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Publisher : Elsevier - Science Direct (الزویر - ساینس دایرکت)
Journal : Journal of Economic Dynamics and Control, Volume 28, Issue 2, November 2003, Pages 209–253
We study the consumption and investment choice of an agent in a continuous-time economy with a riskless asset, several risky financial assets, and two consumption goods, namely a perishable and a durable good with an uncertain price evolution. Assuming lognormal prices and a multiplicatively separable, isoelastic utility function, we provide an explicit Merton-type solution for the optimal strategies for the case where the durable (and all other assets) can be traded without transaction costs. For the case where the durable good is indivisible, in the sense that durable 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 constants View the MathML source. As long as the ratio z of the total current wealth to the value of current durable holdings of the investor is in View the MathML source, it is optimal not to trade the durable. At the boundaries of this interval it is optimal to trade the durable to attain z=z∗. The model is used to examine the optimal substitution between perishable and durable consumption and the importance of the durable price uncertainty and the correlation between the price of the durable good and financial asset prices.
Modern economies oEer an enormous variety of consumption goods. For modeling purposes each good is typically classi0ed either as a perishable good or a durable good. A perishable good cannot be stored and provides utility only at the time of purchase. A durable good provides utility to its owner over a period of time and can be resold so that it also acts as an investment that transfers wealth over time. Traditional models of optimal consumption and investment problems consider either a single perishable consumption good, cf. Merton (1969), or a single durable consumption good, cf. Grossman and Laroque (1990). In this paper, we merge these settings by allowing for 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 setup and 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 how the uncertainty about future relative consumption prices aEects optimal consumption and investment decisions. More speci0cally, we examine the optimal consumption and investment choice of an agent in a continuous-time economy with one riskless and several risky 0nancial assets and both a perishable and a durable consumption good. The durable consumption good is indivisible in the sense that in order to change the stock of the good (beyond the assumed depreciation), the agent must sell his entire current holdings of the good and then buy the desired new stock, which is the case for houses and cars for example. We assume that in doing so the agent must pay transaction costs proportional to the value of the current stock of the durable. The perishable good and the 0nancial assets are traded without transaction costs. The agent extracts utility from the rate of consumption of the perishable good and the stock of the durable good. We study the case where the agent 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 perishable consumption rate and the current stock of the durable, respectively. Furthermore, measured in terms of perishable consumption units, the price of the durable good and the prices of the risky 0nancial assets follow correlated geometric Brownian motions. Our 0rst contribution is to derive an explicit solution to the consumption–investment problem for the case of no transaction costs. The optimal strategy is to keep both the perishable consumption rate, the value of the durable holdings, and the amount invested in each of the risky 0nancial assets as 0xed fractions of wealth. This result generalizes 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 optimal investment strategy combines only the mean-variance tangency portfolio and a durable hedge portfolio, which is the portfolio with the highest possible absolute correlation with the price of the durable consumption good. The optimal strategy for the no transaction costs problem involves continuous rebalancing of the stock of the durable due to Huctuations in 0nancial asset prices and the price of the durable and also due to the physical depreciation of the stock of the durable. With transaction costs, such a strategy is clearly not optimal. Our second contribution is to characterize the optimal consumption and investment policies with transaction costs. We show that the optimal behavior is completely determined by the ratio z = x=(kp) of the total current wealth, x, to the product of the current stock of durable, k, and the current price of durable, p. In addition, using the notion of viscosity solutions, we demonstrate that there are two critical values z¡ 3 z such that it is optimal to refrain from trading the durable good as long as z ∈(z; 3 z). At the boundaries of this interval, it is optimal to trade the durable. When z = z, it is optimal to shift to a lower stock of the durable, and when z= 3z, it is optimal to shift to a higher stock of the durable. In both cases, the optimal transaction is such that, immediately after the transaction, the new value of z, z∗, is in the open interval (z; 3 z). If the initial values are such that z ∈ (z; 3 z), the optimal policy involves an initial transaction to z∗ ∈(z; 3 z). Since there is a rather small loss in utility from deviating a little from the optimal durable consumption level, adjustments of the durable holdings will be infrequent even with small transaction costs. Concerning the optimal investment strategies, we show that the presence of transaction costs on the durable good does not change the fund separating structure of the set of risky 0nancial assets. Our third contribution is to provide numerical results illustrating several important economic eEects of transaction costs in our setting. We 0nd that the relative risk aversion associated with the indirect utility of wealth varies over the interval (z; 3 z) 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∗ (immediately after a durable trade). We show that this variation in risk aversion has a signi0cant impact on the weights of the two separating risky funds in the optimal portfolio. We demonstrate that the no-trade interval (z; 3 z) widens as transaction costs increase and that transaction costs generally have smaller quantitative eEects than in a model with only a durable consumption good (the Grossman and Laroque model discussed below), since the agent in our model will substitute perishable consumption for durable consumption to reduce transaction costs. For example, the expected period of time between durable trades is signi0cantly lower than found by Grossman and Laroque. We 0nd that the perishable consumption propensity, i.e. the optimal rate of perishable consumption as a fraction of total wealth, can vary substantially over the interval (z; 3 z), in contrast to the no transaction costs case where it is kept constant. The precise relation is highly dependent on the exogenous preference parameters and and the endogenously determined relative risk aversion associated with the indirect utility of wealth. For some parameter values, the perishable consumption propensity is a decreasing function of z and, hence, increasing in the price of the durable and decreasing in 0nancial wealth. For other parameter values, the relation is non-monotonic. Finally, we show that the optimal behavior can be highly sensitive to the correlation between the price of the durable good and 0nancial asset prices and, especially, the risk-return relationship of the durable good. The analysis of the paper generalizes that of Grossman and Laroque (1990) who consider the simpler problem with only a durable consumption good that acts as the numeraire good. In their simpler setup, they give a similar characterization of the optimal durable trading strategy and have similar results on risk aversion and the dependence of the no-trade region on transaction costs. Our analysis shows that their approach and conclusions carry over to the more general and economically more appealing framework with two types of consumption goods with an uncertain relative price, but that the quantitative eEects of transaction costs are smaller in the two-good economy. Moreover, our general setting allows us to study the relation between perishable and durable consumption and the impact of the uncertainty of the durable good price and its correlation with 0nancial asset prices on optimal behavior. A few other papers study the implications of durable consumption goods on optimal 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 the durable requires the payment of transaction costs proportional to the change in the stock of durable, not the current stock. Under this assumption it is optimal to keep the ratio z in a closed interval [zl; zu], but at the boundaries the optimal transaction is the minimal needed to keep z in this interval, an eEect also found in models with proportional costs of transacting 0nancial assets, cf., e.g., Davis and Norman (1990). In mathematical 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 the boundaries), contrary to our setup where the changes are infrequent, non-in0nitesimal jumps into the interior of the interval. The reason for this diEerence is that in our model 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 a durable good that cannot be resold once bought. With power utility and lognormal 0nancial asset prices, it is optimal to keep the ratio of wealth to the stock of durable below 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 good where the durable good provides a utility both through current purchases of the good (“status”) and through the current stock of the good (“services”). The investor is not allowed to sell out of his stock of durable and the price of the durable good (in terms of units of the perishable good) is spanned by the 0nancial asset prices, whereas we allow for a durable-speci0c price risk. Using martingale methods for this complete markets optimization problem, they characterize the optimal consumption processes in terms of the state-price density. The rest of the paper is organized as follows. Section 2 describes the details of the model we use. In Section 3, we provide an explicit solution to the utility maximization problem for the special case where the durable consumption good can be traded without transaction costs. For the problem with transaction costs, we derive analytically some important 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 to numerical 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.
نتیجه گیری انگلیسی
We have studied an optimal consumption and investment problem in a setting merging the classical models of Merton (1969) and Grossman and Laroque (1990) so that the agent extracts utility from consumption of both a perishable and a durable consumption good. For the case where the durable good can be traded without transaction costs, we have derived an explicit Merton-style solution to the problem. With transaction costs, we used the notion of viscosity solutions to show that optimal behavior is determined by the ratio z of total wealth to wealth in the durable and that as long as z is in the interval (z; 3 z), the durable is not transacted. At the boundaries, the durable stock is adjusted such that the new value of the ratio z equals a z∗ ∈(z; 3 z). Furthermore, we have illustrated by numerical examples several important economic eEects that do not appear in the simpler one-good models. Our results demonstrate that the optimal perishable consumption rate as a fraction of total wealth can vary substantially over the interval (z; 3 z) and that optimal behavior can be highly dependent on the risk–return relationship of the durable good and its correlation with the 0nancial market prices. Despite the relatively elaborate setup, several extensions are worth considering to make the model more realistic, e.g. introducing a 0nite time horizon, stochastic interest rates, and a stochastic labor income stream. Until now, such features have only been discussed 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 consumption in a stochastic interest rate environment. The dynamics of both interest rates and labor income seem to be important factors for individuals’ transactions in houses and other major durable goods. Of course, such extensions will also increase the complexity of both the analytical and the numerical analysis. Another direction for future research would be to study the general equilibrium eEects of the single agent behavior derived in this paper. In simpler settings, equilibrium eEects of durable consumption goods have already been addressed by, e.g., Grossman and Laroque (1990), Caballero (1993), Marshall and Parekh (1999), and Lax (1999).