Monetary Aggregation Theory and Statistical Index Numbers

1 Hutt (1963) suggested theindex now known as the cur-rency equivalent index.

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31

JANUARY/FEBRUARY 1997

MonetaryAggregationTheory andStatistical IndexNumbersRichard G. Anderson,Barry E. Jones, andTravis D. Nesmith

T

he aggregate quantities of monetaryassets held by consumers, firms, andother economic decisionmakers play

important roles in macroeconomics. TheBoard of Governors of the Federal ReserveSystem publishes monetary aggregates forthe United States that are sums of the dol-lar amounts of monetary assets held by thenonbank public. These assets includecurrency, checkable deposits, money mar-ket mutual fund shares, and savings andtime deposits. Constructing these aggre-gates by summation implicitly assumesthat the owners of the monetary assetsregard them as perfect substitutes. Yet, theobservation that most economic decisionmakers hold a portfolio of monetary assetsthat have significantly different opportuni-ty costs, rather than a single asset with thelowest cost, implies that the owners do notregard these assets as perfect substitutes.

The appropriate method of aggregatingmonetary assets is an important questionin macroeconomics. In general, aggre-gation methods should preserve theinformation contained in the elasticities ofsubstitution, and, in particular, aggre-gation methods should not make strong apriori assumptions about elasticities ofsubstitution. After forming four aggre-gates by simple summation of monetaryassets, Friedman and Schwartz (1970)offered the following caution:

The restriction of our attention to these four combinations seems a less serious limitation to us than our acceptance of the common procedure of taking the quantity of money asequal to the aggregate value of the assets it is decided to treat as money.This procedure is a very special caseof the more general approach [which]consists of regarding each asset as ajoint product having different degreesof “moneyness,” and defining thequantity of money as the weightedsum of the aggregate value of allassets, the weights for individualassets varying from zero to unity witha weight of unity assigned to that assetor assets regarded as having the largestquantity of “moneyness” per dollar ofaggregate value. The procedure wehave followed implies that all weightsare either zero or unity.

The more general approach has been suggested frequently but experi-mented with only occasionally. We conjecture that this approach deserves and will get much more attention than it has so far received. The chief problem with it is how to assign the weights and whether the weights assigned by a particular method will be relatively stable for different periods or places or highly erratic. (pp. 151-2)

Although the microfoundations ofmoney have been widely discussed (see,for example, Pesek and Saving, 1967;Fama, 1980; Samuelson, 1968; andNiehans, 1978), prior to Barnett (1978,1980, 1981) only a few authors, includingChetty (1969), Friedman and Schwartz(1970), and Hutt (1963), had appliedeither microeconomic aggregation theoryor index number methods to monetaryassets.1 Barnett, Fisher, and Serletis (1992)and Belongia (1995) survey the early liter-ature on the subject.

Richard G. Anderson is an assistant vice president at the Federal Reserve Bank of St. Louis. Barry E. Jones and Travis D. Nesmith are Ph.D.candidates at Washington University in St. Louis and visiting scholars at the Federal Reserve Bank of St. Louis. Mary C. Lohmann, Kelly M.Morris, and Cindy A. Gleit provided research assistance.

We offer special thanks toProfessor William A. Barnett,Washington University in St.Louis, for his generous andinvaluable guidance during thisproject. We also thankProfessors W. Erwin Diewert,University of British Columbia,and Adrian R. Fleissig, St.Louis University, for advice andcomments.

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2 Diewert (1980, 1981, 1992,and 1993) surveys the theoryand application of index num-bers in economics.

3 On aggregation theory in eco-nomics, see Green (1964),Samuelson and Swamy(1974), Diewert (1980) andBarnett (1981).


Barnett (1978, 1980, 1981) developeda method of monetary aggregation basedon the assumption that monetary assetsare durable goods in a representative con-sumer’s weakly separable utility function.In this model, the representative consumerchooses, subject to a set of intertemporalbudget constraints, the optimal quantitiesof all the decision variables in its utilityfunction: durable goods, nondurablegoods and services, leisure time, and mon-etary assets. The assumption that current-period monetary assets are weakly separablefrom all other decision variables in theconsumer’s utility function implies theexistence of a monetary aggregate for theconsumer.

Barnett (1987, 1990) developed amethod of monetary aggregation in thecontext of a representative, perfectly com-petitive, non-financial firm. The firm isassumed to maximize profit, subject to aproduction function that contains mone-tary assets. The solution to the firm’soptimization problem produces demandfunctions for the firm’s factor inputs,including monetary assets. If monetaryassets are weakly separable in the firm’sproduction function from all other inputs,then a monetary aggregate will exist forthe firm. Other related research hasdeveloped methods of monetary aggrega-tion based on models of financialintermediaries as multi-product firms(see Barnett and Zhou, 1994; Barnett,1987; Barnett, Hinich, and Weber, 1986;and Hancock, 1985, 1986). Barnett(1987) discusses the conditions underwhich a monetary aggregate will exist fora financial intermediary.

The monetary aggregates that arederived from these microeconomic modelsdo not impose strong a priori assumptionsabout the substitutability of monetaryassets. These methods of monetary aggre-gation are derived from microeconomictheory, which can therefore be used toanalyze the implied monetary aggregates.Precise definitions and statements ofthese results are given in the section ofthis article entitled “Monetary AggregationTheory.”

Theoretical monetary aggregates canbe approximated by statistical index num-bers.2 The monetary services indexes thatwe describe in the next article in this issueof the Review, “Building New MonetaryServices Indexes: Concepts, Data, andMethods,” are such index numbers. Ourindexes are based on the same aggregationand statistical index number theory as theDepartment of Commerce’s real quantityand price indexes, which include grossdomestic product (GDP) and personalconsumption expenditure (PCE), and theirdual price indexes, the GDP and PCEdeflators.

This article contains four sections. Inthe first section, we discuss the conditionsunder which it is valid to aggregate mone-tary assets. In the second section, we dis-cuss the use of statistical index numbers totrack monetary aggregator functions. Inthe third section, we discuss the role of theconsumer’s budget constraint and the the-ory’s implied concept of monetary wealth.In the last section of the paper we examinethe robustness of the theoretical aggregationresults.

MONETARY AGGREGATIONTHEORY

There are two distinct aggregationproblems in economics: aggregation acrossheterogeneous agents, and aggregation ofthe various goods purchased by a singleagent.3 Although this article focuses onaggregation of the monetary assets held bya single representative consumer, webelieve it is relevant to briefly review theissues related to aggregation acrossconsumers and firms.

Aggregation Across HeterogeneousConsumers and Firms

The most common method of develop-ing aggregate demand functions that obeymicroeconomic decision rules is to assumethe existence of a representative agent. Arepresentative consumer is one that maxi-mizes utility—subject to market prices, anaggregate (shadow) expenditure variable,


33

4 In some cases, the aggregateexpenditure variable can bemade independent of prices;this has been called price-inde-pendent generalized linearity(PIGL).

5 Aggregation is possible withoutassuming a representativeagent. See Barnett (1979,1981) and Selvanathan(1991) for a stochasticmethod of aggregation firstsuggested by Theil (1971,1975). Diewert (1980) dis-cusses Hicksian aggregationconditions, although Hicksianaggregation is not valid for allfeasible movements of pricesand quantities. Clements andIzan (1987) discuss stochasticHicksian aggregation.


and a budget constraint—in such a waythat, for each good, the representative con-sumer’s demand function is equal to thesum of the demand functions of allindividual consumers. Such a representa-tive consumer will exist if the demandfunctions for individual consumers satisfyGorman’s (1953) condition that all con-sumers have parallel and linear Engelcurves. In this case, the aggregate expen-diture variable is the sum, or mean, of theexpenditures of all individual consumers,and redistribution effects may be ignoredbecause different expenditure distributionswith the same mean expenditure level willproduce the same aggregate demand foreach good or service. Consequently, wemay construct a model of aggregate con-sumer demand as if all decisions werebeing made by a single consumer.

Muellbauer (1976) introduced a some-what less restrictive set of conditions forexistence of a representative consumer thatcontains Gorman’s result as a special case.Muellbauer’s conditions allow for non-linear Engel curves, but the appropriateaggregate expenditure variable is not, ingeneral, the sum, or mean, of the expendi-tures of the individual consumers. InMuellbauer’s generalization, the aggregateexpenditure variable is a function of the dis-tribution of expenditures and prices.4 Inthis case, mean-preserving changes inexpenditure distributions can have eco-nomic effects. Berndt, Darrough, andDiewert (1977) apply this methodologyusing the income distribution, rather thanthe expenditure distribution. See Deatonand Muellbauer (1980) for more discussion.

The aggregation conditions of Gormanand Muellbauer are highly restrictive andunlikely to be satisfied exactly in any econ-omy. Nevertheless, it often is desirable tointerpret the behavior of aggregate data interms of microeconomic theory. In suchapplications, aggregate models commonlyassume the existence of a representativeconsumer and ignore the potential role ofheterogeneous preferences and distributioneffects in aggregate demand.5

The conditions for aggregation acrosscompetitive firms are weaker than those for

consumers. Consumers’ budget constraintsgive rise to distribution effects that, to jus-tify aggregation, must be ruled out throughrelatively stringent aggregation conditions.Because profit-maximizing competitive firmsare not subject to such budget constraints,these effects are absent. Debreu (1959)showed that, under perfect competition, agroup of optimizing firms can be treated asif it were a single representative firm whichmaximizes profits subject to the sum of theproduction sets of the individual firms.Barnett (1987) discusses aggregation acrossfirms in greater detail.

In empirical research, statistical testsoften reject propositions about the behaviorof quantities and prices that are impliedby representative agent models. This out-come suggests that either the maintainedneoclassical microeconomic demandtheory is incorrect (an unpalatable conclu-sion) or that the hypothesis that aggregatedata behave according to the decisionrules of a single economic agent is false.In the past, the assertion that macroecono-metric models based on aggregate datashould embed decision rules obtainedfrom the solution of representative agentoptimization problems has been controver-sial (see Lucas and Sargent, 1978a,b;Friedman, 1978; Ando, 1981; Kirman,1992). Deaton and Muellbauer (1980)noted the following:

In general, it is neither necessary, nornecessarily desirable, that macroeconomic relations should replicatetheir microeconomic foundation sothat exact aggregation is possible. In-deed, to force them to do so often pre-vents a satisfactory derivation ofmarket relations at all. (p. 149)

Much theoretical macroeconomic researchis, however, conducted with general-equi-librium models that are based on represen-tative agents. The extent to which vio-lations of this assumption weaken ourresults is discussed in the final section ofthis article, titled “Limitations and Exten-sions of Aggregation and Statistical IndexNumber Theory.”


34

6 If the conditions for existenceof a representative agent donot hold for all individuals, theymay nevertheless hold for sub-groups of individuals. In thiscase, separate aggregates maybe needed for different groupsof individuals.

7 Although these conditions aresufficient to permit aggrega-tion, they are not sufficient toallow the behavior of the result-ing aggregates to be analyzedwith microeconomic theorybecause the first condition—that a factorable subfunctionexists—does not restrict theagent’s decision to be a ratio-nal, optimizing microeconomicdecision. For example, the con-dition is not sufficient to guar-antee that the first-orderconditions for the solution tothe agent’s constrained opti-mization problem will resemblethe familiar conditions of neo-classical demand theory.

8 However, the specific reasonthat money is valued cannot beinferred from the form of theutility function. Duffie (1990)reviews general-equilibriummodels in which money haspositive value. Feenstra(1986) derives the specific util-ity functions implied by severalpopular models of moneydemand, including the cash-in-advance model. Fischer(1974) derives a productionfunction that contains moneybalances.

9 See Barnett (1987), pp. 116-20. Note that this modelassumes perfect certainty, andthat all agents are price-takers.


General Conditions forAggregation of Monetary Assets

The general conditions sufficient forthe aggregation of a group of economicdecision variables are: (1) the existence of atheoretical aggregator function defined overthe group variables–that is, the existence ofa subfunction, defined over the group ofvariables, that can be factored out of theeconomic agent’s decision problem; (2) effi-cient allocation of resources over the groupof variables; and, (3) the absence ofquantity rationing within the group of vari-ables. If the underlying price or quantitydata have been previously aggregated acrossagents, an additional assumption is required:(4) the existence of a representative agent.6

These general conditions are sufficientfor theoretically rigorous aggregation of agroup of economic variables and are inno way specific to the monetary aggrega-tion case.7

To facilitate exposition and providethe reader with the strongest and mostelegant linkages between monetaryaggregation and microeconomic theory,the balance of this article relies on theo-retical assumptions that are significantlyless general than those stated above. Wefocus our discussion on the aggregationof monetary assets held by a price-takingrepresentative consumer. This consumer,who is subject to a set of multi-periodbudget constraints, is assumed to maximizean intertemporal utility function in whichcurrent-period monetary assets are weaklyseparable from all other decision variables.This model, which is less general butmore familiar, is sufficient to allow us toaggregate of current-period monetaryassets: (1) The weak separability assump-tion implies the existence of a theoreticalaggregator function that can be definedover current-period monetary assets. (2)The utility maximization implies that theallocation of resources over the weaklyseparable group will be efficient. (3)Quantity rationing is ruled out.

As we noted above, the microeconomicfoundations of monetary aggregation canalso be illustrated in models of profit-maxi-

mizing firms and financial intermediaries(Barnett, 1987). Additional generalizaionsof monetary aggregation theory are alsopossible. In particular, utility (profit)maximization can be replaced by expendi-ture (cost) minimization in other versionsof these models. Expenditure (cost) mini-mization will guarantee that allocation ofresources over the weakly separable blocksis efficient.

The Consumer’s Choice ProblemWe begin by describing a represent-

ative consumer’s intertemporal decisionproblem in which monetary assets appearin the consumer’s utility function; ourpresentation follows Barnett (1978).Monetary assets have been included inutility functions since, at least, Walras(trans. 1954). Arrow and Hahn (1971)show that, if money has positive value ingeneral equilibrium, then there exists aderived utility function containingmoney.8 Thus any model that does notinclude money in the utility function butproduces a motive for holding money inequilibrium is functionally equivalent to amodel that does include money in theutility function. Hence, no generality inmodeling is lost, or gained, by includingmoney in the utility function.

We assume that, in each period, therepresentative consumer maximizesintertemporal utility over a finite planninghorizon of T periods.9 The consumer’sintertemporal utility function in anyperiod, t, is

U(mt,mt+1,...,mt+T;qt,qt+1,...,qt+T; lt..., lt+T;At+T),

where, for all s contained in{t,t+1,...,t+T},

ms = (m1s,...,mns) is a vector of real stocks of n monetary assets,

qs = (q1s,...qhs) is a vector of quantitiesof h non-monetary goods and services,

ls is the desired number of hours of leisure, and


35

At+T is the real stock of a benchmarkfinancial asset, held in the final periodof the planning horizon, at date t+T.

The representative agent is assumed to re-optimize in each period, choosing values of(mt,...,mt+T;qt,...,qt+T; lt,..., lt+T; At,..., At+T)thatmaximize the intertemporal utility func-tion subject to a set of T+1 multiperiodbudget constraints. The set of multiperiodbudget constraints for s contained in{t,t+1,...,t+T} is the following:

where

p*s is a true cost of living index,

ps = (p1s,...,phs) is a vector of prices forthe h non-monetary goods and services,

rs = (r1s,...,rns) is a vector of nominalholding-period yields on the n monetaryassets,

Rs is the nominal holding-period yieldon the benchmark asset,

ws is the wage rate,

As is the real quantity of a benchmarkasset that appears in the utility functiononly in the final period of the planninghorizon, t+T, and

Ls is the number of hours of laborsupplied.

Leisure time consumed by the consumerduring each period is ls = H–Ls, where H isthe total number of hours in a period.

In this model, we assume theexistence of the true cost-of-living index,p*

s; see Barnett (1987). We also assumethat all of the services provided to the

agent by monetary assets, except for theintertemporal transfer of wealth, have beenabsorbed into the utility function, U( ).Note that, even though the benchmarkasset, As, appears in each period’s budgetconstraint, it is included in the utilityfunction only during the final period of theplanning horizon. This is because thebenchmark asset is assumed to furnish nomonetary services to the agent, except inthe final period. During all other periods,the agent uses the benchmark asset onlyto transfer wealth from one period toanother.

To simplify notation, let the vectormt= (m1t,...,mnt) contain all current-periodmonetary assets, and let the vector

xt= (mt+1,...,mt+T;qt,...,qt+T; lt,..., lt+T;At+T)

contain all other decision variables in theutility function. Let the vectors

m*t= (m*

1t,... m*nt) and

x*t= (m*

t+1,...,m*t+T;q*

t,...,q*

t+T; l*t,..., l*t+T; A*t+T)

denote the solution to the consumer’smaximization problem, that is, m*

t is theconsumer’s optimal holdings of current-period monetary assets, and x*

t is theoptimal holdings of all other decision vari-ables.

When the utility function

U(mt,...,mt+T; qt,...,qt+T; lt,...,lt+T; At+T)

is written as U(mt,xt), the first-order condi-tions of this model imply that the marginalrate of substitution between current-periodmonetary assets mit and mjt, evaluated atthe optimum is

∂∂

∂∂

U m x

m

U m x

m

pR r

R

pR r

R

t t

it x xm m

t t

jt x xm m

tt it

t

tt jt

t

t t

t t

t t

t t

( , )

( , ).

*

*

*

*

*

*

==

==

=

−+

−+

1

1

p q w L

r p m p m

R p A p A

is isi

h

s s

i s s i s s i si

n

s s s s s

=

− − −=

− − −

∑

∑

=

+ + −[ ]+ + −[ ]

1

1 1 11

1 1 1

1

1

( )

( )

,*

,*

,

* *



36

10 For earlier treatments in thecontext of business fixed invest-ment, see Jorgenson (1963)and Jorgenson and Stephenson(1967).


The first-order conditions also imply thatthe marginal rate of substitution betweenthe current-period monetary asset, mit, andthe current-period non-monetary good, qkt,at the optimum is

A general relationship in microeconomicoptimization is that, at the optimum, mar-ginal rates of substitution between goodswill be equal to the goods’ relative prices.In these expressions, the “price” (oropportunity cost) of the current-periodmonetary asset mit, is

Barnett (1978) proved this intuitionformally, and we discuss this result inmore detail in the following section.

The User Costs of Monetary AssetsMonetary assets are treated as durable

goods in the model discussed in this article.Similar to other durable goods, monetaryassets appear in the utility function, provideservices to economic agents, and depreciate(but not fully) during each decision period.To aggregate stocks of these durable mone-tary assets, we need their equivalent rentalprices, or user costs.

Diewert (1974, 1980) discusses thegeneral procedure for constructing the usercost of a durable good (or physical capitalasset) from the purchase price of the good,the depreciation rate of the good, and a dis-count factor.10 Intuitively, if an agent(consumer or firm) buys one unit of adurable good at the beginning of a decisionperiod and, later, sells the remaining (non-depreciated) part of the unit at the end ofthe period, the cost to the agent of “renting”

that good for that single period (or, equiva-lently, the cost of the services that the agentreceived from the good during the period)is equal to the difference between the initialpurchase price of the good and the presentvalue of the amount the agent receiveswhen the non-depreciated part of the unit issold. If an explicit rental market for thegood does not exist, the agent’s actions maybe interpreted as if he rented the good tohimself; in this case, the user cost is usuallyreferred to as an implicit, or equivalent,rental price.

Formally, let pt and pt+1 denote themarket prices of a durable good in periods tand t+1, respectively; let

d be the deprecia-tion rate; and let D be the discount factor.The equivalent rental price of the durablegood is

If the depreciation rate equals unity, as itdoes for nondurable goods that are fullyconsumed within a single period, then therental price equals the purchase price.

Barnett (1978) derived the generalform of the equivalent rental price, or usercost, of monetary assets by combining theT+1 budget constraints in the generalintertemporal consumer model, solvingeach equation for As, and recursively sub-stituting the equations backwards in time,beginning with At+T. The general form ofthe discounted nominal user cost, πis, ofeach monetary asset, mi, in each period scontained in{t, t+1,...t+T}, is

where ris is the nominal holding period yieldand the discount factor, ρs, is defined by

ρsu

u t

s

s t

R t s t T=

=

+ + ≤ ≤ +

=

−

∏1

1 11

,

( ), .

πρ ρis

s

s

s is

s

p p r= − +

+

* *( ),

1

1

pp

Dtt−

−( )+( )

+1 1

1

δ.

p R r

R

t t it

t

*

.−( )

+( )1

∂∂

∂∂

U m x

m

U m x

q

pR r

R

p

t t

it x xm m

t t

kt x xm m

tt it

t

kt

t t

t t

t t

t t

( , )

( , ).

*

*

*

*

*==

==

=

−+

1


37


This general form may be specialized to thecurrent-period nominal user cost, πit, of mone-tary asset, mit,

Rt2ritπit = p*t (_____),1+Rt

which may be interpreted as a price of cur-rent-period monetary assets (see Barnett,1978; Donovan, 1978). Note that the cur-rent-period nominal user cost is thepresent value of the interest the agent hasforegone by holding the monetary asset(rather than holding the benchmark asset),discounted to reflect the receipt of interestat the end of the period.

The user cost, πit, represents the equiv-alent rental price of the services providedby a unit of monetary asset, mit. The pro-duct, m*

itπit, represents the optimum expen-diture on monetary asset, mit, in the cur-rent period. The sum

is total optimal expenditure on (the servicesprovided by) current-period monetaryassets. It can be shown that monetaryasset, mit, is implicitly assumed to depreciateat the rate of

which, for non-interest-bearing monetaryassets such as cash, equals the inflationrate (see Fisher, Hudson, and Pradhan,1993).

For consumers, the user costs of mon-etary assets are analogous to the user costsfor other durable goods; for firms, the usercosts of monetary assets are analogous tothe user costs for durable physical capital.Barnett (1987, 1990) derived the user costof monetary assets in the context of aprofit-maximizing manufacturing firmwith a production function that containsmonetary assets and proved that the math-ematical expressions for consumers’ and

firms’ user costs are the same. Becausehouseholds and firms generally facedifferent market interest rates and prices,the values of their user costs will differ.

Barnett (1987) also derived the usercost of monetary assets for a financialintermediary. When such intermediariesare required by statute to maintain non-interest-bearing reserves against theirdeposit liabilities, the mathematical formof their user costs will differ from those ofconsumers and other firms by the size ofthe implicit reserve-requirement tax. Inthe absence of statutory reserve require-ments, the reserve-requirement tax iszero, and financial firms’ user costs formonetary assets have the same form asthose of other firms.11 Finally, note thatthe expressions for both consumers’ andfirms’ user costs can be extended to allowfor the taxation of interest income(Barnett, 1980).

Aggregator Functionsand Two-Stage Budgeting

Monetary aggregates that are consis-tent with the solution to the representativeagent’s decision problem may be derivedby imposing additional structure on themodel. Assume that the intertemporalutility function is weakly separable in thegroup of current-period monetary assets—that is, that the utility function has thefollowing form:

U[u(mt),mt+1,...,mt+T;qt,qt+1,...,qt+T; lt+T,...,lt;At+T],

which may be written as U[u(mt), xt], wherext was defined previously. Note that onlycurrent-period monetary assetsmt = (m1t,...,mnt) are included in the factor-able sub-function u( ), which is called thecategory subutility function, defined overcurrent-period monetary assets. Note alsothat the separability assumption is not sym-metric. Weak separability of current-periodmonetary assets from the other goods andservices included in the utility functiondoes not imply that any other combinationof decision variables is similarly separable.

δ itt it

t

p r

p= − +

+1

1

1

*

*

( )

mit iti

n*π

=∑

1

11 Many countries, includingCanada and Great Britain, donot impose statutory reserverequirements against deposits.Depository institutions inGermany are required to main-tain non-interest-bearingdeposits at the Bundesbankequal to 2 percent of theirtransaction deposits. For theUnited States, Anderson andRasche (1996) estimate thatstatutory reserve requirementsaffected only about 500 depos-itory institutions as of the endof 1995.


38


12 If the category subutility func-tion

u(), is homothetic, simplychoose a first-degree homoge-neous cardinalization of thesubutility function.

Weak separability of the utility func-tion for the representative consumer is animportant assumption because it permitsformulating the consumer’s decision as atwo-stage budgeting problem. It impliesthat the marginal rates of substitutionamong the variables within the weaklyseparable group are independent of thequantities of the decision variables outsidethe group (Goldman and Uzawa, 1964).In our context, weak separability ofcurrent-period monetary assets from theother decision variables in the consumer’sutility function implies that the marginalrates of substitution between current-period monetary assets reduces to

which, evaluated at the optimum, equals

Barnett (1980, 1981, 1987) used this resultto show that the vector of current-periodmonetary assets which solves the consumer’s(weakly separable) intertemporal utility-maximization problem,

m*t = (m*

1t,...,m*nt),

is exactly the same vector that would bechosen by a consumer in solving thefollowing simpler problem, which involvesonly current-period variables,

is the optimal total expenditure on mone-tary assets implied by the solution to theagent’s intertemporal decision problem.

Barnett’s result establishes that, underthis type of weak separability, the represen-tative consumer’s general, intertemporaldecision problem is formally equivalent toa two-stage budgeting problem. In thefirst stage, the consumer chooses theoptimal total expenditure, yt, on current-period monetary assets, and the optimalquantities of the other monetary assets,goods, services, and leisure that appear inthe utility function. In the second stage,the consumer chooses the optimal quanti-ties of the individual current-periodmonetary assets,

m*t = (m*

1t,...,m*nt),

subject to the optimal total expenditure oncurrent-period monetary assets, yt, chosenin the first stage. Interpreted as a two-stage budgeting problem, the second stageof the problem corresponds to maximizingthe subutility function, u( ), subject to theexpenditure constraint implied by the firststage (Green, 1964).

If u( )is first-degree homogeneous, it isa monetary quantity aggregator function.12

The representative consumer will view themonetary quantity aggregate, Mt=u(m*

t), asif it were the optimum quantity of a single,elementary good, which we call current-period monetary services. This allows thefirst-stage decision to be interpreted as thesimultaneous choice, given market pricesand the consumer’s budget constraint, ofthe optimal quantities of (1) current-period monetary services, and (2) all otherdecision variables. It also justifies the useof microeconomic demand theory to studythe behavior of monetary aggregates.

In general, quantity and priceaggregates are said to be dual if the priceaggregate, multiplied by the quantityaggregate, equals the total expenditure(price times quantity) on all individualgoods in the aggregate. Dual to the mone-tary quantity aggregate, Mt, is the dual usercost aggregate, Πt, defined as the unitexpenditure function

where y mt it iti

n

==∑ *π

1

Max u m

m y

m m mn

it it ti

n

=

=

( )

=∑

( , , )

,

1

1

L

subject to π

∂∂

∂∂

ππ

u m

m

u m

m

t

it m m

t

jt m m

it

jt

t t

t t

( )

( )==

=

*

*

.

∂ ∂∂ ∂

u m m

u m mt it

t jt

( ) /

( ) /,


39


where πt=(π 1t ,...,πnt)is the vector of nom-inal user costs of current-period monetaryassets, as defined above.

The consumer is assumed to maximizeU[u(mt),xt] subject to the T+1 multi-periodbudget constraints discussed above. Thefirst-order conditions for this problemimply that the marginal rates of substi-tution between current-period monetaryassets, mit, and current-period non-monetary goods and services, qkt, can bewritten as

Weak separability of current-period mone-tary assets from other decision variables,and first-degree homogeneity of thecategory subutility function, u( ), implythat these expressions can be combinedinto fewer expressions of the form

for all current-period non-monetary goodsand services, qkt. These expressions arethe first-order conditions for the first-stagedecision, and have the interpretation thatMt is the optimal quantity of an elementarygood, monetary services, whose price is

Π t. This result can be generalized, so thatall of the first-order conditions involvingcurrent-period monetary assets can berewritten as first-order conditionsinvolving only the aggregates Mt and Π t .13

These new first-order conditions havestandard microeconomic interpretations.

The final step in the argument is toshow that the budget constraint can berewritten in terms of the aggregates, Mt

and Πt. As we noted earlier, Barnett(1978) showed that the T+1 multi-periodbudget constraints could be combined intoa single budget constraint. It can be shownthat current-period monetary assets enterthis single budget constraint as total expen-diture on current-period monetary assets,

First-degree homogeneity of the functionu( ) implies that the following property,which is called factor reversal, holds as anidentity:

The product of the optimal quantity ofmonetary services, Mt, and its dual usercost Πt, equals the optimal total expenditureon current-period monetary assets. Thebudget constraint, at the optimum, cantherefore be rewritten in terms of theaggregates, Mt and Πt. Because theseaggregates satisfy the conditions for factorreversal, the dual user cost aggregate isimplicitly defined by

The above discussion demonstratesformally that the first-stage decision can beinterpreted as the simultaneous choice ofoptimal quantities of monetary services,Mt, and all other decision variables outsidethe weakly separable block of current-period monetary assets, subject to pricesand a budget constraint, where the price of

Πt it iti

n

tm M=

=∑ * .π

1

Πt t it iti

n

tM m y= ==∑π * .

1

mit iti

n

π=∑

1

.

∂∂

∂∂

U u x

u

U u x

q

p

t

x x

u M u m

t

kt x xu M u m

it

kt

t t

t t

t t

t t

,

,

*

*

*

*( )

( )

( )=

== = ( )

== =

Π

∂∂

∂∂

π

U u m x

m

U u m x

q

p

t t

it x xm m

t t

kt x xm m

it

kt

t t

t t

t t

t t

( ),

( ),.

*

*

*

*

[ ]

[ ]=

==

==

Πt t

ni i t

i

n

E

m u mm m m

=

= =

==∑

( , )

min ( )( ,..., )

π

π

1

11 1

:

13 The exceptions are the first-order conditions that involveonly current-period monetaryassets. These are the first-orderconditions for the second-stageallocation decision.


40


monetary services is given by the dualopportunity cost, Πt. The first-stage deci-sion produces yt=MtΠt, the optimal totalexpenditure on current-period monetaryservices, and this optimal expenditure isallocated among the current-period mone-tary assets in a second-stage decision. Anycurrent-period monetary portfolio substi-tution that does not change the level of themonetary aggregate is irrelevant to otherdecision variables in the model. The mon-etary aggregates, Mt and Πt, thereforecontain all the information about the con-sumer’s portfolio of current-periodmonetary assets that is relevant to otheraspects of the representative consumer’sdecision problem.

STATISTICAL INDEX NUMBERS

In the previous section of this article,microeconomic theory was used to identifymonetary quantity aggregates and their dualuser cost aggregates, for current-periodmonetary assets. In empirical research, nei-ther the functional forms of the aggregatorfunctions nor the values of their parametersare known. If we sought to estimate theaggregator functions directly, we would beforced to make specific assumptions aboutthe functional forms of the utility (produc-tion) or expenditure (cost) functions.

Statistical index numbers are specifica-tion- and estimation-free functions of theprices and optimal quantities observed intwo time periods.14 Unlike aggregatorfunctions, statistical index numberscontain no unknown parameters. A statis-tical index number is said to be exact foran aggregator function if the index numbertracks the aggregator function, evaluatedat the optimum, without error.

The Continuous-Time CaseWe begin our discussion of index-

number theory with the result that theDivisia quantity index (first suggested asan index number by Divisia, 1925) is exactfor the monetary quantity aggregate, Mt.The continuous-time Divisia quantity index,

MDt, is defined for monetary assets by the

differential equation

where, for i=1,...,n,

represents the monetary assets’ expen-diture shares.15

Similarly, the continuous time Divisiauser cost index, Π D

t , is defined by

dlog(Π Dt ) d[log(πit )]_________ = S

n

i=1

wit _________ .dt dt

Note that the continuous-time Divisiaquantity and user cost indexes satisfyfactor reversal; that is, the product of theDivisia quantity and price indexes equalsthe total expenditure on the assets inclu-ded in the index, thus,

MDt ΠD

t = Sn

i=1

m*itπit = yt ,

(see Leontief, 1936).It is possible to describe the path of

the monetary quantity aggregate, Mt, incontinuous time using only the first-orderconditions for utility maximization andthe first-degree homogeneity of the cate-gory subutility function, even though thecategory subutility function, u( ), isunknown. As discussed previously,Mt= u(m*

t ) is the solution to the represen-tative consumer’s optimization problem,

Max u(m) subject to Sn

i=1

miπit = yt ,m=(m1,...,mn)

which includes only current-period mone-tary assets and their user costs. Thefirst-order conditions are

wm

mit

it it

jt jtj

n=

=∑

*

*

π

π1

d M

dtw

d m

dttD

iti

nitlog( ) [log( )]*

==∑

1

14 If consumers and firms are notprice-takers, then it may benecessary to use shadow,rather than observed, prices inthe indexes (Diewert, 1980).An additional problem is thatthe existence of a representa-tive firm in Debreu’s (1959)proof depends on the assump-tion of perfectly competitivemarkets.

15 All logs in this article are natur-al, or base e, logarithms.


41


where λ is the Lagrange multiplier for thebudget constraint. The category subutilityfunction is first-degree homogeneous byassumption, and therefore satisfies Euler’slaw: thus,

To study the time path of the quantityaggregate, we take its derivative withrespect to time:

Dividing this expression by the previousone, we obtain

The last equation above expresses thegrowth rate of the continuous-time Divisiaquantity index. The solution to this differ-ential equation is a line integral that is pathindependent under the maintained assump-tion of weak separability (Hulten, 1973).

We conclude that, in continuous time,the Divisia quantity index is exact for theunknown monetary quantity aggregate, Mt.We emphasize that the exact trackingability of the Divisia index is an implica-tion of economic theory, not an approxima-tion. The arguments in this section arevalid for any first-degree homogeneousquantity aggregator function, and, there-fore, the result is not specific to monetaryaggregation.

The Discrete-Time CaseIn discrete time, the situation is quite

different; there is no statistical indexnumber that is exact for an arbitrary aggre-gator function. Discrete time index-number theory is based on two facts: (1)mathematical functions exist that can pro-vide second-order approximations tounknown aggregator functions; and, (2)statistical index numbers exist that areexact for some of these functions. Animportant class of mathematical functions,locally-flexible functional forms, are able toprovide local second-order approximationsto arbitrary discrete-time aggregator func-tions, in the sense that they can attainarbitrary elasticities of substitution at asingle point (Diewert, 1971). This prop-erty is equivalent to the usual mathematicaldefinition of second-order approximation(Barnett, 1983). Diewert (1976) showedthat there exists a class of statistical indexnumbers, which he called superlative, thatare exact for certain, specific flexible func-tional forms. Thus, superlative indexnumbers are able to provide second-orderapproximations to unknown, arbitrary eco-nomic aggregator functions, in discretetime.

A statistical index number is said to bechained if the prices and quantities used inthe index number formula are the pricesand quantities of adjacent periods, such asmit and mit+1, and said to be fixed base if theprices and quantities used in the indexnumber formula are those of the currentand a fixed base period, such as mit andmi0. Chained superlative index numbershave a distinct advantage over fixed-base

d M

dtw

d m

dt

d M

dt

tit

i

ni t

tD

log( ) log( )

log( ).

*

=

=

=∑

1

dM

dt

du m

dt

u m

m

dm

dt

dm

dt

md m

dt

t t

i m m

it

i

itit

i

it itit

i

t

=

==

=

=

∑

∑

∑

( )

( )

log( )

*

*

*

**

*

=

=

= .

n

n

n

∂∂

λπ

λ π

1

1

1

M u mu(m)

mm

m

m

t tj m mt

jt

n

jt jtj

jt jtj

=

( )

=

=

=

∑

∑

∑

( )*

*

*

*

*

=

=

= .

j=1

n

n

∂∂

λ π

λ π

1

1

∂∂

λπu m

mi

i m mit

t

( ),

* ,== for each

superlative indexes. For a chained superla-tive index, the center of the second-orderapproximation moves in such a way thatthe remainder term in the approximationrelates to the changes between successiveperiods, rather than to the change from thebase period to the current period. As aresult, because changes in prices and quan-tities between adjacent periods are typicallysmaller than changes in prices and quanti-ties between a fixed base period and thecurrent period, the chained index numberis likely to provide a better approximationto the unknown aggregator function than afixed-base index number (Diewert, 1978).Further, Diewert’s (1978, 1980) theoremsprove that if the changes in prices andquantities are typically small between adja-cent periods, chaining will tend to mini-mize the differences among alternativesuperlative index numbers. These lattertheorems are based on numerical analysisand do not require economic optimization.

Many familiar index numbers aresuperlative. Two of the most important arethe Fisher ideal index and the Törnqvist-Theil discrete-time approximation toDivisia’s (1925) continuous-time quantityindex.16 Diewert (1976) showed that theFisher ideal index is exact for a homo-geneous quadratic functional form (see alsoKonüs and Byushgens, 1926). For monetaryaggregation, the Fisher ideal quantity index,MF

t, is defined by

M Ft =M F

t-1

The growth rate of the Fisher idealquantity index is the geometric mean ofthe growth rates of the well-knownPaasche and Laspeyres quantity indexes.The Paasche and Laspeyres indexesprovide only first-order approximations tothe underlying aggregate and are notsuperlative. Until recently, Paasche andLaspeyres indexes were the basis for most

Department of Commerce measures ofeconomic activity, although these measuresare now based on the chained Fisher Idealindex (Young, 1992, 1993; Triplett, 1992).The Laspeyres price index still underliesthe Bureau of Labor Statistics’ ConsumerPrice Index (CPI).17

For monetary aggregation, theTörnqvist-Theil monetary quantity index,Mt

TT, is defined as follows:

Diewert (1976) demonstrated that theTörnqvist-Theil quantity index is exact forthe translog flexible functional form, andthus is superlative.

The dual user cost index, ΠtDual,

defined by the recursive formula

is based on a weak form of factor reversal,and is dual to the Törnqvist-Theil monetaryquantity index.

The monetary services indexes andtheir dual user-cost indexes, which aredescribed in the next article in this issue,“Building New Monetary Services Indexes,”are chained superlative index numbers,employing the Törnqvist-Theil discrete-time quantity index-number formula andits dual-user cost index formula.

MONETARY SERVICE FLOWSAND MONETARY WEALTH

In the preceding section, the Törnqvist-Theil index was shown to measure theflow of monetary services produced by therepresentative consumer’s monetary assets.In this section, we derive an expression forthe stock of the consumer’s monetary wealth.As with all goods and services, changes inthe price of monetary services will affect

Π ΠtDual

tDual

it iti

n

i t i ti

ntTT

tTT

m

m

M

M=

−

=

− −=

−

∑

∑1

1

1 11

1

π

π

*

, ,*

M Mm

mtTT

tTT it

i t

w w

i

n it i t

=

−

−

+( )

=

−

∏111

12 1*

,*

,

m

m

m

m

it iti

n

i t iti

n

it i ti

n

i t i ti

n

*

,*

*,

,*

,

. .

π

π

π

π

=

−=

−=

− −=

∑

∑

∑

∑1

11

11

1 11

16 Mathematically, the Törnqvist-Theil discrete-time index is theSimpson’s rule approximationof the continuous-time Divisiaindex.

17 The Advisory Commission toStudy the Consumer Price Index(1996) recommended that theBureau of Labor Statisticsswitch to using superlativeindex numbers.



42

both the demand for, and supply of, mone-tary services and other non-monetarygoods and services. In addition to substi-tution and income effects, there are wealtheffects associated with monetary assets.

One measure of monetary wealth isthe discounted present value of the repre-sentative consumer’s expected expenditureon monetary services. The measure followsimmediately from Barnett’s (1978, 1987)result that the multi-period budget con-straints for the intertemporal decision,indexed by s contained in {t, t+1,...t+T},

can be combined into a single budget con-straint. Monetary assets enter this singlebudget constraint through the term

where the discount factor,

and the discounted nominal user costs, πis, werediscussed previously. Letting T go to infinityand evaluating Vt at the optimum yields,

Vt = S`

s=t Sn

i=1

πism*is= S

`

s=t

ys ,

where ys is the discounted expectedoptimal total expenditures on monetaryassets in period s.

The economic interpretation of Vt isrelatively straightforward, although its

measurement may be difficult. Vt is thediscounted present value of all current andfuture expenditures on monetary services;in other words, Vt is the stock of monetarywealth. Unfortunately, Vt is an infinite for-ward sum of discounted expenditures and,as such, cannot be directly computed. Tomeasure this definition of the stock ofmonetary wealth, we assume that therepresentative consumer forms staticexpectations. Specifically, we assume thatthe consumer’s mathematical expectationsof all future prices and rates, including thebenchmark rate, equals the current values:for all s contained in {t+1, t+2,...}, Et[ris]=rit and Et[Rs]=Rt, where Et[ ] is the mathe-matical expectation operator based on thetime-t information set. As a result, theconsumer’s expected optimal holdings ofmonetary assets in all future periods areequal to current holdings; that is, Et[m*

is] =m*

it, for all s contained in {t+1,t+2,...}.Under this assumption, Barnett (1991) hasshown that this measure of the stock ofmonetary wealth is equal to the Rotembergcurrency-equivalent (CE) index, CEt, whichis defined as follows:

(see Rotemberg, 1991; and Rotemberg,Driscoll, and Poterba 1995). As a measureof monetary wealth, in this special case,the CE index can be used to study theresponse of consumer behavior to changesin the stock of monetary wealth.

With this interpretation of the CEindex, it is possible for both the Törnqvist-Theil and CE indexes to be contained in thesame model, because they are measures ofdifferent concepts. The Törnqvist-Theilindex measures the current-period flow ofmonetary services. In contrast, the CEindex, under static expectations, measuresthe discounted present value of current andfuture expenditures on monetary services,equal to the stock of monetary wealth.Equivalently, the Törnqvist-Theil index maybe seen as a measure of the demand for aflow of monetary services, and the CE index

CE pR r

Rmt t

t it

tit

i

n

= −

=∑* *

1

ρsus

u

s t

R t s t T=

=

+ + ≤ ≤ + =

−

1

1 111

,

( ),,

Π

Vp p r

m

m

ts

s

s is

si

n

iss t

T

isi

n

iss t

T

= − +

=

+==

==

∑∑

∑∑

* *( )

,

ρ ρ

π

1

11

1

p q w L

r p m p m

R p A p A

is isi

h

s s

i s s i s s i si

n

s s i s s s

=

− − −=

− − −

∑

∑

=

+ +( ) −[ ]+ +( ) −[ ]

1

1 1 11

1 1 1

1

1

,*

,*

,

*,

* ,



43

18 The existence of a corner solu-tion identifies the price indexthat is dual to the simple sumindex as Leontief, that is, thesmallest user cost among theweakly separable block of mon-etary assets. For argumentsagainst the use of simple sumindexes, see Fisher (1922).

19 We also have assumed that theagent’s portfolio is always inequilibrium; see Spencer(1994).



44

as a measure of a term in the consumer’sbudget constraint (Barnett, 1991). The CEindex is able to measure the flow of mone-tary services in one special case. When, inaddition to the assumptions that underlie theTörnqvist-Theil index, the category subu-tility function, u( ), is quasi-linear in amonetary asset whose own rate is alwayszero, the CE index will measure the flow ofmonetary services. Thus, the CE index isstatistically inferior to the Törnqvist-Theilindex as a measure of the flow of monetaryservices (Barnett, 1991).

The simple sum index, SSt, defined as

measures the flow of monetary servicesonly if the representative agent’s indiffer-ence curves for monetary assets are parallellines and, hence, the agent regards all mon-etary assets as perfect substitutes. If theassets, in fact, have different user costs,then this agent would choose a corner solu-tion and hold only one monetary asset inequilibrium—an implication that is bothcounterintuitive and counterfactual.18

In the context of the static-expec-tations model introduced above, the simplesum index may be interpreted as a “stock”variable; it is not, however, a measure ofthe stock of monetary wealth. The followingrelationship is useful in describing thestock concept that the simple sum indexmeasures:

The index can, with this expression, bedecomposed into two terms. The first term,

is the CE index. The second term,

ri t mit* 1 1pt

* Sn

i=1

________ [1+ ________ + __________ +.. .],1 + Rt (1 + Rt) (1 + Rt)2

is the discounted present value of allcurrent and future interest received onmonetary assets, under the assumption ofstatic expectations. Thus, in the context ofthe static-expectations model, the simplesum index may be interpreted as the sum ofthe discounted present value of expendi-tures on monetary services, plus thediscounted present value of interest incomefrom monetary assets; it cannot be inter-preted, however, as a measure of the stockof monetary wealth (Barnett, 1991).

LIMITATIONSAND EXTENSIONSThe discussion in the previous sections ofthis paper has been based on very strongmicroeconomic assumptions. In particular,we have assumed (1) the existence of a rep-resentative agent (consumer or firm), (2)blockwise weak separability of current-period monetary assets, (3) homotheticityof the category subutility function, and (4)perfect certainty.19 In this section, we willdiscuss violations of these assumptions,and recent theoretical advances thatattempt to address such problems.

Representative Agents, WeakSeparability, and Divisia SecondMoments

Assumptions (1) through (4) implythat the chained Törnqvist-Theil discrete-time approximation to the continuous-time Divisia index provides a second-orderapproximation to the unknown monetaryaggregate, Mt. If the conditions that arenecessary for the existence of a representa-tive agent are not satisfied, or if the weak

pR r m

Rtt it it

ti

n*

*−( )=∑

1

SS p m

pR r m

R

pr m

R R R

t t iti

n

tt it it

ti

n

tit it

t t ti

n

=

=−( ) +

++

+( ) ++( )

+

=

=

=

∑

∑

∑

* *

**

**

.

1

1

21

11

1

1

1

1L

SS p mt t iti

n

==∑ * * ,

1

separability of current-period monetaryassets from an agent’s other decision vari-ables is violated during some sampleperiods, then the growth rates of theoptimal quantities of monetary assets,m*

lt,...,m*nt, user costs, πlt ,...πnt,, or expendi-

ture shares, wlt,...,wnt, may contain infor-mation that is not contained in the Törn-qvist-Theil index. Barnett and Serletis(1990) proposed the dispersion dependencytest of the aggregation assumptions basedon the Divisia second moments (variances)of the growth rates of quantities, usercosts, and expenditure shares, as definedsubsequently.20

The log change (growth rate) of theTörnqvist-Theil quantity index is

where for i=1,...n,

are the monetary assets’ between-periodaverage expenditure shares. This expressionis in the form of an average share-weightedmean of the log change of componentquantities (component growth rates).Theil (1967) pointed out that the growthrate of the Törnqvist-Theil quantity indexhas a natural interpretation as the mean ofthe component quantity growth rates,where the average shares induce a validmeasure of probability. Thus, the growthrate of the Törnqvist-Theil quantity indexmay be interpreted as the first moment of adistribution. Similarly, the growth rate ofthe Törnqvist-Theil user-cost index, Pt

T T, isin the form of an average share-weightedmean of component user-cost growthrates,

Finally, the growth rate of the Törnqvist-Theil expenditure-share index, St

TT, is in theform of an average share-weighted mean ofcomponent expenditure-share growth rates,

Thus, the growth rates of the Törnqvist-Theil user-cost and expenditure-shareindexes also can be interpreted as the firstmoments of an underlying probability dis-tribution.22 Theil (1967) showed that thegrowth rates of the three indexes arerelated by the identity

The stochastic interpretation of theTörnqvist-Theil indexes as first momentscan be generalized to higher moments ofthe underlying distributions, which areusually called Divisia higher moments.23

The Divisia quantity growth-rate variance,

is the variance of the growth rates of theindividual quantities. The Divisia user-costvariance,

is the variance of the growth rates of thecomponent user costs. The Divisia expen-diture-share growth-rate variance,

is the variance of the growth rates of thecomponent expenditure shares. Thecovariance of the growth rates of thecomponent user costs and quantities is

Ψ ∆ ∆t it it tTT

i

n

w w S= −[ ]=∑ log( ) log( )

2

1

J w Pt it it tTT

i

n

= −[ ]=∑ ∆ ∆log( ) log( ) ,π

2

1

K w m Mt it it tTT

i

n

= −[ ]=∑ ∆ ∆log( ) log( ) ,* 2

1

w w y

w m w

it iti

n

t

it iti

n

it iti

n

∆ ∆

∆ ∆

log( ) log( )

log( ) log( )*

=

= =

∑

∑ ∑

+

= +

1

1 1

π

∆ ∆log logS w wtTT

iti

n

it( ) = ( )=∑

1

∆ ∆log log .P wtTT

iti

n

it( ) = ( )=∑

1

21π

w w wi t i t i t= + −12 1( ),

∆ ∆log log ,*M w mtTT

iti

n

it( ) = ( )=∑

1

20 Barnett and Serletis (1990)show that aggregation errorintroduces an additive remain-der term in the second-orderdiscrete-time approximation.Some econometric tests forspecification error in linearregression models exploit asimilar correlation betweenregression residuals and nonlin-ear functions of the indepen-dent variables (Ramsey, 1969;Thursby, 1979; Thursby andSchmidt, 1977).

21 Although the continuous-timeDivisia price index is dual to thecontinuous-time Divisia quantityindex, the discrete-timeTörnqvist-Theil price index is notdual to discrete-time Törnqvist-Theil quantity index (Theil,1967). In applications, theTörnqvist-Theil quantity indexshould be used with its dualuser cost index, not theTörnqvist-Theil user cost index.

22 Clements and Izan (1987)developed an alternative inter-pretation of the Divisia index.In a model of statisticalHicksian aggregation, theDivisia price index is interpretedas the generalized leastsquares estimate of the com-mon trend in a price formationfunction, provided that the vari-ances of the individual compo-nent price estimates areinversely proportional to theirexpenditure share. For a cri-tique and extensions, seeDiewert (1995).

23 Diewert (1995) suggests aclass of measures of functionalform error based on the disper-sion of component growth ratesof statistical price indexes. IfDiewert’s approach is general-ized to quantity and share index-es, the Divisia second momentsare the squares of a member ofthis class, so the secondmoments can also be interpret-ed as providing measures of thereliability of an index.


45


.

Theil (1967) showed that these foursecond moments of are related by the fol-lowing identity:

ψt = Kt + Jt +2Γt.

Dispersion-dependency tests based onDivisia second moments for United Statesmonetary data are presented in Barnett andSerletis (1990) and Barnett, Jones, andNesmith (1996). The evidence in thesestudies suggests that Divisia secondmoments do contain economic informa-tion not contained in the growth rates ofthe Törnqvist-Theil quantity, price, andexpenditure share indexes. In otherwords, for at least some time periods,movements in the observed quantities andprices of monetary assets are not consis-tent with the movements that would beimplied by the actions of a representativeagent with a weakly separable utility func-tion. In such cases, Barnett and Serletis(1990) suggest that including Divisiasecond moments in macroeconomicmodels might provide at least a partialcorrection for the aggregation error.24

Homothetic PreferencesThe theoretical results presented in

this paper have been derived under themaintained hypothesis that the categorysubutility function for current-periodmonetary assets, u( ), is homothetic. Ifhomotheticity is violated, then the quan-tity and price aggregator functions are notthe subutility and the unit-expenditurefunctions, respectively, and the Divisiaindex will not track the utility function incontinuous time.

In this section, we extend our previousdiscussion to include economic indexesthat are correct in general, even if homo-theticity is violated. Assuming that therepresentative agent’s utility function is

weakly separable in current-period mone-tary assets, we can define quantity anduser-cost aggregator functions—namely,the Konüs and Malmquist indexes—thatare correct in the absence of homotheticity.

Let u( ) be the agent’s category subu-tility function, which is not necessarilyhomothetic. The monetary quantity aggre-gate may be defined as the distance functiond(m*

t,~u), which is defined implicitly by

mu(_______)= u~

d(m,u~)

The user-cost aggregate, dual to the distancefunction, is the expenditure function eval-uated at the reference utility level,

~u,

defined by

e(πt,u~) =

mm

tin{S

n

i=1

π itmit: u(mt)=u~}(Barnett, 1987; Deaton and Muellbauer,1980). Normalizing these quantity andprice aggregates to equal one in a baseperiod produces exact economic indexes.

The Malmquist quantity index is defined by

d (m*t ,u

~)M(m*

t, m*0, u~) = ________

d(m*0 ,u~)

(Malmquist, 1953), and the Konüs user-costindex is defined by

e(πt,u~

)K(πt,π0,u

~ ) = _______e(π0,u

~)

(Konüs, 1939). Both indexes have beennormalized to equal unity in period t=0,and have been defined in terms of monetaryassets and user costs.

Although these general indexes arecorrect whether or not the category subu-tility function is homothetic, a short-coming is that both depend on thereference utility level,~u. Konüs (1939)showed that the value of the user-costindex, in any period, can be boundedabove and below, albeit at differentreference utility levels. The upper boundin the case of monetary assets is aLaspeyres price index, given by

Γ ∆ ∆

∆ ∆

t it it tTT

i

n

it tTT

w P

m M

= ( ) − ( )[ ]× ( ) − ( )[ ]

=∑{

}.

log log

log log*

π1

24 If aggregation conditions do nothold, additional higher-ordermoments of the index’s distrib-ution could contain additionalinformation. Dispersion depen-dency testing could thereforebe extended to testing othermoments.



46

.

where m*i0 is the optimal quantity of mone-

tary asset i in the base period t=0. The lowerbound is a Paasche price index given by

where m*it is the optimal quantity of mone-

tary asset i in period t. See Frisch (1936)and Leontief (1936) for early discussion ofthese issues. Diewert (1993) reviews theearly history of index-number theory.

It can now be seen why homotheticityis such a valuable property. The Konüsand Malmquist indexes do not requirehomotheticity, but they depend on aspecific reference utility level. It is easilyshown that this dependence vanisheswhen the category subutility function isfirst-degree homogeneous. With first-degree homogeneity, the distance functionis proportional to the utility function, withthe proportionality factor equal to the ref-erence utility level. The Malmquist indexmay thus be written as

First-degree homogeneity also implies that

e(π,~u) = e(π,1)

~u ,

so the Konüs index may be written as

e(πt,1)K(πt,πo,u~)= ______ ,

e(π0,1)

which is independent of the referenceutility level. Further, with first-degree

homogeneity, the Konüs index is boundedby the Paasche and Laspeyres indexes

for any reference utility level, ~u. This is a

formal statement, in the monetary aggrega-tion case, of the often-quoted result thatbase-period-weighted (Laspeyres) priceindexes overstate the true increase in prices,and current-period-weighted (Paasche)price indexes understate the true changein prices. Note that this result is criticallydependent on the assumption of homo-theticity. In general, if homotheticity isviolated, the Paasche price index mayactually exceed the Laspeyres price index(see Deaton and Muellbauer, 1980). Thepotential upward, or downward, biasresulting from the use of Laspeyres andPaasche price indexes is discussed byTriplett (1992).

Although homotheticity producesattractive simplifications of aggregationtheory, it is implausible that any populationis well characterized by an assumption ofidentical homothetic utility functions—Samuelson and Swamy (1974) label this a“Santa Claus” assumption. When homo-theticity is violated, the Törnqvist-Theildiscrete-time approximation to the contin-uous-time Divisia index has been shown totrack the distance function. Caves, Chris-tensen, and Diewert (1982) proved that theTörnqvist-Theil index is superlative in thebroader sense that it can provide a second-order approximation to the Malmquistquantity index, even when homotheticity isviolated. No other statistical index numberis known to have this important property.The monetary services indexes presented inthe next article in this issue of the Review,“Building New Monetary Services Indexes,”which are based on the Törnqvist-Theilindex, should be robust to violations ofhomotheticity.

π

ππ π

π

π

it ii

n

i ii

n t

it iti

n

i iti

n

m

m

K u

m

m

01

0 01

01

01

*

*

*

*

( , , ˜) ,=

=

=

=

∑

∑

∑

∑≥ ≥

M m m ud m u

d m u

u m u

u m u

u m

u m

tt t

t

( , , ˜)( , ˜)

( , ˜)

( ) ˜

( ) ˜.

( )

( )

* **

*

*

*

*

*

00 0

0

= = ⋅⋅

=

π

ππ π

it iti

n

i iti

n t t

m

m

K u m

*

*

*, , ( ) ,=

=

∑

∑≤ [ ]1

01

0

π

ππ π

it ii

n

i ii

n t

m

m

K u m0

1

0 01

0 0

*

*

*, , ( ) ,=

=

∑

∑≥ [ ]


47


Perfect Certainty

Recently, economists have addressedthe problem of constructing monetaryaggregates that include assets with riskyreturns (see Collins and Edwards, 1994;Orphanides, Reid, and Small, 1994; andBarnett, 1994). Prior to this section, thisarticle has included only perfect-certaintymodels, even though some of the monetaryservices indexes presented in the nextarticle in this issue include capital-uncer-tain monetary assets, such as U.S. Treasurybills, with variable returns. Aggregationtheory needs to be extended to includerisky assets.

Extending the theory to include risk-neutral consumers is straightforward.Barnett (1994) has shown that, under riskneutrality, the Törnqvist-Theil discrete-time approximation to the continuous-time Divisia index provides a second-orderapproximation to the unknown aggregatorfunction, where the user costs are definedas the expected value of the nominal cur-rent-period perfect-certainty user costs,

Rt- ritπit = Et {p*t_____}.1 + Rt

Extending the theory further to includerisk-averse consumers is more difficult.Rotemberg (1991) noted that, under riskaversion, Barnett’s (1994) result does nothold because the asset’s user costs are corre-lated with the representative agent’s marginalutility. In related work, Barnett and Zhou(1994) derived a supply-side model of mon-etary aggregation under risk aversion, basedon the profit-maximizing behavior ofdepository financial institutions. Barnettand Liu (1994) and Barnett, Liu, andJensen (1997) discuss a generalized Divisiaquantity index in which the user cost isadjusted to account for risk. The size oftheir adjustment depends on the agent’sdegree of risk aversion and on the covari-ance between the asset’s rate of return andthe agent’s consumption stream. Empir-ically, they find that there is negligibledifference between their generalized Divisia

index and a more standard index, for the setof monetary assets included in the FederalReserve Board’s monetary aggregates. Hence,we believe that the omission of risk adjust-ment in the monetary services indexesconstructed in this research project isunlikely to be empirically relevant.

CONCLUSIONThis paper has surveyed the

microeconomic theory of monetary aggre-gation. This theory is built from theaggregator functions of representativeagents. Applied to a representativeconsumer’s utility function, it generallyrequires that current-period quantities ofmonetary assets be weakly separable fromother assets, goods, services, and leisure.Applied to a representative firm’s produc-tion function, the theory requires thatcurrent-period monetary assets be weaklyseparable from other inputs.

The aggregation theory and methodssurveyed in this article are based onmodels of the optimizing behavior of rep-resentative economic agents. Thesemethods underlie the construction of ourmonetary services indexes and relatedvariables, presented in the followingarticle in this issue, as well as the Depart-ment of Commerce’s measures of realeconomic activity, such as GDP. Becausethese methods assume an optimizing rep-resentative agent, they have a commonfoundation with modern general-equilib-rium business-cycle models. Includingour quantity and dual user cost indexesin empirical models of economic activitydoes not require any stronger assumptionsthan those already embedded in theDepartment of Commerce’s measures ofaggregate economic activity.

REFERENCESAdvisory Commission to Study the Consumer Price Index. Toward a More

Accurate Measure of the Cost of Living, Final Report to the SenateFinance Committee, December 1996.

Anderson, Richard G. and Robert H. Rasche. “Measuring the AdjustedMonetary Base in an Era of Financial Change,” this Review(November/December 1996), pp. 3-37.



48

Ando, Albert. “On a Theoretical and Empirical Basis ofMacroeconometric Models,” Large-Scale Macro-Econometric Models:Theory and Practice, J. Kmenta and J.B. Ramsey, eds., Amsterdam:North-Holland, 1981, pp. 329-67.

Arrow, K. J. and F.H. Hahn. General Competitive Analysis, HoldenDay, 1971.

Barnett, William A . “Exact Aggregation Under Risk,” Washington University,Department of Economics, Working Paper #183, February 1994.

_______. “Reply [to Julio J. Rotemberg],” Monetary Policy on the75th Anniversary of the Federal Reserve System: Proceedings of theFourteenth Annual Economic Policy Conference of the Federal ReserveBank of St. Louis, Michael T. Belongia, ed., Kluwer AcademicPublishers, 1991, pp. 232-45.

_______. “Developments in Monetary Aggregation Theory,” Journalof Policy Modeling (Summer 1990), pp. 205-57.

_______. “The Microeconomic Theory of Monetary Aggregation,”New Approaches to Monetary Economics: Proceedings of the SecondInternational Symposium in Economic Theory and Econometrics,William A. Barnett and Kenneth J. Singleton, eds., CambridgeUniversity Press, 1987, pp. 115-68.

_______. “Definitions of ‘Second Order Approximation’ and ‘FlexibleFunctional Form’,” Economic Letters (vol. 12, 1983), pp. 31-5.

_______. Consumer Demand and Labor Supply: Goods, MonetaryAssets, and Time. Amsterdam: North-Holland, 1981.

_______. “Economic Monetary Aggregates: An Application of IndexNumber and Aggregation Theory,” Journal of Econometrics (Summer1980), pp. 11-48.

_______.“Theoretical Foundations for the Rotterdam Model,” TheReview of Economic Studies (January 1979), pp. 109-30.

_______. “The User Cost of Money,” Economic Letters (vol. 1,1978), pp. 145-9.

_______ and Yi Liu. “Beyond the Risk Neutral Utility Function,”paper presented at the University of Mississippi Conference on DivisiaMonetary Aggregation, October 1994; conference volume, Macmillanforthcoming.

_______ and Apostolos Serletis. “A Dispersion-DependencyDiagnostic Test for Aggregation Error: With Applications to MonetaryEconomics and Income Distribution,” Journal of Econometrics(January/February 1990), pp. 5-34.

_______ and Ge Zhou. “Financial Firms’ Production and Supply-SideMonetary Aggregation Under Dynamic Uncertainty,” this Review(March/April 1994), pp. 133-65.

_______, Douglas Fisher, and Apostolos Serletis. “Consumer Theoryand the Demand for Money,” Journal of Economic Literature(December 1992), pp. 2086-119.

_______, Melvin J. Hinich, and Warren E. Weber. “The RegulatoryWedge between the Demand-Side and Supply-Side Aggregation-Theoretic Monetary Aggregates,” Journal of Econometrics(October/November 1986), pp. 165-85.

_______, Barry E. Jones, and Travis D. Nesmith. “Divisia SecondMoments: An Application of Stochastic Index Number Theory.”International Review of Comparative Public Policy (vol. 8, 1996),pp. 115-38.

_______, Yi Liu, and Mark Jensen. “The CAPM Risk Adjustment forExact Aggregation over Financial Assets,” Macroeconomic Dynamics(May 1997), forthcoming.

Belongia, Michael T. “Weighted Monetary Aggregates: A HistoricalSurvey,” Journal of International and Comparative Economics (vol. 4,1995), pp. 87-114.

Berndt, Ernst R., M.N. Darrough, and W. Erwin Diewert. “FlexibleFunctional Forms and Expenditure Distributions: An Application toCanadian Consumer Demand Functions,” International EconomicReview (October 1977), pp. 551-76.

Caves, Douglas W., Laurits R. Christensen, and W. Erwin Diewert.“The Economic Theory of Index Numbers and the Measurement ofInput, Output, and Productivity,” Econometrica (November 1982),pp. 1393-414.

Chetty, V. Karuppan. “On Measuring the Nearness of the Near-Moneys,”The American Economic Review (June 1969), pp. 270-81.

Clements, Kenneth W. and H. Y. Izan. “The Measurement of Inflation: AStochastic Approach,” Journal of Business and Economic Statistics(July 1987), pp. 339-50.

Collins, Sean and Cheryl L. Edwards. “An Alternative MonetaryAggregate: M2 Plus Household Holdings of Bond and EquityMutual Funds,” this Review (November/December 1994), pp. 7-29.

Deaton, Angus and John Muellbauer. Economics and Consumer Behavior,Cambridge University Press, 1980.

Debreu, Gerard. Theory of Value: An Axiomatic Analysis of EconomicEquilibrium, Wiley, 1959.

Diewert, W. Erwin. “On the Stochastic Approach to Index Numbers,”University of British Columbia Discussion Paper No. DP 95-31,September 1995.

_______.“The Early History of Price Index Research,” Essays inIndex Number Theory, W. Erwin Diewert and Alice O. Nakamura,eds., Vol. 1, Amsterdam: North-Holland, 1993, pp. 33-65.

_______. “The Economic Theory of Index Numbers: A Survey,”Essays in the Theory and Measurement of Consumer Behaviour inHonour of Sir Richard Stone, Angus Deaton, ed., Cambridge UniversityPress, 1981, pp. 163-208.

_______. “Aggregation Problems in the Measurement of Capital,”The Measurement of Capital, Dan Usher, ed., University of ChicagoPress, 1980, pp. 433-539.

_______. “Superlative Index Numbers and Consistency inAggregation,” Econometrica (July 1978), pp. 883-900.

_______“Exact and Superlative Index Numbers,” Journal ofEconometrics (May 1976), pp. 115-45.



49


50

_______. “Intertemporal Consumer Theory and the Demand forDurables,” Econometrica (May 1974), pp. 497-516.

_______. “An Application of the Shephard Duality Theorem: AGeneralized Leontief Production Function,” Journal of PoliticalEconomy (May/June 1971), pp. 481-507.

Divisia, Francois. “L’Indice Monétaire et la Théorie de la Monnaie,”Revue d’Economie Politique (vol. 39, 1925), pp. 980-1008.

Donovan, Donald J. “Modeling the Demand for Liquid Assets: AnApplication to Canada,” International Monetary Fund Staff Papers(December 1978), pp. 676-704.

Duffie, Darrell. “Money in General Equilibrium Theory,” Handbook ofMonetary Economics, Benjamin M. Friedman and Frank H. Hahn, eds.,Vol. 1, Amsterdam: North-Holland, 1990, pp. 81-100.

Fama, Eugene F. “Banking in the Theory of Finance,” Journal ofMonetary Economics (January 1980) pp. 39-57.

Feenstra, Robert C. “Functional Equivalence Between Liquidity Costs andthe Utility of Money,” Journal of Monetary Economics (March 1986),pp. 271-91.

Fischer, Stanley. “Money and the Production Function,” EconomicInquiry (December 1974), pp. 517-33.

Fisher, Irving. The Making of Index Numbers: A Study of Their Varieties,Tests, and Reliability, Houghton Mifflin Company, 1922.

Fisher, Paul, Suzanne Hudson, and Mahmood Pradhan. “Divisia Indicesfor Money: An Appraisal of Theory and Practice,” Bank of EnglandWorking Paper Series No.9 (April 1993).

Friedman, Benjamin. “Discussion [of Lucas and Sargent (1978a)],”After the Phillips Curve: Persistence of High Inflation and HighUnemployment, Federal Reserve Bank of Boston Conference Series,No. 19, 1978, pp. 73-80.

Friedman, Milton, and Anna J. Schwartz. Monetary Statistics of theUnited States: Estimates, Sources, Methods, Columbia UniversityPress, 1970.

Frisch, Ragnar. “Annual Survey of General Economic Theory: The Problemof Index Numbers,” Econometrica (January 1936), pp. 1-38.

Goldman, S. M., and H. Uzawa. “A Note on Separability in DemandAnalysis,” Econometrica. (July 1964), pp. 387-98.

Gorman, W. M. “Community Preference Fields,” Econometrica(January 1953), pp. 63-80.

Green, H. A. John. Aggregation in Economic Analysis: An IntroductorySurvey, Princeton University Press, 1964.

Hancock, Diana. “A Model of the Financial Firm with Imperfect Asset andDeposit Elasticities,” Journal of Banking and Finance (March 1986),pp. 37-54.

_______. “ The Financial Firm: Production with Monetary andNonmonetary Goods,” Journal of Political Economy (October 1985),pp. 859-80.

Hulten, Charles R. “Divisia Index Numbers,” Econometrica (November1973), pp. 1017-25.

Hutt, W. H. Keynesianism - Retrospect and Prospect: A CriticalRestatement of Basic Economic Principles, Henry Regnery Company,1963.

Jorgenson, Dale W. “Capital Theory and Investment Behavior,” TheAmerican Economic Review (May 1963), pp. 247-59.

_______and James A. Stephenson. “Investment Behavior in U.S.Manufacturing, 1947-1960,” Econometrica (April 1967), pp. 169-220.

Kirman, Alan P. “Whom or What Does the Representative IndividualRepresent?” Journal of Economic Perspectives (Spring 1992),pp. 117-36.

Konüs, A.A. “The Problem of the True Index of the Cost of Living,”Econometrica (January 1939), pp. 10-29.

Konüs, A.A., and S.S. Byushgens. “K probleme pokupatelnoi cilideneg” (English translation of Russian title: “On the Problem of thePurchasing Power of Money ”), Voprosi Konyunkturi II(1) (supple-ment to the Economic Bulletin of the Conjuncture Institute), 1926,pp. 151-72.

Leontief, Wassily W. “Composite Commodities and the Problem of IndexNumbers,” Econometrica (January 1936), pp. 39-59.

Lucas, Robert E., and Thomas J. Sargent. “After Keynesian Macroeco-nomics,” After the Phillips Curve: Persistence of High Inflation andHigh Unemployment, Federal Reserve Bank of Boston ConferenceSeries, No. 19, 1978a, pp. 49-72.

Lucas, Robert E., and Thomas J. Sargent. “Response to Friedman” Afterthe Phillips Curve: Persistence of High Inflation and High Unemploy-ment, Federal Reserve Bank of Boston Conference Series, No. 19,1978b, pp. 81-82.

Malmquist, S. “Index Numbers and Indifference Surfaces.” Tradajos deEstatistica (vol. 4, 1953), pp. 209-42.

Muellbauer, John. “Community Preferences and the RepresentativeConsumer,” Econometrica (September 1976), pp. 979-99.

Niehans, Jurg. The Theory of Money, Johns Hopkins UniversityPress, 1978.

Orphanides, Athanasios, Brian Reid, and David H. Small. “TheEmpirical Properties of a Monetary Aggregate that Adds Bond andStock Funds to M2,” this Review (November/December 1994),pp. 31-51.

Pesek, Boris P., and Thomas R. Saving, Money, Wealth, and EconomicTheory, The Macmillan Company, 1967.

Ramsey, J. B. “Tests for Specification Errors in Classical Linear LeastSquares Regression Analysis,” Journal of the Royal Statistical Society,Series B (vol. 31 1969), pp. 350-71.

Rotemberg, Julio J. “Commentary: Monetary Aggregates and TheirUses,” Monetary Policy on the 75th Anniversary of the FederalReserve System: Proceedings of the Fourteenth Annual EconomicPolicy Conference of the Federal Reserve Bank of St. Louis,Michael T. Belongia, ed., Kluwer Academic Publishers, 1991,pp. 223-31.


_______ , John C. Driscoll, and James M. Poterba. “Money,Output, and Prices: Evidence from a New Monetary Aggregate,”Journal of Business and Economic Statistics (January 1995),pp. 67-83.

Samuelson, Paul A. “What Classical and Neoclassical Monetary TheoryReally Was,” Canadian Journal of Economics (February 1968).Reprinted in The Collected Scientific Papers of Paul A. Samuelson,Robert C. Merton, ed., Vol. 3, MIT Press, 1972.

_______ and S. Swamy. “Invariant Economic Index Numbers andCanonical Duality: Survey and Synthesis,” The American EconomicReview (September 1974), pp. 566-93.

Selvanathan, E. A. “Further Results on Aggregation of DifferentialDemand Equations,” The Review of Economic Studies (vol 58 1991),pp. 799-805.

Spencer, Peter. “Portfolio Disequilibrium: Implications for the DivisiaApproach to Monetary Aggregation,” The Manchester School ofEconomic and Social Studies (June 1994), pp. 125-50.

Theil, Henri. Theory and Measurement of Consumer Demand: Volume 1,Amsterdam: North-Holland, 1975.

_______. Principles of Econometrics, Wiley, 1971.

_______ . Economics and Information Theory, Amsterdam: North-Holland, 1967.

Thursby, Jerry G. “Alternative Specification Error Tests: A ComparativeStudy,” Journal of the American Statistical Association (March 1979),pp. 222-25.

_______and Peter Schmidt. “Some Properties of Tests forSpecification Error in a Linear Regression Model,” Journal of theAmerican Statistical Association (September 1977), pp. 635-41.

Triplett, Jack E. “Economic Theory and BEA’s Alternative Quantity andPrice Indexes,” Survey of Current Business (April 1992), pp. 32-48.

Young, Allan H. “Alternative Measures of Change in Real Output andPrices, Quarterly Estimates for 1959-92,” Survey of Current Business(March 1993), pp. 31-41.

_______. “Alternative Measures of Change in Real Output andPrices,” Survey of Current Business (April 1992), pp. 32-48.

Walras, L. Elements of Pure Economics, William Jaff, trans. London:Allen and Unwin, 1954.



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