y*) − 47)3 + ;FzF−1* )2

Z+

(4*(*(z* − y*) − 47)3)2

Y

)

= − 12

(� − r1− [1 − �(1 − �)]��P1)�(��)−1

× (� − r1− [1 − �(1 − �)]��P1)(

(@’=@z)2

Z+

(@’=@y)2

Y

)

is non-positive along these sequences.

It follows from Claim 3.c that there exists a sequence (*n; ;n; 7n) → (∞; 0; 0) suchthat

G(z*;

@’@z

; Z)− G

(y*;−@’

@y;−Y

)

6(

11 − �

− �)

(c�(1−�) − 3c�(1−�)) + (r − �P + � + (� − r1)�

×(��)−1�P1 + [1 − �(1 − �)�2P2])(4*(z* − y*)(*(z* − y*) − 47)3

+ (z* − 1);FzF−1* ) +

12�2P2(Z(z* − 1)2 + Y (y* − 1)2) (B.23)

along this sequence. Moreover, from Claim 1, 3.a, and 3.b the right-hand side of(B.23) converges to zero, and hence (y*; z*)6 0, along the same sequence. This isa contradiction to (B.15) since

(y*; z*)¿ (y∗ − 47=*; y∗) = v(y∗) − 3v(y∗) − ;(y∗)F

= supy∈[�;∞)

{v(y) − 3v(y) − ;yF}¿ 0:

It remains to prove Claim 1–3.


Proof of Claim 1. (a) Because v and 3v have sublinear growth, (y; z) → −∞ fory; z → ∞. Consequently, a constant 3. exists, independent of *, such that

sup[�;∞)2

(y; z) = max[�; 3.]2

(y; z):

(b) Since z* ∈ [�; 3.], (B.18) follows immediately. Moreover, from (a), is bounded.Furthermore,

(y*; z*)¿ (y∗ − 47=*; y∗) = v(y∗) − 3v(y∗) − ;(y∗)F

= supy∈[�;∞)

{v(y) − 3v(y) − ;yF}¿ 0

and consequently

(*(z* − y*) − 47)46 v(z*) − 3v(y*) − ;zF* − (v(y∗) − 3v(y∗) − ;(y∗)F):

Since the right-hand side is bounded, (B.16) follows. Finally, it can be shown thaty* → y∗ and z* → y∗ as * → ∞, implying (B.17).

Proof of Claim 2. The existence of numbers Y and Z such that (B.19)–(B.21) holdfollows from a generalization of Ishii and Lions (1990, Prop. II.3) to the case wherethe controls are not uniformly bounded, see, e.g., Zariphopoulou (1994). To see thatY ¿ 0, note from (B.20) that if this is not the case, i.e. if Y 6 0, then

& 3v(y*)¿G(y*;−@’

@y;−Y

)

= sup3c∈R+ ; 3,∈Rn

{−1

2Y [ 3, �� 3,− 2(y* − 1), ��P1 + (y* − 1)2‖�P‖2]

+1

1 − �3c�(1−�) − @’

@y[(y* − 1)(r − �P + � + [1 − �(1 − �)]‖�P‖2)

− 3c + 3, �(� − r1− [1 − �(1 − �)]��P1)]}

= ∞;

which contradicts the fact that 3v(y*)¡∞. To see that Z ¡ 0, multiply (70) with(1; 1)� from both sides. This yields Y + Z6 ;B¡ 0. As shown above Y ¿ 0 and,consequently, it must be the case that Z ¡ 0.

Proof of Claim 3. (a) follows from Claim 1.b. To show (b) note that Claim 2 impliesthat

Y (y* − 1)2 + Z(z* − 1)2 = (z* − 1; y* − 1)

(Z 0

0 Y

)(z* − 1

y* − 1

)

6 (z* − 1; y* − 1)

(A + ;B −A

−A A

)(z* − 1

y* − 1

)


= 12*2(z* − y*)2(*(z* − y*) − 47)2

+;(z* − 1)2F(F− 1)zF−2* :

From Claim 1 this last expression converges to zero for * ↑ ∞; 7 ↓ 0, and ; ↓ 0.Finally, (c) follows from the fact that for all *¿ 1 we have that

−(

(@’=@z)2

Z+

(@’=@y)2

Y

)

=Y (@’=@y)2 + Z(@’=@y)2

−ZY

=Y ((@’=@z)2 − 2;FzF−1@’=@y + ;2(FzF−1)2) + Z(@’=@y)2

−ZY

→ (Y + Z)(@’=@y)2

−ZY6 0 for ; ↓ 0;

where the inequality comes from Claim 2.

B.3. Proof of Theorem 4.3

De0ne the function g as g(z) =H (z; f; f′; f′′), where f(z) = (z− �)1−�M=(1− �) isthe stopping reward function. Furthermore, de0ne the set G = {z ∈ (�;∞): g(z)¿ 0}.We 0rst show the following lemma:

Lemma B.3. The set G ≡ {z ∈ (�;∞): g(z)¿ 0} is either empty or an open interval.

Proof. Tedious, but straightforward, computations yield that g can be written as

g(z) = M (z − �)−1−�p(z);

where

p(z) =�

1 − �)0(z − �)2 + B(z − �) − 1

2�(1 − �)2�2

P2 + D(z − �)6

and the constants B; D, and 6 are given by

B = −(1 − �)(1 − �))1;

D =(

11 − �

− �)

�1=[1−�(1−�)]−1M 1=[�(1−�)−1];

6 = 1 +�

1 − �(1 − �):

Clearly, g(z)¿ 0 if and only if p(z)¿ 0. Note that p(�)¡ 0, and because 6∈ (1; 2)and )0 ¡ 0, we have that p(z) → −∞ for z → ∞. A standard computation gives


that

p′(z) = 2�

1 − �)0(z − �) + B + 6D(z − �)6−1:

We distinguish between the case where B¿ 0 and the case where B¡ 0.If B¿ 0, the equation p′(z) = 0 has a unique solution z, which is a maximum point

of the function p(z). If p(z)6 0, the set G is empty. If p(z)¿ 0, it follows fromcontinuity that G is an interval around z.

If B¡ 0, the equation p′(z) = 0 can have zero, one, or two solutions. If it has nosolutions, p is decreasing for all z, and the set G is empty. If it has one solutionz; p′(z)¡ 0 for all z except z = z, so G is empty. If it has two solutions, z1 and z2,where z1 ¡z2, it follows that p has a local minimum at z1 and a global maximum atz2. If p(z2)6 0, the set G is empty. If p(z2)¿ 0; G is an interval around z2.

To prove Theorem 4.3, we 0rst show that G ⊆ N. To see this, recall that v(z)¿f(z)for all z. Any z0 with v(z) � f(z) is therefore a local minimum point of the functionv − f with v(z0) − f(z0) = 0. Since v by Theorem 4.2 is a viscosity supersolution ofthe HJB equation, it follows from (B.2) that H (z0; v; f′; f′′) = H (z0; f; f′; f′′) = g(z0)is non-positive.

Since we have assumed that G is non-empty, it follows Lemma B.3 that G is aninterval. We want to show that the set N is also an interval. Assuming the contrary,continuity of v implies that an interval (z′1; z

′2) exists disjoint from G with v(z′i) =

f(z′i); i = 1; 2, and v(z) − f(z)¿ 0 for z ∈ (z′1; z′2). The function v − f will have a

maximum on [z′1; z′2], say at z′, where of course v(z′)¿f(z′). Since v is a viscosity

subsolution of the HJB equation, it follows from (B.1) that

H (z′; v(z′); f′(z′); f′′(z′))¿ 0

and since H is strictly decreasing in its second argument by (31), we have that

g(z) ≡ H (z′; f(z′); f′(z′); f′′(z′))¿ 0;

which is a contradiction.Finally, we will demonstrate that z∗ ∈ (z; 3z). By de0nition of z∗, we have that

f(z∗) = (z∗ − �)1−� M1 − �

= (z∗ − �)1−�(z∗)�−1v(z∗)¡v(z∗);

which implies that z∗ ∈N.

Appendix C. Description of the numerical method

We seek to 0nd a solution to the Hamilton–Jacobi–Bellman equation

0 = max{H (z; v; v′; v′′);

11 − �

(z − �)1−�M − v(z)}

; ∀z ∈S1;


where H (z; v; v′; v′′) is given by the highly non-linear expression

H (z; v; v′; v′′)

=[�(1 − �)

(�P − 1

2[1 − �(1 − �)]‖�P‖2

)− 3&]v(z)

+(

1�(1 − �)

− 1)

�1=(1−�(1−�))v′(z)�(1−�)=�(1−�)−1

+(r − �P + � + (� − r1)�(��)−1�P1 + [1 − �(1 − �)]�2P2)

×(z − 1)v′(z) +12�2P2(z − 1)2v′′(z) − 1

2(� − r1− [1 − �(1 − �)]��P1)�

×(��)−1(� − r1− [1 − �(1 − �)]��P1)v′(z)2

v′′(z);

which is obtained by substituting the optimal scaled controls (32)–(33) into (29).The solution v(z) is a solution to a free boundary equation on the continuation region

(z; 3z) such that v(z)¿ 1=(1 − �)(z − �)1−�M , and with the following value matchingand “smooth pasting” conditions on the boundaries

v(z) =1

1 − �M (z − �)1−�; v′+(z) = M (z = �)−�; (C.1)

v( 3z) =1

1 − �M ( 3z − �)1−�; v′−( 3z) = M ( 3z − �)−�; (C.2)

Here, v′+ and v′− denote the right and left derivatives, respectively. A solution to thissystem is given by (M; z; 3z) so that the equations above are satis0ed, and such that Msolves the equation

M = (1 − �) supz¿�

z�−1v(z):

As Grossman and Laroque (1990) we apply a stepwise numerical procedure to 0ndthe optimal values (M; z; 3z):

(a) Guess M = M0.(b) Solve the free boundary problem, with M = M0, as follows:

(i) Guess z = z0.(ii) Solve the ODE H (z; v; v′; v′′) = 0 using (C.1) as initial conditions until the

value matching part of Eq. (C.2) is satis0ed.(iii) If the smooth pasting condition in (C.2) is satis0ed, a candidate value function

vM0 (z) is found, otherwise repeat steps (i) and (ii).


(c) Compute the implied M∗0 = (1− �) supz∈[z0 ; 3z0] z

�−1 vM0 (z). If M∗0 = M0 the problem

is solved, otherwise go to (a).

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