Appendix A: Definition of ϕ
The marginal propensities to consume ϕ can be written as:
where the functions ϕ1 and ϕ2 are defined as:
Notice that in the logarithmic case when r→1, the ϕs are not functions of factor prices, but only of the parameters θ and δ. In other words, they would depend on the age of the individual but not on time.
Appendix B: The Interest Elasticity of the Money Demand
The interest elasticity of the money demand, η, is defined as follows:
For values of ξ sufficiently close to zero, the elasticity can be approximated by -1/γ(1+i).
Appendix C: The Price Equation
The explicit form of the equation for the price level corresponding to expression (38) is:
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This paper is based on my Ph.D. dissertation at Boston University. I have benefitted from very helpful comments by Larry Kotlikoff, Christophe Chamley, Russell Cooper, Jorge Marquez-Ruarte, Miguel Savastano, and Jose de Gregorio. All remaining errors are my sole responsibility.
Friedman (1968), p. 44.
The following discussion concentrates on the optimal inflation rate in a world in which lump-sum taxation is available. For a discussion on the optimal rate of inflation in a second-best world see, for example, Phelps (1973) and Chamley (1986).
As it will be seen in sub-section IV.2, this is not a robust result since a wide range of values for the inflation rate lead to similar welfare gains.
The reason for using a 55-period model is essentially demographic, since for the United States 55 constitutes a good approximation of the number of years that elapse from the time an individual enters the labor force until he dies.
Different values of γ could have been used for c and m; however, as will be seen in the next section, the selection of a common value yields plausible results in the numerical simulations. Furthermore, a common value of γ ensures that the preferences represented by (1) are homothetic; as a result, the welfare analysis developed in section IV is greatly simplified.
The term ms, s+55 is actually dropped from the lifetime utility function. For a discussion about the plausibility of this assumption see: Orphanides and Solow (1990).
The term “real asset” is used here as a synonym of claims on physical capital.
The nominal interest rate is defined as: i=(l+r)(1+π)-1, and π is the inflation rate: pt/pt-1-1
Of the three equations one is obviously redundant. All three are shown, however, with the purpose of facilitating the discussion.
This assumption is made in order to prevent an intergenerational redistribution of resources. Further discussion on this issue is provided in section IV.1.
The historical averages for the period 1970-90 are 17.6 percent for the Ml-GDP ratio and 6.5 percent for the money base-GDP ratio.
The equivalent variation measure λ indicates the proportion by which the lifetime resources of a newly born individual would have to be adjusted under an inflation rate π0, for him to obtain the same utility that he would have enjoyed if inflation were π*.
The same rate is obtained when the money base is the relevant monetary aggregate.
It should be pointed out that Summers’ results stem from a model in which the Tobin effect is practically imposed by the assumption of a constant savings rate in a world with two assets: money and capital.
The GA is assumed to break even in the sense that the PDV of all its transfers must equal the PDV of the taxes it collects.
As before, λ represents an equivalent variation measure, expressed as a percentage of an individual’s lifetime resources in the original steady state. It is a function of the ratio U1/U0 because the utility function (1) is homothetic.
Given the parameterization of their model, these authors find an optimal inflation rate of -4 percent.
The results reported for C&H are those corresponding to a quarterly cash-in-advance constraint (top panel of their table 2, p. 743). In that case, their implicit money to output ratio is 18.6 percent for the steady state corresponding to an inflation rate of -4 percent, compared to a 25.4 percent in this model’s baseline scenario (Ml).