# Optimal portfolio strategies and the demand for durable goods.

I. Introduction

Very few of the existing models of optimal portfolio behavior are able to accommodate the fact that commodity inventories comprise a significant share of the average household's portfolio. Moreover, no effort has been made to explicitly consider the complications that arise from the durable good nature of most of these commodities.

The purpose of this paper is to develop a model capable of accommodating commodity inventories as part of the utility maximizing portfolio. Furthermore, the analysis will explicitly consider durable goods as consumption and portfolio items.

The failure of most of the existing models to account for commodity inventories in the optimal portfolio is due to the adoption of perfect certainty as the framework of analysis. In such a framework, commodity inventories, in equilibrium, are completely dominated in return by riskless financial assets and, consequently, play no role in the optimal portfolio.

In this paper, it is assumed that future prices are unknown. The consumer decides upon optimal consumption and investment strategies by forming probabilistic expectations about future prices. In an uncertainty framework, the return of investment in commodity inventories is risky and, therefore, they compete against other risky (instead of riskless) assets for a share of the optimal portfolio. (1)

In this model, as will be shown below, optimal portfolio shares are determined by comparing expected marginal utilities among various assets and, thus, positive commodity inventories are possible if they provide relatively high marginal utility.

The argument of the present paper is that future price uncertainty is a sufficient, but not necessary, condition for positive commodity inventories. Storage and transportation costs as well as positive probabilities of rationing. [Zabel, 1977], are other factors that may motivate commodity inventories.

The analysis specifies that one of the commodities consumed is a durable good. From both a theoretical and empirical standpoint, the significance of durable goods as consumption and portfolio items, should be apparent. A durable good provides a stream of services during its lifetime. Consumers derive utility by consuming these services rather than the good per se. When consumers can lease the durable good, purchase demonstrates the desire to consume current services and inventory future services. Hence, durable goods purchase is a joint consumption-investment decision. Because of price uncertainty, the investment part of this decision is risky.

A by-product of the present analysis is the derivation of demand and excess demand functions for the various assets involved. The derived demand and excess demand for new and used durable goods are radically different from those in previous papers. The significance of those functions is further enhanced by the fact that they are a prerequisite for the analysis of the durable goods market under uncertainty. They, also, may provide the basis for speculation as to the determinants of the behavior of consumption expenditures on durable goods during the business cycle.

The next section develops a model in the tradition of those introduced by Hakansson [1970] and Zabel [1977]. Specific results are obtained by using a constant relative risk aversion utility function. For the sake of completeness, four categories of assets have been included in the model: a durable and a non-durable good are the utility yielding real assets, and a riskless bond together with a risky paper claim are the financial assets. Most real world assets can be classified under one of these categories.

II. The Model

The consumer derives utility from a non-durable good N and the services of a durable good A. This implies no demand for durability per se and, given their homogeneity, services of new and used durable goods are perfect substitutes in consumption. For simplicity, assume that the durable good lasts only two periods and has no salvage value. The units of the good are defined such that one unit of the good provides one unit of services each period of its life.

The new durable good D is traded in the primary market, while the used goods U are traded in the Second Hand Market (SHM). The consumer takes the prices prevailing in both markets as given and can buy or sell any desired quantity in these markets. Buying a used good in the SHM is equivalent to renting the good for one period. This justifies the name rental rate for the price of both the used good and the durable good services.

The consumer also has the choice of investing in a risky financial asset F and a (riskless) bond B. Borrowing is allowed and treated as negative investment in these assets. At the beginning of the horizon the consumer starts with a given wealth level w that includes the present value of any non-investment income obtained during the horizon. Initial wealth w equals the value of the consumer's portfolio, which includes inventories of the nondurable good [v.sub.N], the durable good [v.sub.U], the risky asset [v.sub.F] and the bond [v.sub.B].

At the beginning of each period the consumer decides upon optimal order levels [z.sub.N], [z.sub.D], [z.sub.U], [z.sub.F], [z.sub.B], and optimal consumption levels [c.sub.N] and [c.sub.A]. He then proceeds to submit the order which, by assumption, are satisfied immediately. Afterwards, consumption takes place and the non consumed amounts are the inventories at the beginning of the next period. The following equations describe the inventory motion for periods t and t-1: (2)

[Mathematical Expression Omitted]

Equation (2) indicates that the durable goods inventory consists of only the new goods purchased in the previous period, since the used units have expired. It is assumed that the durable good depreciates with time, whether in use or not. (3) Thus for the rational consumer:

[Mathematical Expression Omitted]

i.e., all the available services are consumed or, given that [z.sub.U] can be negative, all excess used units are sold.

Next let [b.sub.jt] be the return per dollar invested in asset j, i.e.,

[Mathematical Expression Omitted]

The bond return is given by [b.sub.Bt] = r = 1+R, where R is the interest rate. Since [p.sub.Ut] is the current rental rate, the difference [p.sub.Dt]-[p.sub.Ut] describes the unit cost of investment in next period's durable good services. The return of the investment in durable goods is:

[Mathematical Expression Omitted]

where ([p.sub.Dt-1]-[p.sub.Ut-1]) is the unit cost of the same investment one period later.

Equation (5) indicates the direct relationship between the current rental rate and the return on the investment in durable goods. Because new durable goods are more expensive than used, the non-negative sign follows. Note that future price uncertainty implies that [b.sub.Nt], [b.sub.Ft], and [b.sub.Ut] are random variables, uncertain at time t.

The consumer's problem is to decide on the expected utility maximizing consumption and portfolio strategies. The consumer's utility function is strictly concave with the following properties:

[Mathematical Expression Omitted]

where c=[[c.sub.N],[c.sub.D],[c.sub.A],[c.sub.F],[c.sub.B]] is the consumption vector, with [c.sub.D]=[c.sub.F]=[c.sub.B]=0.

Consider next the consumer's portfolio adjustments. Let [p'.sub.t] = [[p.sub.Nt],[p.sub.Dt],[p.sub.Ut],[p.sub.Ft],[p.sub.Bt]] be period t's price vector, v = [[v.sub.N],[v.sub.D],[v.sub.U],[v.sub.F],[v.sub.B]] be the inventory vector, with [v.sub.D]=0, and z = [[z.sub.N],[z.sub.D],[z.sub.U],[z.sub.F],[z.sub.B]] be the order vector. At the beginning of period t the consumer's wealth (portfolio) consists of the value of his inventories: w = pv. After allocation to orders: w=p(v+z). The consumer trades out of his inventory and thus some of the z's are positive and some negative. The savings S of the consumer, after consumption takes place, is given by:

S = w-pc = p(v+z-c). (7)

In other words, savings consist of the end of the period inventories evaluated at current prices. When those inventories are evaluated at next period prices, they yield the consumer's wealth at the beginning of the next period, t-1:[w.sub.t-1]=[p.sub.t-1]([v.sub.t]+[z.sub.t]-[c.sub.t]). Use of the expressions for expected returns, (4) and (5), allows the rewriting of next period's wealth as:

[Mathematical Expression Omitted]

Because per dollar returns b are unknown, next period's wealth is uncertain.

The solution of the consumer's investment problem will yield optimal portfolio shares. It is therefore appropriate to express [w.sub.t-1] in terms of portfolio shares. Let [[pi].sub.jt] represent the portfolio share of asset j at the end of period t:

[Mathematical Expression Omitted]

and

[Mathematical Expression Omitted]

Because by definition portfolio shares sum up to one, [[pi].sub.Bt] can be eliminated and next period's wealth written:

[Mathematical Expression Omitted]

While the shares of the goods' inventories are constrained to be non-negative, no such constraint is imposed on the shares of assets F and B, allowing thus for negative shares or borrowing. In order to avoid indeterminacies, Hakansson's [1970] non-easy-money and solvency conditions are imposed. Defining [[theta].sub.j] as any real number:

[Mathematical Expression Omitted]

Condition (10) requires that the risky assets do not totally dominate the bond, while (11) requires that next period's wealth be non-negative.

Dropping the time subscripts and defining f(w;p,v) as the present value of the maximum expected utility over an infinite horizon, the consumer's problem can be written as:

[Mathematical Expression Omitted]

subject to:

[Mathematical Expression Omitted]

and the relation between [[pi].sub.j] and z given in (8). p is a matrix with prices in its diagonal and zeros elsewhere.

Problem (12) can be solved, in principle, to derive optimal consumption levels and portfolio shares as functions of current period wealth, prices and inventories. Regrettably, at this level of generality, a useful closed form solution is impossible. In order to derive meaningful results some structure needs to be imposed on the utility function.

In the existing literature it has been shown that homothetic utility functions, i.e., functions displaying a constant degree of risk aversion [Stiglitz, 1969], yield optimal portfolio shares that are independent of wealth, prices and inventories, and, thus, remain constant throughout the horizon. The following Cobb-Douglas specification is chosen as a representative of the family of homothetic utility functions:

[Mathematical Expression Omitted]

This function satisfies the concavity requirements in (6) and displays a constant degree of relative risk aversion equal to (1-[lambda]).

The solution of the problem in (12) is derived by starting from a one-period horizon problem, proceeding to a two-period problem and then using inductive and limiting arguments to arrive at the infinite horizon.

Consider first the problem with one period remaining in the horizon. The consumer faces the problem: [f.sub.1]([w.sub.1]; [p.sub.1], [v.sub.1]) = [max.sub.c]{u(c)}, subject to: [p.sub.N1][c.sub.N1] + [p.sub.U1][c.sub.U1]=[gamma]w and [f.sub.1]([w.sub.1]; [p.sub.1])=A([p.sub.1])[w.sup.[lambda]] where A([p.sub.t])={[[(1-[gamma]).sup.1-[gamma]] [[gamma].sup.[gamma]]/[[p.sub.Nt.sup.1-[gamma]][p.sub.Ut.sup.[gamma]]}.

Consider next the problem with two periods remaining in the horizon. Use subscripts 2 for the first period and 1 for the last period of this horizon. From the definition of the per dollar returns b it can easily be seen that A([p.sub.1]) = B([b.sub.2])A([p.sub.t]), where B([b.sub.2]) = 1/[[[b.sub.N2.sup.1-[gamma]][b.sub.U2.sup.[gamma]].sup.[lambda]]. The problem the consumer faces at the beginning of the horizon is:

[Mathematical expression omitted]

Using the above derived expression for [f.sub.1], A([p.sub.1]), and (9) to substitute for [w.sub.1], the problem becomes:

[Mathematical expression omitted]

subject to:

[Mathematical expression omitted]

j=N,U,F.

An optimization problem such as (15) can be solved in two stages. First the optimal portfolio shares are determined by solving the portfolio problem, and then the optimal consumption levels are determined.

The consumer's portfolio problem lies in choosing the portfolio shares that maximize the per dollar return of the portfolio, i.e.,

[Mathematical expression omitted]

Hakansson's Lemma [Hakansson, 1970, p. 593] implies that maximum per dollar return k is unique and finite. Furthermore, inspection of (17) reveals that k is independent of wealth and prices. It follows that the optimal portfolio composition is constant across the horizon. The following first order (Kuhn-Tucker) conditions determine the optimal portfolio shares,

[Mathematical expression omitted]

Conditions under which optimal shares are positive, zero or negative are derived below.

Examine first (18c) which determines the optimal share of asset F. It is easily seen that the sign of [[pi].sub.F] (*1) depends directly upon the sign of the expression: [Mathematical Expression Omitted]

The solvency condition implies that the bracketed term is positive. Thus the sign of the expression depends upon the expected sign of the term ([b.sub.F]-r), i.e., the return differential between the risky and the riskless asset. If the expectation of this differential is negative (positive), the consumer borrows (lends) in the risky financial assets market.

Examine next conditions (18a) and (18b) which determine the shares of the commodity inventories. It can be seen that when:

[Mathematical Expression Omitted]

then [[pi].sub.j]*=0, j=N,U. The intuition is that when the expected return differential ([b.sub.j]-r) is non-positive, the consumer is better off by investing in the riskless asset instead of the commodity inventories. The opposite also holds.

Substitution of the optimal portfolio solution in the consumer's problem results in a standard utility maximization problem over an infinite horizon. Use of inductive and limiting arguments and insights developed by Zabel [1991] and Hakansson [1970], results in the following solution:

Proposition (1). Under the assumptions and conditions that justify (12), given that the utility function has the form described in (14) and for [alpha]k<1, (4)

[Mathematical Expression Omitted]

with: Q=[1-[([alpha]k).sup.[1/(1-[lambda])] and [[phi].sub.B]*= 1-[[phi].sub.N]*-[[phi].sub.U]*-[[phi].sub.F]*.

Proof. The proposition is proved by showing that (19a) satisfies (12). Substitution of (19a) into (12) and use of (9) yields:

[Mathematical Expression Omitted]

Passing the expectation operator through and using [Mathematical Expression Omitted]. After the portfolio maximization:

[Mathematical Expression Omitted]

Using (19b) and (19c), it is easily seen that u(c*)=[Q.sup.[lambda]]A(p)[w.sup.[lambda]] and S*=(1-Q)w and thus:

[Mathematical Expression Omitted]

Collecting terms: f(w;p)=[Q.sup.[lambda]-1]A(p)[w.sup.[lambda]]. That is, (19a) satisfies (12). As Zabel [1977] points out, the boundedness of the utility function and the requirement that [alpha]k<1, imply that (19a) is also the only function that satisfies (12).

The derivation of the demand and excess demand functions in (19d)-(19h) follows from S*, (1), (3) and (8).

The behavior of the consumer's inventories can be deduced from (19). First, define critical inventory level [v.sub.i]* as the level at which the optimal order [z.sub.i]* is zero. Consequently:

[Mathematical Expression Omitted]

It follows that when the actual inventory is above (below) its corresponding critical level, the optimal consumer strategy is to sell (buy) an amount equal to their difference. Hence, the sign of the optimal order z depends upon the relative size of the actual and the critical level of inventory.

Expression (19e) is the demand for new durable goods. It indicates that a consumer will place a positive order for new durable goods only if the optimal portofolio share is positive, i.e., consumers purchase new durable goods taking only portfolio considerations into account. Consumption of durable goods services is satisfied, primarily, through used durable goods purchases.

Simple comparative statics show that the demand (order) levels [Z.sub.i] are inversely related to their corresponding prices [p.sub.i]. Furthermore, (19e) shows that the rental rate [p.sub.u] affects the demand for new durable goods directly. This result is opposite of that derived in certainty models. The positive effect of the rental rate on [z.sub.D] stems from the fact that a higher rental rate reduces the cost of investment in new durable goods and thus raises their expected return [b.sub.U].

Finally, (19f) shows that the new durable goods price [p.sub.D] affects the demand for used durable goods directly. Generally, the above effect indicate that when the price of the new durable goods goes up, the consumer will shift to the used goods market, while if the rental rate goes up the consumer will move to the new goods market.

III. Concluding Remarks

The above analysis has expanded the results of Hakansson [1970] and Zabel [1977] by showing that future price uncertainty may motivate consumers to keep commodity inventories even in the absence of storage and transportation costs of rationing.

The results of the model are summarized in the form of demand and excess demand functions. According to these functions, whether the consumer will purchase or sell the various assets depends upon the level of his wealth, the level of current prices, the expected returns and the difference between actual and critical inventory levels. While the results were derived using a specific utility function, any homothetic utility function should yield similar results.

Inclusion of durable goods among the consumption-investment options of the consumer results in two interesting points. First, consumers buy new durable goods only if portfolio considerations allow it. Second, unlike the predictions of the corresponding certainty models, the rental rate affects the demand for new durable goods directly.

In certainty models, such as Parks' [1974], the price of new durable goods equals the discounted sum of rental rates during the goods's lifetime. Therefore the relation between the current rental rate and price is positive. Given the negative relationship between price and new goods' quantity, rental rates affect the demand for new durable goods inversely.

This paper's models lends itself readily to a number of applications. It can be used to examine the probable effects of various types of taxation or taxation changes upon the demand for the various assets. These effects can be studied by analyzing the impact of taxation upon the expected return differentials and prices.

Consider, for instance, the probable effects of abolishing the deductability of interest on consumer loans. This action will reduce the expected after-tax return of durable goods' investment, because of the higher cost of financing. In such a case, it is reasonable to expect that, ceteris paribus, the durable goods' portfolio share will diminish and so will the demand for new durable goods.

Another question which the present analysis provides insight to, is the high volatility of the consumption expenditures on durable goods, as opposed to non-durables and services, during the business cycle. Along with other explanations such as deferred purchase and extended life through more maintenance, the present analysis suggests that portfolio movements can also be a cause.

In particular, changes in expectations simultaneously affect expected returns and portfolio shares. When consumers become more pessimistic, as during recessions, the share of the risky assets will shrink and demand for new durable goods will diminish with consumers shifting to used goods (expenditures on which are not included in the GNP). As a consequence, expenditures on new durable goods will fall by more than expenditures on non-durables or services. An analogous argument holds for expansions.

Numerous extensions of the analysis are possible. The analysis can be extended easily to many goods and assets by substituting the appropriate vectors for the various scalars. Durability can be extended to more than two periods, at the expense of having as many additional prices and inventories as the number of additional periods the durable good lives. This extension will complicate the analysis without altering the results significantly.

An interesting question concerns the nonstochasticity of the consumer's level of wealth or income stream during the horizon. This is an assumption common to all existing consumer models, but it seems to contradict the assertion that the future is unknown. Removal of this assumption may produce some interesting results, but this remains a subject for future research.

(*) Radford University. The author would like to thank Ed Zabel, Jay Marchand, and an anonymous referee for comments and assistance. Any remaining errors are of course the author's responsibility.

(1) It is simplicitly assumed that there are no future markets in which the consumer can costlessly hedge and thus remove the effects of uncertainty. Alternatively, it is assumed that the cost of using future markets is prohibitive for the representative consumer.

(2) Throughout the paper time runs from late to early, i.e., period t is followed by period t-1.

(3) Note that in this paper the non-durable good is infinitely storable (e.g., canned or frozen goods), while the durable good cannot be stored without loss of value (a one year old car is less valuable than a totally new car even if it has not been driven).

(4) This assumption ensures convergence to the optimal solution. It requires that the present value of the maximum portfolio return per dollar invested is less than 1, i.e., the consumer may not double (or more) his money in one period. In the opposite case, one dollar of investment yields an ever increasing stream of income and maximum expected utility is infinite.

REFERENCES

Nils H. Hakansson, "Optimal Investment and Consumption Strategies Under Risk for a Class of Utility Functions," Econometrica, 38, September 1970, pp. 587-607.

Richard W. Parks, "The Demand and Supply of Durable Goods and Durability," American Economic Review, 64, March 1974, pp. 37-55.

Joseph E. Stiglitz, "Behavior Towards Risk with Many Commodities," Econometrica, 37, Octoboer 1969, pp. 660-67.

Edward Zabel, "Consumer Behavior Under Risk in Disequilibrium Trading," International Economic Review, 18, June 1977, pp. 323-43.

Edward Zabel, "Competitive Price Adjustment without Market Clearing," Econometrica, 49, September 1981, pp. 1201-221.

Very few of the existing models of optimal portfolio behavior are able to accommodate the fact that commodity inventories comprise a significant share of the average household's portfolio. Moreover, no effort has been made to explicitly consider the complications that arise from the durable good nature of most of these commodities.

The purpose of this paper is to develop a model capable of accommodating commodity inventories as part of the utility maximizing portfolio. Furthermore, the analysis will explicitly consider durable goods as consumption and portfolio items.

The failure of most of the existing models to account for commodity inventories in the optimal portfolio is due to the adoption of perfect certainty as the framework of analysis. In such a framework, commodity inventories, in equilibrium, are completely dominated in return by riskless financial assets and, consequently, play no role in the optimal portfolio.

In this paper, it is assumed that future prices are unknown. The consumer decides upon optimal consumption and investment strategies by forming probabilistic expectations about future prices. In an uncertainty framework, the return of investment in commodity inventories is risky and, therefore, they compete against other risky (instead of riskless) assets for a share of the optimal portfolio. (1)

In this model, as will be shown below, optimal portfolio shares are determined by comparing expected marginal utilities among various assets and, thus, positive commodity inventories are possible if they provide relatively high marginal utility.

The argument of the present paper is that future price uncertainty is a sufficient, but not necessary, condition for positive commodity inventories. Storage and transportation costs as well as positive probabilities of rationing. [Zabel, 1977], are other factors that may motivate commodity inventories.

The analysis specifies that one of the commodities consumed is a durable good. From both a theoretical and empirical standpoint, the significance of durable goods as consumption and portfolio items, should be apparent. A durable good provides a stream of services during its lifetime. Consumers derive utility by consuming these services rather than the good per se. When consumers can lease the durable good, purchase demonstrates the desire to consume current services and inventory future services. Hence, durable goods purchase is a joint consumption-investment decision. Because of price uncertainty, the investment part of this decision is risky.

A by-product of the present analysis is the derivation of demand and excess demand functions for the various assets involved. The derived demand and excess demand for new and used durable goods are radically different from those in previous papers. The significance of those functions is further enhanced by the fact that they are a prerequisite for the analysis of the durable goods market under uncertainty. They, also, may provide the basis for speculation as to the determinants of the behavior of consumption expenditures on durable goods during the business cycle.

The next section develops a model in the tradition of those introduced by Hakansson [1970] and Zabel [1977]. Specific results are obtained by using a constant relative risk aversion utility function. For the sake of completeness, four categories of assets have been included in the model: a durable and a non-durable good are the utility yielding real assets, and a riskless bond together with a risky paper claim are the financial assets. Most real world assets can be classified under one of these categories.

II. The Model

The consumer derives utility from a non-durable good N and the services of a durable good A. This implies no demand for durability per se and, given their homogeneity, services of new and used durable goods are perfect substitutes in consumption. For simplicity, assume that the durable good lasts only two periods and has no salvage value. The units of the good are defined such that one unit of the good provides one unit of services each period of its life.

The new durable good D is traded in the primary market, while the used goods U are traded in the Second Hand Market (SHM). The consumer takes the prices prevailing in both markets as given and can buy or sell any desired quantity in these markets. Buying a used good in the SHM is equivalent to renting the good for one period. This justifies the name rental rate for the price of both the used good and the durable good services.

The consumer also has the choice of investing in a risky financial asset F and a (riskless) bond B. Borrowing is allowed and treated as negative investment in these assets. At the beginning of the horizon the consumer starts with a given wealth level w that includes the present value of any non-investment income obtained during the horizon. Initial wealth w equals the value of the consumer's portfolio, which includes inventories of the nondurable good [v.sub.N], the durable good [v.sub.U], the risky asset [v.sub.F] and the bond [v.sub.B].

At the beginning of each period the consumer decides upon optimal order levels [z.sub.N], [z.sub.D], [z.sub.U], [z.sub.F], [z.sub.B], and optimal consumption levels [c.sub.N] and [c.sub.A]. He then proceeds to submit the order which, by assumption, are satisfied immediately. Afterwards, consumption takes place and the non consumed amounts are the inventories at the beginning of the next period. The following equations describe the inventory motion for periods t and t-1: (2)

[Mathematical Expression Omitted]

Equation (2) indicates that the durable goods inventory consists of only the new goods purchased in the previous period, since the used units have expired. It is assumed that the durable good depreciates with time, whether in use or not. (3) Thus for the rational consumer:

[Mathematical Expression Omitted]

i.e., all the available services are consumed or, given that [z.sub.U] can be negative, all excess used units are sold.

Next let [b.sub.jt] be the return per dollar invested in asset j, i.e.,

[Mathematical Expression Omitted]

The bond return is given by [b.sub.Bt] = r = 1+R, where R is the interest rate. Since [p.sub.Ut] is the current rental rate, the difference [p.sub.Dt]-[p.sub.Ut] describes the unit cost of investment in next period's durable good services. The return of the investment in durable goods is:

[Mathematical Expression Omitted]

where ([p.sub.Dt-1]-[p.sub.Ut-1]) is the unit cost of the same investment one period later.

Equation (5) indicates the direct relationship between the current rental rate and the return on the investment in durable goods. Because new durable goods are more expensive than used, the non-negative sign follows. Note that future price uncertainty implies that [b.sub.Nt], [b.sub.Ft], and [b.sub.Ut] are random variables, uncertain at time t.

The consumer's problem is to decide on the expected utility maximizing consumption and portfolio strategies. The consumer's utility function is strictly concave with the following properties:

[Mathematical Expression Omitted]

where c=[[c.sub.N],[c.sub.D],[c.sub.A],[c.sub.F],[c.sub.B]] is the consumption vector, with [c.sub.D]=[c.sub.F]=[c.sub.B]=0.

Consider next the consumer's portfolio adjustments. Let [p'.sub.t] = [[p.sub.Nt],[p.sub.Dt],[p.sub.Ut],[p.sub.Ft],[p.sub.Bt]] be period t's price vector, v = [[v.sub.N],[v.sub.D],[v.sub.U],[v.sub.F],[v.sub.B]] be the inventory vector, with [v.sub.D]=0, and z = [[z.sub.N],[z.sub.D],[z.sub.U],[z.sub.F],[z.sub.B]] be the order vector. At the beginning of period t the consumer's wealth (portfolio) consists of the value of his inventories: w = pv. After allocation to orders: w=p(v+z). The consumer trades out of his inventory and thus some of the z's are positive and some negative. The savings S of the consumer, after consumption takes place, is given by:

S = w-pc = p(v+z-c). (7)

In other words, savings consist of the end of the period inventories evaluated at current prices. When those inventories are evaluated at next period prices, they yield the consumer's wealth at the beginning of the next period, t-1:[w.sub.t-1]=[p.sub.t-1]([v.sub.t]+[z.sub.t]-[c.sub.t]). Use of the expressions for expected returns, (4) and (5), allows the rewriting of next period's wealth as:

[Mathematical Expression Omitted]

Because per dollar returns b are unknown, next period's wealth is uncertain.

The solution of the consumer's investment problem will yield optimal portfolio shares. It is therefore appropriate to express [w.sub.t-1] in terms of portfolio shares. Let [[pi].sub.jt] represent the portfolio share of asset j at the end of period t:

[Mathematical Expression Omitted]

and

[Mathematical Expression Omitted]

Because by definition portfolio shares sum up to one, [[pi].sub.Bt] can be eliminated and next period's wealth written:

[Mathematical Expression Omitted]

While the shares of the goods' inventories are constrained to be non-negative, no such constraint is imposed on the shares of assets F and B, allowing thus for negative shares or borrowing. In order to avoid indeterminacies, Hakansson's [1970] non-easy-money and solvency conditions are imposed. Defining [[theta].sub.j] as any real number:

[Mathematical Expression Omitted]

Condition (10) requires that the risky assets do not totally dominate the bond, while (11) requires that next period's wealth be non-negative.

Dropping the time subscripts and defining f(w;p,v) as the present value of the maximum expected utility over an infinite horizon, the consumer's problem can be written as:

[Mathematical Expression Omitted]

subject to:

[Mathematical Expression Omitted]

and the relation between [[pi].sub.j] and z given in (8). p is a matrix with prices in its diagonal and zeros elsewhere.

Problem (12) can be solved, in principle, to derive optimal consumption levels and portfolio shares as functions of current period wealth, prices and inventories. Regrettably, at this level of generality, a useful closed form solution is impossible. In order to derive meaningful results some structure needs to be imposed on the utility function.

In the existing literature it has been shown that homothetic utility functions, i.e., functions displaying a constant degree of risk aversion [Stiglitz, 1969], yield optimal portfolio shares that are independent of wealth, prices and inventories, and, thus, remain constant throughout the horizon. The following Cobb-Douglas specification is chosen as a representative of the family of homothetic utility functions:

[Mathematical Expression Omitted]

This function satisfies the concavity requirements in (6) and displays a constant degree of relative risk aversion equal to (1-[lambda]).

The solution of the problem in (12) is derived by starting from a one-period horizon problem, proceeding to a two-period problem and then using inductive and limiting arguments to arrive at the infinite horizon.

Consider first the problem with one period remaining in the horizon. The consumer faces the problem: [f.sub.1]([w.sub.1]; [p.sub.1], [v.sub.1]) = [max.sub.c]{u(c)}, subject to: [p.sub.N1][c.sub.N1] + [p.sub.U1][c.sub.U1]=[gamma]w and [f.sub.1]([w.sub.1]; [p.sub.1])=A([p.sub.1])[w.sup.[lambda]] where A([p.sub.t])={[[(1-[gamma]).sup.1-[gamma]] [[gamma].sup.[gamma]]/[[p.sub.Nt.sup.1-[gamma]][p.sub.Ut.sup.[gamma]]}.

Consider next the problem with two periods remaining in the horizon. Use subscripts 2 for the first period and 1 for the last period of this horizon. From the definition of the per dollar returns b it can easily be seen that A([p.sub.1]) = B([b.sub.2])A([p.sub.t]), where B([b.sub.2]) = 1/[[[b.sub.N2.sup.1-[gamma]][b.sub.U2.sup.[gamma]].sup.[lambda]]. The problem the consumer faces at the beginning of the horizon is:

[Mathematical expression omitted]

Using the above derived expression for [f.sub.1], A([p.sub.1]), and (9) to substitute for [w.sub.1], the problem becomes:

[Mathematical expression omitted]

subject to:

[Mathematical expression omitted]

j=N,U,F.

An optimization problem such as (15) can be solved in two stages. First the optimal portfolio shares are determined by solving the portfolio problem, and then the optimal consumption levels are determined.

The consumer's portfolio problem lies in choosing the portfolio shares that maximize the per dollar return of the portfolio, i.e.,

[Mathematical expression omitted]

Hakansson's Lemma [Hakansson, 1970, p. 593] implies that maximum per dollar return k is unique and finite. Furthermore, inspection of (17) reveals that k is independent of wealth and prices. It follows that the optimal portfolio composition is constant across the horizon. The following first order (Kuhn-Tucker) conditions determine the optimal portfolio shares,

[Mathematical expression omitted]

Conditions under which optimal shares are positive, zero or negative are derived below.

Examine first (18c) which determines the optimal share of asset F. It is easily seen that the sign of [[pi].sub.F] (*1) depends directly upon the sign of the expression: [Mathematical Expression Omitted]

The solvency condition implies that the bracketed term is positive. Thus the sign of the expression depends upon the expected sign of the term ([b.sub.F]-r), i.e., the return differential between the risky and the riskless asset. If the expectation of this differential is negative (positive), the consumer borrows (lends) in the risky financial assets market.

Examine next conditions (18a) and (18b) which determine the shares of the commodity inventories. It can be seen that when:

[Mathematical Expression Omitted]

then [[pi].sub.j]*=0, j=N,U. The intuition is that when the expected return differential ([b.sub.j]-r) is non-positive, the consumer is better off by investing in the riskless asset instead of the commodity inventories. The opposite also holds.

Substitution of the optimal portfolio solution in the consumer's problem results in a standard utility maximization problem over an infinite horizon. Use of inductive and limiting arguments and insights developed by Zabel [1991] and Hakansson [1970], results in the following solution:

Proposition (1). Under the assumptions and conditions that justify (12), given that the utility function has the form described in (14) and for [alpha]k<1, (4)

[Mathematical Expression Omitted]

with: Q=[1-[([alpha]k).sup.[1/(1-[lambda])] and [[phi].sub.B]*= 1-[[phi].sub.N]*-[[phi].sub.U]*-[[phi].sub.F]*.

Proof. The proposition is proved by showing that (19a) satisfies (12). Substitution of (19a) into (12) and use of (9) yields:

[Mathematical Expression Omitted]

Passing the expectation operator through and using [Mathematical Expression Omitted]. After the portfolio maximization:

[Mathematical Expression Omitted]

Using (19b) and (19c), it is easily seen that u(c*)=[Q.sup.[lambda]]A(p)[w.sup.[lambda]] and S*=(1-Q)w and thus:

[Mathematical Expression Omitted]

Collecting terms: f(w;p)=[Q.sup.[lambda]-1]A(p)[w.sup.[lambda]]. That is, (19a) satisfies (12). As Zabel [1977] points out, the boundedness of the utility function and the requirement that [alpha]k<1, imply that (19a) is also the only function that satisfies (12).

The derivation of the demand and excess demand functions in (19d)-(19h) follows from S*, (1), (3) and (8).

The behavior of the consumer's inventories can be deduced from (19). First, define critical inventory level [v.sub.i]* as the level at which the optimal order [z.sub.i]* is zero. Consequently:

[Mathematical Expression Omitted]

It follows that when the actual inventory is above (below) its corresponding critical level, the optimal consumer strategy is to sell (buy) an amount equal to their difference. Hence, the sign of the optimal order z depends upon the relative size of the actual and the critical level of inventory.

Expression (19e) is the demand for new durable goods. It indicates that a consumer will place a positive order for new durable goods only if the optimal portofolio share is positive, i.e., consumers purchase new durable goods taking only portfolio considerations into account. Consumption of durable goods services is satisfied, primarily, through used durable goods purchases.

Simple comparative statics show that the demand (order) levels [Z.sub.i] are inversely related to their corresponding prices [p.sub.i]. Furthermore, (19e) shows that the rental rate [p.sub.u] affects the demand for new durable goods directly. This result is opposite of that derived in certainty models. The positive effect of the rental rate on [z.sub.D] stems from the fact that a higher rental rate reduces the cost of investment in new durable goods and thus raises their expected return [b.sub.U].

Finally, (19f) shows that the new durable goods price [p.sub.D] affects the demand for used durable goods directly. Generally, the above effect indicate that when the price of the new durable goods goes up, the consumer will shift to the used goods market, while if the rental rate goes up the consumer will move to the new goods market.

III. Concluding Remarks

The above analysis has expanded the results of Hakansson [1970] and Zabel [1977] by showing that future price uncertainty may motivate consumers to keep commodity inventories even in the absence of storage and transportation costs of rationing.

The results of the model are summarized in the form of demand and excess demand functions. According to these functions, whether the consumer will purchase or sell the various assets depends upon the level of his wealth, the level of current prices, the expected returns and the difference between actual and critical inventory levels. While the results were derived using a specific utility function, any homothetic utility function should yield similar results.

Inclusion of durable goods among the consumption-investment options of the consumer results in two interesting points. First, consumers buy new durable goods only if portfolio considerations allow it. Second, unlike the predictions of the corresponding certainty models, the rental rate affects the demand for new durable goods directly.

In certainty models, such as Parks' [1974], the price of new durable goods equals the discounted sum of rental rates during the goods's lifetime. Therefore the relation between the current rental rate and price is positive. Given the negative relationship between price and new goods' quantity, rental rates affect the demand for new durable goods inversely.

This paper's models lends itself readily to a number of applications. It can be used to examine the probable effects of various types of taxation or taxation changes upon the demand for the various assets. These effects can be studied by analyzing the impact of taxation upon the expected return differentials and prices.

Consider, for instance, the probable effects of abolishing the deductability of interest on consumer loans. This action will reduce the expected after-tax return of durable goods' investment, because of the higher cost of financing. In such a case, it is reasonable to expect that, ceteris paribus, the durable goods' portfolio share will diminish and so will the demand for new durable goods.

Another question which the present analysis provides insight to, is the high volatility of the consumption expenditures on durable goods, as opposed to non-durables and services, during the business cycle. Along with other explanations such as deferred purchase and extended life through more maintenance, the present analysis suggests that portfolio movements can also be a cause.

In particular, changes in expectations simultaneously affect expected returns and portfolio shares. When consumers become more pessimistic, as during recessions, the share of the risky assets will shrink and demand for new durable goods will diminish with consumers shifting to used goods (expenditures on which are not included in the GNP). As a consequence, expenditures on new durable goods will fall by more than expenditures on non-durables or services. An analogous argument holds for expansions.

Numerous extensions of the analysis are possible. The analysis can be extended easily to many goods and assets by substituting the appropriate vectors for the various scalars. Durability can be extended to more than two periods, at the expense of having as many additional prices and inventories as the number of additional periods the durable good lives. This extension will complicate the analysis without altering the results significantly.

An interesting question concerns the nonstochasticity of the consumer's level of wealth or income stream during the horizon. This is an assumption common to all existing consumer models, but it seems to contradict the assertion that the future is unknown. Removal of this assumption may produce some interesting results, but this remains a subject for future research.

(*) Radford University. The author would like to thank Ed Zabel, Jay Marchand, and an anonymous referee for comments and assistance. Any remaining errors are of course the author's responsibility.

(1) It is simplicitly assumed that there are no future markets in which the consumer can costlessly hedge and thus remove the effects of uncertainty. Alternatively, it is assumed that the cost of using future markets is prohibitive for the representative consumer.

(2) Throughout the paper time runs from late to early, i.e., period t is followed by period t-1.

(3) Note that in this paper the non-durable good is infinitely storable (e.g., canned or frozen goods), while the durable good cannot be stored without loss of value (a one year old car is less valuable than a totally new car even if it has not been driven).

(4) This assumption ensures convergence to the optimal solution. It requires that the present value of the maximum portfolio return per dollar invested is less than 1, i.e., the consumer may not double (or more) his money in one period. In the opposite case, one dollar of investment yields an ever increasing stream of income and maximum expected utility is infinite.

REFERENCES

Nils H. Hakansson, "Optimal Investment and Consumption Strategies Under Risk for a Class of Utility Functions," Econometrica, 38, September 1970, pp. 587-607.

Richard W. Parks, "The Demand and Supply of Durable Goods and Durability," American Economic Review, 64, March 1974, pp. 37-55.

Joseph E. Stiglitz, "Behavior Towards Risk with Many Commodities," Econometrica, 37, Octoboer 1969, pp. 660-67.

Edward Zabel, "Consumer Behavior Under Risk in Disequilibrium Trading," International Economic Review, 18, June 1977, pp. 323-43.

Edward Zabel, "Competitive Price Adjustment without Market Clearing," Econometrica, 49, September 1981, pp. 1201-221.

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Author: | Roufagalas, John |
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Publication: | Atlantic Economic Journal |

Date: | Mar 1, 1991 |

Words: | 3693 |

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