Divisia Index, Inflation, and Welfare

Abstract

This work addresses a usual criticism in the literature of the welfare costs of inflation, related to the fact that some items in the relevant definition of money pay interest, while others do not. We show that the problem can be solved by using a Divisia index of monetary services as the welfare measure.

This work aims at investigating, in the search for an adequate measure of the welfare costs of inflation, the fact that some assets included in the theoretically relevant definition of money pay interest, while others do not. The problem is widespread in the literature. Many analyses define money as a non-interest-bearing asset held by households but, at the same time, use M_α(α ≥ 1) as the respective empirical counterpart.

Referring to the calculation of the welfare costs of inflation, Marty (1999, p. 46) notes that:

if M₁ is used as a relevant money supply, some correction must be made for the interest paid on portions of M₁.

Lucas (2000) also voices concern regarding this fact. Indeed, the connection between the welfare costs of inflation and the Divisia indices of monetary services has been conjectured by this author (p. 270):

I share the widely held opinion that M₁ is too narrow an aggregate for this period [the 1990s], and I think that the Divisia approach offers much the best prospects for resolving this difficulty.

However, Lucas (2000) does not develop the link between the theoretical measures of the welfare costs of inflation and the Divisia index of monetary services. By developing such a link here, we establish a framework that validates his conjecture.

Our primary purpose in this paper is not empirical. We are solely interested in providing a theoretical investigation of how one could think about the welfare costs of inflation in a model in which different monetary aggregates are used for transacting purposes.

Our results build directly on Simonsen and Cysne's (2001) work, which in turn draws upon the work of Lucas (2000). We extend Simonsen and Cysne's original results by investigating situations in which: (1) the opportunity costs of all monetary assets are allowed to vary; (2) interest rates are endogenously determined in a general-equilibrium setting; and (3) financial innovations are taken into consideration. As in these previous works, our economy is a deterministic one. Further extensions, including the analysis of risk, are suggested in Section 7.

We present our results in two consecutive steps. In the first (Section 5), we assume that the government (consolidated here with the Central Bank) issues all the types of money, either interest-bearing or non-interest-bearing, setting the respective interest rates. Although the objective here is more of a didactic nature, this part of the analysis can be useful as a proxy for situations faced by high-inflation economies,¹ or in the case of a banking system facing legal restrictions.

In a second step (Section 6), we close the model by assuming that interest rates on monetary assets are determined by a competitive banking system. Instead of fixing the interest rates, the government is then supposed to fix the reserve requirements on each asset.

The remainder of this work is organized as follows. Section 1 presents the model. Section 2 is used to define three different versions of the Divisia index of monetary services and prove their path independence. Section 3 demonstrates how well the Divisia indices approximate the welfare cost of inflation.

Section 4 shows that financial innovations have a direct negative impact on the welfare costs of inflation. The result sheds light on the reasons for the large demand for financial innovations in countries subject to high rates of inflation. Section 5 exemplifies the use of the different welfare measures investigated here in applied work and briefly discusses the case when the assets' demand functions are not known by the researcher.

Section 6, as previously mentioned, closes the model by introducing a banking system and allowing interest rates to be competitively determined. In this section we also establish sufficient conditions under which the monetary base emerges as the adequate aggregate to be used in the calculations...