Developments in monetary aggregation theory

Developments in Monetary Aggregation Theory William A. Barnett, University of Texas at Austin We survey the recent advances in monetaryaggregation theory as well as the earlier literatureon the subject. We discuss the limitations in the currentstate of our knowledge about monetary aggregationand speculate about productive areas for future research.

1. INTRODUCTION The fields of aggregation and index number theory both have long histories dating from the tum of the century, with particularly dramatic developments occurring in the 1920s, 1930s, and 1970s. The use of the techniques from those fields in the production of official government data began in the 1920s, and since that time nearly all data used in economics have been produced from methods generated from those literatures. For example, the Department of Labor's Consumer Price Index is a Laspeyres price index, real GNP is a Laspeyres quantity index , the Commerce Department's consumption-implicit Price Deflator is a Paasche price index , and virtually all of the quantity aggregates available from government agencies (the Department of Agriculture, the Department of Commerce, etc.) are Laspeyres quantity indexes , which often are supplied along with their dual Paasche price indexes. The use of those index numbers acquired solid aggregation-theoretic foundations in microeconomic theory in research produced during the past two decades . Nevertheless, there remains one conspicuous exception . The monetary quantity aggregates supplied by nearly every central bank in the world are not based upon index number or aggregation theory. At first glance that may seem surprising, but there are reasons-both good ones and bad ones.

IA. The Bad Reasons Most of the bad reasons result from the observation that all component monetary quantities are denominated in dollars, so some persons Address correspondence to William A. Barnett , The University of Te xas at Austin , The IC' Institute, 28 \5 San Gabriel Avenue , Austin , TX 78705--4081. I am indebted to David Laidler and J. Huston McCullo gh for their comments on this article .

JOUTMI of Policy Modeling \2(2):205-257 (1990) © Society for Policy Modeling, \990

205 0161-8938/90/$3.50

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therefore have long felt that there was no problem associated with "adding dollars and dollars." Monetary aggregation therefore seems to such persons to be different from adding " apples and oranges," which almost everyone has long suspected was somehow wrong. However, the commonality of expenditure units is irrelevant. Expenditure on any good is denominated in dollars, yet no one would add the quantity of subway trains in a city to the number of bicycles to acquire the city 's aggregate flow of transportation services. The reason is that the unit of denomination is irrelevant to the decision of whether to aggregate quantities by addition. The correct criterion is whether the component quantities are perfect substitutes. One does not" add apples and oranges" (or subway trains and bicycles) , because they are not perfect substitutes. Commonality of units is neither necessary nor sufficient for aggregation by simple summation. Furthermore, nominal units of all goods, not just monetary assets, are denominated in dollars. Real balances are unitless. A related error is produced by observing that the stock of money appears to be the simple sum of nominal units. Simple sum aggregation would indeed be the correct method of measuring the nominal accounting stock of money, if none of the components yielded interest , as once was the case with MI . However, when some components yield interest, the simple sum stock discounts to present value both the flow of expenditure on monetary services and the investment yield on the components. But the economic capital stock of money is the discounted value of the expenditure on monetary services alone-excluding the investment yield, which has never been viewed as a monetary service, whether by monetarists, Keynesians, or new-classical economists. Hence, even as a measure of stock, the simple sum aggregates confound the monetary service flow and the investment motive. As mentioned above, there also were good reasons for delaying the conversion of monetary aggregates to those produced by aggregation and index number theory. However, those valid issues are far more subtle than the denominational illusion just described. The purpose of this article is to discuss the good reasons and to describe the recent evolution of research designed to deal with those real problems. The primary successful and unsuccessful contributions to the field are presented and discussed. The remaining unsolved issues are described, and the fruitful directions for future evolution of that research are considered.

lB. The Good Reasons Until relatively recently, the fields of aggregation theory and statistical index number theory were developing independently. Aggrega-

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tion-theoretic aggregation over goods was a part of microeconomic theory. The theory produced conditions under which decisions could be solved in two stages such that the first-stage decision was in aggregate quantities and aggregate prices, and such that the aggregates could not be distinguished behaviorally from elementary goods. As a result of that perfect shadow-world decision, the quantity and price aggregates produced from aggregation theory are viewed as the exact aggregates. An example is the Koniis "true cost of living" index produced from the consumer's cost function. From the standpoint of construction of official governmental data, the problem with the exact aggregates of aggregation theory is that they depend upon unknown aggregator functions, which typically are utility functions, production functions, cost functions, or distance functions. Such functions can be used only if econometrically estimated. Hence, the resulting exact quantity and price indexes, once estimated, become estimator- and specification-dependent. This dependency is troublesome to government agencies, which therefore have always viewed aggregation theory as being solely a research tool. Statistical index number theory, on the other hand , provides quantity and price indexes that are directly computable from quantity and price data, without the need to estimate unknown functions. Such index numbers, whether price or quantity indexes, always depend jointly upon all prices and all quantities , but not upon any unknown parameters. The exact aggregates of aggregation theory , however, depend upon unknown parameters , and either solely upon quantity data (if a quantity aggregate) or solely upon price data (if a price aggregate). In a sense, index number theory introduces joint dependency upon prices and quantities in exchange for lack of dependency upon unknown parameters . Examples of such statistical index numbers are the Laspeyres, Paasche, Divisia , Tornqvist, and so on. Despite the obvious usefulness of such statistical index numbers, those indexes until recently could not be produced directly from aggregation-theoretic microeconomic theory. As a result, statistical index numbers were judged primarily by their known "good" and "bad" properties, none of which were their order of approximation to the exact aggregates of aggregation theory.' The resulting lack of a direct link between aggregation theory and index number theory produced a division between some index number

' An exception is the Divisia index in continuous time, which has long been known to have the remarkable ability to track any arbitrary unknown aggregator functions without any error at all.

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theorists and some aggregation theorists. In particular, some aggregation theorists viewed index number theory as somewhat ad hoc and not based upon solid theoretical foundations. In addition, the lack of a direct link between index number theory and aggregation theory resulted in the production of a tremendous number of proposed index number formulas in index number theory. Many index numbers were produced that had a large number of "good" properties. Although most of the reputable index numbers that appeared in the literature tended usually to move closely together, the proliferation of proposed index numbers tended to reinforce the impression that index number theory was somewhat ad hoc and only loosely related to theory. These difficulties in the index number theory literature were paralleled in the monetary theory literature by a proliferation of proposed ad hoc monetary "weighting" schemes, such as weighting by bid-ask spreads, turnover rates, and so forth. The appearance of arbitrariness and theoretical looseness in that research undoubtedly contributed to the fact that central banks have never used index numbers in constructing their monetary aggregates. Virtually all other government data-producing agencies, on the other hand, were willing to overlook those difficulties, since those agencies correctly viewed the problems of simple sum or arithmetic average aggregation as being far worse than the problems that existed in the professional index number literature. In addition, monetary aggregation was the one area in which it was not immediately evident how to apply statistical index number theory. The reason was that quantity indexes depend upon prices as well as quantities, and it was not clear what to view as being the "price" of a monetary asset. For the same reasons discussed above, the fact that all monetary assets are denominated in dollars does not imply that the price of any monetary asset is one dollar. For example, the opportunity cost of foregoing the services of a non-interest-bearing asset is clearly not the same as the opportunity cost of foregoing the services of an interest-bearing asset. That problem was only recently solved by Barnett (1978, 1980a), when he derived the formula for the user cost of demanded monetary services.' More recently Barnett (1987) has done the same for .supplied monetary services.' Hence, until recently the link between statistical index number theory and microeconomic ag-

'For a related less formal derivation producing the same result, see Donovan (1978). 'A similar, but not quite identical, result has been produced by Hancock (1985, 1986a,b), who explicitly introduced transactions costs into the user cost formula,

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gregation theory appeared to be even weaker in monetary quantity aggregation than in other areas of quantity aggregation. The loose link between index number theory and aggregation theory was turned into a solid one by Diewert (1976). He defined the class of "superlative" index numbers, which approximate arbitrary exact aggregator functions up to a third-order remainder term. Statistical index number theory became part of microeconomic theory, as aggregation theory had been for decades. With the introduction of Barnett's results on the user cost of the services of monetary assets, the scene was set for the introduction of index number theory into monetary economics." The case for using statistical index number theory in monetary aggregation is now as strong as in aggregation over any durable good. The remaining difficulties and issues are also the same. In the last sections of this article, we discuss those remaining issues .

Ie. The Link Between Aggregation Theory and Index Number Theory As just discussed, the direct relevancy of index number theory to monetary economics was produced from two developments. The first was Diewert's introduction of superlative index numbers, which closed the gap between index number theory and aggregation theory. The second was Barnett's derivation of the user cost of monetary asset services, which closed the gap between monetary theory and economic aggregation theory. In Sections 2-9, we survey Barnett's most recent results connecting monetary theory with economic aggregation theory. That connection is established in the case of three classes of decision makers: consumers, who use monetary services; financial intermediaries, which produce monetary services; and manufacturing firms, which use financial services. In those same sections , we also briefly review the theory relevant to Diewert's successful merging of index

'For empirical applications, see Barnett (l980b, 1981b, 1982b, 1983a, 1984, 1987); Barnett et aI. (1984); Barnett and SerIetis (l989a); Barnett and Spindt (1979, 1982); Batchelor (1989); Belongia and Chalfant (1989); Belongia and Chrystal (1988); SerIetis (I 987a ,b, 1988a,b); Cagan (1982); Ewis and Fisher (1984, 1985); Barnett and Hinich (1989); Barnett et aI. (1986); Barnett, Hinich, and Vue (1989); Cockerlinc and Murray (1981); Dorfman (1983); Drake (1989); Fase (1985); Fayyad (1986a ,b); Fisher and Serletis (1989); Serletis and Robb (1986); Marquez (1986); Ishida (1984); and Swofford and Whitney (l986a,b, 1988). For a survey of alI of that empirical research, see Barnett, Fisher, and Serletis (1989). For a survey of the theory , see Barnell (1982a, 1983b) and Barnell et aI. (1981).

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number theory with economic aggregation theory-as those results relate to monetary theory. 2. THE CONSUMER In this section we formulate a consumer 's decision problem over goods and monetary assets. The decision will be structured in a manner that provides immediate applicability of the relevant literature on economic aggregation over consumer goods. Two versions of the decision problem will be provided. The first version will use a finite planning horizon and will be an extension of Barnett's (1978, 1980a, 1981a) formulation. The second version will use an infinite planning horizon and will produce the same aggregationtheoretic results as the finite horizon version. The reason is that both versions produce the same current period aggregator function over monetary assets, and both versions produce the same user cost formulas for monetary assets.

2A. Finite Planning Horizon The following variables are used in the decision problem of consumer + I, ... , s, ... , t + T, where T is the number of periods in the consumer's planning horizon: x, = vector of planned consumption of goods and services during period s, p, = vector of goods and services expected prices and of durable goods expected rental prices during period s, IDs = vector of planned real balances of monetary assets during period s, r, = vector of expected nominal holding period yields for monetary assets during period .'I, As = planned holdings of the benchmark asset during period .'I, R, = the expected one-period holding yield on the benchmark asset during period .'I, L~ = planned labor supply during period s, Is = k - L~ = planned leisure demand during period s, where k = total hours available per period, w, = the expected wage rate during period s, and I,. = other expected income (government transfer payments, profits of owned firms, etc.) during period .'1. 5 The benchmark asset is defined to provide no services other than its yield, R" which motivates holding of the asset solely as a means of accumulating wealth. As a result, R, is the maximum expected holding period yield in the economy in period s; and the benchmark asset is c, solved by the consumer in period t for the periods t, t

51f the expected demand for the consumer's labor is not equal to planned supply, then "', is the expected shadow price ofleisure. See Barnett (l980a, 1981a).

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held to transfer wealth between multiperiod planning horizons, rather than to provide liquidity or other services. The consumer's intertemporal utility function in period t is blockwise weakly separable as follows. 6

-

u, == U,(u(m,) , u.; I(m ,+ I) , .. . , u,+y(mr t T);L" . . . V(X,) ,

V"

-

,L' +T~

(2.1)

,(x,+I), .. . , V, +l{xa T);A,i T)'

Then dual to the functions v and VL (s = t + 1, ... ,1 + T) there exist current and planned true cost of living indexes , p~ and p: (s = t + I, . .. , t + T), which can be used to deflate nominal to real values. Assuming continuous replanning at each t, the consumer's decision is to chose (m., ... mt+T, I" ... , I t + T; x, ... , x t + T; At + T) to maximize u, subject to the T + 1 budget constraints P:x,

= w,L;

+

t [(I

+

ri,,_,)P's- Im,., _. - p:m .,]

I)P~- 'A '_I -p:A,l

[O+R,

+ I.,

+

(2.2)

for s = t, t + 1, . . . , t + T. Let (m:, ... , T , It, ... , I~+T; xt, ... , Xt+T; At+T) be the solution to that constrained optimization problem. Following the procedures in Barnett (l980a, 1981a), it then can be shown that mt is also the solution for m, to the following current period conditional decision:

m:+

(2.3)

maximize u (m .) subject to

'IT,' m ,

= y"

where 'Ii, = ('iiI" "" 'lint)' is the vector of monetary asset nominal user costs 'IT

«

r:

R, - T» Pt* 1 +R,'

(2.4)

and Yt = 1f,'m~. The function u is assumed to be monotonically increasing and strictly concave. Decision problem (2.3) is in the form of a conventional consumer decision problem, and hence the literature

"Regarding the existence of this derived utility function. into which the consumer's transactions technology has been absorbed , see Arrow and Hahn ( t97 1). Feenstra ( 1986), Phlips and Spinnewyn (1982) , and Samuelson and Sato (1984) ,

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on aggregation theory and index number theory for consumers is immediately available.

2B. Infinite Planning Horizon Barnett (1987) has reformulated the consumer's decision problem by using an infinite planning horizon. He shows that the same current period conditional decision problem, (2.3), is acquired . As a result, the exisitng literature on aggregate and index number theory will remain relevant. He replaced the finite planning horizon intertemporal utility function, (2.1), by the infinite horizon intertemporally separable function u,

~ ( 1+1 ~ ) =~,

s -,

-

U(u(m,), L" x.),

(2.5)

where £ is the consumer's subjective rate of time preference , assumed to be constant to assure Strotz-consistent planning ." The consumer selects the sequence (m., I S! x.), s = t, t + I, . . . , to maximize (2.5) subject to the sequence of constraints , (2.2), for s == t, t + I, . .. The upper limit, t + T, to the planning horizon no longer exists. Barnett (1987) showed that the solution for m, also solves problem (2. 3). As a result, we again find that we can use the conventional neoclassical decision, (2.3) , and thereby all of the existing literature on aggregation over goods consumed. The results above do not explicitly incorporate taxes. Nevertheless all of those results would remain valid if we converted to after-tax yields .

3. SUPPLY OF MONETARY ASSETS BY FINANCIAL INTERMEDIARIES Monetary asset'I generally are either primary secunnes, such as currency and Treasury bills, or assets produced through the financial intermediation of financial firms. In this section, we develop a model of production by financial intermediaries under perfect certainty, or equivalently under risk neutrality with random variables replaced by their expectations. It will be shown that the model can be manipulated into the conventional neoclassical form of production by a multiproduct firm. As a result, the existing literature on output aggregation becomes 7The same current period conditional decision would result if ~ were not constant. If intertemporal separability were not assumed, expectations could be endogenizcd through rational expectations , as in Attfield and Browning (1985).

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213

immediately applicable to the construction of a neoclassical money supply function for aggregated money. Consider a financial intermediary that makes only one kind of loan, yielding and produces , through financial intermediation, a vector of monetary asset real balances , fLr- The finn uses c, real units of excess reserves, in the form of currency , as a factor of production." Real balances , such as fL, and c., are defined to equal nominal balances divided by P:, defined in Section 2. The finn also uses as inputs the vector of labor quantities L, and the vector z, of other factor quantities . The vector of reserve requirements is k., where kit is the reserve requirement applicable to j..Lit with 0 $ k; $ I for all i. The firm's efficient production technology is defined by the transformation function F(fl.t, z; L" c.; k,) = 0. 9 The finn's technology equivalently can be defined by its efficient production set (also called the production possibility efficient set)

R"

S(k,) = {(u"z"L" c,)

2:

O:F(u"z"L,;k,) = O}.

Since our model contains only one kind of primary market loan, yielding R" the federal funds rate therefore also must equal RI • However , the discount rate, being regulated, can differ from R I • Let R~ be the discount rate during period t, and define

R

t

= min

{R"R~} . We

assume that required reserves are never borrowed from the Federal Reserve, but could be borrowed in the federal funds market." Excess reserves can be borrowed from either source . Let W, be the vector of wage rates corresponding to labor quantities L" and let p, be the vector of yields paid by the finn on fl., . Also let

8We treat monetary assets produced by financial intermediation to be outputs of financial intermediaries , and we thereby implicitly assume that the user costs of such assets are positive. Hancock (1985, 1986a,b) postulates that some such assets can be inputs to financial intermediaries if the corresponding user costs are negative . That possibility is not excluded by the formulation presented below, although we shall not explicitly discuss the probably unusual case of negative user costs . "The transformation function, F, is strictly quasiconvex in (.... t z" L" c,). In addition , iJFliJL" < 0, iJFliJ z < 0, and iJFl iJc, < 0, since L" Z" and c, are inputs. Conversely , iJFliJ!L" > 0, " since Il., consists of outputs . All factors , z, are purchased at the start of period t for use during period t, and the firm must pay for those factors at the start of the period. The exception is labor, L" which receives its wages at the end of period I for labor quantities supplied to the firm during period t. Interest on produced monetary assets ...., is paid at the end of the period, and interest on loans outstanding during period t is received at the end of period I. '<>-rhis assumption of " perfect moral suasion" could easily be weakened or removed.

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W.A. Barnett

_ * (I 'Y" - p,

k,,)R, - p" I +R '

(3.1)

t

and let -

'Yo, -- p,*~ 1 +R . ,

(3.2)

Let the vector of all variable factor quantities be at = (Z;,L; .c.)', and let P, = (a;, w; /(1 + Rt ) , 'Yot)'. Barnett (1987) showed that variable profits take the conventional form P, = p,;'Y, -

a;p"

(3.3)

and the firm's variable profit maximization problem takes the conventional form of selecting (flt,aJ E S (k.) to maximize (3.3). Also see Hancock (1987). Hence the existing literature on output aggregation for multiproduct firms becomes immediately applicable to aggregation over the produced monetary assets flt as well as to measuring value added and technological change in financial intermediation. Observe that "'lot is the nominal user cost of excess reserves (real balances of currency), and "'lit is the nominal user cost price of produced monetary asset flit Hence Pt is the vector of factor prices corresponding to the factor at. Important contributions to this literature on financial intermediation have been made by Hancock (1985, 1986a,b), Fixler (1989), and Fixler and Zieschang (1989a,b,c).

3A. Properties of the Model Observe that variable revenue can be written in the form t

,

b

p,,'Y, = p,,1T, -

pt R,k;p" I

+ R, '

where 'lT~ =

pt (R, - p,,)/(I + R,)

(3.4)

has the same form as the monetary asset user cost formula, 7r it, for consumers, (2.3). Clearly 7r~ would equal "lit if k, = O. As a result, it is evident that pt R.k; ....,;(1 + R,) is the present value at the beginning of the period of the tax pt R.k; fl, "paid" by the financial intermediary at the end of the period as a result of the existence of reserve requirements. The tax is the foregone interest on uninvested required reserves.

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215

3R. Separability of Technology If the user costs, "(" all moved proportionally , then we could use Hicksian aggregation to aggregate over the firm's joint monetary supplies, f.1,. But since that proportionality assumption is not typically reasonable for monetary asset user costs , aggregation over outputs is possible only if outputs are separable from inputs in the financial firm's technology . Hence, in order to establish the existence of an output aggregate , we shall assume that there exist functions f and H such that F (f.1" z" L" c,;k,) = H(ftf.1,;k,),z"L"c,). There will exist a function g such that fll1 ,;k,) = g(z"L"c,)

(3.5)

is the solution for flJ1 ,;k,) to H(flf.1,;k,),z"L"c,) = O. The function flJ1,;k ,) is called the factor requirements function, since it equals the right-hand side of (3.5), which is the minimum amount of aggregate input required to produce the vector J1,. The function g(z"L"c,) is the production function, since it equals the left-hand side of (3.5), which is the maximum amount of aggregate output that can be produced from the inputs (z"L"c,). Hence, f is both the factor requirements function and the outputs aggregator function, while g is both the output production function and the inputs aggregator function. II

4. DEMAND FOR MONETARY ASSETS BY MANUFACTURING FIRMS In addition to consumer demand for monetary assets , there also is demand for monetary assets by manufacturing firms, which we define to be any firms that are not financial intermediaries. In this section, we formulate the decision problem of such a manufacturing firm when monetary assets enter the firm's production func-

"We assume that j is strictly convex and linearly homogeneous in tA,. In addition , it follows from our assumptions on the derivatives of the transformation function , F, that g is monotonically increasing in all of its arguments and f is monotonically increasing in tA,. We assume that g is locally strictly concave in a neighborhood of the solution to the first-order conditions for variable profit maximization . In addition , it follows from the strict quasiconvexity of the transformation function, F , that g is globally strictly quasiconcave .

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W.A. Barnett

tion." The firm is assumed to maximize the discounted present value of its profit flow subject to its technology. The firm's intertemporal technology over its T-period planning horizon is defined by its transformation function: (4.1)

where for t :S S :s t + T we define: 8s = vector of planned production of output quantities during period s, Es = vector of planned real balances of monetary assets held during period s, and K s = vector of planned use of other factors during period s. 13 The firm's decision problem is formulated in period t for periods t, t + 1, . . . , s, ... , t + T, where T is the number of periods in the firm's planning horizon. During period s, the firm's profits are 'I' =

fi;v; -

K:~; + ~

[(l +

",-I)

P7-,€i.<-1 - P7€i,] ,

(4.2)

where Vs = vector of output expected prices and ~s = vector of prices of the factor K s . 14 Define the discount factor, Ss> such that 8s = 1.0 for s = t and s-(

es

t,

=

II (l

+

R a ) for t

+ 1

:S S ::::; t

+ T +

1. Let Vs =

v,;e s and

a=t

=

~,;es be the discounted present values of prices

Vs

and ~s>

respectively.

"Since manufacturing technology typically does not depend upon money balances, our production function is actually a derived production function acquired by absorbing the finn's transactions technologies in factor markets into the production function. The existence of such a derived production function follows from the same analysis used to prove the existence of a derived utility function containing monetary balances. See Arrow and Hahn (1971), Phlips and Spinnewyn (1982), and Samuelson and Sato (1984). For explicit derivations of the derived production function based upon a Baumol-Tobin transactions model or a vending machine model, see Fischer (1974). Fischer (1974, p. 532) concludes that real balances can be entered into production or utility functions unless a "deeper explanation of the demand for money" is needed. Since we are not seeking to explain why people demand money,but rather how they behave when money has value, we have no need for a deeper explanation for motives. "The transformation function, n, is assumed to be strictly quasiconvex. In addition afl/aS" > 0, afl/SEO u < 0, and afl/ilKj , < O. l4To simplify the notation, we assume that consumers and manufacturing firms have access to the same monetary assets, so the expected nominal holding period yields on E, can be viewed as being r., defined in Section 2A. Real balances, E" are defined to equal nominal balances divided by p~, defined in Section 2A. In principle, the price deflator could differ for different economic agents, although we do not permit that complication to arise. For a rigorous treatment of separate input, output, and consumption price deflators, see Fisher and Shell (1972, essay 2; 1979, 1981).

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217

Barnett (1987) has shown that the discounted present value of variable profits is I t T

'l'~

r+ T

t +-T

= 2: 8.: v, - 2: Ki - 2: e; TI"

(4. 3)

where the user cost of Ei . is T) is = (p ~ /6 s ) - (l + r is)p'1' /6s + I' Clearly (4.3) is in conventional form. In addition, the user cost , T) is, in the current period s = t is

_ * (R,-r,,)

T),,-P,

I+R,'

(4.4)

which is in familiar form (see (2.3) and (3.4) above. With the conventional decision problem of maximizing variable profit, (4.3), subject to (4.1), we have immediate access to the existing literature on aggregation over factors of production, in order to aggregate over the monetary assets E, demanded by the manufacturing firm.

4A. Separability of Technology For the same reason discussed in Section 3B, we shall require that technology be separable , although this time separability will be assumed in current monetary assets used as inputs (by the manufacturing firm), rather than in monetary assets produced as outputs (by the financial intermediary) . In particular, we assume that there exists functions a and B such that n(o" . .. , o' +T> E , E, i-T. K " . . •• K,+T) = B(o/, , o/+T> aCE,), E'+I. ···' E,+T> K"" , K,+1) ' Let (8;t', . .. , 8tH' E;t' , E;t'+ T' K;t', . . . , K;t'+T) be the solution to maximizing (4.3) subject to (4.1), and let b, = E;t" '1),. Then it follows that E;t' must also be the solution for E/ to the current period conditional decision: maximize a(e,) subject to TI,' er

=

b.;

(4.5)

which is in the same form as (2.3) for the consumer. The function aCE,) is assumed to be monotonically increasing and strictly concave in E, . The large literature in aggregation and index number theory based upon the conventional consumer decision (2.3) is immediately applicable to decision (4.5) and hence to aggregation over E,.

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W.A. Barnett

5. AGGREGATION THEORY UNDER HOMOGENEITY The theory of aggregation over goods directly produces unique, exact results when the aggregator function is linearly homogeneous. In that case, the growth rates of the aggregation-theoretic price and quantity aggregates are independent of selected reference levels for utility, prices, or quantities. In addition, the dual quantity and price aggregates then behave in a manner indistinguishable from that of an elementary good. In this section, we discuss that most elegant of situations. In the next section, we present recent theory relevant to aggregation in the nonhomothetic case. SA. Consumer In this section, we seek to produce the exact aggregation-theoretic aggregate over the monetary asset quantities m, of Section 2. As shown in Barnett (l980a, 1981a), the exact quantity aggregate is the level of indirect (i. e., optimized) utility.

M;

= max {u(m,):1T;m,=y,}, mt

(5.1)

so u is the aggregator function that we assume to be linearly homogeneous in this section. Dual to any exact quantity aggregate exists a unique price aggregate, which aggregates over the prices of the goods. Hence there must exist an exact price aggregate over the user costs Tr,. As shown in Barnett (l980a, 198Ia), the consumer behaves relative to the dual pair of exact quantity and price aggregates as if they were the quantity and price of an elementary good. As a result, the aggregate is empirically indistinguishable from an elementary good. One of the properties that an exact dual pair of price and quantity aggregates satisfies is Fisher's "factor reversal" test, which states that the product of an exact quantity aggregate and its dual exact price aggregate must equal actual expenditure on the components. Hence, if IY(Tr,) is the exact user cost aggregate dual to M~', then rrC(Tr,) must satisfy rr'(1T,) =

~~.

(5.2)

Barnett (I 983b) has shown that (5.2) can equivalently be written TI'"(Tr,) = (max{u(m,): Tr;m, = 1})-1, which clearly depends only upon 'TT" mt

as would be expected of a price aggregate. Although (5.2) provides a valid definition of TI~, a direct definition (not produced indirectly through M~ and Fisher's factor reversal test)

DEVELOPMENTS IN MONETARY AGGREGATION THEORY

219

is more informative and often more useful. The direct definition depends upon the cost (or expenditure) function, E, defined by E(uooTr,)

=

rm~ {n;m,:u(m,)

=

Uo } .

It can be proved (see, e.g ., Shepard,

1970, p. 144) that

IT' (1T,) =

E(l ,1T,) =

~ {1T;m,:U(m,) = I} ,

(5.3)

which is often called the unit cost or price function. The duality between and n~ is evident from (5.1) and (5.3), which use dual decision problems. We now have the fundamental aggregation-theoretic tools for aggregating over goods within the decision of a consumer. The reason that M~' and II~ are called aggregates is that (M~',rr~") can be used to decompose the consumer's decision into a two-stage budgeting process. In the first stage, M~ is treated as an elementary good with price II;' within the intertemporal utility maximization decision . In particular , M~' appears in place of u(m,) within the intertemporal function , U,. Having solved for M~ in the first stage, the consumer then solves for m, from the second stage decision, (2.3), with u, determined from M~n:· . For all possible nonnegative values of prices and wealth, the two-stage decision will produce the same solution as the original complete decision defined in Section 2A or Section 2B. Since the firststage decision in the aggregates has the usual neoclassical form, (M~' ,n~) are behaviorally indistinguishable from the quantity and price of an elementary good. The details of the two-stage budgeting theorem are available in Barnett (1980a, 1981a) and Green (1964, Theorem 4) .

M~

58. Manufacturing Firm Since the decision problems (2.3) and (4.5) are in the same form, the aggregation theory in Section SA for consumer demand for monetary assets is immediately applicable to aggregation over monetary asset demand by a manufacturing firm. The only change is in the interpretation of the quantity aggregator function. With a consumer, the quantity aggregator function is the category subutility function, u. With a manufacturing firm, the quantity aggregator function is the category subproduction function, aCE,) . Clearly in both cases, the quantity aggregator function is the objective function of the corresponding conditional decision problem, (2.3) or (4.5) . However, in the case of demand by a manufacturing firm, a particularly interesting interpretation of the derivable two-stage budgeting process is available, as has

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been observed by B1ackorby et al. (1978, p. 210). In our case , the first-stage decision is for the firm to solve for the profit-maximizing output supplies and factor demands, but with financial asset services demand determined only in the aggregate. In the second stage, the firm solves for the cost-minimizing financial asset portfolio at fixed aggregate financial services flow from the portfolio. See Barnett (1987) for details and the relevant proofs. It follows from the production analog of (5.1) that the firm's solution value for a-;: = a( must equal its exact monetary quantity aggregate

En

(5.4)

The corresponding exact economic price aggregate clearly is I1{(y),), which is acquired from (5.3) with the symbols from the consumer's decision replaced in the obvious manner by the corresponding symbols from the firm's decision.

5C. Financial Intermediary The aggregation theory relevant to aggregating over the outputs of the multiproduct financial firm is analogous to that for aggregation over the financial inputs of the manufacturing firm, but with the manufacturing firm's cost function replaced by the financial firm's revenue (or " benefit" ) function. In that manner we can produce a two-stage decision for the financial intermediary. In the first stage the firm solves for profit-maximizing factor demands and the profit-maximizing quantity of aggregate financial assets produced . In the second stage , the revenue-maximizing vector of individual financial asset quantities supplied is determined at fixed aggregate financial asset quantity supplied. See Barnett (1987, Section 5.3) for details. The financial firm's revenue function is

R* (lr,,'Y,;k,)

= m,t/x {1-l:'Y,fil-l,:k.)

= g(lr')}

(5.5)

By Shephard's (1970, p. 251) Proposition 83 it follows that there exists a linearly homogeneous output price aggregator function r such that R* (a/l'Y,;k,) = r (')',)g(a,). The exact economic output quantity aggregate for the financial firm is

DEVELOPMENTS IN MONETARY AGGREGATION THEORY M~

= fl ....~;k,) =

221 (5.6)

g(an,

r

when ... is the profit-maximizing vector of monetary assets produced, is the profit-maximizing factor demands. The corresponding and output price aggregate is r~ = f( 'Y J. It is easily shown that

a:

[('V,) =

m,:: {11:'V,;f{....,;k,)

=

I}.

(5.7)

which is the unit revenue function. It can be shown (see Barnett, 1987) that

M~ = 'W~ {fll1,;k') :I1:'V' =

n'V,)g(a ,)}

(5.8)

Comparing (5.7) and (5.8), we can see the clear duality between the decision problems . As usual, the exact quantity and price aggregates of economic theory are true duals . By nesting weakly separable blocks within weakly separable blocks, a hierarchy of nested exact aggregates can be produced. See Barnett (l980a, 1981a) for the construction of such a hierarchy of exact monetary assets for a representative consumer. The analogous nesting is possible for the manufacturing finn or financial intermediary .

6. INDEX NUMBER THEORY UNDER HOMOGENEITY The results in Section 5 provide unique exact economic aggregator functions for aggregating over monetary assets demanded or supplied by each of the three classes of economic agents . Each of the resulting monetary quantity aggregates depends only upon the component monetary asset quantities and the form of the aggregator function . To use such an aggregate, it is necessary to select a parameterized econometric specification for the aggregator function and estimate its parameters. Estimating aggregator functions and exploring their properties plays an important role in the aggregation theory literature. However, for many purposes the most useful aggregates are those produced without the need to estimate unknown parameters. In that case, nonparametric approximations to the unknown aggregator functions are needed. The production of such nonparametric approximations is the subject of index number theory . Index number theory eliminates the need to estimate unknown parameters by using both prices and quantities simultaneously along with approximation techniques often resembling revealed preference theory , in order to estimate economic quantity aggregates, which depend only

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upon quantities (and not prices) . Similarly index number theory uses both prices and quantities simultaneously in order to approximate economic price aggregates, which depend only upon prices (and not quantities) .

6A. Consumer and Manufacturing Firm We present the index number theory only for the consumer in this section. The corresponding results for the manufacturing firm could be acquired immediately by changing notation in the obvious way. If m~ is acquired by solving (2.3), then u(mt) is the exact monetary aggregate M~. In continuous time, ~ = u(mt) can be tracked without error (see Barnett, 1983b, for a proof) by the Divisia index, which provides M~ as the solution to the differential equation d1ogM~/dt =

L sitdlogmi~/dt,

(6.1)

where Sir = 'TTi,mt/u t is the ith asset's share in expenditure on the total portfolio 's service flow. In continuous time the Divisia index, under our conventional neoclassical assumptions, is perfect. However in discrete time many different approximations to (6.1) are possible. The most popular discretetime approximation to the Divisia index is the Tornqvist-Theil approximation (often called the Tornqvist index), which is just the Simpson's rule approximation: log M; - log M;_ , =

L

Sit (log

m% - log mr,-I) ,

(6 .2)

where Sit = (l/2)(Sit + Su- 1)' In discrete time we shall call (6.2) simply the Divisia index. Recently a very compelling reason has appeared for using (6.2) as the discrete-time approximation to the Divisia index. Diewert has defined a class of index numbers, called "superlative" index numbers, which have particular appeal in producing discrete-time approximations to M~ = u(mn. Diewert defines a superlative index number to be one that is exactly correct for some quadratic approximation to u. The discrete Divisia index (6.2) is in the superlative class, since it is exact for the translog specification for u. The translog is quadratic in the logarithms. As a result, if the translog specification is not exactly correct, the discrete Divisia index (6.2) has a remainder term that is third-order in the changes, since quadratic approximations possess third-order remainder terms. With weekly or monthly monetary asset data, the Divisia index (6.2) is accurate to within three decimal places,

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which typically is smaller than the data's round-off error (see Barnett, 1980a).15 The Fisher ideal index is another popular element of Diewert's superlative class, and it was proposed for use with monetary data, along with the Divisia index , by Barnett (l980a, 1981a). The Fisher ideal index is exact for the square root of the quadratic specification for u. Usually the Fisher ideal index and the Divisia index are identical to within round-off error with monetary data. Nevertheless, on theoretical grounds the Divisia index is usually preferred by index number theorists, largely as a result of its uniquely attractive properties in the nonhomothetic case to be discussed in Section 8 below. When the quantity aggregate , M~, is acquired from the Divisia index, the dual user-cost price index, called the implicit Divisia price index, is acquired from Fisher's factor reversal test, 'IT; = u,lM:'. Since relative user costs usually vary more than relative quantities, use of the Divisia index to produce the quantity aggregate, with the price aggregate produced from factor reversal, is preferable to the converse . See Allen and Diewert (1981) . When M;' is produced from the discrete Divisia index , it is easily shown that the implicit Divisia price index , produced from factor reversal, is superlative in the Diewert sense.

6B. The Financial Intermediary The financial intermediary's output aggregation is produced from a decision that does not have the same form as the decision that produced the Divisia index for the consumer or manufacturing firm. Monetary output aggregation is produced by solving the financial intermediary's second-stage decision for fl.7 and substituting it into f to acquire M ~ = fi.J1.1 ; k.). That second-stage decision is to select u, to maximize JL: "f, subject to f (JL,;k,) =

(6 .3)

M7.

Barnett (1987) has proved that the Divisia index tracks M~ without error in continuous time, so long as ....7 is continually selected to solve (6.3) at each instant, t. In particular, if J.lt solves (6.3) continually at each instant t E To> then for every t E To> d 10gM~ / dt = 2: d log

st

r

i

fl.t / dt, where the asset's share in the financial intermediary's revenue from production of monetary services is = "titfl.tl-Y:tJ1.r.

st

'SIn addition, Star and Hall (1976) derived an analytic expression for the error in the discrete Divisia approximation to the exact continuous-time Divisia index, He found that the error was always small if the shares do not fluctuate wildly.

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Hence the Divisia index is equally as applicable to aggregating over the monetary assets produced by the financial intermediary as over the monetary assets purchased by the consumer or manufacturing firm. In addition, Simpson's rule again produces the Tornqvist-Theil discretetime approximation log M~ - log M~_, =

2: s~ (log JLt

- log JLt,- ,),

(6.4)

where st = (112) (st + st-l)' Futhermore, if the input requirement function, f, is translog, then the discrete Divisia index (6.4) is exact in discrete time (see Diewer, 1976, p. 125). Hence (6.4) is a superlative index number. The reason for preferring the Divisia index over the other superlative indexes will be evident from the nonhomothetic case below. While the Divisia monetary quantity index takes the same form for all three economic agents, the financial firm's monetary output aggregate nevertheless is distinguishable from the monetary aggregates for the consumer or manufacturing firm, because only the financial intermediary's aggregate depends upon reserve requirements, k.. The financial intermediary's monetary asset user costs, 'Yt> each depend directly in (3.1) upon the "reserve requirement tax" produced by nonzero k, when no other interest is paid on required reserves. The output quantity aggregate, having been produced from the Divisia index, the dual price aggregate is produced from output reversal as follows: C = fJ.t''Y/M~. The user-cost price index produced in that manner is called the implicit Divisia price index. 7. AGGREGATION THEORY WITHOUT HOMOTHETICITY As we have seen in Sections 4 and 5, aggregation theory and index number theory provide readily derived results when aggregator functions are linearly homogeneous. However, linear homogeneity is a strong assumption, especially for a consumer. As a result, despite the elegance of the theory produced under linear homogeneity, extension of aggregation and index number theory to the nonhomothetic case can be very important empirically. The quantity and price aggregates presented in the sections above were produced from duality theory under the assumption that the category utility (or production) function defined over the component quantities is linearly homogeneous. However, in recent years it has been shown in the literature on duality theory that the quantity and price aggregates produced under that homogeneity assumption are special

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cases of a more general pair of dual functions that are applicable without any homotheticity assumptions on tastes or technology. In this section we provide the more general aggregation theory, as it applies to the three economic agents postulated in Sections 2, 3, and 4 above.

7A. The Consumer and the Manufacturing Firm As we have seen in the homogeneous case, the monetary aggregation theory applicable to the manufacturing firm is identical to that of the consumer, since the conditional decision problems producing the aggregates are identical. Bence, in this section we shall use (2.3) with the understanding that the appropriate change of notation would be used if our results were applied to the manufacturing firm rather than the consumer. However, we now no longer assume that u is linearly homogeneous, or even homothetic. As we have seen, under linear homogeneity of u, the quantity aggregator function is the category subutility function, u, itself. However, if u is not linearly homogeneous, u clearly cannot serve the role of the quantity aggregator function, since a quantity aggregator function that is not linearly homogeneous does not make sense. If every component quantity is growing at the same rate, then any sensible quantity aggregate would have to grow at that same rate. But that is the definition of linear homogeneity. Hence the theoretically appropriate quantity aggregator function should be linearly homogeneous in the component quantities, m., and should reduce to the category subutility function, u, in the special case of linearly homogeneous u. The corresponding price aggregator function should be the true dual to the quantity aggregator function, should be linearly homogeneous in the component prices, and should reduce to the unit cost function (5.3) in the case of linearly homogeneous category subutility, u. It has been shown in recent literature that the duals that serve those purposes are the distance function at fixed utility level (as the quantity aggregate) and the cost function at fixed utility level (as the dual price aggregate). When normalized to equal unity at base period prices and utility, the result is the aggregation-theoretic Malmquist quantity index and the Koniis true cost of living index, respectively. Before we can present these results, we must define the distance function d(u o , in,) at UO" That function can be defined in implicit form to be the solution to the equation u(m/d(u",m,)) =

Un'

(7.1)

for preselected fixed reference utility u.: The distance function at fixed u; is indeed linearly homogeneous,

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W.A. Barnett

monotonically increasing, and strictly concave in m, even when u is not linearly homogeneous. Hence, when u is linearly homogeneous, u has exactly the same properties that d always has, regardless of whether u is linearly homogeneous. In addition, when u is actually linearly homogeneous, the distance function becomes proportional to the utility function. Hence, when u is linear homogeneous, the quantity aggregate produced from the distance function will always grow at exactly the same rate as the quantity aggregate u(mf) that we already have derived in the case of linearly homogeneous u. In order to acquire the dual price aggregate, we need only observe the following two relationships: (7.2)

and (7 .3)

Equations (7.2) and (7.3) demonstrate that the distance function, d. and the cost function, E, are duals. As we therefore might expect, the price aggregate in general is the cost function as a function of 'n, at a fixed reference utility level u.; At fixed reference u.; the cost function has all of the correct properties for a price aggregator function, including linear homogeneity in rr, In addition, the cost function is monotonically increasing and concave in 1ft' Hence, regardless of whether u is linearly homogeneous, the cost function has all of the same properties that the unit cost function has when u is linearly homogeneous . In addition, if category subutility really is linearly homogeneous , we get the same price aggregate growth rates by using the cost function as we did in Section 5, when we used the unit cost function , since the two functions are then proportional to each other. In aggregation theory, exact economic aggregates are converted to exact economic indexes by dividing by a base period value of the aggregate. The natural ways thereby to convert (7.2) and (7.3) into exact indexes are M" C(m'2' m'l; uo ) = diu ., m'z)ld(uoom'l) and [lkc( 'n' 2' 'Tr't; uo) = ei»; 'n ,z>!E(uO' 'Tr,). The price index [lkc('n'2' 'Tr'I; u o ) is the famous Konus (1924) true cost of living index. The dual quantity index M"C(m,z' mIl ; uo ) is the Malmquist (1953) index. At this point, it is clear why the case of linear homogeneity of u is so important. Without linear homogeneity of utility , the exact quantity

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227

and price aggregator functions , M"'Cand Il", although unique and based upon very elegant duality theory, nevertheless depend upon a reference utility level, U o ; and the index provides no information about how to choose the reference utility level. The base period need not be [lor [ 2' Hence the aggregation theory is equally applicable for any selection of UO ' It can be shown that M'" and Il" are independent of u; if and only if category utility is linearly homogeneous (see Diewert, 1981, sec . 3). Otherwise exact economic monetary quantity and price aggregates depend upon the reference utility surface relative to which the indexes are defined.

7B. The Financial Intermediary As may be expected from our results in the homogeneous case, the results in Section 7A for the consumer and manufacturing firm apply to aggregation of financial intermediary output with a nonhomogeneous factor requirement function, if we replace the cost function by its output analog, the revenue function , and if we replace the category subutility function , u, (or aCE,) for the manufacturing firm) by its output analogue, the factor requirements function . We present those results in this section. By analogy with Equation 7.1, define the financial firm's output distance function implicitly to be the value of D(f.L" a, ,k,) that solves fifJ./D(fJ." a,; k,); k,) = g(a o ) for preselected reference input vector a o • Then the exact monetary quantity output aggregate for the financial intermediary is Mb( J1,; a M k.) = D(J1" a o ; k.), and the corresponding Malmquist economic output quantity index is Mmb ( J112 ' J1,\; a o , k.) = D(f.1'2 ' a ,,; k,)ID(J1'J' a ,,; k,) . The corresponding dual Konus monetary output price aggregate is produced by replacing the cost function in Section 7A by its output analog, the revenue function (5.5). Hence the true output price aggregate is f('Y,; a o , k ,) = R*(a o "I,; k.), and the corresponding Konus k true financial output price index is r ('Y'2' "III; a" k.) == R*(an> 'Yt2;k,)/

s -i«: "I,,; k.). 8. INDEX NUMBER THEORY UNDER NONHOMOGENEITY There are a number of published approaches to producing nonparametric approximations to the Malmquist input or output quantity index. Yet virtually all of them result in selection of the discrete (Tornqvist) Divisia index . Sec , for example, Theil (1968), Klock

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W.A. Barnett

(1967), and Diewert (1976). In fact it is the nonhomogeneous case in which the Divisia index stands out as being the uniquely best element of Diewert's superlative class. 16 In addition, it recently has been shown that the Divisia index is not only exact for anonhomothetic translog aggregator function but remains appropriate even when tastes or technology are changing over time (see Caves et aI., 1982a,b).

9. AGGREGATION OVER CONSUMERS AND FIRMS The aggregation theory presented above is for individual decision makers: either one consumer, one manufacturing firm, or one financial intermediary. We have not discussed aggregation over individual decision makers. However, a large literature exists on that subject. The subject of aggregation over consumers has been heavily researched, and the results are well known. The most important of those results are surveyed in Barnett (1983a, 1983b) and will not be repeated here. Two different literatures exist on aggregation over firms. One literature resembles that for aggregation over consumers, but a more specialized literature exists on the derivation of production transformation surfaces for an industry or the entire economy. Sato (1975) has provided an excellent survey of that literature. One of his more interesting results (on p. 283) is that a constant-retums-to-scale economywide production function exists if there is a social utility function. As a result, aggregation over consumers can itself produce perfect aggregation over firms. Other results on aggregation over firms are surveyed in Diewert (l980a, pp. 464-470). Particularly useful results are those of Bliss (1975, p. 146) and Debreu (1959, p. 45), who found that if all firms are competitive profit maximizers, then the group of firms can be treated as a single firm maximizing profits subject to the sum of the individual firms' production sets. That result provides a very simple means of aggregating over firms. All of the theory presented above is directly linked to deterministic microeconomic aggregation theory. However, the Divisia index can be acquired in another manner, which permits easy aggregation over goods, consumers, and firms jointly. This approach, called the atomistic approach and championed by Theil (1967) and Clements and Izan I·When Denny asked whether it is possible to acquire superlative indexes other than the Divisia in the nonhomothetic case, Diewert (I 980c, p. 538) replied: "My answer is that it may be possible, but I have not been able to do it." More recently, Caves et al. (l982b, p. 1411) have proven that the (discrete) Divisia index "is superlative in a considerably more general sense than shown by Diewert. We are not aware of other indexes that can be shown to be superlative in this more general sense."

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(1981), treats the expenditure weights in the Divisia index as the probabilities of drawing the corresponding quantity log changes. Then the Divisia quantity index is just the mean of the distribution of log changes. We thereby can aggregate over all economic agents, including firms and consumers jointly , by computing the share-weighted average of quantity growth rates across economic agents as well as across goods. See Barnett (1987) for a proof.

10. EARLIER RESEARCH The prior sections of this paper have outlined the currently available microfoundations for monetary aggregation. We now are in a position to be able to consider potentially useful areas for future research on this subject and for extension of the currently available results. We also can assess the merits and historical importance of earlier contributions to this literature relative to the development of the currently available theory. We first shall consider those earlier contributions, and then we shall speculate about possible future refinements and extensions of our knowledge in these areas .

lOA. The Chicago School The Monetary Economics Workshop at the University of Chicago has had a long-standing interest in monetary aggregation, especially during the years when that workshop was run by Milton Friedman. Many of the approaches tried by participants at that workshop resulted in dissertations , which are mentioned in Friedman and Schwartz (1970, pp. 151-152). The conclusions produced by that line of research are well summarized by Friedman and Schwartz (1970, pp. 151-152) when discussing the potential generalization of the simple sum aggregates to index numbers: This [summation] procedure is a very special case of the more general approach. In brief, the general approach consists of regarding each asset as a joint product having different degrees of "moneyness," and defining the quantity of money as the weighted sum of the aggregate value of all assets, the weights for individual assets varying from zero to unity with a weight of unity assigned to that asset or assets regarded as having the largest quantity of " moneyness" per dollar of aggregate value. The procedure we have followed implies that all weights are either zero or unity. The more general approach has been suggested frequently but experimented with only occasionally . We conjecture that this approach deserves and will get much more attention than it has so far received.

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Although the Friedman group clearly understood the nature of the problem, none of the members of that group possessed expertise in index number theory or aggregation theory. In addition, that group's work preceded Diewert 's (1976) critical breakthrough in index number theory and Barnett's (1978) and Donovan's (1978) derivations of the user cost of monetary services. As therefore might be expected, in retrospect, the Chicago School's group never succeeded in determining how to apply aggregation or index number theory to monetary aggregation . In addition, the approaches considered by that group had a distinctly circular appearance. Those approaches usually involved estimating the parameters of a parameterized function of component quantities. The parameters often were estimated to maximize the correlation between the monetary aggregate and GNP, or to minimize the variability of velocity, and so forth. Clearly an economist whose prior view is that money "should" have constant velocity can produce data in that manner to confirm his prior view. On the other hand, a person who believes that velocity should be unstable can confirm that theoretical view by estimating the parameters of an aggregator function such as to maximize the variability of velocity . In principle that procedure is very unattractive. Data production should be logically prior to the testing of theory. By contrast, the Tornqvist Divisia index advocated above contains no parameters. Alternatively if one were to estimate the parameters of the aggregationtheoretic aggregator function directly , rather than to use the Divisia index as a nonparametric approximation, then the relationship between the aggregate and GNP would not be part of the estimation criterion. Recall that the exact aggregator function is produced either by a utility function or a production function. Hence, estimation of the parameters of the aggregator function necessarily involves producing best "fit" for the derived system of demand or supply functions generated by the utility or production function . No final targets of policy are involved in that estimation procedure . Since a hierarchy of nested weakly separable blocks can exist, aggregation and index number theory can produce multiple monetary aggregates, such as Divisia MI, M2, and M3. All can be rigorously produced from theory. Once those admissible data series have been produced , it is entirely reasonable to select between them by determining which data series best suits one 's purposes. However , the series themselves should not be constructed to suit the application. One would not construct a price index of the cost of living to maximize the fit of a model of price formation.

DEVELOPMENTS IN MONETARY AGGREGATION THEORY

23\

Another problem associated with the work done by the Chicago group was that the estimated aggregator functions often were linear, which implies perfect substitutability. 17 In short, the Chicago group understood the relevant issues and anticipated the thrust of the research to follow. However, the group's work preceded the developments in index number theory and in monetary theory that made possible the later rigorous connection between monetary theory, aggregation theory, and index number theory. Hence, none of the monetary aggregation approaches attempted by the Chicago group were rigorously founded in theory.

lOB. The Chetty Approach Chetty (1969) appears to have been the first to recognize the direct relevancy of microeconomic aggregation theory to monetary aggregation, since he was the first to estimate a neoclassical aggregator function embedded within a constrained optimization decision. The decision problem used by Chetty was to maximize u(m .. . . . , mn ) . subject to "" LJ

-m, - =

, \+

ri

W,

(10.1)

where u is a C.E.S. utility function, m, is the dollar value of the i'h liquid asset held at the beginning of the next period, r. is the interest rate on liquid asset i , and W is the discounted value of this period's financial wealth. Then the exact quantity aggregate is u(m) , where m = (m l , . . . ,m n ) . The exact price aggregate should be dual to the quantity aggregate and should be the unit cost function. However, observe that the "prices," 1/(1 + r;), are not the user costs presented above for the consumer or firm. The reason is that Chetty's model is appropriate only in a two-period world. In a muItiperiod world, the user costs measure the price of a unit of the services of an asset. Chctty' s "prices" do not measure the opportunity cost of acquiring a unit of the service flow and hence are not user costs or "rental prices. ,, 18 In addition, Chetty did not recognize the duality between quantity and price indexes, and hence advocated the use of a price aggregate that was not

" In that tradition, see. for example, Laumas (1969). " For a detailed and accurate critique of Chelly's approach. see Donovan (1978, pp. 679680).

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the correct dual to u(x) . In addition he proposed no nonparametric approximation (such as the Divisia index) to u(x). Many papers later were published by others using Chetty's appreach." While Chetty's work was seminal in first recognizing the valid aggregation-theoretic link between the exact aggregate for a consumer and his utility function , the behavioral model that he used was too primitive to produce useful aggregates. While two-period lifetimes are useful in overlapping generations models in theory, the only way to produce an empirically relevant current period decision is by assuming intertemporal blockwise weak separability in a multiperiod intertemporal decision, as we have done above. The result is the necessary use of user cost prices, as we have seen.

lOCo The British Columbia School The next steps in the evolution of the use of aggregation and index number theory in monetary aggregation were the steps that led directly to the developments surveyed above in Sections 1-8. Although Barnett, who was not at the University of British Columbia, played the central role in these recent developments , the approach that produced these latest results nevertheless perhaps best can be called the British Columbia School approach . The reason is the path-breaking nature of the early contributions of Erwin Diewert at the University of British Columbia (UBC). Diewert (1976) defined the class of superlative index numbers. That discovery unified index number theory and aggregation theory and provided an understanding of the order of approximation involved in the use of statistical index numbers as nonparametric approximations to exact aggregator functions . Diewert (1974) also introduced Jorgensonian user cost prices for consumer durables into consumer demand systems modeling . At the same time that Barnett (1978) produced a formal mathematical derivation of the user cost price of monetary services, Donal Donovan (1978) produced the same formula by reasoning through the form that Diewert's user cost for a consumer durable would take, if the good were a monetary asset rather than a consumer durable good. Donovan was a student at UBC when he generated that result. More recently, Diana Hancock ( 1985; 1986a,b), also while a student at UBC, incorporated transactions costs into the user cost formula for monetary services produced by financial intermediaries and contributed " Perhaps the best of those papers was Barth et al. (1977).

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much to the empirical work on supply side monetary aggregation . Furthermore, important new advances in aggregation theory recently have been produced at UBC by Blackorby and Schwonn (1983, 1984a,b, 1985). A case could be made for identifying this research tradition with Canada , rather than only with UBC, as a result of the path-breaking early work on index number theory produced at Ottawa by Afriat (see, e.g. , Afriat, 1977) and the recent important empirical research at Calgary and McMaster in monetary aggregation by Serletis (1987a ,b) and Serletis and Robb (1986). In any case , it is clear that Canadian economists , especially at British Columbia, have played a strikingly important role in the development of neoclassical aggregation and index number theory; and that research tradition has produced many valid microeconomic tools that are directly relevant to aggregation over monetary assets.

11. ALTERNATIVE APPROACHES The tradition outlined above is derived directly from neoclassical microeconomic theory and hence is produced from the implications of optimizing behavior by economic agents. The neoclassical approach has a long history in economics and has been developed to a high level of sophistication and refinement. The developments surveyed above in Sections 1-8 are produced from that approach and therefore are comparably sophisticated and refined. Nevertheless, there have always been economists who have chosen not to subject their research to the discipline imposed by neoclassical rigor. The same is true in monetary economics. In this section , we survey and discuss some of the better known ad hoc approaches to monetary aggregation.

11A. The Preference Independence Transformation The neoclassical approach surveyed above produces monetary aggregates that measure the flow of monetary services produced by the component assets. The monetary services of a liquid asset are then defined to be all of the services valued by the asset holder other than the interest rate yielded by the asset. 20 The behavior of the asset holders reveals the aggregate service flow valued by the asset holders .

"'If an implicit interest rate is used, then the "monetarized" services valued by the implicit interest rate are removed from the aggregate.

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However, for some purposes it may be useful to be able to evaluate the quantity of a single service, such as liquidity, produced by an asset stock. Since it rarely is possible to measure directly the quantity of one service produced by a multiservice asset, the evaluation of the quantity of one service depends upon the ability to untangle the jointness in service production by some inference procedure. One such approach is that proposed by Barnett et al. (1981, Section 5). Their approach is a direct application of Theil's preference independence transformation . In the Lancaster sense, goods are assumed to produce characteristics (or services). The preference independence transformation reveals the quantities of those services as functions of the quantities of the goods or assets. For purposes of monetary aggregation; the approach has two limitations. Although the approach can indeed produce a measure of liquidity, it does so under the assumption that characteristics (i.e., services) are preference-independent (i.e., completely strongly separable). A philosophical case can be made for that view, since complete strong separability is equivalent to the complete untangling of jointness, and therefore does indeed produce separate services, which become measurable. Nevertheless, it is not clear that monetary services need to be preference-independent. If they are not preference-independent, then the orthogonalization produced by the preference independence transformation may reveal characteristics that are " deeper" than the usually understood services of monetary assets. Hence, it may be very difficult correctly to interpret the identities of the characteristics uncovered and measured by the preference independence transformation. Furthermore, the approach requires the specification of a utility function and the estimation of its parameters. No nonparametric approximation exists. This difficulty substantially limits the potential usefulness of the approach to data-producing government agencies, which typically want their data to be estimator- and specificationindependent.

lIB. The Roper and Turnovsky Approach Although not based upon microeconomics, aggregation theory, or index number theory, Roper and Turnovsky's (1980) approach to monetary aggregation is particularly interesting, since it is based upon macroeconomic theory. In particular, Roper and Turnovsky use a simple IS-LM Keynesian macroeconomic model to produce the controltheoretic solution for the optimum policy that maximizes a particular policy objective function. They then solve for the monetary aggregate

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that will produce that optimum policy if the aggregate's growth rate is stabilized in terms of a constant growth rate rule. Hence, the "optimum monetary aggregate" equates maximization of the policy objective function with stabilization of the growth rate of the aggregate. Microeconomic aggregation theory produces aggregates that "exist," in the sense that they can be factored out of the economy's structure such that the economy's structure becomes a composite function of the aggregator function. In this sense, the Roper and Tumovsky aggregate does not "exist," since it is not produced from blockwise separability of any structural function. Hence, the Roper and Tumovsky "optimum aggregate" does not measure a real structural economic variable. Nevertheless, the Roper and Tumovsky monetary aggregate acquires its existence from the policy objective function rather than from the measurement of a structural aggregate. It would be wrong to interpret the Roper and Tumovsky monetary aggregate as measuring monetary services, liquidity, or "money" in any meaningful sense. However, their aggregate produces a very interesting approach to operationalizing an optimum control policy solution in a potentially practical manner, if no structural change occurs along the solution path. If the resulting aggregate were erroneously interpreted to measure "money," then the approach would share the circularity problems discussed in Section lOA with respect to some of the Chicago School attempts.

He. The Latent Variables

Approach

Spanos (1984) proposed viewing money as an unmeasured latent variable and used the MIMIC (multiple-indicator multiple-cause) model to produce a measure of money. It is difficult to know just what to make of that approach. On the one hand the model proposed by Spanos does not appear to be convincing. But on the other hand his results are very interesting and suggest that there may be more to the approach than is immediately evident. To begin with, it is not clear that money should be viewed as a latent variable. As we have seen above in Sections 1-8, the identity of the exact aggregation-theoretic monetary aggregate can be defined rigorously, and index number theory and aggregation theory produce well-understood means of estimating the exact aggregate (or aggregates, if nested at various levels of aggregation). The identity of the exact monetary aggregate is no more mysterious than that of the true cost of living index, and there is no more reason to treat money as a

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latent variable than the true cost of living index. In addition, the specification of Spanos's model is linear and unrelated to neoclassical economic theory, which would require that linearly aggregated component assets be perfect substitutes. In short, a monetary aggregate is an aggregate and not a latent variable. Hence, it is aggregation theory and index number theory that provide the relevant tools, not the linear MIMIC model. Nevertheless, Spanos's approach generates results that are surprisingly similar to those produced by the Divisia index. Perhaps a more sophisticated nonlinear variant of the MIMIC model might in fact produce the Divisia index exactly. Hence, there may be some currently unclear relationship between Spanos's approach and aggregation theory.

un,

Turnover Rate Weighting

When applying aggregation and index number theory to produce macroeconomic results, it is critically important to assure that the behavioral theory upon which the index numbers are based is consistent with one's model. Many examples of such inconsistency exist in the literature on monetary aggregation. For example, the official simple sum monetary aggregates are consistent with aggregation theory if and only if the components are indistinguishable perfect substitutes." Yet many models in which the official aggregates are used are structured in a manner that is not consistent with that assumption. A more recent example is the aggregate M Q , proposed by Spindt, which uses the Fisher ideal index to produce a monetary aggregate and a velocity aggregate. However, turnover rates, rather than the user costs or rental prices appearing in the usual Fisher ideal index, are treated as dual to quantities. The theory on which Spindt's aggregates are based is presented in Spindt (1985). Nevertheless, it is easily seen from the theory in Sections 5 and 6 above that his results are consistent with the relevant index number theory only if economic agents solve one of the following two possible decision problems: select m, to maximize M(m,) subject to m.v, = p;X"

(11.1)

or "This result follows from the fact that an aggregator function, which is a category utility or production function, is linear if and only if the indifference curves or isoquants are linear.

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select v, to maximize V(v,) subject to m~v, = p:X"

(11.2)

where VI is the vector of turnover rates corresponding to the vector of monetary asset quantities, m, The resulting monetary aggregate is M(m t) and the resulting velocity aggregate is V(vl ) . Either of the above two behavioral hypotheses would be consistent with Spindt's use of index number theory. In addition, if either of those two decision problems is applicable, and if M and V are linearly homogeneous , it follows from factor reversal that M(m,)V(v t ) = prX/l which is the "quantity equation. " However, it is not at all clear how either of those two decisions could be embedded in a sensible way into a jointly rational decision over goods and monetary assets for either a consumer or firm under any kind of separability assumption . In addition, if (11.1) is solved, then VI must be treated as exogenous to the decision maker. Alternatively, if (11.2) is solved, then m, must be treated as exogenous to the decision maker. Neither possibility is reasonable , since both turnover rates, VI and monetary asset quantities demanded , m, are in actuality selected by monetary asset holders. Spindt used Diewert' s (1976) result that the Fisher ideal index is exact for the square root quadratic aggregator function . However, Spindt overlooked the fact that Diewert's proof applies only for optimizing behavior. With the use of turnover rates rather than user costs, the optimizing behavior required for use of Diewert's theorem must be either (11. 1) or (I I. 2) above. Without rigorous internally consistent behavioral theory, a Fisher ideal index in monetary quantities and turnover rates is just an arbitrary combination of component quantities and turnover rates. An infinite number of such functions of monetary quantities and turnover rates exist , and they can produce any arbitrary growth rate for the aggregate from the same component data. As a result, Spindt's aggregate is entirely arbitrary. In short, rolling one's own index number , without access to the lOO-year-old cumulative literature on quantity and price (not turnover rate) indexes, is a hazardous venture. In fact, in general it is worth observing that statistical index numbers are defined in the index number literature to be functions only of quantities and prices (not of turnover rates, bid-ask spreads , etc.) ." The reason is that all of the relevant theorems and results in the literature on aggregation and index number theory depend upon that definition. " Sec Diewert (1976. 1978. 1980a. 1980b. 1981) and Samuelson and Swamy (1974).

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Furthermore, the early "atomistic" or "test" approach of producing ad hoc indexes without theory has been completely discredited, although even that obsolete tradition used functions only of quantities and prices. 23 Not only is M Q groundless in theory, but its use in practice is clearly circular. Turnover rates are functions derived from the demand function for a monetary asset. Hence, the use of MQ as data in estimating demand for money functions obviously is internally contradictory.

12. RECENT EXTENSIONS A few important extensions of the theory presented in Sections 19 recently have appeared. These extensions of our knowledge about monetary aggregation paral1e1 the state-of-the-art research in aggregation and index number theory and hence are solidly and validly grounded in microeconornic theory. Perhaps the most ambitious extension is that of Poterba and Rotemberg (1987). They introduced risk aversion into the decision problem. The formulations presented in Sections I -9 above require either perfect certainty or risk neutrality. The introduction of risk aversion into aggregation and index number theory has rarely been attempted, since risk aversion requires expected utility or expected profit maximization. The resulting integration greatly complicates the implications of quantity aggregation, since the necessary blockwise weak separability assumption on functional structure no longer necessarily produces a simplifying two-stage decision. In addition, there is not yet any known analog to statistical index number theory in the case of risk aversion, since no known nonparametric approximations exist to the exact parametric aggregator function. Poterba and Rotemberg estimated the aggregation-theoretic exact aggregator functions under rational expectations, as is the only approach available in the case of risk aversion, but they did not then compute the implied values of the estimated aggregates. The implied values of the monetary aggregate have been called the "exact theoretical rational expectations" monetary aggregate (or just the estaimted Theoretic aggregate) by Barnett, Hinich, and Yue (1989), who generated the implied data through a generalized method of moments estimation of the aggregator function. Poterba and Rotemberg's ap2JSee, for example, Diewert (1981, p. 180) and Samuelson and Swamy (1974, pp. 566, 567, 573). These conclusions apply regardless of whether the aggregation is in consumption or production. See, for example, Diewert (1981, p. 199) on the "revealed production" approach.

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plication of aggregation theory with ex.pected utility maximization is correct and is a very useful approach to research on monetary aggregation. Because of the limitations in the present state of aggregation theory with risk, Poterba and Rotemberg were not able to produce a nonparametric approximation, and hence their aggregates are specification- and estimator-dependent, as is also the case with the results produced by Barnett , Hinich , and Vue (1989). This deficiency surely will limit the usefulness of that approach as a means of producing monetary aggregate data, especially by central banks, although the approach remains useful as a research tool and as a means of monitoring the quality of the Divisia approximation. It would be interesting to know whether Poterba and Rotemberg's parametrically estimated monetary aggregates differ appreciably from the corresponding Divisia monetary aggregates. It is possible that our assumption of risk neutrality produces aggregates of adequately high quality. If not, then fundamental research is needed to product nonparametric approximations to Poterba and Rotemherg ' s theoretically correct monetary aggregator function . In fact the ability of the Divisia index to track the Poterha and Rotemherg aggregates has been explored by Barnett , Hinich, and Vue (1989), who found that the Divisia aggregate tracks the estimated Theoretic aggregate well, except during the eight months that immediately followed the Federal Reserve's monetarist experiment. That period not only was one of unusual uncertainty about monetary policy but also was the period during which money market deposit accounts and super-NOW accounts first were introduced. It is not surprising that the incorporation of risk aversion into the estimation procedure was important during that period of unusual risk in financial markets. We believe that the approach proposed by Poterba and Rotemberg and further developed and applied by Barnett, Hinich , and Vue will be the source of much additional research in the future. A number of other important extensions and applications recently have appeared. Serletis (1987a,b) , Serletis and Robb (1986), and Hancock (1985, 1986a,b) have produced some very high quality applications of this literature in a manner that makes full use of the theoretical restrictions and assumed functional structure. Most earlier applications used conventional older tests, which did not make full use of the implications of the monetary aggregation theory. Hancock's work deals with supply side aggregation, whereas the work by Serletis and by Serletis and Robb is on the demand side. Marquez (1986) and Ewis and Fisher (1984, 1985) introduced foreign currency substitution into the monetary aggregation theory and applied the resulting mi-

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croeconomic foundations empirically. Hancock (1985, 1986a,b) introduced transaction costs into the supply side user costs, and Barnett et al. (1986) tested the statistical significance of the regulatory wedge, which produces the divergence between the demand side and supply side Divisia monetary aggregates. 24 Ishida (1984) produced an especially noteworthy investigation of the merits of the Divisia monetary aggregates as a substitute for the Bank of Japan's official simple sum monetary aggregates. The dynamic behavior of the Divisia monetary aggregates has been studied by Barnett and Chen (1988), Barnett and Hinich (1989), Barnett, Hinich, and Yue (1989), and Hoa (1985).

13. POSSIBLE REFINEMENTS Even with a given index number formula, there always exists the possibility of improving upon the quality of the aggregate by refining the manner in which the component data are acquired and produced. The same is true for the Divisia monetary aggregates. 25 For example, the benchmark rate, R" is the maximum expected available holding period yield during period t. In theory it has been argued that R, is tne rate of return on human capital, since human capital is completely illiquid in a world without slavery. However, regular measurements on that rate of return are not available. An approximation or proxy is needed. Clearly the current procedures of maximizing over a group of about 15 rates of return each period is not the best possible approach to measuring R,. When rates of return are regulated, implicit rates of return often are more appropriate than the explicit regulated rate of return. In the current Divisia monetary aggregates, an implicit rate of return is used for commercial bank demand deposits. The measurement of implicit rates of return always can be further refined. The principle involved is to impute an implicit rate of return to any services that should not be viewed or evaluated as monetary services by the monetary aggregate. For example, free dishes supplied by a bank in return for opening a passbook account at a regulated rate should be evaluated monetarily and used to augment the explicit rate of return to create a higher implicit 24Since the supply side Divisia aggregates have never before been published, we have attached them as an appendix to this article. The data are those used by Barnett, Hinich, and Weber (1986). The corresponding demand side aggregates have been published as an appendix to Barnett (1984). See Fayyad (l986a) for weekly data. 25For a presentation of the current procedures along with a tabulation of the current weekly Divisia monetary aggregates data base, see Fayyad (I 986a).

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rate of return. Similarly all yields could be measured net of transactions costs. The price index used to deflate nominal to real balances was treated as the same price aggregate for all economic agents in Sections 1-9. However, in principle different price deflators exist for each of the three categories of economic agents as well as for inputs and outputs. For discussion of that potentially important refinement, see Fisher and Shell (1972, 1979, 1981). Using the observations in Diewert (1980a, pp. 478-479) , extension of all of the results in Sections 1-9 to include taxes is straightforward. See Coen and Hickman (1970 , p. 299) for discussion of the approaches to dealing with some further complications, such as differences between borrowing and lending rates, the existence of more than one lending rate, risk-induced dependency upon debt/equity ratios, and so forth. For a list of areas needing such extensions in the aggregation theory literature, see Diewert (l980b, p. 265). The selection of monetary asset groups to be used as components of monetary aggregates should be based upon tests for blockwise weak separability. No such tests were used to produce the component groupings in the Federal Reserve's MI , M2, or M3. Some initial results from weak separability tests are available in Fayyad (1986a,b) and Serletis (l987a). However, the available procedures for testing weak separability have been shown to perform poorly, and new statistical approaches are needed. 26 No use has yet been made in monetary aggregation of the literature on aggregation over economic agents, as surveyed in Section 9 above. 14. POTENTIAL FUTURE EXTENSIONS

There are many potential future extension s and applications of the microeconomic foundations surveyed in Sections 1-9 above . We conclude this article with a speculative discussion about a few such fruitful areas for future research. 14A. Macroeconomic and General Equilibrium Theory

In terms of their relationship with final targets of policy , the demand side Divisia monetary aggregates are the most relevant. Those aggregates measure the economy's monetary service flow, as perceived by

'·See Barnell and Choi (1989) about the deficiencies of the usual tests. See Blackerby, Schworrn, and Fisher (1986) for a potentially successful new approach.

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the users of monetary assets. As a result, it might seem that we would be interested only in the consumer and the manufacturing firm, whose decisions are analogous and pose little difficulty. This convenient situation would arise, for example, if we were seeking monetary aggregates to be used as indicators. However, frequently money plays a more complex role in economics. For example, a money market appears in most macroeconomic models and in many recent general equilibrium models. Policy simulations with macroeconometric models usually require a modeled money market. In addition, targeting a monetary aggregate requires information about the supply function for the aggregate. All of the objectives would be facilitated if the supply side Divisia monetary aggregates from the financial firm along with the transmission mechanism from central bank instruments were incorporated into a full model of the economy. Producing a cleared money market in a Divisia monetary aggregate along with a transmission mechanism operating through that market is an objective towards which the above research is tending. However, much research remains before such a closed model can be constructed. If a model included only a demand side Divisia monetary aggregate, then closing a complete model would require modeling the supply of every component in the Divisia monetary aggregate. Market clearing would occur at the level of disaggregated component quantities, which then could be substituted into the Divisia demand monetary aggregate. While the resulting aggregate could be informative and useful, much of the potential simplification is lost by the need to produce completely disaggregated supply functions. However, incorporating both supply and demand aggregates in a model, as would be needed to produce a market for an aggregate, has been given very little consideration in aggregation theory. Many difficult issues arise. What do we do if the financial firm's technology is separable in different blockings of monetary assets from those that appear in the consumer's utility function or the manufacturing firm's technology? How do we deal with the fact that currency and primary monetary securities appear as components of the consumer's and the manufacturing firm's monetary demand aggregate, but not necessarily as components of the financial firm's monetary output aggregate? What happens to the ability to close the money market if homotheticity applies on one side of the money market, but nonhomotheticity requires use of a Malmquist index on the other side of the market? One potential problem is immediately evident from the results in the prior sections. The user cost prices of monetary assets produced by the financial intermediary are different from those for the consumer

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(or manufacturing firm). The reason is the existence of reserve requirements, which produce an implicit tax on financial intermediaries, but not on consumers or manufacturing firms. Similarly, income taxes produce' 'wedges" between user costs on the supply and the demand sides of the money markets. As a result, even when the markets for every component are cleared , the Divisia demand aggregate and the corresponding Divisia supply aggregate need not be equal, since the different user costs can produce different weights, although all corresponding component quantities nevertheless may be equal. Another potential problem results from the fact that in some parts of some macroeconomic models, monetary wealth rather than the monetary service flow is relevant. This could be the case, for example, in producing an argument of a consumption function. Then it is necessary to compute the expected discounted present value of the service flow. The expected service flow in a future period is measured by its expected Divisia index. The simple sum aggregates are acquired from discounting only if the investment yield is discounted to present value along with nominal expenditure on the service flow. However, the interest yield on an asset has never been viewed as a monetary service in macroeconomic theory. For a discussion of the appropriate use of capital stocks and capital service flows, see Usher (1980, pp. 17-18). For the formula for the discounted service flow (net of the investment yield), see Barnett, Hinich , and Yue (1989). Further important issues that could be addressed by the methodology surveyed in this article are those discussed in Laidler (1971, 1982, 1985). 14B. The Utility Production Function

The supply of money function used in monetary theory often has not been a neoclassical supply function, in the traditional sense. Instead, the supply of money function often has been an equilibrium condition or reduced-form equation relating money to the monetary base through a multiplier. 27 As seen from our model of the financial intermediary, the relationship between such an equation and monetary " Such equations often have been associated with Brunner and Meltzer (1964) and Johannes and Rasche (1979) . In fact, on econometric grounds a strong ease can be made for that sort of procedure with all durable good s. See Deaton and Muellbauer (1980 , pp . 350-351) , who advocate solving a two-equation durable demand and supply system to produce the single-equation specification to be estimated. In our case , we thereby would solve for the market-clearing value for the dual user cost aggregate . That solution function then would be substituted back into either the demand or supply function for the quantity aggregate.

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production technology is not clear. In fact, although high-powered money is a factor of production for the financial intermediary , the total monetary base, which also includes currency held by the consumer and manufacturing firm, does not appear in the financial intermediary's decision. Problems produced by this fact have been discussed by Saving (1977, p. 294). In addition, as described above, the demand Divisia aggregate need not equal the corresponding supply Divisia aggregate. Nevertheless , a straightforward interpretation can be given to the conventional multiplier-type " supply" function as a utility production function. As defined by Samuelson (1968), a utility production function is acquired by replacing the quantities in a utility function by their reduced-form equations . By applying that procedure to the consumer's monetary category subutility function and analogously to the manufacturing firm's monetary category subproduction function, equations can be produced that describe the equilibrium monetary service flows as functions of exogenous variables, including central bank instruments." In continuous time, Sato (1975, p. 283) has shown that the left-hand side of a utility production function can be measured exactly by a Divisia index. This possibility has not yet been used in monetary economics. 14C. Velocity Function The aggregation theory provided above can be used to derive the velocity function . The separability of the utility function can be used to produce a current period category utility function of the form v(u(m,), v(x /), I,), Then a consistent current period conditional decision can be defined. That decision is to select (m., X" I I) to -

maximize v(u(m,), v(x,), L) subject to 1l'; m,

+

p;x,

+

-

wl.; = Y"

(14.1)

where Y, is total current period expenditure on goods, monetary services, and leisure. The variable Y is preallocated from a prior-stage intertemporal allocation decision. 29 However, decision (14.1) can itself be solved in two stages. The first stage is to solve for (M;',x"LJ to I

"'For some related discussion. see Saving (1977. pp. 300--301). ""See Barnett (l980a, 1981a).

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maximize v(M; ,X" L,) subject to M;n;

+ p~X, +

IV,L, = Y"

(14.2)

where X, is the goods quantity aggregate over X" The solution function for M~ will be of the form M~ = 'I'(Y II~,p~ ,w,), If v is linearly " form M;' = Y,<\>(Il~ ,p~ ,w,), homogeneous, then 'I' can be written in the or M; (
Y"

(14.3)

so that (<\>(II;,pt,w r) ) -1 is velocity-relative to the " income" variable Y" For some purposes it would be useful to solve (14.2) conditionally upon I, with W;L, subtracted from both sides ofthe constraint in (14.2). Then the income variable would become Yt = Y, - W;L, and w, would not appear as a variable in velocity. In addition, v could be the current period utility function for a representative consumer under the conditions for aggregation over consumers. Since the second-stage decision for the manufacturing firm is analogous to that for the consumer, (14.3) could even be produced after aggregation over consumers and manufacturing firms jointly. It is particularly interesting to observe that the velocity function depends only upon the tastes of consumers and the technology of manufacturing firms. The velocity function does not depend upon the technology of the financial finn, or upon Federal Reserve policy, or upon the money multiplier . As a result, the velocity function is not affected by structural change in the banking sector. In contrast, the velocity of the monetary base depends jointly upon money demand and money supply and hence is affected by structural change in the banking industry . Observe that the aggregation-theoretic velocity function, <\>, is not the usual one, since the relevant income variable, Yt , or Y~, is not GNP. Hence, the value of <\> has never been computed.

14D. Other Potential Extensions The log change of the Divisia index can be viewed as the mean of the component log changes with the expenditure shares treated as the probabilities . The analogous second moments can be used to produce Divisia variances and covariances. This idea was originated by Theil (1967) and was used in an elementary application with monetary data by Barnett et al. (1984). The use of Divisia second moments was

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introduced into monetary economics in a systematic fashion by Barnett and Serletis (1989b). The neoclassical financial intermediary model presented in Section 3 can be used to produce monetary asset output aggregates, as we have shown above. As a result, the literature on estimating value added and technological change becomes immediately available. Both concepts are potentially very valuable, since technological change in banking is often viewed as important, and value added in banking is related to the distinction between inside and outside money. For a survey of relevant methods for measuring technological change and value added, see Barnett (1987, Sections 10 and 11).

15. EMPIRICAL ISSUES As discussed in footnote 4 as well as in various sections of this article, the aggregation-theoretic approach surveyed here has produced a large and rapidly growing body of empirical research. While the objective of the current review is to survey only the theory, Barnett, Fisher, and Serletis (1989) have surveyed the voluminous empirical results that now are available, from many countries and relative to many empirical criteria. However, certain of these empirical criteria seem to be particularly relevant to the objectives of this conference volume. In this section, we briefly summarize those criteria, which are raised in the general case by Laidler (1971, 1982, 1985).

15A. Structural Change A major issue in the monetary economics literature in recent years has been the effect of structural change upon the stability of the demand for money and supply of money functions. The weight of the evidence from the studies mentioned in footnote 4 and surveyed by Barnett, Fisher, and Serletis (1989) is that the use of Divisia monetary aggregates rather than simple sum aggregates produces functions that are much more stable than the functions produced from the simple sum aggregates, but at the cost of greater interest elasticity. When Divisia monetary data are used, the economic variables in the models explain the demand for and supply of monetary services adequately, and tests for structural change reject the hypothesis of shifts in the economy's structure. When simple sum monetary data are used, the interest elasticities of the demand and supply functions are usually lower than those found with the Divisia data, but the functions estimated with simple sum monetary aggregate data are subject periodically to unexplained structural shifts. It is perhaps not surprising that greater explanatory power of eco-

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nomic variables tends to produce greater elasticity to variations in those variables. In fact, it is my belief that conclusions reached with the simple sum aggregates regarding structural change are not tenable. The structural innovations usually observed to have occurred in money markets are widely viewed to have been produced by spikes in nominal interest rates produced in response to surges in inflationary expectations. However , nominal interest rates already are variables in the equations purported to have shifted. Since structural shifts by definition are produced by omitted variables, the claim that a function has shifted in response to a shift in an included variable is a contradiction . This contradiction is not observed with Divisia data. "Quantity theorists" are sometimes divided into "new quantity theorists" and "old quantity theorists," where the old quantity theory is characterized as being based upon the assumption that velocity is a constant, and the new quantity theory is represented as permitting one or more interest rate variables in the velocity function. Although results with the "new quantity theory" are reported in Barnett and Serletis ( I989a) , I find the distinction between the new quantity theory and nonquantity theories to be difficult to define. Hence , the results just mentioned do not clearly confirm or contradict the new quantity theory. However, it is clear from the high interest elasticities of demand found with the Divisia monetary aggregates that the Divisia monetary aggregates do not confirm the old quantity theory . Since there is no reason in theory to believe that velocity is or should be a constant, I consider the search for monetary aggregates producing constant velocity to be misguided. 30 It should be observed that the conclusions just mentioned are acquired with the Divisia data, only if innovations producing the introduction of new monetary assets (such as the introduction of money market deposit accounts and super-NOW accounts) are introduced into the Divisia index in accordance with valid index number procedures involving the imputation of a reservation price during the period immediately prior to the innovation. The proper procedures for introducing new components into an aggregate are discussed in Diewert (l980a, 1981).

ISH. Regulation There are circumstances under which simple sum aggregation is justifiable over certain subsets of monetary aggregates, although never "'Cagan' s (1982) findings are an exception to the conclusion s in this section , since Cagan finds that Divisia M I is the aggregate that has the most nearly constant velocity.

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over an aggregate that includes currency as a component. Paradoxically the breadth of the span of assets over which simple sum aggregation is justified increases as regulation either increases to severe levels or decreases to nearly total deregulation. In the middle range of regulation, which has characterized the U.S. economy in recent years, there are few groups of monetary assets over which simple sum aggregation can be justified. Simple sum aggregation can be justified only over those assets that have nearly equal user costs, (2.4). Under severe regulation of all monetary assets, the benchmark asset's yield may be very large relative to the own rates of return on monetary assets. In this case, even though the own rates may not be the same for all monetary components, the user costs nevertheless may be nearly equal. The Divisia index then can be approximated by the simple sum index, since the Divisia index converges to the simple sum index as the user costs approach each other. Whether these circumstances ever existed in the United States is not clear, since there is considerable evidence that demand deposits always have yielded a substantial implicit rate of return to corporate depositors. However, even if those circumstances did once obtain, those days clearly are long gone. The other possibility, which is of more potential interest, is complete deregulation. In this case, the rates of return on many monetary assets become similar, since rate differentials and the corresponding differentiation of the competing monetary products have to some degree been produced by regulation. Similarly yielding, closely substitutable, deregulated assets, of which there are many in Canada, can be aggregated by simple summation. But since currency's yield is zero, the user cost of currency is much greater than that of such deregulated monetary assets. Hence, while some subgroupings of monetary assets can be aggregated by simple summation under deregulation, no monetary aggregate containing currency can be fully aggregated by simple summation.

15C. Controllability Procedures for controlling the Divisia monetary aggregates are necessarily fundamentally different from those for controlling the simple sum aggregates. As seen in the earlier sections of this article, the simple sum aggregates (unless all of their components are indistinguishable perfect substitutes) do not enter anywhere into the decisions of any economic agents within the economy and hence are not variables

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within the structure of the economy. The Divisia monetary aggregates, on the other hand, track separable subfunctions within the structure of the economy and hence measure variables that exist within the economy. There is only one possible way to control the sum of imperfect substitutes, and that is to control each individual component asset. The imposition of reserve requirements on component assets-a tendency that has been growing in recent years in the United States-is intended precisely to produce such direct controllability of each component .asset. While the same method is equally as applicable to controlling the Divisia aggregates, such direct control over components is not necessary. Any procedure that alters the economy's general equilibrium (including that of nominal variables) will alter the economic variables being equilibrated. Since the Divisia monetary aggregates do indeed measure such existing structural variables, control of the Divisia monetary aggregates can operate through the value of the aggregate itself, as if it were an elementary asset. The evidence on controllability found in the articles cited in footnote 4 and in Barnett , Fisher, and Serletis (1989) support that VIew. Whether the money supply should be controlled is a separate issue that requires consideration of the properties of the entire macroeconomy. The theory and empirical evidence above demonstrate only that a control procedure operating directly through the value of the monetary aggregate is available with the Divisia aggregates, but is not available with the simple sum aggregates. Whether or not the money supply is actively controlled by the central bank , the Divisia monetary aggregates can serve as indicators. Since the simple sum aggregates do not measure service flow variables that exist within the economy's structure, the simple sum aggregates cannot serve the role of indicators . The simple sum aggregates discount to present value both expenditure on the monetary service flow and also the investment yield produced by interest-yielding monetary components. The confounding of fundamentally different motives accounts for the fact that the simple sum aggregates are not found anywhere within the structure of the economy. But both the service flow, measured by the Divisia monetary aggregates, and their discounted values are found within the structure of the economy: the former as variables within tastes and technology and the latter as accounting stocks appearing in constraints. See Barnett, Hinich, and Yue (1989) for details regarding the measurement of the discounted present values evaluating the stock of money.

W.A . Barnett

250

16. CONCLUSION Much progress in monetary aggregation theory recently has been made. Much remains to be done.

APPENDIX 1. Supply Divisia Monetary Aggregates (January 1969-July 1984) Date

8DMI

5DM2

8DM3

Date

8DMI

8DM2

8DM3

1/69 2/ 69 3/69 4/69 5/69 6/69 7/69 8/69 9/69 10/69 11/69 12/69 1/70 2/70 3170

100.0 100.3 100.7 101.0 101.1 101.3 101.1 100.9 100.8 101.1 101.5 101.8 102.3 102.2 102.6 103.4 104.5 105.8 107.0 108.3 109.1 109.3 109.2 109.5 110.2 111.3 112.2 112.9 113.9 114.6 115.2 115.6 116.2 116.8 117.3 117.6 118.0 118.6 119.3

100.0 100.3 100.7 100.9 101.0 101.3 101.3 101.2 101.5 101.6 102.0 102.4 102.5 102.0 102.0 102.2 102.9 103.9 104.8 105.9 107.0 107.7 108.3 109.2 110.2 111.5 112.9 114.3 115.6 116.5 117.6 118.5 119.7 120.7 121.8 122.6 123.5 124.6 125.6

100.0 100.3 100.6 100.7 100.8 101.0 100.8 100.7 100.8 100.9 101.3 101.7 101.8 101.3 101.3 101.6 102.3 103.2 104.3 105.5 106.7 107.6 108.5 109.6 110.7 112.4 113.9 115.0 116.4 117.4 118.5 119.4 120.5 121.8 123.1 124.1 124.9 126.2 127.1

4/72 5/72 6/72 7/72 8/72 9/72 10/72 11 /72 12/72 1/73 2/73 3/72 4/73 5/73

120.1 121.1 122.1 123.3 124.3 125.3 126.4 127.5 128.8 130.0 130.4 130.7 131.6 132.7 133.7 134.4 135.0 135.6 136.2 136.9 137.9 139.1 140.3 141.4 142.2 142.7 143.0 143.5 144.1 144.8 145.7 146.7 147.6 148.2 148.9 150.0 150.1 151.3 153.2

126.5 127.5 128.8 130.4 131.8 133.3 134.7 136.0 137.4 138.6 139.2 139.5 140.3 141.4 142.5 143.1 143.6 143.8 144.3 145.0 146.1 147.0 147.7 148.6 149.2 149.5 150.0 150.5 151.0 151.7 152.6 153.5 154.4 155.3 156.5 158.3 159.7 161.7 164.2

128.2 129.7 131.2 132.8 134.4 135.8 137.2 138.9 140.3 141.6 142.8 143.6 144.6 145.8 146.9 147.6 148.1 148.4 148.8 149.7 150.7 151.7 152.5 153.4 154.0 154.3 154.7 155.3 155.9 156.6 157.5 158.4 159.3 160.2 161.3 162.5 163.2 164.8 166.6

4170 5/70 6/70

7170 8170 9170 10/70 11/70 12/70 1171 2/71 3171 4/71 5171 6/71 7171 8/71 9171 10/71 11171 12/71 1/72 2/72 3/72

6173 7/73 8/73 9/73 10/73 11/73

12173 1174 2174 3174 4/74 5/74

6174 7/74

8174 9174 10/74

11174 12/74 1/75 2/75 3/75

4175 5175 6175

DEVELOPMENTS IN MONETARY AGGREGATION THEORY

251

APPENDIX 1 (continued ) Date

SDMI

SDM2

80M3

Date

SOMl

SOM2

80M3

7175 8175 9175 10175 11175 12/75 1176 2176 3176 4/76 5176 6176 7176 8/76 9176 10/76 11176 12176 1/77 2/77 3/77 4/77 5/77 6177

153.8 154.6 155.3 155.6 156.7 157.0 157.8 158.9 159.7 160.9 161.9 162.2 163.0 164.0 164.6 166.0 166.7 168.1 169.5 170.6 171.6 172.9 173.6 174.7 176.0 177.1 178.4 180.1 181.7 183.5 185.5 186.6 187.8 189.0 190.4 191.4 192.5 193.8 195.7 197.1 198.8 200.3 201.2 201.3 202.3

166.0 167.4 168.8 169.8 171.4 172.8 174.5 176.8 178.3 180.1 182.2 183.0 184.4 186.5 188.3 190.6 192.7 195.0 197.1 198.9 200 .7 202.5 204.1 205.3 207.1 208.6 210.0 211.6 213 .2 214.8 216.6 217.9 219.1 220.2 221.4 222.5 222.8 223.6 225 .1 226.1 226.7 226.8 226.5 226.1 226.5

168.0 169.1 170.6 171.5 173.2 174.1 175.3 177.1 178.6 180.3 182.1 183.0 184.4 186.2 187.6 189.2 191.1 193.3 195.0 196.8 198.5 200.1 202.1 203.6 205.6 207.5 209.2 211.0 213.1 215.0 216.9 218.5 220.3 221.8 223 .6 224 .8 225.4 226.6 228.1 229.2 230.0 230.4 230.2 229.8 230.3

4179 5179 6179 7179 8/79 9/79 10/79 11/79 12/79 1180 2/80 3/80 4/80 5/80 6/80 7/80 8/80 9/80 10/80 11/80 12/80 1181 2/81 3/81 4/81 5/81 6/81 7/81 8/81 9/81 10/81 11 /81 12/81 1182 2/82 3/82 4/82 5/82 6/82 7/82 8/82 9/82 10/82 11182 12/82

204.1 206.4 208.8 210.8 212.2 213.2 213.7 213.7 214.4 216.4 217.8 218.3 216.8 217.9 220.0 222.6 225.9 228.9 231.4 233.0 232.2 236.8 238.4 238 .8 241.0 241.9 242.8 243.9 244.1 244.5 245.2 248.0 251.5 255.5 256.1 256.6 257 .5 258.8 259.7 260.2 262.5 265.4 268.6 271.5 273.4

226.9 227.6 229.5 230.4 231.3 231.5 230.7 229 .5 229.5 229.9 230.0 229.1 226.6 226.7 229.8 233.2 236.6 238.9 240.6 241.6 239.8 239.4 239.2 239.2 240.5 240.1 240.2 240.7 240.5 240 .2 239.8 242.1 244.7 247.4 247.7 248.3 249.2 250.1 251.3 252.0 254.5 256.7 259.1 261.5 216.1

230.8 231.6 233.5 234.5 235.5 235.9 235.3 234.1 234.2 234.7 234.9 234.2 231.8 232.0 234.6 237.7 241.1 243.2 245.0 246.2 244 .8 244.8 244.9 244.9 246.2 246.0 246.4 247.2 247.2 247.1 246.9 249.2 251.9 254.3 254.9 255.8 256.9 257.8 259.3 260.3 263.3 265.4 268.0 270.4 269.3

7177

8177 9/77 10/77 11/77 12177 1178 2/78 3178 4178 5/78 6178 7178 8/78 9/78 10/78 11/78 12/78 1/79 2/79 3/79

252

W .A. Bamett

APPENDIX 1 (continued) Date

SDM}

SDM2

SDM3

Date

SDM}

5DM2

8DM3

1/83 2/ 83 3/ 83 4/83 5/83 6/83 7/83 8/83 9/83 10/83 11/83 12/83

273.2 274.3 275.9 276.4 280.4 282.4 283.7 284.8 285.7 286.9 288.0 289.2

258.7 259.9 261.0 261.8 263.8 265.5 266.6 267.6 268.4 270.1 271.7 272.7

265.0 265.1 265.9 266.7 268.5 270.5 271.3 272.7 274.0 275.8 278.2 278.8

1/84 2/84 3/84 4/84 5/84 6/84 7/84

292.0 293.5 294.9 295.4 298.5 301.1 301.1

273.8 275.2 276.5 277.5 279.1 280.9 281.9

280.6 283.0 284.9 286.7 289.1 291.4 292.9

Note: SDMi designates the supply Divisia Mi index, for i = I, 2, 3.

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