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In litigation involving personal injury, wrongful death, job discrimination, and breach of employment contracts, questions concerning the estimates of future earnings and the choice of an appropriate discount rate arise. We address the question of whether, in order to estimate the growth of future earnings and then to discount future earnings, it is necessary to forecast the inflation rate. Beginning with the neoclassical theory of the firm and intertemporal utility maximization theory we derive results which show that it is not necessary to forecast the rate of inflation; if discrete growth rates are combined with a discrete growth model or, if continuous growth rates are combined with a continuous growth model, the inflation rate cancels out. Our results obviate the need to forecast the inflation rate.

I. Introduction

In litigation involving personal injury, wrongful death, job discrimination, and breach of employment contracts, expert witnesses often are called upon to calculate the present discounted value (PDV) of an estimated stream of future earnings. Such calculations raise any number of practical questions concerning the estimates of future earning and the choice of an appropriate discount rate. Analysts advocate using long-term government bond rates, Treasury bill rates, or rates available on annuities (Edward, 1975; Harris, 1983; Carpenter et a1., 1986). While the issue over which discount rate is most appropriate is important, this note addresses a more fundamental theoretical issue. We address the question of whether, in order to estimate the growth of future earnings and then to discount future earnings, it is necessary to forecast an inflation rate. In other words, should nominal earnings be discounted by (some appropriate) nominal rate of interest or can real earnings be discounted by a real rate of interest. Beginning with the neoclassical theory of the firm and intertemporal utility maximization theory we show that it is not necessary to forecast the rate of inflation; if discrete growth rates are combined with a discrete growth model or, if continuous growth rates are combined with a continuous growth model, the inflation rate cancels out. Thus, the nommal and real approach lead to precisely the same estimate of PDV.

While the above proposition has been denied in the literature (for a recent example, see Abraham, 1988), this note traces the reason for the denial to the application of growth rates derived for continuous time periods to discrete time growth models. Typically, textbooks in economics and finance express the relationship between the nominal rate of interest (observable market rates) and the real rate as:

(1) R = r + i

And, analysts often simply express the growth rate in earnings as:

(2) G = g + i

Where:

R = nominal interest rate

r = real interest rate

i = anticipated rate of inflation

G = growth rate of nominal earnings

g = growth rate of labor productivity

What is too rarely made clear is that this expression of Fisher's Equation and this expression of earnings growth rate are valid only at the limit of continuous compounding. When these expressions are applied to data defined for discrete periods (annual earnings, for example), using a discrete time growth model, differences do arise between the calculations ofPDV of nominal earnings discounted by R and real earnings discounted by r. But, the difference stems from a conceptual error, not because of some inherent superiority of nominal rates ove'r real rates (or vice versa) as the growth and discount rates.

In Section II.A we derive the discrete growth rate for earnings for the neoclassical theory of the firm. In addition, the discount rate for discrete time is derived from intertemporal utility maximization theory. The two rates are combined with a discrete time growth model. In Section II.B the two rates are derived for continuous time. These two rates are combined with a continuous time growth model. In the models of Sections II.A and II.B it is seen that the inflation rate cancels out. In Section II.c we show that if the continuous growth rates are combined with a discrete growth model, (or vice versa) it would be erroneously concluded that the inflation rate does not cancel out and, therefore, must be forecasted. Section III contains the conclusions.

II. The Model

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Dr. Gary R. Albrecht has more than 25 years of experience specializing in Economic Forecasting and Forensic Economics. The Director of Econometric Modeling at the University of Kansas, his research has been published in the Journal of Forensic Economics, Journal of Legal Economics, Trial Briefs, and The Earnings Analyst.

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