a monetary theory of nominal income

8/12/2019 A Monetary Theory of Nominal Income

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A Monetary Theory of Nominal IncomeAuthor(s): Milton FriedmanSource: Journal of Political Economy, Vol. 79, No. 2 (Mar. - Apr., 1971), pp. 323-337Published by: The University of Chicago PressStable URL: http://www.jstor.org/stable/1832113.

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A MonetaryTheoryof NominalIncome

MiltonFriedmanUniversity of Chicago

This paperconstructsa simplemodel or analyzing hort-term conomicfluctuations hat bypasses the problemof the divisionof a changeinnominal ncomebetween pricesand output. The model combinesoneelement romIrvingFisher (thedifference etween he nominaland thereal interestrate) and one elementfrom John Maynard Keynes (thedetermination f market nterestrates by speculatorswith firmlyheldanticipations)with two empiricalassumptions (a unit real incomeelasticity of demand for money and an exogenousexcess of the realinterest rateover the real secularrate of growth).The result s a modelconnectingcurrent nominal income with currentand prior nominalquantitiesof money.

In a recent paper on A Theoretical Framework for Monetary Analysis,I outlined a simple model of six equations in seven variables that wasconsistent with both the quantity theory of money and the Keynesianincome-expenditure theory (Friedman 1970). The difference between thetwo theories is in the missing equation-the quantity theory adds anequation stating that real income is determined outside the system (theassumption of full employment ); the income-expenditure theory addsan equation stating that the price level is determined outside the system(the assumption of price or wage rigidity).

The present addendum to my earlier paper suggests a third way tosupply the missing equation. This third way involves bypassing thebreakdown of nominal income between real income and prices and usingthe quantity theory to derive a theory of nominal income rather than atheory of either prices or real income. While I believe that this third way

This article is a paper given to the Sheffield Money Seminar on September 13, 1970.It will be included in the volume of proceedings, Monetary Theory and Monetary Policyin the 1970's, edited by G. Clayton, J. C. Gilbert, and R. Sedgwick (London, 1971),with comments by Sir Roy Harrod and Professor T. Wilson and a summary of thegeneral discussion.323

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324 JOURNAL OF POLITICAL ECONOMYis implicit in that part of my theoretical and empirical work on moneythat has been concerned with short-period fluctuations, I have not here-tofore stated it explicitly. This third way seems to me superior to theother two ways as a method of closing the theoretical system for the pur-pose of analyzing short-period changes. At the same time, it shares someof the defects common to the other two ways that I listed in the earlierpaper.

1. The Simple ModelTo repeat the model from my earlier paper, it is given by:

MD P.l(ir); (1)MS= h(r); (2)MD= Ms; (3)

C = f (p, r); (4)

I = (r) (5)= C+4 (or, alternatively,S = =73); (6)

where MD = quantity of money demanded; MS, quantity of money sup-plied; Y = nominal income; P = price level; r = interest rate; C = con-sumption; and I = investment are the seven variables to be determinedsimultaneously.Equations (1)-(3) summarize the monetary sector; equations (4)-(6),the savings and investment sector.The simple quantity theory adds the equation

p = YO, (7a)which enables equations (4)-(6) to be solved for the interest rate. Equa-tions (1)-(3) then yield an equation relating the price level to thenominal quantity of money.The simple income-expenditure theory adds the equation

P = PO) (7b)which enables equations (1)-(3) to define one relation between the inter-est rate and real income (Hicks's LM curve) and equations (4)-(6) todefine a second such relation (Hicks's IS curve).' Their simultaneoussolution gives the interest rate and real income.

1 Keynes distinguishedbetween the price level of products and the wage rate andallowed for a changein the ratioof the one to the otheras output changed,even before



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A MONETARY THEORY 3252. A Third WayThe third way that I want to suggest is somewhat more indirect.a) Demandfor MoneyAs a first step, assume that the elasticity of the demand for money withrespect to real income is unity. We can then write (1) in the equivalentform: MD = Y.l(r) , (la)where I have used the same symbol 1to designate a different functionalform. This enables us to eliminate prices and real income separately fromthe equations of the monetary sector.This assumption cannot, so far as I am aware, be justified on theoreti-cal grounds. There is no reason why the elasticity of demand for moneywith respect to per capita real income should not be either less than oneor greater than one at any particular level of income, or why it should bethe same at all levels of real income. However, we have much empiricalevidence that indicates that the income elasticity is not very differentfrom unity. The empirical evidence seems to me to indicate that theelasticity is generally larger than unity, perhaps in the neighborhood of1.5-2.0 for economies in a period of rapid economic development, and of1.0-1.5 for other circumstances. Other scholars would perhaps set itlower. More important, the present theory is for short-term fluctuationsduring which the variation in per capita real income is fairly small. Giventhat the elasticity is unlikely to exceed 2.0, no great error can be intro-duced for such moderate variations in income by approximating it byunity.2b) Savings and Investment FunctionsAs a second step, it is tempting to make a similar assumption for thesavings and investment functions, that is, to write:

C = Y f(r), (4a)or C = Y f(r, Y), (4b)and I = Y-g(r), (5a)the point of full employment. However, this change in relative price plays no im-portant role in the aspects of his theory that are relevant to our purpose, so I sim-plified the model by taking prices rather than wages as rigid-a simplification that hasbeen widely used. However, I should have referred explicitly to this simplification inthe earlier paper. I am indebted to an unpublished paper by Paul Davidson for myrecognition that my earlier exposition on this point may have been misleading.

2 Of course, considerations such as these can at most be suggestive. The real testof the usefulness of this, and the later assumptions, is in the success of the resultingtheory in predicting the behavior of nominal income.



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326 JOURNAL OF POLITICAL ECONOMYwhich would eliminate any separate influence of prices and real incomefrom the savings-investment sector also. However, this is an unattractivesimplification on both theoretical and empirical grounds. Theoretically,it dismisses Keynes's central point: the distinction between expendituresthat are independent of current income (autonomous expenditures) andexpenditures dependent on current income (induced expenditures). Em-pirically, much evidence suggests that the ratio of consumption to incomeover short-term periods is not independent of the level of measured in-come (eq. [4a]), or of the division of a change in income between pricesand output (eq. [4b]). The extensive literature on the consumption func-tion rests on this evidence.c) InterestRatesA more promising route is to combine a key idea of Keynes's with a keyidea of Irving Fisher's.

The idea that we take over from Keynes is that the current marketinterest rate (r) is largely determined by the rate that is expected toprevail over a longer period (r*). There exists, Keynes argues, a sub-stantial body of asset owners who have firmly held views about the rateof interest and who force the current rate into conformity with their an-ticipations. This is the basic idea behind Keynes's short-run liquiditytrap (Leijonhufvud 1968, pp. 158, 405, 411; Friedman 1970, pp. 212-14).Carrying this idea to its limit gives:

r= r*. (8)The idea that we take over from Fisher is the distinction between thenominal and the real rate of interest:

r = p + (I dP) (9)where p is the real rate of interest and [(1/P)(dP/dt)] is the percentagechange in the price level. If the terms r and [(1/P)(dP/dt)] refer to theobserved nominal interest rate and observed rate of price change, p isthe realized real interest rate. If they refer to permanent or antici-pated values, which we shall designate by attaching an asterisk tothem, then p* is likewise the permanent or anticipated real rate.Combine (8) and the version of (9) that has asterisks attached to thevariables. This gives: ( 1 dP\ *1r=p*+ (pd) '(10)which can be written as:

r = P*+(v )_(_%*= p*_g*+QXdY), (11)



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A MONETARY THEORY 327where y = Y/P = real income, and g* = [(1/y)(dy/dt)*] = perma-nent or anticipated rate of growth of real income, that is, the secularor trend rate of growth.

Let us now assume thatP- * = ko, (12)(or [7c])

that is, that the difference between the anticipated real interest rate andthe anticipated rate of real growth is determined outside the system. Asthe designation on the right indicates, this equation is the counterpartfor the third way of the full-employment and rigid-price assumptions ofthe simple quantity theory and the simple Keynesian income-expendituretheory.There are two ways that assumption (12) can be rationalized: (1) thatover a time interval relevant for the analysis of short-period fluctuations,p* and g* can separately be regarded as constant; (2) that the two canbe regarded as moving together, so the difference will vary less thaneither. Of course, in both cases, what is relevant is not absolute con-stancy, but changes in p* - g* that are small compared to changes in[(1/P)(dP/dt)*], and hence in r.

1. The stock of physical capital, the stock of human capital, and thebody of technological knowledge are all extremely large compared withannual additions. Physical capital is, say, of the order of three to fiveyears' national income; annual net investment is of the order of one-tenth to one-fifth of national income or 2-8 percent of the capital stock.Let the capital stock be subject even to very rapidly diminishing returnsand the real yield will not be much affected in a few years' time. Similarconsiderations apply to human capital and technology.

If we interpret g* as referring to growth potential, then a roughly con-stant yield on capital, human and nonhuman, and a slowly changingstock of capital imply a slowly changing value of g* as well.

Empirically, a number of pieces of evidence fit in with these assump-tions. We have interest rate data over long periods of time, and theseindicate that rates are very similar at distant times, if the times comparedhave similar price behavior (Gupta 1964). More recently, the FederalReserve Bank of St. Louis has been estimating the real rate, and theirestimates are remarkably stable despite very large changes in nominalrates.

Similarly, average real growth has differed considerably at any onetime for different countries-compare Japan in recent decades withGreat Britain-but for each country has been rather constant over con-siderable periods of time.2. Let s* = the fraction of permanent income which is invested. Thenthe permanent rate of growth of income as a result of this investmentalone will be equal to see p*. Empirically, the actual rate of growth tends



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328 JOURNAL OF POLITICAL ECONOMYto be larger than this product, if s* refers only to what is recorded ascapital formation in the national income accounts. One explanation, fre-quently suggested, is that recorded capital formation neglects most in-vestment in human capital and in improving technology and that allow-ance for these would make the relevant s* much higher than the 10 or20 percent that is the fraction estimated in national income accounts,both because it would increase the numerator of the fraction (invest-ment) and decrease the denominator (income) by requiring much ofwhat is commonly treated as income to be treated as expenses of main-taining human capital and the stock of technology. In the limit, as s*approaches unity, p* approaches g*, so p* - g* = 0.3 Without going tothis extreme, p* -g*= (1-s*)p*. (13)The preceding argument suggests that p* is fairly constant, and subtract-ing g* decreases the error even further.Empirically, it does seem to be the case that p* and g* tend to varytogether, although in the present state of evidence, this is hardly morethan a rough conjecture.d) The Alternative ModelIf we substitute equation (la) for equation (1), keep the original equations(2) and (3), and substitute equation (12) in equation (11) to replace theremaining equations of the initial simple model, we have the followingsystem of four equations:

MD = Y l(r); (la)MI= h(r); (2)MD= MS; (3)

r = ko+ (Y d)* (14)At any point of time, [(1/ Y) (dY/dt) *], the permanent or antici-pated rate of growth of nominal income is a predetermined variable,presumably based partly on past experience and partly on considerations

outside our model. As a result, this is a system of four equations in thefour unknowns, MD, MS, Y. and r.Prices and quantity do not enter separately, so the set of equationsconstitutes a model of nominal income.I An argumentjustifying this equality on a purelytheoretical level has been devel-oped ingeniouslyand perceptively by Stephen Friedberg n some unpublishedpapersthat take Frank H. Knight's capital theory as their starting point. This equality isalso a key implication of Von Neumann's general equilibriummodel (Von Neumann1945, p. 7).



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A MONETARY THEORY 329It will help to clarify the essence of this third approach to simplify itstill further by assuming that the nominal money supply can be regardedas completely exogenous, rather than a function of the interest rate,4and

to introduce time explicitly in the system. Let M(t) be the exogenouslydetermined supply of money. We then have from (la), (2), and (3)YQ() I(r) (15)

or Y(t) = V(r) *M() , (16)where V stands for velocity of circulation. This puts the equation instandard quantity theory terms, except that it does not try to go behindnominal income to prices and quantities. Equations (14) and (16) thenconstitute a two-equation system for determining the level of nominalincome at any point in time. To determine the path of nominal incomeover time, there is needed in addition some way to determine the antici-pated rate of change of nominal income. I shall return to this below.Although the symbolism in the demand equation for money ([la] or[16]) is the same as in the two other specializations of the general model,there is an important difference in substance. Both the simple quantitytheory and the income-expenditure theory implicitly define equilibriumin terms of a stable price level; hence real and nominal interest rates arethe same. The third approach, based on a synthesis of Keynes and Fisher,abandons this limitation. The equations encompass equilibrium situa-tions in which prices may be rising or falling. The interest rate that entersinto the demand schedule for money is the nominal interest rate. As longas we stick to a single interest rate, that rate takes full account of theeffect of rising or falling prices on the demand for money.

3. The Saving-Investment SectorWhat about equations (4)-(6), which we have so far completely by-passed? Here the interest rate that is relevant, if a single rate is used, isclearly the real not the nominal rate. If we replace r by p, these equationsbecome

C 1( y ); (4')I= g(p) (5')Y = C + I (6)757J P.

4 Alternatively, we could write (2) as Ms = H-m(r), where H is high-poweredmoney and m(r) is the money multiplier.



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330 JOURNAL OF POLITICAL ECONOMYIf we were to accept a more restricted counterpart of equations (8) and(12), namely,

p = p* Po (17)that is, the realized real rate of interest is a constant, then these equationswould be a self-contained consistent set of five equations in the five vari-ables, C/P, I/P, Y/P, p, p*. Equation (17) would give the real interestrate. Equation (5') would give real investment and equations (4') and(6), real income. The price level would then be given by the ratio of thenominal income obtained from equations (14) and (16) to the real incomegiven by equations (4'), (5'), (6), and (17). The two sets of equationscombined would be a complete system of seven equations in seven vari-ables determining both real and nominal magnitudes.Such a combination, if it were acceptable, would be intellectually veryappealing. Over a decade ago, during the early stages of our comparisonof the predictive accuracy of the quantity theory and the income-expen-diture theory, my hopes were aroused that such a combination mightcorrespond with experience. Some of our early results were consistentwith the determination of the real variables by the multiplier, and thenominal variables by velocity. However, later results shattered this hope(Friedman and Meiselman 1963). These empirical findings are reinforcedby theoretical considerations.The major theoretical objections are twofold. First, it seems entirelysatisfactory to take the anticipated real interest rate (or the differencebetween the anticipated real interest rate and the secular rate of growth)as fixed for the demand for money. There, the real interest rate is at besta supporting actor. Inflation and deflation are surely center stage. Sup-pressing the variations in the real interest rate (or the deviations of themeasured real rate from the anticipated real rate) is unlikely to introduceserious error. The situation is altogether different for saving and invest-ment. Omitting the real interest rate in that process is to leave outHamlet. Second, the consumption function (4') seems to me highly unsat-isfactory, especially once we take inflation into account. Wealth, antici-pations of inflation, and the difference between permanent and measuredincome seem to me too important and too central to be pushed off stagecompletely.

Hence, for both empirical and theoretical reasons, I am inclined toreject this way of marrying the real and the nominal variables and toregard the saving-investment sector as unfinished business, even on thehighly abstract general level of this paper.

4. Some Dynamic ImplicationsIn equation (14), which determines r, we have so far taken [(1/Y)(dY/dt)*] as a predetermined variable at time t and not looked clearly at



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A MONETARY THEORY 331its antecedents. It is natural to regard it as determined by past history.If it is, we can write (16) as

Y(t) = V[Y(T)]M(Q), T < t1 (18)where V is now a functional of the past history of income, Y(T) for T < t.However, the past history of income in its turn is a function of the pasthistory of money, thanks to equation (16) for earlier dates. Hence, wecan also write equation (16) as

Y(t) = F[M(T)]-M(t), T < t (19)where F is a functional of the past history of money. There is also im-bedded in these equations the value ko, that is, the assumed fixed valueof the difference between the anticipated real interest rate and the secularrate of growth of output, so equations (18) and (19) must be interpretedas depicting the movements of nominal income around a long-term trendon which ko,and its components, p* and g*, adjust to more basic long-term forces--fundamentally for both, changes in the quantity of re-sources available (human and nonhuman) and in technology.A specific example may help to bring out the dynamic character ofthis simple model. Take logarithms of both sides of equation (16) anddifferentiate with respect to time. This gives

1 dY _ 1 dV 1 dM _ I dVdr + 1 dM (20)Y di V dl M dt V dr d+ MdtReplace [(1/V)(dV/dr)] by s (to stand for the slope of the regression oflog V on r), and dr/dt by the derivative of the right-hand side of equa-tion (14):

dYd= sd I dY + I dM (21)Y di dt BY dt J M dtAssume that the anticipated rate of growth of income is determined by

a simple adaptive expectations model:d (1dYy AT dV /dV\dt Y d ) LY- Ydt )] (22)

Substitute equation (22) in equation (21) and solve for [(1/Y)(dY/dt)].The result isI d _ IddYM*+ 1 1 dM 1 dY (23)Y dt YF dt I -AOs lM dt VY dt

Subtract [(1/M)(dM/dt)] from both sides, and equation (23) can also bewritten 1 dV Os[ 1 dM (IdVV] (24)V t h1-A -LM di \Y di 2



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332 JOURNAL OF POLITICAL ECONOMYAssume that 0 < O3s 1.5 Equations (23) and (24) give a very simpleand appealing result. If the rate of change of money equals the antici-pated rate of change of nominal income, then nominal income changes atthe same rate as money-we are in the simple quantity equation world.If the rate of change of money exceeds the anticipated rate of change ofnominal income, so will the actual rate of change of nominal income,which will also exceed the rate of change of money-velocity is increasingin a boom ; conversely, for a contraction or recession, interpretedas a slower rate of growth in the actual than in the anticipated rate ofgrowth of income.

Note that this way of introducing a pro-cyclical movement in velocityis an alternative or complement to the approach I suggested in an earlierarticle on The Demand for Money (Friedman 1959, 1969). I thereinexplained the pro-cyclical movement of velocity by the difference be-tween measured and permanent income. The two approaches are notmutually exclusive-as I indicated in my earlier article, when I left roomfor interest rate effects on velocity (Friedman 1969, pp. 130-36). In thepresent context, the simplest way to introduce both effects would be torewrite (la) as

MD = Y*l(r) , (lb)where Y* is permanent nominal income. To complete the system, equa-tion (3) must be replaced with a more sophisticated adjustment mecha-nism involving Y-otherwise the system, with Y* treated as determinedby the past history of Y, would be overdetermined. Equation (31) of myearlierpaper (Friedman 1970, p. 226) is such a more sophisticated mecha-nism. Hence, proceeding farther along this line would simply duplicatepart of my earlier paper.

5. Comparison of the Three ApproachesNone of the three simple theories-the simple quantity theory, thesimple income-expenditure theory, the simple monetary theory of nomi-nal income-professes to be a complete, fully worked out, analysis ofshort-term fluctuations in aggregate economic magnitudes. All are to beinterpreted, rather, as frameworks for such analyses, establishing thebroad categories within which further elaborations will proceed.

The simple quantity theory puts in center stage the relation at eachpoint in time between a particular flow (the flow of spending or income);and a particular stock (the quantity of money); the simple-income ex-penditure theory, the relation at each point in time, between two com-ponents of the flow of income (autonomous and induced spending); the

I This is the condition for dynamicstability of the system (see Cagan 1956).



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A MONETARY THEORY 333simple monetary theory of nominal income, the relation between the flowof income at each point in time and the past history of the quantity ofmoney.

In the prior article, I listed six elements that the first two approacheshad in common and that indicated the main unresolved problems. Thethird approach differs significantly in some of these elements:

1. It does not, as they do, analyze short-run adjustments in terms ofshifts from one static equilibrium position to another (Friedman 1970,p. 221). It embodies a dynamic adjustment process.

2. It does not, as they do, regard each equilibrium position as char-acterized by a stable levelof prices or output (ibid., p. 221). It encorn-passes steady growth in prices or output as long-run equilibrium posi-tions.

3. It does not regard interest rates as adjusting instantaneously to anew equilibrium level (ibid., p. 222) because it allows for a change ininterest rates along with a change in the anticipated rate of change ofprices. However, it does neglect the effect of other factors on interestrates (the saving-investment process stressed by the quantity theory; theeffect of changes in the nominal quantity of money stressed by the in-come-expenditure theory) except as they affect the course of nominalincome and, in consequence, the anticipated rate of change of prices.4. It does, unlike the other approaches, give an explicit role to an-ticipations about economic magnitudes (ibid., p. 222). The differencesbetween anticipated and actual magnitudes are the motive force behindthe short-run fluctuations.

5. Like the others, it fills in the missing equation by an assumptionthat is not part of the basic theoretical analysis (ibid., p. 222). Theassumption (that speculators determine the interest rate in accord withfirmly held anticipations, and that the difference between the permanentreal interest rate and the secular growth of output can be taken as aconstant for short period fluctuations) is intermediate between the othersin its link to economic theory. It is not as clearly linked to a well-devel-oped body of theory as the simple quantity approach is to the Walrasianequations of general equilibrium, yet it has more of a link to theory thandoes the rigid-price assumption of Keynes. Further, like the quantityapproach and unlike the income-expenditure approach, there is a theo-retical link between the short-run model and the long-run model (ibid.,p. 222).

6. The chief defect that this model shares in common with the othertwo is that none of the three has anything to say about the factors thatdetermine the proportions in which a change in nominal income will, inthe short run, be divided between price change and output change(ibid., p. 222)-the topic with which most of the rest of my prior article



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334 JOURNAL OF POLITICAL ECONOMYdeals. The one advantage in this respect of the third approach is that itdoes not make any assertion about this division as both the others do.It is, as it were, orthogonal to that issue and can therefore be more easilylinked to alternative theories about that division.

6. Correspondence of the Monetary Theory of Nominal Incomewith Experience

I have not, before this, written down explicitly the particular simplifica-tion I have labeled the monetary theory of nominal income-althoughMeltzer has referred to the theory underlying Anna Schwartz's and myMonetary History as a theory of nominal income (Meltzer 1965, p.414).6 But once written down, it rings the bell, and seems to me to cor-respond to the broadest framework implicit in much of the work that Iand others have done in analyzing monetary experience. It seems to mealso to be consistent with many of our findings. I do not propose here toattempt a full catalog of the findings, but I should like to suggest a num-ber and, more important, to indicate the chief defect that I find with theframework.

One findingthat we have observed is that the relation between changesin the nominal quantity of money and changes in nominal income is al-most always closer and more dependable than the relation betweenchanges in real income and the real quantity of money or between changesin the quantity of money per unit of output and changes in prices.7 Thisresult has always seemed to me puzzling, since a stable demand functionfor money with an income elasticity different from unity led me to expectthe opposite. Yet the actual finding would be generated by the approachof this paper, with the division between prices and quantities determinedby variables not explicitly contained in it.

Another broad finding is the pro-cyclical pattern of velocity, whichcan be rationalized either by the distinction between permanent andmeasured income or, as in the approach of this paper, by the effect ofchanges in the anticipated rate of change of prices.

On still another level, the approach is consistent with much of thework that Fisher did on interest rates, and also the more recent work by

6 However, he referred o it as a long-run heoryof nominalincome, whereas thetheory of this paper is intended to be a short-run theory. I accept much of whatMeltzersays about the theory underlying a MonetaryHistorybut also disagreewithmuch of it-in particular, the way he introduces real income and changes in realincome into the analysis. This is strictly ad hoc and renders the asserted theory alogically open and underdeterminedheory.I However, Walters reports a different result for Britain for the period since theend of World War I-a closer relationwith pricesin the interwarperiodand with realoutput in the post-World War II period (Walters 1970, p. 52).



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A MONETARY THEORY 335Anna Schwartz and myself, Gibson, Kaufman, Cagan, and others. Inparticular, the approach provides an interpretation of the empirical gen-eralization that high interest rates mean that money has been easy, in thesense of increasing rapidly, and low interest rates, that money has beentight, in the sense of increasing slowly, rather than the reverse.Again, the approach is consistent with the importance we have beenled to attach to rates of change in money rather than levels and, in par-ticular, to changes in the rate of change in explaining short-term eco-nomic fluctuations.

The approach is consistent also with the success of the equations con-structed by Anderson and Jordan at St. Louis relating changes in nominalincome to current and past changes in the quantity of money (Andersonand Jordan 1968).The chief defect of the approach is that it does not give a satisfactoryexplanation of the lags in the reaction of velocity and interest rates atturning points in monetary rates of change. We know, for example, thatwhen the rate of growth of the quantity of money declines, the rate ofchange of income will not show any appreciable effect for something likesix to nine months (for the United States) on the average. During thisinterval, interest rates typically continue to rise, indeed generally at anaccelerated pace. After the interval, both velocity and interest rates startto decline.

This result is not necessarily inconsistent with the approach of thispaper. Suppose that prior to the decline in the rate of monetary growththe system was not in full equilibrium, so that the actual rate of growthof nominal income [(1/Y)(dY/dt)] was higher than the anticipated rateof growth [(1/ Y) (d Y/dt) *]. Then, even the new rate of monetary growthcould be higher than [(1/Y) (dY/dt)*], implying from equation (24) afurther rise in velocity; from equation (23), a larger actual than antici-pated rise in nominal income; from equation (22), a further rise in[(1/Y)(d Y/dt)*]; and from equation (14), a further rise in the nominalinterest rate. These would continue until [(1/Y)(dY/dt)*] had risen toequality with the new rate of monetary growth.However, this reaction would imply a slower rate of rise in velocityanctinterest rates than prior to the monetary turning point, whereas myimpression is that the opposite often occurs. More important, even if thesystem is not in full equilibrium prior to a decline in the rate of monetarygrowth, the decline in monetary growth, if large enough, will make thenew rate of monetary growth less than [(1/Y)(dY/dt)*]. In that case,equations (24), (23), (22), and (14) would produce a decline in velocityand in interest rates contemporaneous with the decline in the rate ofmonetary growth. Yet the lag in reaction is highly consistent and, inparticular, seems to be independent of the size of the change in the rateof monetary growth.



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336 JOURNAL OF POLITICAL ECONOMYAccordingly, I believe that the movements of velocity and interest

rates in the first nine months or so after a distinct change in the rate ofmonetary growth cannot be satisfactorily explained by the approach ofthis paper. If these periods were cut out of the historical record, my im-pression is that the model would fit the rest of the record very well-notof course without error but with errors that are on the modest side asaggregate economic hypotheses go.Periods just after turning points can, I believe, be explained best byincorporating two elements omitted from the mode] of this paper. Thefirst is a revision of our equation (3) to allow for a difference betweenactual and desired money balances, as in equation (31) of my prior paper(Friedman 1970, p. 226). The second is a weakening of equation (8) topermit a stronger liquidity effect on interest rates.

7. ConclusionThis paper has neglected completely many important issues-the dis-tinction between short and long interest rates, between different conceptsof income, between currency and deposits, and between demand and timedeposits, the role of government spending and taxing, and so on and on.It has kept to the high and rarefied, but hopefully not arid, level ofMV = PT, and C = a + b Y. On that level it suggests a more satisfac-tory simple model for analyzing short-term economic fluctuations thaneither the simple quantity theory which takes real output as determinedoutside the system and regards economic fluctuations as a mirror imageof changes in the quantity of money or the simple Keynesian income-expenditure theory which takes prices as determined outside the systemand regardseconomic fluctuations as a mirror image of changes in auton-omous expenditures.That more satisfactory model, which I have labeled a monetary theoryof nominal income, is not new. It is implicit and parts of it explicit inmuch work in the field of money of the past two decades. Its key ele-ments are:

1. A unit elasticity of the demand for money with respect to realincome.

2. A nominal market interest rate equal to the anticipated real rateplus the anticipated rate of change of prices, kept at that level by specu-lators with firmly held anticipations.

3. A difference between the anticipated real interest rate and the realsecular rate of growth determined outside the system.

4. Full and instantaneous adjustment of the amount of money de-manded to the amount supplied.

These elements are borrowed mostly from Irving Fisher and John



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A MONETARY THEORY 337Maynard Keynes. Together they yield a simple two-equation systemthat determines the time path of nominal income but has nothing to saydirectly about the division of changes in nominal income between pricesand quantity.ReferencesAnderson, Leonall C., and Jordan, Jerry L. Monetary and Fiscal Actions: ATest of Their Relative Importance in Economic Stabilization. Federal Re-serve Bank of St. Louis Review (November 1968), pp. 11-23.Cagan, Phillip. The Monetary Dynamics of Hyperinflation. In Studies in theQuantity Theory of Money, edited by Milton Friedman. Chicago: Univ.

Chicago Press, 1956.Friedman, Milton. The Demand for Money: Some Theoretical and EmpiricalResults. J.P.E. 67 (August 1959):327-51. Reprinted in Friedman 1969.. The Optimum Quantity of Money and Other Essays. Chicago: Aldine,1969.. A Theoretical Framework for Monetary Analysis. J.P.E. 78 (March/April 1970) :193-238.Friedman, Milton, and Meiselman, David. The Relative Stability of the Invest-ment Multiplier and Monetary Velocity in the United States, 1897-1958.Stabilization Policies. Englewood Cliffs, N.J.: Prentice-Hall, 1963.Gupta, Suraj. Expected Rate of Change of Prices and Rates of Interest. Ph.D.dissertation, Univ. Chicago, 1964.Leijonhufvud, Axel. On Keynesian Economics and the Economics of Keynes. NewYork: Oxford Univ. Press, 1968.Meltzer, Allen H. Monetary Theory and Monetary History. SchweizerischeZeitschriftfir Volkswirtschaftund Statistik (1965), pp. 404-22.Von Neumann, John. A Model of Economic Equilibrium. Rev. Econ. Studies,vol. 13. no. 33, pt. 1 (1945-46).Walters, A. A. A Survey of Empirical Evidence. In Money in Britain, 1959-1969, edited by David R. Croome and Harry G. Johnson. London: OxfordUniv. Press, 1970.

a monetary theory of nominal income

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