The monetary aggregates used by most central banks are simple-sum indexes in which all monetary components are assigned the same weight: in which is one of the monetary components of the monetary aggregate. The summation index implies that all monetary components contribute equally to the money total, and it views all components as dollar-for-dollar perfect substitutes. It has been argued that such an index does not weight such components in a way that properly summarizes the services of the quantities of money. There have been many attempts at weighting monetary components within a simple-sum aggregate. An index can rigorously apply microeconomic- and aggregation-theoretic foundations in the construction of monetary aggregates. That approach to monetary aggregation was derived and advocated by William A. Barnett and has led to the construction of monetary aggregates based on Diewert's class of superlative quantity index numbers. The new aggregates are called the Divisia aggregates or Monetary Services Indexes. Salam Fayyad's 1986 PhD dissertation did early research with those aggregates using U.S. data. This index is a discrete-time approximation with this definition: Here, the growth rate of the aggregate is the weighted average of the growth rates of the component quantities. The discrete time Divisia weights are defined as the expenditure shares averaged over the two periods of the change for, where is the expenditure share of asset during period, and is the user cost of asset, derived by Barnett, which is the opportunity cost of holding a dollar's worth of the th asset. In the last equation, is the market yield on the th asset, and is the yield available on a benchmark asset, held only to carry wealth between different time periods. In the literature on aggregation and index number theory, the Divisia approach to monetary aggregation,, is widely viewed as a viable and theoretically appropriate alternative to the simple-sum approach. See, for example, International Monetary Fund, Macroeconomic Dynamics, and Journal of Econometrics. The simple-sum approach,, which is still in use by some central banks, adds up imperfect substitutes, such as currency and non-negotiable certificates of deposit, without weights reflecting differences in their contributions to the economy's liquidity. A primary source of theory, applications, and data from the aggregation-theoretic approach to monetary aggregation is the in New York City. More details regarding the Divisia approach to monetary aggregation are provided by Barnett, Fisher, and Serletis, Barnett and Serletis, and Serletis. Divisia Monetary Aggregates are available for the United Kingdom by the , for the United States by the , and for Poland by the . Divisia monetary aggregates are maintained for internal use by the , the , the , and the .
