# A Stochastic Inventory Decision Model and the Holding of Family Wealth

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Ivan F. Beutler (1976) ,"A Stochastic Inventory Decision Model and the Holding of Family Wealth", in NA - Advances in Consumer Research Volume 03, eds. Beverlee B. Anderson, Cincinnati, OH : Association for Consumer Research, Pages: 155-160.

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http://acrwebsite.org/volumes/9191/volumes/v03/NA-03

The decision process of managing family wealth is modeled. Stochastic inventory theory provided the underlying framework of the model by which family assets are viewed as inventories of wealth. The impact of family saving behavioral patterns upon asset management is also shown. Explicit functions predicting equilibrium cost and short term family asset holdings are derived.

INTRODUCTION

Optimal family wealth holdings and consumption patterns over the life cycle have long been of interest to the social scientist. [Fisher (1930) provided micro foundations for much of modern macro theory of aggregate saving, wealth and consumption. The focus here is at the micro level for individual households.] The consumption/wealth problem is a complex multiperiod decision process. This paper addresses only part of the overall problem by focusing on the holding of family wealth. This focus provides a unique understanding of consumer behavior different from that obtainable through a direct study of consumption. The focus of this paper can be understood by considering the portfolio problem which each family or household must solve. [The solution may range from following an elaborate plan to that of simple default behavior.] Given that a family has a certain amount of wealth, that wealth must be held in some asset or combination of assets. The typical family holds a certain amount of their wealth in the form of currency and demand deposits. The remainder of their wealth is usually held in some combination of short and long term financial assets plus real assets, such as consumer durables and real estate. [Such factors as human wealth are obviously not considered in the management of wealth as used here.] The key question here is, what determines the combination, type and magnitude of assets held by the household? It is of interest to know what economic forces have a major bearing on the process, the quantitative magnitude of these forces and their role in shaping household portfolio behavior.

BACKGROUND LITERATURE

Keynes (1936) described three motives for holding cash; the transaction, precautionary and speculative motives. These motives have since served as the major underlying framework for most of the literature concerning the demand for money. Hicks (1935) recognized the potential importance of detail (such as transaction cost which he called "frictions") in an optimization-oriented approach to money demand.

Nearly two decades after Hicks stated that "frictions" were potentially important, Arrow, Harris and Marschak (1959), Whitin (1953) and others recognized "Inventory Theory" as a potential tool and began to apply it in asset management theory. [Inventory theory goes back to the mid 1920's when development apparently began independently by several groups. Arrow (1958) makes reference to these beginnings.] Whitin's work in inventory management theory served as a suggestion to Baumol that it may have important application to the transaction demand for cash. The resulting article by Baumol (1952) is now a bench mark of the first inventory theoretic approach to cash management. However, the Baumol model employed a fairly simplified certainty construct. More recently others have expanded the Baumol certainty model to include various dimensions of household asset management. [For example see Feige and Parkin (1971), Hamburger (1966), Sastry (1970), and Sprinkle (1969). Santomero (1974) sets forth a certainty model invoking the assumption of a uniform rate of change in the level of assets. From his model is then derived the intraperiod average balance of currency, demand deposits, saving deposits and goods purchased as a function of the rates of return and transaction costs.]

Miller and Orr (1966) introduced an inventory stochastic model for cash and asset management of the firm. The general flexibility and stochastic nature of the Miller-Orr model make it an important take-off from Baumol. Thus it was through an interest in the financial management of the firm at the micro level that the theory was further developed. Eppen and Fama (1969) have further expanded on the model within the context of the firm. However, to the writer's knowledge no serious attempt has been made to implement the structure into a theory of family asset management. [Orr (1971) Chapter 7, makes several innovative suggestions concerning household demand for money and further suggests that the basic model developed by himself and Miller might be modified for the household setting.]

THE STOCHASTIC INVENTORY THEORETIC DECISION MODEL

The family wealth or asset management decision process is viewed as an inventory problem. Each family has a certain amount of wealth which may be held in the form of cash, short term financial assets ("shorts"). long term financial assets ( longs ) or real assets. [Cash is defined here to include currency and demand deposits.] These four assets plus consumer credit sources are viewed as inventories subject to stochastic fluctuation due to cash inflow and outflow, consumption/saving decisions, and adjustments between the inventory assets.

Assumption One

A major assumption of the model is that each family has an asset manager who acts to minimize the expected cost of managing household assets by following certain policy rules of behavior.

Rule 1. (h-z) Policy Rule. The family asset manager is assumed to follow an (h-z) policy rule. This management policy is illustrated in Figure 1. Cash balances are allowed to fluctuate freely until either an upper bound h or a lower bound zero is reached. When cash accumulates to the upper bound h, the asset manager effects a portfolio transfer of (h-z) dollars into the short term asset account. When cash balances fall to the zero boundary, z dollars are transferred from the short term asset ("shorts") into cash, restoring the cash balance to level z. Likewise, when the upper bound H is reached in the shorts account, (H-Z) dollars are transferred into long term assets ("longs") or the purchase of real assets. [Consumer credit debt repayment is assumed to take place via installment payments. Thus credit repayments are made out of the cash account.] When the value of short term assets is reduced to the lower bound Z_{o}, short term assets are restored to a level Z either by the use of credit, or by the liquidation of long term financial assets or real assets. The source used depends on three factors: 1) the transaction cost of liquidating either "longs" or real assets versus using credit, 2) the cost of credit compared to the differential return on longs versus real assets, and 3) the cost of credit compared to the differential return on long and short term assets. [The transaction cost of liquidating "longs" will depend on the dollar amount of funds transferred. The transaction cost will be infinite if a positive level of long term assets does not exist. The cost of credit will depend on the risk class into which the particular family fits as viewed by creditors. It should be noted that the return on real assets such as consumer durables may be an implicit return as well as financial.] The short term asset lower bound Zo, will vary from family to family and depends on the household's present-future orientation.

AN ASSET MANAGEMENT POLICY RULE (h-z).

In short, the family asset manager is assumed to respond to cash balance x, short term asset X, and long term asset and real assets Y, situations as follows:

Rule 2. (c-s) Policy Rule. The (c-s) policy rule takes into account two types of family saving behavior patterns not accounted for as yet in the model. These saving patterns may affect cash balances substantially. Manning (1960) has reported three distinct family saving practices. The first practice is modeled above by the (h-z) policy rule. 1) Residual Saving -this is the traditional view of saving where money left over after personal consumption expenditures have been taken from disposable income is considered to be a residual balance. When these residual balances have accumulated to a certain level and are transferred into an interest bearing asset it is considered, by definition, to be residual saving. 2) Fixed Commitment Saving - fulfilling a commitment to save or set aside funds before consumption demands are met. 3) Special Receipt Saving - Setting aside special receipts or income (such as a tax refund).

Committed saving and the saving of special receipts require separate treatment in the model. These two practices give rise to the (c-s) management policy (committed and special receipt saving). All (c-s) managed funds which are used for the purchase of short term assets, by-pass the cash balance account and go directly for the purchase of "shorts". This is so by definition of committed and special receipt saving. Furthermore, for the purpose of this model, it is assumed that, when these (c-s) short term assets are liquidated, purchase synchronization is perfect. [Perfect synchronization assumes that the purchase of goods and services from (c-s) short term assets is coincidental with (c-s) liquidation. Thus the assumption is, that of all short term assets liquidated over time, a dollar amount equal to (c-s) saving goes directly for the purchase of goods and services.] Thus (c-s) saving effects both the mean and variance of short term asset holdings but, the cash balance stationary density function remains unaltered.

Assumption Two

A second assumption of the model is that the level of each asset inventory is stochastic in nature such that it can be treated as a random variable.

Cash Balance Asset, a Random Variable-x. To keep the model manageable, a simple Bernoulli transition rule is assumed to describe changes in the cash balance random variable x. Whenever a state exists, the Bernoulli assumption is that during the succeeding state of the process, cash balances will increase with probability p, decrease with probability q, and that p + q = 1. Thus the state-to-state transitions can be defined as

By invoking the steady state condition, the limit as t approaches infinity, the transition rule can be

characterized by the following difference equations.

The equation listed above is a homogeneous difference equation which can be readily recognized by normalization,

P_{x+1} - 1/qP_{x} + p/qP_{x-1} = 0

The auxiliary equation is of the form

m^{2} - (1+p/q) m + p = 0

Letting p/q = r, the roots of the auxiliary equation can be easily seen.

(m_{1} = 1)(m_{2} = r)

Hence the general solution form of the homogeneous equation can be solved to obtain the stationary density function describing the random variable x. [For derivational detail see Beutler (1974).] The resulting function (15) is a triangular, probability density function illustrated

in Figure 2. [By the assumption of nonpersistent drift (r = p/q = 1), the density function given in (1) is considerably simplified from the more general form. EQUATION] The nature of the random variable x (cash balances) is described by the density function, which has, as its parameters, the policy control limits z and h. This density function contains all the information about x which is necessary to derive the expected asset management cost function.

Short Term Assets, a Random Variable-(X). It is also of interest to know the stochastic nature of the random variable X. The level of short term asset holdings has its origin from two sources. The first source is the (h-z) policy rule. An accumulation and liquidation of "shorts" behaving in accord with this rule creates a component of the short term asset which behaves as a random variable, denoted u. The (c-s) policy rule creates a second component of short term asset holdings which also behaves as a random variable, denoted s.

Short term asset behavior is thus described as the sum of two random variables, so that X = s + u. Since s is exogenous to the model, s and u are taken to be stochastically independent. The stochastic nature of the random variables u and s, can be formulated as separate asymmetric Bernoulli processes. These may then be pooled to determine the stochastic nature of X. However, the error introduced into the mean balance estimator, by using a single-step symmetric Bernoulli estimate (H+Z)/3 to represent the complex pooled process, has been shown to be remarkably small and virtually independent of the values H and Z. [Feller (1966) and Orr (1974) have shown that small error results when complex pooled process mean estimates are replaced by simpler approximation estimates from a single process. Furthermore, Miller and Orr (1968) have shown (H+Z)/3 to be a reasonably good estimate of X regardless of the level at which h and z, the cash balance policy parameters, are set and regardless of the level of H and Z.] Thus, the short term asset mean value of the pooled process X = u + s, may be estimated by equation (2).

E(X) = (H+Z)/3 (2)

Since u and s are taken to be independent random variables, the variance of X, s_{X}^{2}, is the sum of the two variances s^{2}_{X}^{2}=s^{2}_{s} + s^{2}_{u}. First consider the variance of u, s_{u}^{2}.It is given by the following equation. [In words the variance of u reads s^{2} = E (u-u)^{2} = [expected increase in "shorts"]^{2} (probability of am increase) + [expected decrease in "shorts"]^{2} (probability of a decrease).]

s^{2}_{u} = s^{2}/z(h-z) [(h-z)(z/h)]^{2} z/h + [(z/h)(h-z)]^{2} (h-z)/h =s^{2} (3)

Thus the variance of the periodic movement between "shorts" and cash is s^{2}. That the variance of this short term asset component u, and the cash balance asset x, both equal s^{2} is no trivial matter. Since the short term asset variance does not depend directly upon z and h, the derivation of optimal policy controls is much less difficult. This will become clear when the probability of a portfolio adjustment between "shorts" and "longs" is considered in the following section.

A second contribution to deviation in short term asset holdings is the variance of the random variable s. The variance of this exogenous variable is denoted s^{2}_{c-s}. Thus, the total variance of the short term asset is

s^{2}_{X} = s^{2} + s^{2}_{c-s} (4)

Assumption Three

So far two aspects of the model have been established; 1) each family asset manager follows an (h-z) and (c-s) policy rule of behavior, and 2) the stochastic nature of the asset inventories is characterized as a symmetric Bernoulli process. Now the additional assumption is made that asset managers choose the optimal policy controls z, h; Z and H so as to minimize the expected cost of asset management. This requires that an expected cost function be formulated and minimized with respect to z, h, Z, and H as expressed in general form in the following equation.

Minimize E(c) = T_{c}P(T_{c}) + i_{l}E(x) + aG(x) + T_{s}P(T_{s}) + (i_{l} - i_{s}) E(X) (5)

where:

E(c) = The expected cost of asset management.

P(T_{c}) = The probability of a transfer between short term and cash balance assets.

E(x)= The expected level of cash balances.

G(x) = The cost function representing the price which households place on holding wealth in a liquid form. A form more subject to impulsive, short run purchases which may be suboptimal in the longer run.

P(T_{s}) = The probability of a transfer between short term assets and credit or long term assets.

E(X)= The expected level of short term assets.

T_{c} = A lump sum transfer cost associated with transfers between the cash balance and short term assets. This cost includes the value of management time and other indirect costs, as well as, direct outlays, such as gasoline for a trip to the bank, postage and brokerage fees.

i_{l} = The earning rate on long term assets which re- presents the interest opportunity cost of holding cash.

i_{s} = The earning rate on short term assets.

a = The rate or price which households attach to the holding of liquid wealth.

T_{s} = A lump sum transfer cost associated with transfers between short term assets and credit, long term assets, or real assets.

From the cost function T_{c}, i_{l}, a, T_{s}, and (i_{l} - i_{s}) are all fixed price parameters which do not vat, with respect to the level of the policy controls z, h, Z and P(T_{c}), E(x), G(x), P(T_{s}), and E(X) are each stochastic variables, the value of which depends on the level at which the policy controls are set.

The probability of a transfer from the cash account into the short term asset account is denoted P(T_{c-s}). Likewise, a transfer from "shorts" into the cash ac- count is denoted P(T_{s-c}). The probability of a T_{c-s} transfer during any given time period is equal to the probability of a state h-1 occurrence, times the probability of a transfer, given the state h-1. [The probability of a state h-1 occurrence during a given time period is equal to the number of transactions which occur in the cash account during that period, times Ph-1. The unit of change in the simple Bernoulli process is s^{2}. For detail on this point see Wold (1946). The probability of a transfer, given the occurrence of the state h-l, equals 1/2. Thus we have P(T_{c-s}) = s^{2} [2/h(h-z)] (1/2) = s^{2}/h(h-z).] Similar reasoning holds for the probability of a T_{s-c} transfer. The total probability of a transfer, irrespective of the direction, is equal to P(T_{c-s}) plus P(T_{s-c}) as shown in equation (6).

P(T_{c-s}) = s^{2}/h(h-z) (6a)

P(T_{s-c}) = s^{2}/zh (6b)

P(T_{c}) = P(T_{c-s}) + P(T_{s-c}) = s^{2}/z(h-z) (6c)

The average cash balance or mean value of x is given by equation (7).

The first two components of the expected cost function, equation (5), are seen to be readily stated in explicit form, equation (6) and (7), once the stationary distribution function of equation (1) is known.

Procedure for derivation of the third cost component G(x), is not as clearly defined as were the first two. There is no research, to the author's knowledge, which suggests the explicit form the G(x) cost function should take. Thus for lack of a better form, G(x) is assumed here to he a simple positive function of the expected level of cash balances,

G(x) = E(x) = (h+z)/3. (8)

[Families which do not have a residual saving motive of the G(x) type, will have a price vector equal to zero. In this event the aG(x) cost component will fall out of the expected cost function.]

The last two cost components of equation (5) are related to the holding of short term assets. Since the variance of movement in the shorts account is s^{2}_{X} = s^{2}_{ + }s^{2}_{c-s}, given by equation (4), the expected first passage time for the shorts balance from Z to either 0 or H is Z(H-Z)/(s^{2}_{ + }s^{2}_{c-s}). The steady-state probability of a first passages which signals a purchase or sale of longs, is the inverse of the first passage time (s^{2}_{ + }s^{2}_{c-s})/Z(H-Z). Thus the explicit function of expected costs, resulting from a portfolio transfer between "shorts" and "longs", real assets or credit, is

P(Ts) = s^{2}_{X}/Z(H-Z). (9)

[Equation (9) may appear ordinary, but a very significant point should be noted. The probability of a transfer between "longs" and "shorts" is seen to depend only on the level of Z, H and ^{2}_{X}. It does not depend directly on h or z. The minimization problem would he much more difficult if h and z came into the cost expression (9).]

The expected level of short term asset holdings is given by equation (2) as E(X) = (H+Z)/3. However, this expression was derived without taking into account the reserve level of short term asset Z_{o} held by families. Z_{o} will vary from household to household, but is assumed to be a constant for any given family. The fifth and final component of the expected cost function is thus

E(X) = (H+Z+Z_{o})/3. (10)

The expected cost minimization problem stated explicitly in terms of the policy controls h, z, H and Z follows directly from equation (5) through (10).

The optimization object (as stated by equation ( 5 )) is to set the policy parameters at an optimal level, denoted z*, h*, Z* and H*, such that the expected cost of asset management is minimized.

The necessary conditions for a minimum are given by equation (12).

EQUATIONS (12a), (12b), (12c) and (12d)

The above first order conditions are set equal to zero and solved simultaneously for the optimal level of each policy control variable. [For details of this solution see Beutler (1974). Let h=kz and H=KZ. The second order conditions for a minimum solution are met so long as k 2.7 and K 2.7. With these constants greater than 2.7, the Hessian Matrix formed from equation (13) is positive definite. The positive definite Hessian Matrix implies that the expected cost function is strictly convex and thus the first order conditions lead to a global minimum solution.]

In terms of the optimal policy control limits, the expected cash balance is equal to (h*+ z*)/3. Substituting into this expression the value of h* and z* from equation (13), gives the average cash balance x.

The average cash balance x, is a measure of average family cash holdings. Likewise, the average holding of short term financial assets is given by X. [These results are equivalent to those of Miller and Orr (1966) with the exception of Z_{o} in equation (16) and the variance of short term assets equal to s^{2}_{ + }s^{2}_{c-s}instead of simply s^{2}.]

Although the mechanics of the above optimization problem get involved, the essence of the problem is straight forward. By minimizing the expected cost function, the model-predicted, optimal policy controls are given by equations (13) and (14). The model predictions of average asset holdings, are also given by equations (15) and (16). Thus the __basic three assumptions__ of the model culminate in equations (13) through (16) to provide; 1) the optimal level at which policy controls would he set to minimize management cost, and 2) an explicit expression of family asset holdings in equilibrium given basic information about the economic and family environment.

MODEL IMPLICATIONS

The optimal policy control limits z* and h*, are seem. from equation (13), to vary directly with the cash balance variance (s^{2}). On a priori grounds, it seems correct that a family with larger (rather than smaller) cash inventory variation will hold larger (rather than smaller) cash balances. Thus, a family which has large variation in cash balance holdings over time, will optimally maintain a high z and h level. This result is intuitively appealing and provides a facet of the theoretic model which can be tested against the data. Note that s^{2} serves to generalize the model by providing a mechanism whereby a wide range of management behavior is explained through a single parameter. Family differences such as stages in the life cycle, income level, and expenditure patterns all have a likely impact on s^{2} which in turn, accounts for a wide range of management behavior. [For the purpose of expedition, a is assumed to equal zero here.] For example, households which synchronize cash inflow and outflow, by the use of credit and other payments, will reduce cash balance variations. In fact, s^{2} is a measure of inflow-outflow synchronization.

The mean cash balance x and the optimal policy control limits (z and h ), are seen to be an increasing function of the transaction cost (T) and a decreasing function of the opportunity interest rate (i) and the management price of holding cash (a). When a family accumulates short term assets with a residual saving motive coupled with the interest earning incentive, will be positive. A positive a lowers the optimal policy controls (z* and h*). This creates more frequent transfers between short term assets and the cash accounts, generating greater residual saving. If no residual saving motive exists, beyond the interest incentive, e will equal zero and average cash holdings will be independent of this motive. The direction of impact which these variables (T, i and a), have on cash balances is in accord with economic theory.

DISCUSSION

This paper represents a focus on consumer behavior which is not a direct act of consumption but rather a closely related process of asset management for the purpose of postponed consumption. The unique contribution of the stochastic inventory model set forth here, is that the family wealth and asset management decision process is characterized in a somewhat complex, but manageable, world including uncertainty. In contrast, the Baumol model and extensions thereof, assume a simple world of certainty where cash inflows take the form of income at a constant rate placed directly into an interest bearing asset. The result being that portfolio transfer costs and the interest opportunity cost of holding cash are the only variables perceived as relevant in explaining the decision process. On the other hand, the asset management of family wealth as modeled here, more completely exposes the actual environment and decision process by including a form of uncertainty, family saving behaviors, and family propensity to manage (cash/saving).

REFERENCES

Kenneth J. Arrow, "Historical Background" in Arrow, Karlin and Scarf (ed.), __Studies in the Mathematical] Theory of Inventory and Production__ (Stanford: Stanford University Press, 1958).

Kenneth J. Arrow, Theodore Harris, and Jacob Marschak, "Optimal Inventory Policy," __Econometrica__, 19, 1951, 250-272.

Robert J. Barrow and Anthony M. Santomero, "Household Money Holdings and the Demand Deposit Rate," __Journal of Money; Credit and Banking__, 4(2), 1972, 397-413.

William J. Baumol, "The Transactions Demand for Cash: An Inventory Theoretic Approach," __Quarterly Journal of Economics__, 66, 1958, 545-556.

Ivan F. Beutler, __Family Cash Management-A Stochastic Inventory Theoretic Model__, Ph.D. Dissertation, 1974.

Gary D. Eppen and Eugene F. Fama, "Cash Balance and Simple Dynamic Portfolio Problems with Proportional Costs," __International Economic Review__, 10, 1969, 119-133.

William Feller, An Introduction to Probability Theory and Its Applications (New York: John Wiley & Sons. Vol. I (discrete sample spaces) 1951, Vol. II (continuous sample spaces), 1966).

Irving Fisher, __The Theory of Interest__ (New York: Augustus M. Kelly, Publishers, 1965 - Reprinted from the 1930 edition).

J. M. Keynes, __The General Theory of Employment, Interest and Money__ (New York: Harcourt-Brace & Co., 1936).

Sarah L. Manning and Ethel L. Vatter, "Financial Management Practices of Families with Steady or Fluctuating Incomes," Cornell University Agricultural Experiment Station, __Bulletin 997__, New York State College of Home Economics, Ithaca, New York, 1964.

Merton H. Miller and Daniel Orr, "The Demand for Money by Firms: Extensions of Analytic Results," __Journal of Finance__, 23, 1968, 735-759.

Merton H. Miller and Daniel Orr, "A Model of the Demand for Money by Firms," __Quarterly Journal of Economics__, 80, 1966, 413-435.

Daniel Orr, __Cash Management and the Demand for Money__ (New York: Praeger, 1970).

Anthony M. Santomero, "A Model of the Demand for Money by Households," __Journal of Finance__, 42, 1974, 89-101.

Rama A. S. Sastry, "The Effect of Credit on Transactions Demand for Cash," __Journal of Finance__, 25, 1970, 777-781.

Case M. Sprenkle, "The Uselessness of Transactions Demand Models," __Journal of Finance__, 24, 1969, 835-847.

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