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# Divisia monetary aggregates

The monetary aggregates currently in use by the Federal Reserve (and most other central banks around the world) are simple-sum indices in which all monetary components are assigned a unitary weight, as follows

$M_\left\{t\right\}=sum_\left\{j=1\right\}^\left\{n\right\}x_\left\{jt\right\}$

where $x_\left\{jt\right\}$ is one of the $n$ monetary components of the monetary aggregate $M_\left\{t\right\}$. This summation index implies that all monetary components contribute equally to the money total and it views all components as dollar for dollar perfect substitutes. Such an index, however cannot, in general, represent a valid structural economic variable for the services of the quantity of money.

Over the years, there have been many attempts at properly weighting monetary components within a simple-sum aggregate. With no theory, however, any weighting scheme is questionable. Recently, however, attention has beenfocused on the gains that can be achieved by a rigorous use of microeconomic- and aggregation-theoretic foundations in the construction of monetary aggregates. This new approach to monetary aggregation was derived and advocated by William A. Barnett (1980) and has led to the construction of monetary aggregates based on Diewert's (1976) class of superlative quantity index numbers. The new aggregates are called the Divisia aggregates or Monetary Services Indexes. Early research with those aggregates using American data was done by Salam Fayyad.

The Divisia index (in discrete time) is defined as

$log M_\left\{t\right\}^\left\{D\right\}-log M_\left\{t-1\right\}^\left\{D\right\}=sum_\left\{j=1\right\}^\left\{n\right\}s_\left\{jt\right\}^\left\{*\right\}\left(log x_\left\{jt\right\}-log x_\left\{j,t-1\right\}\right)$

according to which the growth rate of the aggregate is the weighted average of the growth rates of the component quantities. The original continuous time Divisia index was derived by Francois Divisia in his classic paper published in French in 1925 in the "Revue d'Economie Politique". The discrete time Divisia weights are defined as the expenditure shares averaged over the two periods of the change

$s_\left\{jt\right\}^\left\{*\right\}=frac\left\{1\right\}\left\{2\right\}\left(s_\left\{jt\right\}+s_\left\{j,t-1\right\}\right)$

for $j=1,..., n$, where

$s_\left\{jt\right\}=frac\left\{pi _\left\{jt\right\}x_\left\{jt\left\{sum_\left\{k=1\right\}^\left\{n\right\}pi _\left\{kt\right\}x_\left\{kt$

is the expenditure share of asset $j$ during period $t$, and $pi _\left\{jt\right\}$ is the user cost of asset $j$, derived in Banett (1978),

$pi _\left\{jt\right\}=frac\left\{\left(R_\left\{t\right\}-r_\left\{jt\right\}\right)\right\}\left\{\left(1+R_\left\{t\right\}\right)\right\}$

which is just the opportunity cost of holding a dollar's worth of the $j$th asset. In the last equation, $r_\left\{jt\right\}$ is the market yield on the $j$th asset, and $R_\left\{t\right\}$ is the yield available on a 'benchmark' asset that is held only to carry wealth between multiperiods.

The Divisia approach to monetary aggregation represents a viable and theoretically appropriate alternative to the simple-sum approach, which is unfortunately still in use by some central banks. Barnett, Fisher, and Serletis (1992), Barnett and Serletis (2000), and Serletis (2007) provide more details regarding the Divisia approach to monetary aggregation. Divisia Monetary Aggregates are available for the United Kingdom by the [http://www.bankofengland.co.uk/statistics/ms/current/index.htm Bank of England] and for the United States by the [http://research.stlouisfed.org/msi/index.html Federal Reserve Bank of St. Louis] .

References

* [http://econ.tepper.cmu.edu/barnett/Welcome.html Barnett, William A.] "The User Cost of Money". "Economics Letters" (1978), 145-149.

* Barnett, William A. "Economic Monetary Aggregates: An Application of Aggregation and Index Number Theory", "Journal of Econometrics" 14 (1980), 11-48.

* Barnett, William A. and Apostolos Serletis. "The Theory of Monetary Aggregation". Contributions to Economic Analysis 245. Amsterdam: North-Holland (2000).

* Barnett, William A., Douglas Fisher, and Apostolos Serletis. "Consumer Theory and the Demand for Money". "Journal of Economic Literature" 30 (1992), 2086-2119.

* [http://www.econ.ubc.ca/diewert/hmpgdie.htm Diewert, W. Erwin.] "Exact and Superlative Index Numbers". "Journal of Econometrics" 4 (1976), 115-146.

* [http://econ.ucalgary.ca/serletis.htm Serletis, Apostolos.] "The Demand for Money: Theoretical and Empirical Approaches". Springer (2007).

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