Capital Budgeting and Inflation

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American Finance Association

Inflation and Capital BudgetingAuthor(s): Charles R. NelsonReviewed work(s):Source: The Journal of Finance, Vol. 31, No. 3 (Jun., 1976), pp. 923-931Published by: Blackwell Publishing for the American Finance AssociationStable URL: http://www.jstor.org/stable/2326436 .

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THE JOURNAL OF FINANCE - VOL. XXXI, NO. 3 JUNE 1976

INFLATION AND CAPITAL BUDGETINGCHARLES R. NELSON*

THE LITERATUREON capital budgeting has not generally reserved a specific role forthe rate of price inflation in the analysis of capital investments. In the taxless worldof most textbook discussions of the subject, inflation would presumably onlyaugment both future cash flows and discount rates by comparable amounts, thushaving no effect on present value calculations. Under the U.S. tax code, however,after-tax present values are not neutral with respect to different rates of inflation,because depreciation charges are based on historical costs. This complication mayhave been of little practical importance during much of the post-war period, butrates of inflation such as those experienced in recent years will generally be amaterial factor in present value calculations.In order to focus on the role of the tax effects of inflation in the capitalbudgeting decisions of the firm it is necessary to hold constant both real pretaxcash flows associated with alternative investment projects (including capital costs)and real discount rates. The question of whether real pretax cash flows and realdiscount rates are in fact systematically related to the rate of inflation is esentiallyan empirical one and goes beyond the scope of the present paper. Some evidence tothe contrary in the case of cash flows is given by Kessel and Alchain [5]. Thehypothesis that real discount rates are unrelated to the rate of inflation, of course,goes back to Fisher [3, 4] and enjoys support from a large body of empiricalliterature.1 The analysis presented here does not distinguish between actual andexpected future rates of inflation or cash flows. Whether the reader wishes toregard the discussion as an analysis under perfect certainty or prefers to interpretfuture magnitudes as certainty equivalents accompanied by appropriate discountrates is immaterial for present purposes. The paper is organized around fivepropositions dealing with the impact of inflation, ceterisparibus, on aspects of thecapital budgeting decision relating to the optimal level of investment, the choice oftechnology, the ranking of competing projects, optimal durability, and replacementpolicy.

PROPOSITION A. The optimal level of capital investmentwill dependin general onthe rate of inflation. The amount invested will typicallybe smaller the higher the rateof inflation.*AssociateProfessor f BusinessEconomicsandFinance,GraduateSchoolof Business,UniversityofChicago.This paper was writtenduringa visitingresidenceat the GraduateSchool of Business,

Universityof Washington. wouldlike to expressmy appreciationo thatinstitution or providingmewithresearchacilitiesduringmyvisit.I would alsolike to expressmay appreciationo LarrySchallandNancyJacobforhelpingto stimulatemy interest n thistopic, although heybearno responsibilityorthefinal product. am alsograteful o DouglasVickers, he referee, orhelpfulcomments.Theresearchwas supportedn partby the NationalScienceFoundationundergrantGS-34501.1. See, for example,Yohe and Karnosky 6) and Fama(1).

923


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924 The Journal of Finance

L.

1\ X

p=p2FIGURE 1

This proposition is easily demonstrated with the simple example of an invest-ment of I dollars producing a cash flow of X dollars one period later. The projectwould be fully depreciated in one period so the taxable profit would be X- I andthe tax owed T X - I) where T is the corporate tax rate. Let the real discount ratebe r so that with no inflation the present value of the project is

-I+ X- T(X-I)I + rIf, however, the rate of inflation over the life of the project will be p, then thepresent value becomes

PV(p) =-I+ ( ()X - lT (I +p)X-I)I (-T)X TI(l + r) (+ r)(l +p)


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Inflation and CapitalBudgeting 925The optimal level of investment will be given by the solution to

aPV(P) (1 - ) aX + _T_ __aI ( + r) aI (1+r)(1 +p)

which will depend on p. Using the approximation (1 +p)- 1(I -p) we haveaPV(P) (1 -iT) ax T ___T- +aI (1+r) I 1+r P1+r

so that the effect of a higher anticipated rate of inflation is a downward shift in themarginal present value schedule by amount pT/(1 + r). The situation is illustratedin Figure 1 for rates of inflation P1and P2with P2>P . It is clear that the optimalinvestmentI*(p) will be smaller the higherthe anticipatedrate of inflation.PROPOSITION . The rate of inflation will influence the firm's choice of technolo-gies of productionthroughits choice of a capital! labor ratio. Higher rates of inflationwill typicallybe associated with lower capital! labor ratios.We now consider how anticipated rates of inflation will affect the choice of anoptimal combination of labor and capital inputs which in practice amounts to achoice among production technologies. The framework of Proposition A is ex-tended to note that output X in constant dollars will be a function f( ) of I and

labor input L. Discounted revenue for a competitive firm will beRevenue - (1 -T)Pf(I,L)(I1+r)

where P is the price of output assumed to vary according to the rate of generalprice inflation. Discounted costs can be written asCost= -l+W( T) L + I - 1 ICs-(l+r) (Il+r)(I+p

W(- -T) (I + r)( +P) II +r (l +r)(l+p)

where the wage rate W (paid at the end of the production period) is also assumedto vary with the rate of inflation. Now the "price" of labor is W(I - T)/(1 + r) perunit while the "price" of investment is {(I + r)(l+p) - T}/(I+ r)(l +p) per unit(dollar) which depends on p. It is clear that different p correspond to differentfactor price ratios and will in general lead to different choices of I and L.The situation is illustrated graphically in Figure 2. A typical isorevenue curve isindicated by R1 and does not depend on p. A typical cost schedule is representedby C(p1) which does depend on p. The decision problem can be thought of as oneof maximizing revenue at each alternative level of cost and then finding the level ofcost which maximizes the difference. IfP2 is larger thanpI then cost schedule C(p2)


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926 TheJournal of FinanceL

I 4 V \c(p2) \ ( lFIGURE 2

would be relevant and isorevenue curve R2 would represent the highest attainablerevenue. The shift in capital/labor ratio can be decomposed into that due to thechange in the slope of C( ) (a "substitution" effect) and that due to a shift in theposition of C ) (an "income" effect). In the "normal" case where the substitutioneffect dominates, higher levels of p will typicallybe associated with lower capital/la-bor ratios.PROPOSITION C. The net present value ranking of mutually exclusive investmentprojects will depend in general on the rate of inflation.When investment projects have lives greater than one period the impact of therate of inflation on net present values will depend on how depreciation charges aredistributed over the lives of alternative projects. The ranking of projects whichproduce different streams of depreciation charges against taxable income willtherefore depend on the rate of inflation. To demonstrate that this is true in generalwe need to introduce notation as follows. Let X, represent net cash flow beforetaxes in constant dollars generated by the project in year t and let depreciation becalculated by the declining balance method at the rate of 8 for each tax year.Taxable profit in year t for a project costing I dollars in year zero would thereforebe

?T _ - y0 _ 8) _R\-I


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Inflation and Capital Budgeting 927If the rate of inflation were zero the net present value of such a project would be

PV= -I+ z ,T{ ,6l-) I1=1 (1+ r)where r denotes the constant real discount rate, and the life of the project isassumed to be infinite for algebraic simplicity. Now, if the general level of pricesand costs rises at rate p, assumed to be constant, then current dollar cash flowsbecome (1 +p)'X, and the net present value is

00 Xt 8PV(p) =I+ (I1-T) E ( ) + ( )1+)(-using the formula for summing geometric series and the fact that the factors (1 +p)tcancel in the numerator and denominator of the second term. The third term is thecontribution which depreciation charges make to net present value through taxsavings and is easily seen to depend on the depreciation rate a and to be smallerthe higher the rate of inflation p.To see how different rates of inflation would alter the ranking among alternativeprojects it is convenient and reasonably realistic to impose a simple pattern ofdecay on the cash flows generated by any given project. In particular the rate offlow is assumed to decay at rate X starting in period one so that the sequence ofcash flows is (1 - X)X, (1 - X)2X,.... When this is the case, PV(p) simplifies to afunction of p and X given by

(1-T)(1-X)X TSIPV(p,X)=-I+ r+X +(1r)(1Ip-(-#)where again the third term is the contribution which tax savings from depreciationmake to net present value.To illustrate how rankings among projects will depend on p consider hypotheti-cal projects I and II where the tax rate T =.5, the real interest rate r is .05, andwhere

XI= .10

I= .10(1 - XI)XI= 90 (first year cash flow)

I=300whileXII=.5

(1 -X,,)X,, = 165 (first year cash flow)I= 150


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928 The Journal of FinanceProject I requires a larger initial outlay than II and generates a smaller initialcash flow, but does so with a slower rate of decay. While X and a need not havebeen exactly identical, it is not unrealistic to assume that a project with thecharacteristics of I would be depreciated at a lower rate than II. Table 1 comparesnet present values for these two projects over a range of rates of inflation.

TABLE 1NET PRESENT VALUES OF HYPOTHETICAL

PROJECTS

Rate ofInflation Project I Project II.00 100.00 68.18.10 58.82 57.25.15 48.78 53.00.20 41.67 49.34.50 22.22 34.88

1.00 12.50 23.44

The reader will note that Project I has a higher present value than Project II at lowrates of inflation, but that the ordering is reversed at rates of inflation above 10%(11.1 to be exact). The variation across rates of inflation is entirely due to variation inthe capitalized value of future tax savings from depreciation charges. These savingsare realized more slowly under Project I and therefore their present value is erodedmore severely by higher rates of inflation.PROPOSITIOND. Net present value rankings of mutually exclusive projects whichdiffer with respect to durability will depend on the rate of inflation. Typically, rankingswill change infavor of projects with lowerdurabilityat higher rates of inflation.Investment projects which generate a declining stream of cash flows will gener-ally be replaced after some interval depending on the specific pattern of cash flows,replacement cost, scrap value, and so forth. Projects which produce more rapidlydeclining cash flows, in other words are less durable, will typically be replacedmore often.Where durability and therefore replacement interval differs among competing

alternative projects, the rate of inflation will be an important determinant of netpresent value rankings. Higher rates of inflation will typically alter rankings infavor of less durable projects, because their depreciation cost will be restated interms of current dollars at more frequent intervals.To focus on the role of durability in the choice among projects and abstract fromreplacement policy for a given project (considered explicitly under Proposition E)consider the following example. The problem is to choose one of two different


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Inflation and Capital Budgeting 929types of electric generating plants producing a specified power output. Project a isa thermonuclear reactor which is subject to almost no physical deterioration. Oncebuilt, it produces the specified power output in perpetuity, but is depreciated fortax purposes at rate 8. Project /3 on the other hand generates electric power usingthe residual heat from an underground nuclear blast. The heat generated issufficient for only one year at which time a new blast is required. The cost of eachnew blast is written off in one year. The net present value of Project a will be

PVa= -,a+ -r (1+.r)(1 +p) (1 8) aThe net present value of Project /3 on the other hand is given by

00 (l+ )' (I +P) 'X- T {(l +P)tx - (I +p)'- lipt=O (1+ r)t( +p)t t=i (1 + r)t(1+p)t

I + r (I3 (- T)X T___r r +r(1 +1)Iwhere X is the same for both projects and is the outlay in constant dollarsrequired to produce a new nuclear blast each year. The first term of both presentvalue expressions is the present value of all future investment outlays (which in thecase of Project /3 ecessarily involves the real discount rate), the second term is thepresent value of future cash flows, and the third term is the present value of futuretax savings from depreciation charges. The third terms differ in algebraic formbecause the cost of replacing Project a is restated each year in current dollars.It is clear from comparison at the third terms of the two present value expres-sions that the ranking of the two projects will depend on the rate of inflation.Suppose for purposes of illustration that

Ia= 1,5001I=1008 =.10

X=200then the net present values corresponding to various rates of inflation are given inTable 2.At low rates of inflation Project a is preferred. However, successively higher ratesof inflation decrease the tax savings occurring under a more rapidly than thoseunder /3until the balance shifts in favor of /3 t p equal to 7.14. Very high rates ofinflation ultimately reduce the tax savings from both projects to the point wheretheir present values are dominated by the first two terms in each expression.Because in this illustration the sum of the first two terms is $500 for a but(-$100) for /3,the balance finally shifts back in favor of a at rates of inflationabove 33.3.


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930 TheJournal of FinanceTABLE 2

NET PRESENTVALUES OF HYPOTHETICALPROJECTSWITH REPLACEMENT

Rate ofInflation Project a Project f3.00 1,000.00 900.00.05 870.37 852.38.10 794.12 809.09.20 708.33 733.33.30 661.29 669.23.50 611.11 566.67

1.00 562.50 400.00

PROPOSITION. Replacementpolicy will depend in general on the rate of inflation.The higherthe rate of inflation the more likely will replacementbe deferredto a futureperiod.In many real situations the lifetime of a project is not prescribed by technologyas in the case of the hypothetical projects discussed under Proposition D, butrather is a variable under the control of the firm. The optimal lifetime will dependon the structure of costs as the installation ages, scrap value, and details of the taxcode under which the firm operates. The possible combinations of these factors arefar too varied to permit a comprehensive treatment, but the basic role of inflationin replacement policy can be outlined as follows.A firm can continue to operate its present installation for another year or replaceit with a new installation.2 New and old installations generate the same grossrevenues, but the old installation becomes more costly to operate as it ages. Theafter-tax cost of operating the old installation for another year is C in constantdollars, while the present value of future after tax costs for a new installation lesstax savings from future depreciation charges is PVC(p), a function of the rate ofinflation as we have seen before. We assume in the calculation of PVC(p) that theinstallation will be replaced at appropriate intervals. Replacement is called for thisyear as opposed to next if

(l+p)c (1+p)PVC(p) >PVC(p)+(1 +'r)(1 +p) (1 +r)(1 +p)orC > rPVC(p)

2. See Fama and Miller (2), pp. 130-134 for a lucid discussion of the replacement problem.


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Inflation and Capital Budgeting 931where C is assumed to be paid at the end of the current period. Since thecomponent of PVC(p) which is a function of the rate of inflation is just the portiondue to future tax savings from depreciation, and the present value of these savingsvaries inversely with the rate of inflation, then PVC(p), which is net cost, variesdirectly with the rate of inflation. Thus, some installations which would be replacedat a given point in time, for a given rate of inflaion, would not be replaced if therate of inflation were higher. Of course, the calculated value of PVC depends onwhat is assumed about future replacement policies, but higher inflation rates willwork against currentreplacement in any case.

CONCLUDINGEMARKSWhile a general discussion of the role of inflation on capital budgeting can onlydraw on illustrative examples, it seems apparent that at current rates of inflationthe related tax effects must be an important factor in many present value compari-sons arising in practice. Further, unless relative factor costs, prices, and discountrates have changed in very particular ways, it seems very probable that the ceterisparibus effects discussed in this paper are indicative of actual distortions whichhave been occurring as a result of inflation. One of the strongest arguments for"indexing" of accounting costs would be the elimination of such distortions.

REFERENCES1. E. F. Fama."ShortTerm InterestRates as Predictors f Inflation."Mimeo, Universityof Chicago,1974(forthcomingn American Economic Review).2. and M. H. Miller.The Theory of Finance. New York:Holt, Rinehart,and Winston, 1972.3. I. Fisher.Appreciationand Interest. New York: The MacmillanCo., 1896.4. . The Theory of Interest. New York: The MacmillanCo., 1930.5. R. A. Kesseland A. A. Alchian."TheMeaningandValidityof the Inflation nducedLag of WagesBehindPrices."American Economic Review (March, 1960),45-66.6. W. P. Yohe and D. S. Karnosky."InterestRates and Price Level Charges,1952-1969."FederalReserve Bank of St. Louis Review (December,1969), 18-38.

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Capital Budgeting and Inflation

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