This entry on monetary aggregation will appear under that title in The New Palgrave Dictionary of Economics, 2nd edition, edited by Steven Durlauf and Lawrence Blume. The entry provides an up-to-date overview of state-of-the-art research on monetary aggregation and index number theory, from its origins in 1980 to the current time. At the end of this dictionary entry, emphasis is placed on ongoing research on extensions to risk and to multilateral aggregation within multicountry areas, such as the euro area. Research on monetary aggregation theory has been especially successful in solving the 'puzzles' that have appeared in the monetary economics literature over the past 35 years.monetary aggregation, Divisia index, money demand, monetary policy, dictionary, Divisia monetary aggregates

William Barnett

English

Research Papers in Economics

 
 
 
Monetary Aggregation 
 
 
 
 
 
William A. Barnett 
University of Kansas 
 
March 21, 2005 
 
 
 
 
 
 
 
Forthcoming in Steven N. Durlauf and Lawrence Blume (eds.), The New Palgrave Dictionary of 
Economics, 2
nd edition, Macmillan 
 
 
 
 
 
JEL Classifications:  E41,G12,C43,C22 
  1 
1.  Introduction 
  Aggregation theory and index-number theory have been used to generate official 
governmental data since the 1920s.  One exception still exists.  The monetary quantity 
aggregates and interest rate aggregates supplied by many central banks are not based on index-
number or aggregation theory, but rather are the simple unweighted sums of the component 
quantities and quantity-weighted or arithmetic averages of interest rates.  The predictable 
consequence has been induced instability of money demand and supply functions, and a series of 
“puzzles” in the resulting applied literature.  In contrast, the Divisia monetary aggregates, 
originated by Barnett (1980), are derived directly from economic index-number theory.  
Financial aggregation and index number theory was first rigorously connected with the literature 
on microeconomic aggregation and index number theory by Barnett (1980,1987).  A collection 
of many of his contributions to that field is available in Barnett and Serletis (2000).  
  Data construction and measurement procedures imply the theory that can rationalize the 
procedure. The assumptions implicit in the data construction procedures must be consistent with 
the assumptions made in producing the models within which the data are to be used.  Unless the 
theory is internally consistent, the data and its applications are incoherent.  Without that 
coherence between aggregator function structure and the econometric models within which 
aggregates are embedded, stable structure can appear to be unstable.  This phenomenon has been 
called the “Barnett critique” by Chrystal and MacDonald (1994).  
2.  Aggregation Theory versus Index Number Theory   
  The exact aggregates of microeconomic aggregation theory depend on unknown aggregator 
functions, which typically are utility, production, cost, or distance functions.  Such functions 
must first be econometrically estimated.  Hence the resulting exact quantity and price indexes 
  2become estimator and specification dependent.  This dependency is troublesome to governmental 
agencies, which therefore view aggregation theory as a research tool rather than a data 
construction procedure. 
  Statistical index-number theory, on the other hand, provides indexes which are computable 
directly from quantity and price data, without estimation of unknown parameters.  Such index 
numbers depend jointly on prices and quantities, but not on unknown parameters.  In a sense, 
index number theory trades joint dependency on prices and quantities for dependence on 
unknown parameters.  Examples of such statistical index numbers are the Laspeyres, Paasche, 
Divisia, Fisher ideal, and Törnqvist indexes.   
  The loose link between index number theory and aggregation theory was tightened, when 
Diewert (1976) defined the class of second-order “superlative” index numbers.  Statistical index 
number theory became part of microeconomic theory, as economic aggregation theory had been 
for decades, with statistical index numbers judged by their nonparametric tracking ability to the 
aggregator functions of aggregation theory.   
  For decades, the link between statistical index number theory and microeconomic 
aggregation theory was weaker for aggregating over monetary quantities than for aggregating 
over other goods and asset quantities.  Once monetary assets began yielding interest, monetary 
assets became imperfect substitutes for each other, and the “price” of monetary-asset services 
was no longer clearly defined.  That problem was solved by Barnett (1978,1980), who derived 
the formula for the user cost of demanded monetary services.  Subsequently Barnett (1987) 
derived the formula for the user cost of supplied monetary services.  A regulatory wedge can 
exist between the demand and supply-side user costs, if nonpayment of interest on required 
reserves imposes an implicit tax on banks. 
  3  Barnett’s results on the user cost of the services of monetary assets set the stage for 
introducing index number theory into monetary economics. 
3.  The Economic Decision 
  Consider a decision problem over monetary assets that illustrates the capability of the 
monetary aggregation theory.  The decision problem will be defined so that the relevant 
literature on economic aggregation over goods is immediately applicable.  Initially we shall 
assume perfect certainty. 
 Let  m t′  = (m1t, m2t, … , mnt) be the vector of real balances of monetary assets during 
period t, let rt be the vector of nominal holding-period yields for monetary assets during period t, 
and let Rt be the one period holding yield on the benchmark asset during period t.  The 
benchmark asset is defined to be a pure investment that provides no services other than its yield, 
Rt, so that the asset is held solely to accumulate wealth.  Thus, Rt is the maximum holding period 
yield in the economy in period t. 
 Let  yt be the real value of total budgeted expenditure on monetary services during period t.  
Under simplifying assumptions for data within one country, the conversion between nominal and 
real expenditure on the monetary services of one or more assets is accomplished using the true 
cost of living index on consumer goods.  But for multicountry data or on data aggregated across 
heterogeneous regions, the correct deflator can be found in Barnett (2003,2005).  The optimal 
portfolio allocation decision is: 
  
 maximize  u(mt)       (1) 
 subject  to  π t′ mt = yt,  
 
  4where π t = (π ′ 1t ,…,πnt) is the vector of monetary-asset real user costs, with 
  πit = 
ti t
t
R r
1R
−
+
.       (2) 
This function u is the decision maker’s utility function, assumed to be monotonically increasing 
and strictly concave.  The user cost formula (2), derived by Barnett (1978, 1980), measures the 
foregone interest or opportunity cost of holding a unit of monetary asset i, when the higher 
yielding benchmark asset could have been held.  
  To be an admissible quantity aggregator function, the function u must be weakly separable 
within the consumer’s complete utility function over all goods and services.  Producing a reliable 
test for weak separability is the subject of much intensive research by an international group of 
econometricians.  See, e.g., Jones, Dutkowsky, and Elger (2005), Fleissig and Whitney (2003), 
and De Peretti (2005).  Two approaches exist.  One approach uses stochastic extensions of 
nonparametric revealed preference tests, while the other uses parametric econometric models.     
 Let   be derived by solving decision (1).  Under the assumption of linearly homogeneous 
utility, the exact monetary aggregate of economic theory is the utility level associated with 
holding the portfolio, and hence is the optimized value of the decision’s objective function: 
* mt
 M t = u( ).       (3) 
* mt
4.  The Divisia Index 
  Although equation (3) is exactly correct, it depends upon the unknown function, u.  
Nevertheless, statistical index-number theory enables us to track Mt exactly, without estimating 
the unknown function, u.  In continuous time, the exact monetary aggregate, Mt = u( ), can be 
tracked exactly by the Divisia index, which solves the differential equation 
* mt
  5 
*
i
log log
=∑
ti
it
dM dm
s
dt dt
t        (4) 
for Mt, where  
 
* π
=
it it
it
t
m
s
y
 
is the i’th asset’s share in expenditure on the total portfolio’s service flow.  In equation (4), it is 
understood that the result is in continuous time, so the time subscripts are a shorthand for 
functions of time.  We use t to be the time period in discrete time, but the instant of time in 
continuous time.  The dual user cost price aggregate Πt = Π( t π ), can be tracked exactly by the 
Divisia price index, which solves the differential equation 
 
i
log log ti
it
dd
s
dt dt
Ππ
=∑
t .       (5) 
The user cost dual satisfies Fisher’s factor reversal in continuous time: 
  ΠtMt = π t m ′ t.       (6) 
  As a formula for aggregating over quantities of perishable consumer goods, that index was 
first proposed by Francois Divisia (1925) with market prices of those goods inserted in place of 
the user costs in equation (4).  In continuous time, the Divisia index, under conventional 
neoclassical assumptions, is exact.  In discrete time, the Törnqvist approximation is: 
 
**
i
log log (log log ) −= − ∑ t t-1 it it i,t-1 MM s mm ,       (7) 
where  
  () =+ it it i,t-1
1
2
ss s . 
In discrete time, we often call equation (7) simply the Divisia quantity index.  After the quantity 
  6index is computed from (7), the user cost aggregate most commonly is computed directly from 
equation (6).     
  Diewert (1976) defines a ”superlative index number” to be one that is exactly correct for a 
quadratic approximation to the aggregator function.  The discretization (7) to the Divisia index is 
in the superlative class, since it is exact for the quadratic translog specification to an aggregator 
function.  With weekly or monthly monetary data, Barnett (1980) has shown that the Divisia 
index growth rates, (7), are accurate to within three decimal places.  In addition, the difference 
between the Fisher ideal index and the discrete Divisia index growth rates are third order and 
comparably small.  That third-order differential error typically is smaller than the round-off error 
in the component data. 
5.  Prior Applications 
  Divisia monetary aggregates were first constructed for the United States by Barnett (1980), 
when he was on the staff of the Special Studies Section of the Board of Governors of the Federal 
Reserve System, and now are maintained by the Federal Reserve Bank of Saint Louis in its data 
base, called FRED.  See Anderson, Jones, and Nesmith (1997), who produced the Divisia data 
for FRED.  A Divisia monetary-aggregates data base also has been produced for the UK by the 
Bank of England.  An overview of Divisia data maintained by many central banks throughout the 
world can be found in Belongia and Binner (2000,2005) and in Barnett, Fisher, and Serletis 
(1992), along with a survey of empirical results with that data.  The most extensive collection of 
relevant applied and theoretical research in that area is in Barnett and Serletis (2000) and Barnett 
and Binner (2004). 
6.  The State of the Art 
  6.1.  Multilateral Aggregation 
  7  The European Central Bank is implementing a multilateral extension of the Divisia 
monetary aggregates for monetary quantity and interest rate aggregation within the Euro area.  
This aggregation is multilateral in the recursive sense that it permits aggregation of monetary 
service flows first within countries, followed by aggregation over countries.  The resulting 
aggregation will be in a strictly nested, internally consistent manner.  The multilateral extension 
of the theory was produced by Barnett (2003,2005).  This extension was produced under three 
increasingly strong sets of assumptions, (1) with the weakest being produced from 
heterogeneous agents theory, (2) followed by the somewhat stronger assumption of existence of 
a multilateral representative agent, and (3) finally with the strongest being the assumption of 
existence of a unilateral representative agent.  The intent is to move from the weakest towards 
the strongest assumptions, as progress is made within the European Monetary Union towards its 
harmonization and economic convergence goals.  Since Barnett’s three assumption structures are 
nested, construction of the data under the most general heterogeneous countries approach would 
continue to be valid, as the stronger assumptions become more reasonable and are attained 
within the Euro area.  
  6.2.  Extension to Risk 
  Extension of index number theory to the case of risk was introduced by Barnett, Liu, and 
Jensen (2000), who derived the extended theory from Euler equations, rather than from the 
perfect-certainty first-order conditions used in the earlier index number theory literature.  Since 
that extension is based upon the consumption capital-asset-pricing model (CCAPM), the 
extension is subject to the “equity premium puzzle” problem of smaller than necessary 
adjustment for risk.  We believe that the undercorrection produced by CCAPM results from its 
assumption of intertemporal blockwise strong separability of goods and services within 
  8preferences.  Barnett and Wu (2005) have extended Barnett, Liu, and Jensen’s result to the case 
of risk aversion with intertemporally nonseparable tastes.   
  The extension to risk is likely to be especially important to countries in which its residents 
hold significant deposits in foreign denominated assets, since exchange rate risk can cause rates 
of return on monetary assets to be subject to non-negligible risk.  With the recent trend towards 
financial integration in many parts of the world, exchange rate risk is likely to grow in 
importance in monetary aggregation.  In many countries, the largest holder of foreign 
denominated deposits is the central bank itself.  Within the United States, the extension to risk is 
highly relevant to the so called “missing M2” episode of the early 1990s, when substitutability 
among small time deposits, stock funds, and bond funds produced “puzzles.” 
7.  Duality 
  User cost aggregates are duals to monetary quantity aggregates.  Either implies the other 
uniquely.  In addition, user-cost aggregates imply the corresponding interest rate aggregates 
uniquely.  The interest-rate aggregate rt implied by the user-cost aggregate Πt is the solution for 
rt to the equation: 
 
tt
t
R r
1R
−
+
 = Πt. 
Accordingly any monetary policy that operates through the opportunity cost of money (i.e., 
interest rates) has a dual policy operating through the monetary quantity aggregate, and vice 
versa.  Aggregation theory implies no preference for either of the two dual policy procedures, or 
for any other approach to policy, so long as the policy does not violate principles of aggregation 
theory.   
8.  Conclusion 
  9  Aggregation theory is about measurement, and has little, if anything, to say about the 
choice of policy instrument, such as the funds rate or the base.  But accurate measurement, 
through proper application of aggregation theory, has much to say about the transmission of 
policy, modeling of structure, and the measurement of intermediate targets (if any) and final 
targets. 
  Policies that violate aggregation theoretic principles include the following oversimplified 
approaches:  (1) inflation targeting that targets one arbitrary consumer-good price as a final 
target, while ignoring all other consumer goods prices, rather than targeting the true cost-of-
living index over all consumer goods prices, (2) interest rate targeting that analogously targets 
one arbitrary interest rate as an intermediate target, while ignoring all other interest rates, rather 
than targeting the aggregation-theoretic interest-rate or user-cost aggregate over a weakly-
separable collection of monetary assets, (3) monetary quantity targeting that targets a simple-
sum monetary aggregate as an intermediate target, rather than the aggregator function over a 
weakly-separable collection of monetary assets, and (4) policy simulations using money-demand 
or money-supply functions containing simple-sum monetary aggregates or quantity-weighted 
interest-rate aggregates.  The measurement defects in the above four cases are unrelated to the 
choice of the funds rate or monetary base as an instrument of policy.  Unlike intermediate 
targets, final targets, and variables in models, the chosen instruments of policy tend to be highly-
controllable, disaggregated variables, presenting few serious measurement problems.     
  The objective of the Divisia monetary aggregates is measurement of the economy’s 
monetary service flow and its dual opportunity cost (user cost) and implied interest rate 
aggregate, not advocacy of any particular policy use of the correctly measured variables.  But all 
uses of data are adversely affected by improper measurement, and a long series of “puzzles” in 
  10monetary economics have been shown to have been produced by improper measurement.  See, 
for example, chapter 24 of Barnett and Serletis (2000). 
 
REFERENCES 
Anderson, Richard G., Barry E. Jones, and Travis Nesmith (1997), “Building New Monetary 
Services Indexes:  Concepts, Data and Methods,” Federal Reserve Bank of St. Louis Review, 
January/February, pp. 53-82.   
 
Barnett, W. A. (1978), “The User Cost of Money,” Economics Letters, vol 1, pp. 145-149.   
 
Barnett, W. A. (1980), “Economic Monetary Aggregates:  An Application of Aggregation and 
Index Number Theory,” Journal of Econometrics, vol 14, pp. 11-48.   
 
Barnett, W. A. (1987), “The Microeconomic Theory of Monetary Aggregation,” in: W. A. 
Barnett and K. J. Singleton (eds.), New Approaches in Monetary Economics, Cambidge 
University Press, Cambridge, pp. 115 – 168.   
 
Barnett, William A. (2003), “Aggregation-Theoretic Monetary Aggregation over the Euro Area, 
when Countries are Heterogeneous,” European Central Bank Working Paper No. 260.  
 
Barnett, William A. (2005), “Multilateral Aggregation-Theoretic Monetary Aggregation over 
Heterogeneous Contries,” Journal of Econometrics, forthcoming. 
 
  11Barnett, William A. and Jane Binner (2004), Functional Structure and Approximation in 
Econometrics, North Holland, Amsterdam. 
 
Barnett, William A. and Apostolos Serletis (eds.) (2000), The Theory of Monetary Aggregation, 
North Holland, Amsterdam.  
 
Barnett, William A. and Shu Wu (2005), “On User Costs of Risky Monetary Assets,” Annals of 
Finance, vol. 1, no. 1, January, pp. 35-50.  
 
Barnett, W. A., Douglas Fisher, and Apostolos Serletis (1992), “Consumer Theory and the 
Demand for Money,” Journal of Economic Literature 30, pp. 2086-2119. 
 
Barnett, W. A., Y. Liu and M. Jensen (2000), “CAPM Risk Adjustment for Exact Aggregation 
over Financial Assets”, Macroeconomic Dynamics 1, pp. 485-512. 
 
Belongia, Michael T. and Jane M. Binner (2000), Divisia Monetary Aggregates:  Theory and 
Practice, Palgrave, London.  
 
Belongia, Michael T. and Jane Binner (2005), Money, Measurement, and Computation, 
Palgrave, London. 
 
Chrystal, A. K. and R. MacDonald (1994), “Empirical Evidence on the Recent Behaviour and 
Usefulness of Simple-Sum and Weighted Measures of the Money Stock,” Federal Reserve Bank 
  12of St. Louis Review, vol. 76, pp. 73-109.  
 
De Peretti, P. (2005), “Testing the Significance of the Departures from Utility Maximization,” 
Macroeconomic Dynamics, June, vol 9, no 3, forthcoming. 
 
Diewert, W. Erwin (1976), “Exact and Superlative Index Numbers,” Journal of Econometrics, 
vol. 4, pp. 115-145.  
 
Divisia, Francois (1925), “L’Indice Monétaire et la Théorie de la Monnaie,” Revue d’Economie 
Politique 39, pp. 980-1008. 
 
Fleissig, A. and G. Whitney (2003), “A New PC-Based Test for Varian’s Weak Separability 
Conditions,” Journal of Business and Economic Statistics, 21, pp. 133-144. 
 
Jones, B., D. Dutkowsky, and T. Elger (2005), “Sweep Programs and Optimal Monetary 
Aggregation,” Journal of Banking and Finance 29, pp. 483-508. 
 
 
  13

A New PC-Based Test for Varian’s Weak Separability Conditions,”

Aggregation-Theoretic Monetary Aggregation over the Euro Area, when Countries are Heterogeneous,” European Central Bank Working Paper No.

and Apostolos Serletis (eds.) (2000), The Theory of Monetary Aggregation,

Building New Monetary Services Indexes: Concepts, Data and Methods,” Federal Reserve Bank of St. Louis Review, January/February,

CAPM Risk Adjustment for Exact Aggregation over Financial Assets”,

Consumer Theory and the Demand for Money,”

Divisia Monetary Aggregates: Theory and Practice,

Economic Monetary Aggregates: An Application of Aggregation and

Exact and Superlative Index Numbers,”

Multilateral Aggregation-Theoretic Monetary Aggregation over Heterogeneous Contries,”

Testing the Significance of the Departures from

The Microeconomic Theory of Monetary Aggregation,”

Monetary Aggregation

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Monetary Aggregation

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