Optimum consumption and portfolio rules with money as an asset

Journal of Banking and Finance 7 (1983) 231-252. North-Holland Publishing Company

O P T I M U M CONSUMPTION AND PORTFOLIO RULES WITH MONEY AS AN ASSET Patrice PONCET* ESSEC, 95021 Cergy Pontoise, France Received May 1981, final version received January 1983 This article generalizes Merton's optimum consumption and portfolio rules in continuous time by introducing money as a capital asset and allowing for uncertain inflation. Assuming that prices are log-normally distributed, a three-funds theorem is derived and the introduction of money is shown not to change the form of the standard inflation-adjusted CAPM but to change the market price of risk. The individual's consumption-portfolio problem is completely solved under uncertain inflation if his utility function is iso-elastic in its arguments. Comparative statics are used to assess the influence of changes in exogenous parameters on the individual's optimal rules.

1. Introduction

Capital market theory and monetary theory have developed largely independently of one another. The standard Capital Asset Pricing Model (CAPM) is derived in a frictionless economy: to the extent that there exists an asset whose nominal return is positive and certain, money is dominated as a portfolio asset; since, by assumption, there are no transaction costs, the riskless asset is perfectly liquid, and the money asset is void of economic usefulness and then is not introduced into the analysis. On the other hand, the recent works in monetary theory, whether they pertain to the optimal quantity of money, the super-neutrality of money or the relationship between inflation and unemployment, for example, do not fully incorporate what we know about microeconomic behavior under uncertainty. One purpose of this paper is to provide a way to bridge partially the gap between the two by deriving a demand for money (along with optimum demands for consumption and for other assets) from the individual's decisions under uncertainty, while keeping the standard assumptions of perfect financial markets. On a priori grounds, two approaches are conceivable: the need for using money may be derived from the explicit introduction of transaction costs in *I would like to thank my colleague R. Portait for his numerous valuable comments and suggestions. I also benefited from helpful discussions with B. Dumas, E. Losq, M. Subrahmanyam and P. Sercu. I bear sole responsibility for any remaining errors. 0378-4266/83/$3.00 © Elsevier Science Publishers B.V. (North-Holland)

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the model, or else money may be included as an argument in the individual's utility function. We have adopted the latter approach, because of the formidable technical problems associated with the first one. Therefore, we assume that, although it bears no nominal interest, money is demanded because the household derives direct utility from the flow of services (which are related, in some unspecified way, to transaction costs not included explicitly in the analysis) it provides. In fact, all results and relationships of interest are derived in a world where the rate of inflation is uncertain, i.e., when the future purchasing power of money is random. This is so because the demand for future money balances from households, which depends on their future random wealth, and the future supply from the government, which supposedly depends on future economic activity, are uncertain. Chen and Boness (1975), and Friend, Landskroner and Losq (1976) built models of capital asset pricing under uncertain inflation but did not introduce money in their models. In addition, the first model dealt with decisions of firms, not individuals, while the second considered only the individual's portfolio choices, ignoring entirely his consumption decision. Another class of papers has included money in the analysis of financial markets. For instance, Levhari and Liviatan (1976) and Liviatan and Levhari (1977) examine the risk premium which should be paid on nominal bonds because of inflation risk. Their analysis however is not formulated in terms of the CAPM, does not involve risky stocks, assumes quadratic utility functions and focuses on the macroeconomic consequences of issuing government indexed bonds. Landskroner and Liviatan (1981) analyze the relationship between the risk premia paid on nominal assets as compared with a real riskfree asset and the sources of inflation, real or monetary. They derive CAPM-like equilibrium premia by using quadratic utility functions and imposing macroeconomic constraints relating to the sources of inflation but, in contrast to this study, they do not tackle the problem of the individual's optimal consumption and portfolio-choice rules over time. In a recent contribution, Fama and Farber (1979) investigated the respective role of money and nominal bonds in the individual's portfolio decisions and in the pricing of nominal bonds, given the rate of return on a riskless real bond and given the results of the standard CAPM. In contrast, this paper provides a generalization of the CAPM under uncertain inflation with money as an asset. In addition, it derives explicit solutions for optimal consumption and portfolio-choice rules for some cases and then analyses the way those rules are changed when the main exogeneous parameters are altered. Finally, Breeden (1979), extending a previous work by Grauer and Litzenberger (1979), derives an intertemporal capital asset pricing model when consumption-goods prices and investment opportunities are both

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stochastic, which is a setting more general than ours. However, he cannot deal explicitly with the specific problems of uncertain inflation and demand for money holdings and he cannot provide a closed-form solution for the optimal consumption path since he introduces no restrictions on the utility functions. The paper is organized in the following manner. The setting and the main assumptions of the model are presented in section 2. The individual maximizes his expected utility over an infinite horizon and trading in assets takes place continuously. From the optimization problem faced by the household, consumption and portfolio-composition rules are derived and analyzed in section 3, which also yields a separation theorem and a generalized CAPM with money and uncertain inflation. Up to this point, the only restriction imposed on the individual's utility function is that it be concave in consumption and real money balances. However, in order to obtain closed-form solutions for real consumption and the real demand for money, the utility functions are further restricted to the iso-elastic class, a sub-set of the Hyperbolic Absolute Risk Aversion family. This is done in section 4, where our results are shown to generalize those obtained by Merton (1971). By using the tools of comparative statics, we assess, in section 5, the influence on optimal consumption and portfolio rules of a change in the exogeneous parameters and, in particular, those under the control of the monetary authorities. The last section broadens somewhat the scope of the paper and indicates some avenues for further research.

2. Setting and assumptions We make the standard assumptions of a perfect capital market which have been extensively discussed in the literature. 1 In addition, the market is open all the time and hence trading in assets takes place continuously. Following Merton's pioneering work, we will use the continuous time mode of analysis, which imposes only a minimum requirement on the utility functions (namely, to be concave) and assumes a return structure consistent with limited liability. For simplicity, it is assumed that there is only one commodity whose future price (in terms of money), P, is uncertain. 2 There are (m+ 1) assets in the economy: the first (m-1) are risky assets, the ruth is non-interest-bearing money and the (m+ 1)th is the nominally riskless asset, whose instantaneous rate of return is r s. Fiat money is issued by the government which buys some amounts of the 1See, for example, Jensen (1972) and Merton (1973). 2The model can be generalized to multi-commodity price uncertainty, as done in Losq (1977). This will be discussed further in the concluding section of the paper.

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commodity and 'transforms' them into public services. The authorities may also issue the nominally riskless bond for that purpose. It is further assumed that the prices of the risky securities, V, and the commodity price P are generated by It6 processes, i.e.,

dVi/Vi=ridt +aidzi,

i= 1,...,m-- 1,

dP/P = r~dt+ a~dz~,

(1) (2)

where r i is the non-random instantaneous expected percentage change in price of asset i per unit time, a 2 its instantaneous variance per unit time, r~ the instantaneous, non-random, expected rate of inflation, a,,2 its instantaneous variance, and dzi and dz~ are Wiener processes. Since the r~, r~, tr~ and a,, are supposed to be constant, price changes are assumed to follow a geometric Brownian motion and, hence, prices V~and P have stationary and lognormal distributions, which implies that they cannot be negative. In continuous time, if we assume for simplicity that all income is derived from capital gains on assets, a the budget (or wealth) constraint is given, in nominal terms, by

d W = ~ e i ( r i d t + a i d z i ) W + ( 1 - ~i=1 7i)

(3)

where ai{i = 1,..., m - 1} is the proportion of total nominal wealth W invested in risky asset i, ~,~ the proportion of wealth invested in money, ( 1 - ~ ' a i ) the proportion invested in the riskless asset and c is the amount of real consumption per unit time. Noting that r,~ =am = 0 and ~ ei=~,~ + ~ ,m - 1 ~i, (3) becomes, after rearranging

~ri-ry)W+(1-~m)ryW-Pc dr+ ~ ~iWaidz v

dW=

(4)

Note that, although (4) is expressed in nominal terms, it is real wealth W/P which is relevant to the individual's decisions in absence of money illusion. The optimization problem faced by an individual is formulated as oO

max E o ~ e-Pt U[c(W, P; O, re(W,P; t)] dr, (c, ~)

0

(5)

aSee Merton (1971). The last section of the paper will briefly examine the case where wages are earned by the individual.

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subject to W(0)=Wo, the wealth constraint (4) and m=Ctm(W/P),where E is the conditional expectation operator, conditional on the knowledge of W(0) and P(O), where U is assumed to be a Von Neumann-Morgenstern utility function strictly concave in c, real consumption, and m, real money balances, and where p is the individual's rate of time preference. According t.o the neoclassical view, the household derives direct utility from money, whose services are proportional to the real amount held. Money presumably reduces transaction and information costs associated with exchanges of the commodity, although those costs are ignored for analytical convenience. 4'5 Capital markets, however, are assumed to be free from such costs and frictionless. Note that, for analytical convenience, a time-additive utility function U and an infinite time horizon have been selected. 6 Also, it is important to note that the current security prices V~ are absent from the objective function (5) and the wealth constraint (4) because of the stationarity assumption embedded in (1). In contrast, the wealth constraint being expressed in nominal terms, the commodity price level P appears in (4) and then influences c(.) and m(.) in (5).

3. Optimal consumption and portfolio rules We derive now the individual's optimal paths of real consumption and investment over time. Given the assumptions of the previous section, the only state variables are the nominal wealth W(t) and the price level P(t). The optimal rules c(W,P;t) and ~i(W,P;t) are obtained by using the techniques familiarized by Merton. Noting that, in the absence of money illusion, the indirect-utility-of-wealth function oo

J(W/P) = max Eo ~ e-P' U(c, m)dt 0

is homogeneous of degree zero in nominal wealth and commodity price, we 4In this respect, it would be preferable to have a multi-commodity rather than a single commodity economy (but see footnote 2). 5Fama and Farber (1979) present an alternative formulation which is formally identical to ours: the individual derives utility from consumption alone but 'produces' consumption with a production function utilizing two inputs: real money and the commodity. 6The infinite time horizon case yields essentially the same results as the finite time horizon case but is simpler to solve [see Merton (1969)]. In addition, this assumption takes care of the problem of terminal money balances, a standard difficulty in finite-horizon models involving the explicit use of money. Indeed, the question pertains to the disposition of the money balances after the termination of the individual's economic life (if there is no overlapping generation). If they are not destroyed (after having provided for their liquidity services), the opportunity cost of holding them is zero. If they are destroyed, this cost is 100%.

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236

obtain 7

[{m~l

o~i(ri --

dp(c, o~; W, P) - U(c, m) - p J + J ~

rf -- ¢Ti~) -~-(rf -- n) -- O~mrf

} W-- Pc

]

W2[ra-lm-1 ra-1 ] +J''-2-L ~ ~1 ~it~jtTiJJr-tT~t2 2 ,Y__.~,a,~ ,

(6)

where a u denotes the covariance between the rates of return on risky assets i and j, ai,~ the covariance between the rate of return on i and the rate of inflation and where the subscripts denote partial derivatives, for example

Jw = ~J/dW. The continuous-time version of the Bellman fundamental equation of optimality is then 0 = max ~b(c,~; W, P)

(7)

(c, a)

and the first order conditions are

(8)

dp~= 0 = U~(c*, m * ) - PJw, 1

,

dp~,,= 0 =-ff U,,(c , m * ) - ryJw,

(9)

[m-1

flPai"-O=Jw(ri--rf--ffin)+

)

JwwWI ~'J O~O'ij--Gin '

i= 1,...,m-1. (10)

Note that (10) forms a system linear in ~*. Those conditions can be rewritten

Vc(,)=e'Jw,

(11)

r f = U,,( * )/P" J w = Um( * )/Uc( * ) - #( * ),

(12)

1

m--1

m--1

j = 1,..., m - 1,

(13)

where R = -Jww" W/Jw is the Arrow-Pratt measure of the individual's relative

7A brief explanation of the mathematics of continuous time is given in appendix B for the readers not familiar with them.

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237

risk aversion and v o is the ith row-jth column element of the inverted ( m - 1) x ( m - 1) variance-covariance matrix of returns [air] - 1 A number of conclusions can be formulated. - - U n l e s s we specify more narrowly the individual's utility function, we cannot determine explicitly his optimal consumption rule nor the exact proportions ~,, and ( 1 - ~ ' 0 c i ) of his wealth that he will choose to invest in money balances and in the riskless asset, respectively, since we cannot solve analytically for c and m ( = ~ , , ( W / P ) ) from (11) and (12). However, the demand for the s u m of money and nominal bonds is determinate, since * is known from eq. (13). This is so because this sum plays the role of the riskless asset in the traditional analysis. This result is contained implicitly in Fama and Farber (1979). - - A t the micro level where any individual is by assumption a price taker, the demand for money, from eq. (12), is such that the ratio of the marginal utilities of money and of the commodity, U m ( * ) / U c ( * ) - l ~ ( * ) , which depends on the individual's preferences and tastes, is equal to the rate of return on the nominally riskless bond, a market determined variable. Since money is not destroyed in the consumption process, it is an asset the nominal (holding) cost of_which is r I, the (instantaneous) nominal return on an asset whose degree of riskiness is identical to that of money. In other words, holding one unit of money reduces the end-of(infinitesimal) period value of the portfolio by r I. But the instantaneous opportunity cost of consuming (destroying) one unit of the commodity is, in nominal terms, one (doUar). 8 Hence ry is (also) the ratio of the (instantaneous) opportunity costs of money and consumption, which must be equal, at the optimum, to #(.), the ratio of marginal utilities. 9 - - T h i s result and the Fama-Farber interpretation of eq. (12) discussed in footnote 9 suggest that the demand for money ct* can be viewed as the sum of two components: the first one, ~*rI is the 'consumption' portion while ~ * ( 1 - r l ) is the 'investment' portion, i.e., the 'bond equivalent' of money, to use the Fama-Farber terminology. In that sense, money, as an argument introduced in the individual's utility function, can be viewed as a commodity whose (relative) price is r I in the same way as the real good Sin a discrete-time model, the left-hand side of (12) would be rl/(1 +rl) in lieu of r s, the opportunity cost of consumption being then 1 + ri instead of 1. 9Fama and Farber give another, illuminating, interpretation of (12). They (rightly) view the nominal bond market as a market for forward contracts on money which allows investors to separate their (consumption) decision on the amount of money to hold for consumption purposes and their (investment) decision on the amount of risk (due to the randomness of the purchasing power of money) to bear. They (1979, p. 643) conclude that 'through (this) market the purchasing power risk of the money supply can be allocated across investors in accordance with their willingness to bear risk in exchange for a return, and independently of their demands for current and future money services' (italics provided). For a brief discussion of what happens in the absence of such a market, see our text below (at the end of this section 3).

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P. Poncet, Optimum consumption and portfolio rules with money

has a (relative) price P. This interpretation will prove useful in section 4 below for the discussion of the price index problem. Also, since this interpretation makes our model really a two-goods rather than a singlegood model, it makes clear why the generalization to multi-commodity price uncertainty hinted at in footnote 2 is straightforward. - - T h e demand for risky assets ~, {i= 1,...,m-.1} is formally the same as in the standard case with uncertain inflation but without money. 1° So that for a given probability distribution of risky returns, the introduction of money would affect only the demand for the risk-free asset, which would be lowered, were the coefficient of relative risk aversion R unaffected. It is important to note, however, that, in general, R will change because the J(.) function will be altered. Hence the need for money changes the 'technology' of consumption and alters all the individual's decisions. - - A s is well known, even when inflation is random, but in absence of money, provided all investors agree on the real investment opportunity set (i.e., have homogeneous expectations) the so-called 'separation' or 'twomutual-funds' theorem holds: there exist a unique pair of 'mutual funds' such that, independent of preferences, wealth distribution (and time horizon when it differs among individuals), 11 investors are indifferent between choosing portfolios from these two funds or from among the original m assets (whether one of them is nominally riskfree or not). x2 If we combine the money asset and the nominally riskless asset in a sub-portfolio, so that the proportion invested in that sub-portfolio is ( 1 ~,~-x ~,), we observe, from (13), that the separation property holds true. But since the proportion of money relative to the riskless asset depends on the individual's preferences (i.e., his utility function), it is clear that the above mentioned two-funds theorem must be extended to a three-funds separation theorem. We can therefore state the following:

Theorem 1. When prices are log-normally distributed and trading takes place continuously, any investor, independently of his wealth, time horizon and preferences, would be indifferent to the reduction of his investment opportunity set from the original assets to three basic portfolios ('mutual funds'), one of them being solely the money asset. If the growth rate of the commodity price level is not random, so that the real return on the nominally riskless asset (and on 1°See Friend et al. (1976), and particularly their Eq. (9), which is our Eq. (10). 11Remember that, for simplicity, we have assumed an infinite time horizon. This result holds even with a finite time horizon, which is not necessarily the same for all individuals. ~2If inflation is not random and there exists a riskless asset, one 'fund' contains only the riskless asset and the other only risky assets. With uncertain inflation, since the nominally riskless asset has a stochastic real return, this is no longer true although the two-funds property remains valid; see Black (1972).

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239

money) is certain, then one fund contains only risky assets, one other only the riskfree asset, and the third only money. 13 Theorem 1 is thus a particular case of the (S + 2)-fund theorem of Merton (1973) and Breeden (1979) where S stands for the number of additional (other than the random nature of risky assets returns) sources of uncertainty. - - B y aggregating (10) over investors k={1,...,N}, weighted by their relative wealth Ogk,an equilibrium for the ith risky asset is derived:

ri-rf-ffin=R°

~k ~j

O)kO~jktTij--17in

=°fR° ffiM--o~°/'

(14)

where M is the market portfolio of all nominally risky assets weighted by their relative market value, a° =--~,k~jOgkCtjk is the ratio of the value of the risky assets to total value of all assets including money, R °= 1/~(09k/Rk) is the harmonic mean of the individual Rk'S and R°a ° is the market price of risk. 1. Then, if we formulate the CAPM in terms of the excess return on the nominally risky market portfolio M over the riskfree return rl, we obtain the same form as for the standard CAPM adjusted for inflation. Formally:

rr M - ry -- ~Mn7

ri = rf + tri~+ L-~M ~-~u,--~ J (triM--tr,J °~°)

cov ,OVM VM

(15) 0~°

--

"

With this formulation, the expected return on any (nominally) risky asset i is still equal to the sum of three components: the return on the nominally riskfree bond, the covariance between the return on i and the rate of inflation and the market price of risk times the covariance between the return on i and the return on the nominally risky market portfolio adjusted (partially, because M is not the entire portfolio) for inflation. Eq. (15) looks like the standard inflation-adjusted-CAPM derived by XaTo see this, let tr;~=0 in eq. (13). It is then clear that the ratio of the demands for risky assets is independent of preferences (i.e., of R) and the same for all investors. So that the risky component of each individual's portfolio will be the market portfolio containing all risky assets. A similar result would also obtain under uncertain inflation provided a real riskless asset (an index bond) exists. In that case, the second fund would consist solely of the index bond; see S. Fischer (1975). 1*Which, from (14), is equal to o:°R°=(rM--rf--trMt)/[ir2--(ffMJo~°)].For more details and a discussion, see Friend et al. (1976).

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P. Poncet, Optimum consumption and portfolio rules with money

Friend et al. (1976) 15 but this could be misleading, as we shall see after a short digression. We know that it is possible to express r~ in terms of the excess return on the total market portfolio (including all assets) over the riskless return ry. Let us suppose for a moment that money does not exist. Call M' the total market portfolio whose return is then rM,=o~°rM+(1--~°)rf. Therefore, we have: aM,,~=~°tru,, tr~,=~°2tr 2 and tr~M,=Ct°triM. It follows that (15) can be rewritten: r

r i - - r f -~- o-ilt W

_2

L tru'-tru',~

J

(15')

where, of course, s ° has disappeared. (15') is the standard C A P M adjusted for inflation with the market price of risk computed from the return on the entire portfolio. If we now introduce the money asset, whose (equilibrium) rate of return is

not determined by the market but is exogeneously set by the authorities at zero, the return on M', the total market portfolio, is ru,=ct°ru+(1-~°-ct~t)rs+ s°'0, where ~,~ is the proportion of money in the market portfolio M'. We still have: aM,== ~°au,~, o.2, = ~t°2tr2 and triM,----Or°filM" Eq. (15) then becomes

ri=r y + tr,~+ [r~'--r Y - - ~ trM, '= + ~ ct~'rf _

(15")

which is different from (15'). There is an extra term in ~.rc because money is not evaluated by the market according to (15'). If it were, given that its nominal risk is the same as that of the riskless asset, i.e., zero, its return would be r I and this extra term would disappear. 16 In other words, this version of the CAPM differs from the standard one because current money holdings are evaluated a s / f they were a non-marketable asset, iv i.e., are held, not for an explicit market-determined pecuniary return, but for an implicit return which depends on the individual's preferences. Hence, we can state the following:

Corollary. When prices are log-normally distributed, the introduction of money as a capital asset does not change the form of the C A P M adjusted for inflation but changes the market price of risk. 1SOur eq. (15) is formally identical to their eq. (12). 16~° F m I is, in fact, ct~(rI - 0) where 0 stands for the nominal return on money. If the latter is rl, this extra term vanishes. This is (implicitly) the case in (15') where money and the riskless asset are (implicitly) perfect substitutes, i.e., have the same risk and the same return, making their distinction useless. 17For the introduction of non-marketable assets in the CAPM, see Mayers (1973).

P. Poncet, Optimum consumption and portfolio rules with money

241

In eq. (15), ~o is the ratio of risky to all assets including money instead of the ratio of risky to all assets excluding money as in Friend et al. (1976). The market takes explicit recognition of the fact that the (fiat) money supply is part of the individuals' total wealth. It follows that the market price of risk in [(15")] is greater than in the standard C A P M [(15')]. A final word of caution may be in order: it may seem at first glance, looking at eq. (15), that were the rate of inflation known with certainty, we would get exactly the standard CAPM, because ~° then would disappear from (15) [let tri,,=aM,~=O ]. This of course is incorrect, as it is clear from (15"), where the extra term in ~ r f would not vanish, is - - S i n c e expected returns rj are expressed in nominal rather than real terms, it is not possible to express eqs. (15") directly under the form of a 'two-beta' asset pricing model, while in essence the risk premia do depend on two betas: the first one would be the traditional beta measuring the correlation of the jth asset return with the market portfolio return and the second one the correlation of the jth asset return with an asset perfectly correlated with the inflation rate. This result is thus similar to those of Merton (1973) and Breeden (1979). Also, by defining the individual's risk tolerance (or aversion) from his direct utility function for consumption U(.) rather than from his indirect utility function J(.), we could obtain Breeden's single- (consumption) beta model, where the relevant aggregate for computing betas is aggregate consumption rather than aggregate (marketable) wealth. - - N o w , let us briefly mention the case where there is no nominally riskless asset. In this framework, since r I does not exist, eq. (12) disappears and eqs. (13) are modified so that the term U , . ( . ) / P . J w = U , , ( . ) / U c ( , ) =- It(*) is substituted for rf. 19 Since It(.) depends on individual preferences, we obtain a slightly :STo reconcile this finding with (15), recall that the market price of risk in (15) is R°~ °, whether inflation is random or not. 19Eq. (6) becomes

WZ[-m- 1 ,,-:

+'--T/z z,

m-:

z

-]

/

so that the first order conditions are

(8')

uc(*) = e.J,, --~Um(*)+ Jw(ri-ain)+ Jww

ot~rij--crin

=0

Dividing (10') by Jw and using (8') and the definition of R yields

r~- a,= Uc(*)

(10')

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242

different CAPM with #=~kOgk#k(*) substituted for r: in eqs. (14) and (15), where # is a simple weighted (by the relative wealth Ok) average of the individual #k(*)" The structure of asset returns is thus preserved, although the benchmark return for no (nominal) risk is the unknown /~ instead of the k.nown r:. In addition, the absence of riskfree nominal lending and borrowing implies the very important fact that the investors cannot reallocate across themselves, given the amount of money each one of them needs for consumption purposes, the risk associated with the randomness of the purchasing power of money.

4. Closed form solutions for special utility functions Because of Theorem 1, we can assume without loss of generality that there is only one risky asset, with expected return r, variance 0-2 and ~ its proportion in the individual's portfolio. Then, from eqs. (6) and (7), we have 0 = U(c, m ) - pJ + J w [ { o ~ ( r - r f - 0-,.,~)+ ( r : - re)- o~.,rf} W - P'c] W2

+ Jww--~-- [ ¢z20"2+ 0-2 _ 2~0-,,,]

(16)

and from eq. (13) [eqs. (11) and (12) being unchanged] c~= (r -- r f -- 0-~.)/ R "tr 2 + 0-rlt/ 0-2.

(17)

As shown in Merton (1971) in a world without inflation and where money is ignored, if, in addition to the log-normality of prices assumption, the individual's utility function is assumed to belong to the HARA (Hyperbolic Absolute Risk Aversion) family, then the optimal consumption and portfolio rules can be derived explicitly. Unfortunately, it turns out that when inflation is uncertain, it is not possible to solve (16) unless we further restrict the utility function to the isoelastic class. 2° Recall that all members of this family can be expressed, when U is a function of consumption c alone, as U(c)

1-~ y

2

1

2°It can be shown that, when the rate of inflation is not random, the consumption-portfolio problem is completely solved in this model with HARA utility functions taking the form U(c,m) =((1-V)/V)[(2/(1-v))f(e,m)+q] ~. The results, expressed in real rather than nominal terms, are otherwise similar to those of Merton (1971): the demand functions are linear in (real) wealth but their elasticities are different from one. In addition, provided f(c,m) is homogeneous of degree one in c and m, the demand for real balances is proportional to real consumption, the factor of proportionality depending on r:, the nominally risk free rate.

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243

where 7 and 2 are parameters characteristic of individual preferences subject to the restrictions y < 1 and 2 > 0 . The coefficient of relative risk aversion R =-Jww" W/Jw = 1 - y is then constant. Since there are two arguments in the utility function, however, one convenient way of specifying U(.) is

=

c% 1-~

with

0
and

e>y. 21

(18)

The solution to (16), using (11), (12), (17) and the assumption (18) is

J

J \7/'

where 6 - ( 1 - y ) = R > 0 , k=-(1-e)/ery and O = r : - r r + ( r - r r - y a , ~ ) 2 / 2 6 a 2 (yf/2)tr 2. Note that J(.) is iso-elastic in real wealth (W/P). 22 Solutions for c*, m* and 0~* are

c , = e(p-?O) W__ e--7 P '

(19)

m*= ( 1 - e ) ( p - y O ) W-= k.c* ry(e- y) e ~* =

r - r : - %=

trr~

~a2

+ " ~ - --

and

0c*=(1-e)(P-70), r: ( e - y)

r - rf + ( g - 1) tr,~ -

t~O.2

(20)

r - r : - y tr,~ -

~0"2

(21)

We can therefore state the following:

Theorem 2. Under the assumptions of the model, the household's consumption-portfolio problem is completely solved even though the future price level is not known with certainty, provided that the household's utility function is isoelastic in its two arguments. 21This restriction is necessary for consumption and the demand for money to be positive at the optimum. It is related to the desirability of c vis-a-vis m. Since the direct source of utility is real consumption, the utility of money stemming only from the reduction of transaction costs, e is likely to be closer to one than it is to zero. In addition, empirical evidence suggests that R is close to 2, making ~ equal to - 1. So this restriction should not be a problem. 22Merton (1971) has shown, in the no-money no-inflation case, that J(V¢) is HARA in W if and only if U(.) is HARA in consumption. It can be shown that the same theorem applies with real wealth W/P when U is a function of several arguments but is homogeneous of degree one in those arguments. In particular, in our case of uncertain inflation, J(W/P) is iso-elastic in W/P if and only if U(.) is iso-elastic in consumption and money.

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P. Poncet, Optimum consumption and portfolio rules with money

Eqs. (19)-(21) show that, for a given ry, the demands for consumption and money are proportional, that their real wealth elasticities are equal to one and that the distribution of the portfolio between money, riskless and risky assets depends only on tastes, not on the level of wealth. Those results are not surprising since our assumption of iso-elastic utility function implies 'homothetic' indifference curves and then unitary demand elasticities for the goods (here the commodity and money) entering the utility function. Also, this assumption solves the so-called index problem. Samuelson and Swamy (1974) proved that a globally valid price index that does not depend on the individual's nominal budget exists if and only if his indifference curves are homothetic. We have previously indicated that the fraction ~ * r I . ( W / P ) of optimal money holdings a * ( W / P ) represents the 'consumption' element. Then we could define a globally valid (for one individual) price index Z combining the commodity and money with their respective prices P and rs: Z = ~ 2 cOkPk, where cok is the budget share of 'good' k(~ g o)!,= 1) and Pk is the 'price' of good k. Here, using eqs. (19) and (20) and since the total budget devoted to 'consumption' is c*+ rfm*, we have Z = eP + ( 1 - e) r I,

where Z appears to be (also) independent of risk-aversion. Obviously, if the different individuals all had the same taste parameters, e and (1-e), the price index Z relevant for the whole economy would be the cost of the universal consumption basket. 23 It may be useful to compare the various demands when inflation is random and when it is not. Setting tr,2 = a,,~=0, eqs. (19)-{21) become c* -

e(p -

e--?

W

P'

m* = (1 - e)(p - tO') w ri(e- y) -if'

(19')

(20')

where 0'=- r I - rc + (r - r s ) Z / 2 & r 2, ct = (r - r f)/rtr 2

(21')

Comparing, firstly, (21) and (21') shows that the demand for risky assets 2aBreeden (1979) defines two price indices without requiring that individuals share the same tastes nor have homothetic preferences. However, his two price indices are only locally, not globally, valid.

P. Poncet, Optimum consumption and portfolio rules with money

245

increases, decreases, or remains the same, depending on the sign of o.,," and the magnitude of the coefficient of relative risk aversion, 6. Empirical estimates for the United States seem to indicate that 6 is close to two, making ( 6 - 1 ) positive, z4 Empirical estimates of o.,," are unambiguously negative: 25 the riskless bond performs relatively better than the risky asset as a hedge against inflation so that the demand for the latter decreases. How does uncertain inflation influence consumption and the demand for money? Comparing (19)-(20) to (19')--(20') shows that the answer depends solely on the respective magnitudes of 0 and 0', or, more exactly, of y0 and y0'. From the definitions of 0 and 0', we have ~6 2

0' = 0 + -z-o.,, q 2

(r--rf)To.r,"

2 o-rTr 2

6o.2

2 6 o -2,

so that

y(O, O)=yz[ar,'(r--rf--(y/2)o-r,') 6o-2-] ao.2

-5- J'

(22)

Note that r-ri-(y/2)ar,'=r-ri+((f-1)/2)a,~ is positive since, from (21), r-ry+(6-1)at," is positive (and r - r i > 0 ). Hence, if a,," was positive or zero, i.e., if the rate of return on the risky asset and the inflation rate were positively correlated, y0' would be greater than y0. This, in turn, would imply that consumption and the demand for money increase when the future price level becomes random, z6 When a,," is negative, however, results concerning consumption and the demand for money are no longer unambiguous, although the demand for risky asset decreases (provided 6 > 1). Indeed, the sign of the bracketed term in eq. (22) depends on the relative magnitudes of the terms multiplying Or, and a,', 2 respectively. It seems however extremely likely that, apart from those countries or circumstances in which the inflation rate is very highly unpredictable, the term in a,,2 is small. Therefore, unless a,," is itself very small (in absolute value), the bracketed term is likely to be negative, making consumption and the demand for money smaller. The economic interpretation of that finding is straightforward." the proportion of the risky asset in the portfolio being smaller and that of the riskless bond being larger, 24See, for instance, Friend, Lanskroner and Losq (1976). Of course, as is well known, if the coefficient 6 was equal to one (implying a logarithmic utility function), (21) and (21') would be identical: individuals would choose their portfolios as if the future level of price were fixed, i.e., they would be indifferent to changes in their future opportunity set when making their current choices, thereby exhibiting a myopic behavior. The same result would obtain, obviously, if the rate of return on the risky asset and the rate of inflation were not correlated at all. 25See, for instance, Bodie (1976), Hong (1977), Jaffee and Mandelker (1976) and Nelson (1976). It seems that inflation adversely affects returns on stocks mainly through additional tax burdens. 26If the taste parameter 6 = 1 - y is held constant.

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P. Poncet, Optimum consumption and portfolio rules with money

the expected return on this part of the portfolio (excluding money) is lower, so that consumption - - and the demand for real money balances which is proportional to it - - decreases. As a conclusion up to this point, it may be useful to summarize the main results of this section: when asset prices satisfy the geometric Brownian motion hypothesis and the individual's utility function is HARA, the consumption-portfolio problem is completely solved when only one good ('the' commodity) enters as an argument in the utility function [see Merton (1971)]. When two (or more) arguments appear in this function (here, commodity and money), the problem is completely solved only if there is no uncertainty on the future price(s) of the other argument(s). When uncertainty does exist, the problem cannot be solved unless we further restrict the utility function to the iso-elastic type. In that case, the index problem associated with the presence of money as an argument in the direct utility function is also solved. How letting the real rate of return on money be uncertain influences the households' various demand functions cannot be assessed by the theory alone, the results depending upon the magnitude of the coefficient of relative risk aversion ~ and upon the sign of ar~.

5. Comparative statics The purpose of this section is to assess, by the means of comparative statics, the influence on the various demand rules of a modification in the exogenous parameters, and particularly those under the control of the authorities. Since Merton (1969) has already opened the way (in the isoelastic, no inflation and no money case) we will focus on what may be considered as new. 27 We will examine successively three types of modifications, relative to the individual's tastes and preferences, the risk or return of the assets, and the rate of inflation, respectively. Since the magnitude and sign of y ( 5 0 and < 1), and then the magnitude of 3(= 1 - y > 0) play often a significant role, we will occasionally distinguish strong risk averters ( y < 0 , 6 > l ) from weak risk averters [0 < (y, 3) < 1]. To avoid tedious repetitions and to save space, appendix A gives partial derivatives of the marginal propensity to consume (MPC) out of real wealth and of ~, the proportion of wealth invested in the risky asset, with respect to the main parameters. 27The tools of comparative statics cannot be used in the general case where the forms of the U(.) anti J(.) functions are not known. Indeed, even though eq. (17) holds, R is an unknown function which, in general, will depend on all financial parameters. Hence we are restricted to the iso-elastic case where R =(1--~,) = 6 is a constant.

P. Poncet, Optimum consumption and portfolio rules with money

247

When the individual's rate of time preference p gets larger, his consumption, and the proportion of money in his portfolio, ~,., both increase. Since the proportion ~ of risky assets is unaffected, the demand for the riskless asset decreases (in the absence of money, the latter is not modified since c~ does not change). When e, measuring the relative capacity of the commodity (vis-a-vis money balances) to provide utility, increases, ~ does not change but the impact on consumption and ~,, is ambiguous, depending on the exact utility function, i.e., on ~,: it may be that consumption is enhanced (the demand for money will be too, but to a lesser extent) because it yields relatively (to money) more satisfaction but it may be, on the contrary, that the individual needs to consume less (and demands lesser amounts of money) to obtain the same level of utility. The influence of o.2 and r, the risk and return on the risky asset, is easy to analyze. ~ is higher in an economy where r is larger and 0.2 smaller. The impact on consumption, however, depends upon the relative risk aversion coefficient b. Consider an increase in r or a decrease in 0-2. For a weak risk averter, consumption (and the proportion of money) will decrease because he will be induced to save more to benefit from the higher expected return (i.e., the substitution effect dominates the wealth effect). A strong risk averter, however, will increase his consumption (the wealth effect dominates). In that case, since ~ and 7,, get larger, the demand for the riskless bond decreases. By changing its monetary policy, the government can affect the average rate of inflation n or its variance 0.,~. 2 If the latter is increased, the proportion ~ of the risky asset invested in the portfolio does not change but consumption (and the demand for money) increase and the demand for the riskless bond decreases, whatever the magnitude of b. This is so because risk averters (whether strong or weak) will hedge against the larger uncertainty of the purchasing power of money by consuming more today and less tomorrow. As for the influence of a change in n, we must be very careful in interpreting the derivatives shown in appendix A. Indeed, n should be made endogeneous rather than exogeneous because the price level depends on the demand for consumption. Therefore, the model should be closed by a set of equilibrium equations making P and rf (in addition to 1") determined endogeneously. With that strong restriction, it is clear that, unlike the standard model where n does not affect ~ nor consumption, n here influences consumption (and the demands for money and for the riskless asset), provided (rl-Tr) and (r-n) are affected. This is so because the opportunity cost of money is altered, hence the demand for it and consequently the optimal composition of the entire portfolio and the optimal consumption rule.

248

P. Poncet, Optimum consumption and portfolio rules with money

6. Concluding remarks As indicated at the outset of the paper, it is possible to further generalize the model by introducing many commodities, and therefore many more sources of uncertainty. The main results of this research would be unaffected; in particular, we would have a (2+n) mutual-funds theorem instead of a three-funds theorem if n commodities, with random future prices, were introduced; also, the individual's problem could be completely solved (only) in the iso-elastic case. Stochastic investment opportunities could also be introduced, by making the ri and tri of eqs. (1) dependent of some state variables, including, for instance, the riskless rate r:. The difficulty here would be ideally to have the stochastic process governing ry be determined within the model. The model can also be generalized by explicit consideration of the fact that income does not come exclusively from assets but also from wages. As shown in Merton (1971), the individual would simply add to his financial wealth W/P his human wealth; the latter would be the lifetime flow of wage income (y, if constant) capitalized at the market riskfree rate r: O.e., y/rf in the infinite time case) if this wage income were certain. If the latter were stochastic, then the capitalization rate would be larger or smaller than r: depending upon the sign of the covariance between wage income and the market return. In conclusion, this model generalizes the optimal consumption and portfolio-choice rules under uncertainty when money is considered as a capital asset. Minimal conditions sufficient to obtain a separation theorem and a generalized CAPM have been derived, with another source of uncertainty (namely the random purchasing power of money) explicitly introduced. Further research should try to complete the model by making the future price level and the nominally riskless rate of interest endogeneous. At least, it seems hardly appealing to consider inflation without explicit recognition of the demand for and supply of money.

P. P o n c e t , O p t i m u m

c o n s u m p t i o n a n d p o r t f o f i o rules w i t h m o n e y

249

Appendix A: Signs of partial derivatives

Partial derivatives a

Weak risk averters 0 < (~, 3) < 1

Strong risk averters ~ < O, 6 > 1

OMPC/Op=e/(e-y)

+

+

~/~p = o

o

a M P C / O e = - ~(p - yO)/(e - 7)2

_

0=/~ = 0

o +

0

0

O M P C / O r = - e~ct/(e - ~)

-

+

Oo~/Or = 1/~o"2

+

+

a M C P / a t r 2 = e~30"2/2(~-- ~)

+

~/~9o-2= -~ta 2

-

_

8 M P C / O n = e~/(e -- y)

+

-

8~/Or~= 0

0

8 M P C / & r 2 = e,~y2/2(e, -- )~)

+

O~/&r~= 0

0

0 +

0

a M P C stands for marginal propensity to consume (out of wealth). Since the demand for money is proportional to consumption, the sign of the partial derivative 0 ~ J 0 x is that of a M P C / O x .

Appendix B: Some mathematics of continuous time We will sketch briefly the way to derive eqs. (6) and (7) in the text. Let us define the indirect-utility-of-wealth function: GO

I(W(t),P(t),t)= max E t~ e - p x U ( c , m ) d x Ic(t), ~'(t)l

(B.1)

t

which is the maximum expected utility of lifetime consumption and real money holdings as of time t and where the expectation E, is conditioned on W(t), P(t) and t. Let us revert now to a discrete time setting. From (B.1), we have for a small At l(W(t),P(t), t)-~ max {e-P' U(.)At + Et[I(W(t + At), P(t + At), t + At)]}. to,~j (B.2)

Let us determine the expectation Et. By expanding in Taylor's series and neglecting terms in At of order higher than one, we have I[W(t + At), P(t + At), t + At] ~ I(W(t), P(t), t) + Iw(. )Aw + Ip(. )AP + I,(. )At + 2[Iww(.)(Aw)2 + 2lwp(.)AwAP + Ipp(.)(AP) 2] + 0((At)2),

(B.3)

250

P. Poncet, Optimum consumption and portfolio rules with money

where A W is given by eq. (4) in the text and AP by eq. (2), substituting the differential operator d by the operator A. It follows then that E(AW) =Im~ 1 ~i(r,- rs)W + (1-o~m)riW-P "clAt, E(AP) =P'n'At, m-1 m-1

~ ~,~j%W 2At+O((At)2),

E((AW)2) = ~ 1

1

m-1

E(A W" AP)= ~., o~a~,~P.WAt + O((At)2), 1

E((AP):)=P:a~At + O((At):). Substituting those results into (B.3) yields

E{I[W(t + At),P(t + At),t + At]} ~l(I4(t),P(t),t)+ I,At + I ~ I ~ ' z,(r,-rl)W+(1-Ctm)rfW-P'clAt m--1 m-1

+ IpP'~zAt + I ~ ~ 2

1

~ otiotjaijW2dt 1

+IwP ~t °~iai'~P"W'At +-2IppP2a2At +O((At)2)"

(B.4)

Substituting (B.4) into (B.2), cancelling I(W,,P,t) on both sides, dividing through by At and taking the limit as At~O yields

U(.) + I t + IpPn

0 = max e c)- tP(') p t =¢max ~ ,[

~i(ri - ry) W + (1 - otm)riW- P'c I m-1 1 2 21 E O~itXjlTiJW2 all-IwP E °~i~TinP"WJI--2lpp P 17nA" 1 1

m-1 m-1

+2Iww E

1

(a.5)

P. Poncet, Optimum consumption and portfolio rules with money

251

Now, we can greatly simplify (B.5) in two ways: Firstly, we can eliminate the explicit time dependance of ~(.) by defining oO

J(W/P)=eP'I(W,P,t)=maxE, ~e-P~s-')U(c,m)ds t

o0

=max E o ~ e-P~U(c,m)dx. 0

Secondly, we can use the result that, in the absence of money illusion, I(.), and then J(.), must be homogeneous of degree zero in nominal wealth and commodity price. Thus, by Euler's theorem

WIw+PIn=O

or

Ip=-(W/P)Iw.

Ipw=-(1/P)[Iw+ WIww]

and

Hence

Ipp=(W/P)2Iww.

Now, multiplying (B.5) by ep' and then applying (B.6) to

(B.6)

J(W/P) yields

0 = m a x ¢(c, oc; W,P) [c, a]

W2 I-m'z~"- x

"-~

11

Eq. (B.7) is the combination of eqs. (6) and (7) in the text.

References Black, F., 1972, Capital market equilibrium with restricted borrowing, Journal of Business, July. Bodie, Z., 1976, Common stocks as a hedge against inflation, Journal of Finance, May. Breeden, D.T., 1979, An intertemporal asset pricing model with stochastic consumption and investment opportunities, Journal of Financial Economics 7, Sept. Chert, A.H. and A.J. Boness, 1975, Effects of uncertain inflation on the investment and financing decisions of a firm, Journal of Finance 30, May. Fama, E.F. and A. Farber, 1979, Money, bonds and foreign exchange, The American Economic Review, Sept. Fischer, S., 1975, The demand for index bonds, Journal of Political Economy 83, June. Friend, I., Y. Landskroner and E. Losq, 1976, The demand for risky assets under uncertain inflation, The Journal of Finance, Dec.

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P. Poncet, Optimum consumption and portfolio rules with money

Grauer, F.L.A. and R.H. Litzenberger, 1979, The pricing of commodity futures contracts, nominal bonds and other risky assets under commodity price uncertainty, Journal of Finance, March. Hong, H., 1977, Inflation and the market value of the firm: Theory and tests, Journal of Finance, Sept. Jaffee, J. and G. Mandelker, 1976, The 'Fisher effect' for risky assets: An empirical investigation, Journal of Finance, Proceedings, May. Jensen, M., 1972, Capital markets: Theory and evidence, Bell Journal 2. Landskroner, Y. and N. Liviatan, 1981, Risk premia and the sources of inflation, Journal of Money, Credit and Banking 13, May. Levhari, D. and N. Liviatan, 1976, Government intermediation in the indexed bonds market, American Economic Review Papers and Proceedings 66, May. Liviatan, N. and D. Levhari, 1977, Risk and the theory of indexed bonds, American Economic Review 67, June. Losq, E., 1977, Commodity price uncertainty and capital market equilibrium, unpublished manuscript (McGill University, Montreal). Mayers, D., 1973, Non marketable assets and the determination of capital asset prices in the absence of a riskless asset, The Journal of Business 46, no. 2, April. Merton, R., 1969, Lifetime portfolio selection under uncertainty: The continuous time case, The Review of Economics and Statistics, Aug. Merton, R., 1971, Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, Dec. Merton, R., 1973, An intertemporal capital asset pricing model, Econometrica, Sept. Nelson, C.R., 1976, Inflation and rates of return on common stocks, Journal of Finance, Proceedings, May. Samuelson, P.A. and S. Swamy, 1974, Invariant economic index numbers and canonical duality: Survey and synthesis, American Economic Review 64, Sept. Sercu, P., 1981, Mean-variance asset pricing with deviations from purchasing power parity, unpublished doctoral dissertation (Katholieke Universiteit, Leuven).