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Appendix CALCULATION OF THE DISCOUNTED VALUE OF ECONOMIC BENEFITS FROM RESEARCH In trying to obtain some notion of the possible economic benefits from research, we have assumed that the annual production or the annual savings resulting from research in a particular area will follow a growth curve of the form Vit research = bB [ (1 + a) t- (1 + a') t I (l) where Vi:~.ese~Ch is the value of benefits resulting directly from research during a particular year, t, in the future; b is the fraction of the annual savings or annual increased production that can be credited directly to research; a' is the rate of growth of production or savings without benefit of research; a is the rate of growth with research; and B is the annual pro- duction or the annual savings, minus the production or savings that would occur without research, at the time, T. when (1 +a)T_~1 {a')T= 1. We may make the following substitutions in (~) or or cat - anda= T T v (2) o~ ~ 0.7 if a' is small compared with a To judge the value of research expenditures in comparison with other possible ways of employing the same money and human effort, we may reduce the anticipated future economic benefits from research to their "present worth," that is, their value at the present time. This is equal to the immediate return on an investment at compound interest that would yield the same future return as the research. The rate of interest, i, is called the discount rate. The discounted value (present worth) of the benefits from research during the year, t, is twit research = bB :( T) ( T) ~ ~ ~ ~ and the total discounted benefits during the period from t = 0 (the present time to t = t will be DVit; research bB :(1+ ~ )t-(1 + at] ~ 47

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If the research expenditures during any year, t, are cat, then the net discounted benefits from the research are t=t Whit research = ~ ~ bB t( + T) ( T) ] ~ (t - i) ~ =o For simplicity, in calculation we may replace Equation 1 by the exponential equation Vit research = b (`V~ - VO) = bB (`eat- 1), (3) where Via is the annual production or the annual savings at any time, t; VO is the production at the present time (in the case of savings, or new industries, VO = 0), B is the annual production at time T minus the .693 present production, or, alternatively, the annual savings, and a =-. The expression is "normalized" by assuming that a' is zero. When t = 0, Vie - VO = 0; when t = T. Vie - VO = B; when t = 2 T. Vat- VO = 3B. The total value of the benefits from research received over the period from t = 0tot = tis t Vit research = bB J (eat - 1) dt o bB - (eat-1-at). a The value of net benefits discounted back to the present time is t t NVit research = bB J (e(a-r)t-e-rt) dt- J Ct e-rt dt O O where i = 1-e-r. If cat is constant from year to year and equal to R. the solution is Nitwit research = bB [- Pea- _ I) +- (e-rt- 1) ~ -- (1-em) and the benefit-cost ratio is ,_ ~ _ (efa-r) ~- 1) + 1 (e-rt- 1) | R ~l-e-rt) L a-r r When T is 20 years and i = 0.1 (r = 0.105) this becomes bB 0.12T / 13.~fi \ R .693 - 0.1T1, 014e T -1 J 1 A ten per cent discount rate takes into account the considerable uncer- tainty of research results; that is, the fact that the research investment is a fairly risky one. 48

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In addition to research expenditures, capital investment and institu- tional changes will usually be required if the anticipated benefits are to be realized. Since the capital investment will be amortized over some depre- ciation period, it must be discounted differently than the research expenditures. The assumptions of an exponential growth curve for production or savings, and of a constant level of research expenditures, are conservative; that is, they tend to reduce the discounted values, compared with assump- tions of straight-line growth and a rising curve of research expenditures. Implicit in this treatment are the assumptions that the values of B. T. and b can be estimated from other considerations, and that, above a certain level of research effort, these quantities are more or less independent of the amount of research expenditures. The first assumption is a very shaky one. The above formulation is similar to the usual method employed by economists to aid in decision-making about investments. It is assumed that the funds would be committed at the present time, and therefore a comparison is made with other possible uses of present funds. But in the case of federal expenditures for research, the decision is actually made, at least in part, from year to year by the Administration and the Congress, and an alternative formulation may be more realistic. If we assume that each year of research is equally important in pro- ducing the economic benefits attained during all subsequent years, we may discount the benefits back to the times when the research is carried out, rather than to the present time. Hence, the discounted value of the benefits during the year, t, is Writ research = bB ~ t( T ) ( T ) ~ ~ where, as before, i is the discount rate, while n is the time interval in years from the time when research is done to the time when the benefits are received. The discounted average value of the annual benefits resulting directly from research becomes D Vi ~ research =-~ { ~ ( T ) ( T ) ~ t ~ (] o If the research expenditures vary with time, the net benefits can be computed by subtracting the costs of each year's research, multiplied by the appropriate discounting factor, from the total benefits. Let Cn be the cost of research during the nth year before the year when the benefits are received, then the net discounted benefits attained during a particular year, t, are 49

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Writ research = bet ~( 1 + T ) -( 1 + T ) t ~ (t + id 1 en (1 + i) ~ __$ t ~(1 + i) n and the average of the net discounted benefits is NVit research = ~ ~( T ) ( T ) ~o 1 en (1 + i) ~ _ ~ t ~(~1 + i) n If the annual research expenditures are constant and equal to R. then the ratio of average discounted annual benefits to costs is DVit research = _ ~[(1 + _ )t- (1 ~ and the net of the average discounted annual J ~ t O (1 + i) n enefits minus costs is N/it research T [( T ) ( T ) ~ l ~ (] + i) In exponential form, benefits minus costs is [J V i t ~ e seal ch t t >` bB ((eat- 1)( I ie-rtdt)dt o ) 1-e-rt) aft. Expanding in series, neglecting higher-order terms, and integrating, we have, approximately whence if ~ = 0.1 bBat / atrt \ _ 1+--1 DVit ,e~e~2~c~l ~ ~ 33 / 0.693 ~.693t O.lt \ = Bbt 1 ~ - 1. DVit research AT ~ 3T 3 J The assumption that each year of research contributes equally to the benefits attained during all subsequent years tends to lower the discounted values in later years about as rapidly as the assumption of a constant lag between research and benefits of approximately ten years. ~0