A Monetary Model of Nominal Income Determination

A Monetary Model

of Nominal Ineome DeterminationLEONALL C. ANDERSEN

II)URING the past several years the Federal Re-serve Bank of St. Louis has presented a number ofempirical studies demonstrating a strong and pre-dictable response of nominal gross national product(GNP) to changes in the nation’s money stock. Thesestudies found that changes in the secular trend ofthe money stock are the prime determinant ofchanges in the secular trend of GNP. They also foundthat short-run changes in GNP are related to similarchanges in the money stock. On the other hand,changes in Government spending were found to haveonly a temporary short-run influence on changes inGNP. A number of other studies, using variations ofthe approach of the St. Louis studies, have yieldedsimilar empirical relationships.

The St. Louis studies related changes in nominalGNP directly to current and past changes in themoney stock and in high-employment Governmentexpenditures. No underlying set of relationships wasspecified. Instead, the empirical relationship was pre-sented as the reduced-form of an unspecified struc-ture of the economy.

This article sets forth a theoretical model consist-ing of a set of postulated relationships which form abasis for the relationship of nominal income (GNP)to the money stock and Government expenditures.The theoretical properties of this model are similar tothe empirical relationships found in the St. Louisstudies. The parameters of the model are estimatedfor the period from first quarter 1955 to fourth quarter1973. Empirical tests fail to reject the theory as anexplanation of nominal income determination in thesample period.

THEORETICAL MODELThis section develops a model of nominal income

determination which is based primarily on a prominent

NOTE: This presentation was given as part of the LawrenceH. Seltzer Memr,rial Lecture Series at Vtrayne State LTniver~sity, Detroit, Michigan, May 21, 1975.

theory of the influence of changes in the money stockon income) At times, however, assertions differentfrom those in the prominent theory are made. It isalso asserted that the responses of holders of moneybalances to changes in the dependent variables are inrelative terms; that is, rates of change or percentdiscrepancies. The model, therefore, is expressed inlog-linear form,

Des-ired Nom-inal Money BalancesThe theory underlying this study is based on the

assertion that households and business firms desirenominal money balances to conduct market transac-tions and to hold as a store of value. Specifically, thedesired amount of nominal money balances, in theaggregate, depends on the perceived price of newlyproduced final goods and services, perceived nominalincome, the expected rate of inflation, yields on alter-native financial assets, and the technical efficiency ofthe economy’s system of making money payments.2

Since holders of money balances do not have perfectinformation regarding market opportunities, and sincethere exist costs of both collecting information andconducting market transactions, the desire to holdmoney balances depends on the perceived, instead

1For statements of this theory, see Milton Friedman, “TheDemand for Nlonev: Some Theoretical and Empirical Re-sults,” Journal of Political Economy (August 1959), pp.327-351; “A Theoretical Framework for Monetary Analysis,”Journal of Political Economy (March/April 1970), pp. 193-238; and “A Monetary Theory of Nominal Income,” Journalof Political Economy (March/April 1971), pp. 323-337.

2For various asserted fol-ms of the money demand fnnction,see David E. W. Lajdler, The Demand For Money: 2’he-0-Ties and Evidence (Scranton: International Textbook Com-pany, 1969). For an excellent statement of the “conventionalview” of the money demand function, see Stephen M.Goldfeld, “The Densand For Money Revisited,” BrookingsPapers on Economic Activity — 3 (1973), pp. 577-638. Alsosee John T. Boornsan, “The Evidence on the Demand forMoney: Theoretical Formulations and Empirical Results, inJohn T. Boorn3an and Thomas M. Havrilesky, Money Sup-ply, Money Demand, and Macroeconomic Models (Boston:

Allyn and Bacon, Inc., 1972), pp. 248-291, and StephenRousseas, Monetary Theory (New York: Alfred A. Knopf,1972).

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FEDERAL RESERVE BANK OF ST. LOUIS JUNE 1975

of the actual, price of goods and services and nominalincome. Given the influences of the other factors, theamount of nominal money balances demanded is posi-tively related to the perceived price of newly pro-duced final goods and services. Given the influencesof both perceived price and the other factors, theamount of money balances demanded is positively re-lated to the amount of goods and services that per-ceived nominal income can purchase.

It is asserted that the response of holders of moneybalances is of the same magnitude with regard to bothperceived price and the amount of goods and servicesthat perceived nominal income can purchase. Thedesire for nominal money balances, therefore, is form-ulated in terms of perceived nominal income only(Equation 1) .~

A continuing rise in the price of goods and servicesresults in a given amount of nominal money balancesheld as a store of value commanding, over time, adecreasing amount of goods and services, Given theinfluences of the other factors, the amount of moneydemanded is negatively related to the rate of inflationexpected to prevail in the future (Equation 1) .~

Other financial assets are substitutes for money bal-ances as a store of value, and they yield a rate ofreturn. These rates of return, with constant, per-ceived prices for goods and services, are measured interms of real rates of interest. To the holder of afinancial asset, the real rate of interest is the annualpercent increase in the amount of goods and services,at current prices, that a given dollar value of a finan-cial asset will command over the term to maturity.With the influence of other factors held constant, theamount of money demanded is negatively related toreal rates of return on alternative financial assets.

It is asserted that the dominant substitutes formoney as a store of value are short-term financial

3lgnoring the other factors,

In N-1 a1

In ~P + a2

In a1

In ~P ~ a~In Y~— a., In pP~

By assertion a1

a~ therefore, In M* ~a1

In Y~.Friedmanin “A Monetary Theory of Nominal Income” asserts that forall practical purposes a

1= a

2= 1. In this study the only

assertion made is that aj a2

- The validity of this assertionis tested later in the study and the hypothesis that the tworesponses are identical cannot he rejected. Friedman’s asser-tion that they both equal unity is also tested and is rejected.

41t is important to note that there are two different influenceson the amount of money balances demanded related to theprice of newly produced final goods and services. One is thepositive influence of the perceived price level for the cnrrentperiod, and the other is the negative influence of the ex-pected rate of change in the price level over the relevantfuture,

Page 10

assets and that the time horizon upon which expecta-tions of the rate of inflation are based is the samefor money balances as for these assets. It is generallyaccepted that the nominal rate of interest on a finan-cial asset is the sum of the real rate of return and therate of inflation expected over the term of the financialasset. The nominal short-term interest rate, therefore,is presumed to capture the negative influence on theamount of money demanded of both the real short-term interest rate and the expected rate of inflation(Equation 1).

The technical efficiency of the system of makingmoney payments is defined as the average amount ofmoney balances required to be held to conduct a givennominal volume of transactions. With the influencesof other factors held constant, the amount of moneydemanded is negatively related to the technical ef-ficiency of the system of making money payments(Equation 1). In other words, the smaller the amountof money balances technically required to be held toconduct a given dollar volume of transactions, thesmaller is the amount of money demanded.

(1) Its NI * (t) = a0

In E(t) ±aj In Y~(t) -~-a., In r(t).

a0

<0, a~> 0, and a2

<0.desired nominal money balances.

E = technical efficiency of the payments system.perceived level of nominal income.

= nominal short—term interest rate.

Adjustment of Spending byHouseholds and Business FirmsThe nominal stock of money is assumed to be ex-

ogenous, determined by the monetary authorities.Therefore, holders of money balances, in the aggre-gate, cannot adjust their holdings when a discrepancyoccurs between actual and desired money balances,Instead, it is asserted that individual holders of moneybalances attempt to add to (reduce) money balancesby reducing (increasing) their rate of spending ongoods and services. Such an adjustment continues un-til desired money balances are brought into equalitywith actual money balances.

It is assumed that the rate of change in the dollarvolume of spending by households and firms on newlyproduced final goods and services from both domesticand foreign sources is proportional to the percent dis-crepancy between actual and desired money bal-ances.5 It is also assumed that the response of spend-ing is distributed over time (Equation 2).

5This specification of the adjustment process is a marked de-parture from that specified in other studies. Some specify

FEDERAL RESERVE BANK OF ST. LOUIS

(2) d In Yd = A[ln MU) — In M°U)]-

M = actual nominal money balances.

d In Yd = rate of change in spending by households and

dt business firms for newly produced, final goodsand services.

A = speed of adjustment. 0 < A u os~The larger is A,the faster is the speed of adjustment. A = n isinstantaneous adjustment.

Co-mbining Desired MoneyBalances and Adjustment Process

The preceding assertions regarding the demandfor nominal money balances and the adjustment proc-ess are now combined by substituting equation (1)into equation (2). The result is that the rate of changeof nominal spending by households and business firmsis related to the level of actual nominal money bal-ances, the technical efficiency of the payments sys-tem, the level of the nominal short-term interest rate,and the perceived level of nominal income.

(3) d In Yd = A [In MU) — a0

In EU) — a5

In ~~U)— a2

In rU)]-

cit

Definition of Nominal Income

Nominal income is defined as total value added(that is, factor payments) in the domestic productionof new final goods and services. Total nominal spend-ing on domestic product is also equal, by definition, tovalue added. Nominal income, accordingly, is equalto nominal spending for both domestic and foreignproduct by domestic households, business firms, andall units of government, plus nominal spending byforeigners for domestic product, less domestic nomi-nal spending for foreign produced goods and services.

(4) YU) = YdU) * CU) -~- X(t) — 114(t).

Y = nominal income.= spending by household and business firms for both

domestic and foreign product.C = spending by all units of government for both domestic

and foreign product.X = spending by foreigners for domestic product.

IM = domestic spending for foreign product.

JUNE 1975

Domestic Spend-ing for Foreign P-roduct

Domestic nominal spending for foreign product isassumed to be a constant proportion of total nominalspending in the economy.

IMU)(5)8(0= =8.

yd (t) ±CU) + XU)

Dynamic Form of the Model

The model, up to this point, involves the changein the level of income in response to changes in thelevel of the money stock, or in Government spending,or in foreign spending on domestic product, or in theshort-term interest rate, or in the technical efficiencyof the payments system. All of these variables, how-ever, are continuously changing over time. The modelis differentiated with respect to time to reflect theresponse of households and business firms to thosevariables which are continuously changing.

In its dynamic form, the rate of -change in desiredmoney balances is related to the rates of change inthe technical efficiency of the payments system, in theshort-term interest rate, and in perceived income.

dInM° dInE dInY~ dlur(1) a

0±aj +a

2-

dt dt dt dt

The change in the rate of change in nominal spend-ing by households and business firms is proportionalto the discrepancy between the rate of change inactual money balances and the rate of change in de-sired money balances.(2’) ml2 In yd = X[ d In 14 — dIn

dt2

dt cit

The statement combining the adjustment processand the factors influencing desired money balances isaccordingly modified.

d2InY

4dlnM dInE dIn Y~

(3) A[ —a0

—a1dt2 dt dt dt

d In r

dt

Substituting equation (5) into (4) and differentiat-ing with respect to time, the rate of change in nominalincome is equal to the weighted sum of the rates ofchange of spending for product by domestic house-holds and business firms, domestic units of govern-ment, and foreigners.(4~

)dIn Y = wi(t) d In \‘d ±W2(t) d InC + w3U) d InX

dt dt dt dt

\Vi(t) = [1 8] Yd(t) w2U) = [1--5]YU) YU)

and \V.3

(t) = [1—8]YU)

that the actual money stock changes in response to a dis-crepancy between actual and desired money balances; othersspecify that “real” money balances change; and others spe-cify that the short-term interest rate changes. See referencesin fn. 2 for specific examples of these other specifications.For discussion of the importance of the specification of theadjustment process in enspirical research, see: A. A. Walters,“The Demand for Money — The Dynamic Properties of theMultiplier,” Journal of Political Economy (June 1967), pp.293-298, and D. R. Starleaf, “The Specification of MoneyDemand — Supply Models Which Involve the Use of Dis-tributed Lags,” Journal of Finance (September 1970), pp.743-760.

Page Ii


Summary of Variables

The endogenous variables of the model are thechange in the rate of change in nominal spending byhouseholds and business firms, the rate of change indesired money balances, and the rate of change innominal income. It will also be developed in sub-sequent analyses that the rate of change in the per-ceived level of nominal income is an endogenousvariable related to past rates of change in nominalincome. The exogenous variables are the rates ofchange in actual money balances, in the short-terminterest rate, in the technical efficiency of the pay-ments system, in Government spending on goods andservices, and in foreign spending on domestic product.

The model is one of partial analysis, inasmuch asthe nominal short-term interest rate is an exogenousvariable, There is no provision for feed-back effectson the rate of change in nominal income. These effectswould emanate from changes in exogenous variableswhich change the rate of change in the short-terminterest rate, either by induced changes in the realinterest rate or in the expected rate of inflation,

Conditions for Dynamic Equilibrium

Dynamic equilibrium is defined as the state inwhich all of the variables are changing at constantrates.6 This occurs when the rates of change in actualand desired money balances are equal. Three condi-tions are required for this equality to be fulfilled —

changes in the rates of change in nominal spendingby households and business firms and in nominal in-come equal zero (E-i and E-2), and the rates ofchange in nominal income and in the perceived levelof nominal income are equal (E-3).

(h-i) d2

In yd = o.dt2

(E-2) d2 In Y = o.dt2

(E-3) dInYP = dinYcit cit

Dynamic Equilibrium State

The relationships of the model (Equations 3’, 4’,and the weights) together with the three dynamicequilibrium conditions (Equations E-i through E-3)are used to solve for the dynamic equilibrium state ofthe endogenous variables.~ In the dynamic equili-brium state, taking into consideration the postulatedsigns of the coefficients, the rate of change in nominalspending by households and business firms is posi-tively related to the rates of change in actual moneybalances, in the technical efficiency of the paymentssystem, and in the nominal short-term interest rate,and is negatively related to the rates of change ingovernment spending and exports (Equation ES-i).The rate of change in nominal income is positivelyrelated to the rates of change in actual money bal-ances, in the technical efficiency of the paymentssystem, and in the nominal short-term interest rate(Equation ES-2).

dlnYd 1 dInM dIn E cllnr(ES-i) = [ — a0

—a2dt WjU)ai dt dt dt

— Wa(t) dInG — WU) dIuXdt dt

ES’ dInY 1 dm14 dInE dlnr-2) dt dt ao dt —a2 dt

A Change in Dynamic Equilibrium State

An existing dynamic equilibrium state is disturbed,in this model, by two types of events. One type is theinitial creation of a discrepancy between the rates ofchange in actual and desired money balances. Factorsinitiating such a discrepancy are changes in the rateof change in actual money balances or changes in therate of change in desired money balances. This latterchange can result from changes in the rate of changein either the technical efficiency of the paymentssystem or in the short-term interest rate. The othertype of equilibrium disturbance is a change in therate of change in government spending or in foreignspending for domestic product, which initiates achange in the rate of change in nominal income. Fol-lowing both types of disturbance, a new equilibriumstate is achieved when the rate of change in nominalincome has changed to the extent that the rate ofchange in desired money balances is brought intoequality with the rate of change in actual moneybalances.

~This analysis and the one immediately following is for partialequilibrium only inasmuch as the interest rate is exogenous.The model only considers the influence of exogenous shockson the choice between money and new production and noton the choice between money and existing assets.

6The definition of equilibrium as constant rates of change isa nsarked departure from the standard literature in monetary-economics, other than some growth models. The standardanalysis defines equilibrium as constant keels of the endo-genous variables. The equilibrium conditions for the standardanalysis would be that desired and actual levels of nioneybalances are equal, that the rates of change in spending byhouseholds and business firms and in nominal income equalzero, and that the perceived level of nonsinal income equalsthe actual level. Thus, in the standard analysis there is bothstock and flow equilibrium. In the dynamic foms in thisstudy, however, equilibrium is defined as constant rates ofchange in both stock and flow variables.

Page 12


For example, starting from dynamic equilibrium(that is, all variables changing at constant rates) as-sume there is a maintained increase in the rate ofchange in actual money balances, with the rates ofchange in the other independent variables remainingas they were. First, there occurs a positive discrep-ancy between the rates of change in actual and de-sired money balances, resulting in an increase in therate of change in nominal spending by householdsand business firms. As a result, there is an increase inthe rate of change in nominal income. This latter in-crease, in turn, results in an increase in the rate ofchange in perceived nominal income and thereby inthe rate of change in desired money balances.

As a result, the discrepancy between the rates ofchange in actual and desired money balances is nar-rowed and the rate of change in spending by house-holds and business firms decreases. This decreasereduces the rate of change in nominal income fromwhat it was initially. This process continues untilthere is equality between the rates of change in actualand desired money balances. In the new dynamicequilibrium state, the rates of change in spending byhouseholds and business firms and in nominal incomeare higher than their starting equilibrium values (ES-iand ES-2).

In another example, starting from equilibrium, as-sume there is a maintained increase in the rate ofchange in government spending, with rates of changein the other independent variables remaining thesame. The first response is an increase in the rate ofchange in nominal income which produces an increasein the rate of change in perceived nominal income.This results in an increase in the rate of change indesired money balances.

As a consequence, there is a negative discrepancybetween the rates of change in actual and desiredmoney balances, resulting in a subsequent decrease inthe rate of change in spending by households andbusiness firms. This is acconspanied by a decrease inthe rates of change in nominal income and in per-ceived nominal income, which, in turn decreases therate of change in desired money balances. Thisprocess continues until the rate of change in desiredmoney balances equals that of actual money balances.

In the new dynamic equilibrium state the rate ofchange in nominal income is the same as it was be-fore the increase in the rate of change in governmentspending (ES-2). The rate of change in spending byhouseholds and business firms has been permanentlydecreased (ES-i), offsetting fully the initial increase

in the rate of change in nominal income generated bythe increase in the rate of change in governmentspending.

The length of time required for achieving the newequilibrium state depends on (1) the speed of re-sponse of the rate of change in spending by house-holds and business firms to a discrepancy between therates of change in actual and desired money balancesand (2) on the length of time required for the rateof change in perceived nominal income to respondfully to past rates of change in actual nominal income,The time path to a new equilibrium — whether it

oscillates or moves only in one direction — dependson the response characteristics of the relationshipscomprising the model.

EMPIRICAL FORM OF THETHEORETICAL MODEL

The theoretical model consists of three relation-ships. One is for the change in the rate of change inspending by households and business firms (Equation3’). The other two are the identity for the rate ofchange in nominal income (Equation 4’), and theweights involved in this identity. The theoreticalmodel is next given an empirical form which pro-vides the basis for testing whether or not the theorycan be accepted. This section presents the empiricalmodel and the following section uses it to test thetheory.

Data ProblemsA major data problem is involved in developing

the empirical form of the model — the theory is interms of changes in time which are infinitesimallysmall while data on the variables are available onlyfor discrete points in time separated by a month or aquarter of a year. A linear approximation in discretetime is used — the first difference of natural loga-rithms of a variable between two discrete points intime is presumed to approximate its rate of change(Appendix).

Another data problem exists inasmuch as there areno measurements of the technical efficiency of thepayments system or of the perceived level of nominalincome. It is assumed that, on average, the technicalefficiency of the payments system increases at a con-stant rate (Appendix).8 The second problem is solved

5Examples of clevelopnsents which are usually presumed tohave increased the efficiency of the payments system (areduced amount of snoney balances required to carry out agiven volume of nsoney payments) over the sanspie period

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FEDERAL RESERVE BANK OF ST. LOUIS

by employing a procedure presented by Philip Cagan.°According to this procedure, the rate of change inperceived nominal income can be approximated bya weighted average of past rates of change in actualnominal income (Appendix). In this procedure, theweights sum to unity.

Estimated Parameters of the Model

Only the parameters of the relationship explainingthe change in the rate of change in spending forproduct by households and business firms are esti-mated; the other relationships are identities. Two ver-sions of this relationship are estimated. One is basedon the assertion that desired money balances are posi-tively related to perceived price and the amount ofgoods and services that perceived nominal incomecan purchase, with no assertions regarding the magni-tudes of response. The other one is based on theassertion that the magnitude of response of desiredmoney balances is the same with regard to perceivedprice as with regard to the amount of goods andservices that perceived nominal income can purchase— the perceived income formulation used in thetheoretical discussion. The general forms of these tworelationships are as follows:

(A) tin Y~’— tin Y~1

= a0

+ a5

AIn Mt + a2

tin r~

w~tin P~.t+ a4

~ wi tin Qt-~+ at

(B) AIn Y~ — AInY~1

=b0

+ h1

tin M5

+ b2

Ale r~

+h3

~w-t AInYet +Et

= change in the rate of change in spendingby households and business firms forproduct.

= response to the average rate of change inthe technical efficiency of the paymentssystem.

tinM

t = rate of change in nominal money balances.

tin r~ = rate of change in nominal short-terminterest rate.

= weighted sum of past rates of change inprice.

= weighted sum of past rates of change inthe amount of goods and services thatnominal income can purchase.

wt Abs Yet = weighted sum of past rates of change in= I nominal income.

= a random error term.

are the introduction of faster means of transferring fundsand the adoption of insproved methods of managing moneybalances. See George Garvy and Martin H. BIyn, The Veloc-ity of Money (New York: Federal Reserve Bank of NewYork, 1969), pp. 207-218, for further discussion of this point.

OPhillip Cagan, “The Monetary Dynamics of Hyperinflation,”Stndie,s in the Quantity Theory of Money, ed. Milton Fried-man (Chicago: University of Chicago Press, 1956), pp. 3141.

Page 14

JUNE 1975

Ordinary least-squares regressions, using quarterlydata from first quarter i955 to fourth quarter 1973,are used to estimate the parameters of equations Aand B. Spending by households and business firms ismeasured by the sum of consumption plus investmentin the national income accounts. Nominal income ismeasured by nominal GNP.1° The price level ismeasured by the GNP deflator, and the amount ofgoods and services that nominal income can purchaseis measured by nominal GNP divided by the pricedeflator. Two definitions of money are used. One(M1) is the sum of demand deposits and currencyheld by the nonbank public. The other one (M2) isM1 plus time and savings deposits at commercialbanks, other than large, negotiable certificates of de-posit. The short-term interest rate is measured by the4- to 6- month commercial paper rate. Two zero-onedummy variables are included for the average influ-ence of major strikes on income, with D1 equaling onefor the quarter of a strike and D2 equaling one forthe quarter following a strike.11

A test was performed for a structural change in themodel (Appendix). For M, such a change is allegedby some analysts to have occurred after the fourthquarter of 1966 when a change in the trend growthof the ratio of GNP to M1 is observed. A similarchange in the trend growth of the ratio of GNP to M2occurred after the fourth quarter of 1961. In bothcases the structural change hypothesis was rejected.A statistical procedure (Appendix) was next used toselect the appropriate number of lagged changes inthe price level, nominal income, and in the amountof goods and services that nominal income can pur-chase; four lagged terms were deemed appropriate.The estimated parameters of equations A and B arereported in Tables I and II.

EMPIRICAL MODEL AS A TESTOF THE THEORY

The empirical model is used to test whether or notthe theory of nominal income determination advancedearlier can be accepted. The part of this theory re-garding the determination of nominal spending byhouseholds and business firms carries specific implica-

101n national income terms, the definition of nominal incomeused in this study is the suns of consumption, investment,government expenditures, and exports less imports. Thissum should be adjusted for depreciation and indirect busi-ness taxes, to be identical to value added. It is assumed thatvariations in these two magnitudes will have little influenceOn the outcome of this study.

11The snajor strikes in the sample period were: steel 111/1959,autos IV/1964, and autos IV/1970.

AIn Y~— Am Y~1

a0

and b0

n

w AIn Pet

w~AIn Q,.i


Table II

Regression ResultCEquation (B)

In do pandont

—1.930 —1.886(—3./44) (—3.616)

ED, 2.380 2.041(4.326) (3.630)

AIn M- ./01 .572(4.527) (4.309)

AIn r~ .020 .030(2.183) (2.974)

Al1 V — .782 — .781(—5.242) (—5.157)

AIr Vi. 2 - .274 -- .225‘21641 ~- 1.759)

Ak Vt.j .226 .197(1.846) (1.592)

Ak YL 4 . .309 .3152.739) ( 2.765)

Constant 1.152 .911(3.785) (2.792)

.580 .570

SEE .864 .874OW 2.036 2.049

Nu:ntjer$ Ii parail LhL”.”. ar4’ t—valLw,.

Table I

Regression RosultCEquation ~A)

IndependentVariable

1.996 1.947I 3.765) ( 3.637)

02 2 259 1.921(3.9/8) (3.321)

Ale M’ .685 .534(4.0031 3.814)

Am r .022 .032(2.251~ (3.096)

Ak Ot - .797 .799I 5.250) I 5.182)

Alp Q~2 -. .249 -. 2081--L905) ( 1.580)

Alp 0.3 .263 .238(2.051) (1.842)

Alp 0’ — .278 .. .3012.341) (—2.518)

An P., .915 . .8112.103) ( 1.855)

Ale P - .558 . .5921.193) (—1.254)

Am P1

.143 .232

.2981 I .479)

AIn P~. .251 .274.606) I .657)

Constant 1.074 /72(3.078) (2.135)

R2

.569 .561

SEE .875 .883OW 2.009 2.037

it. p. .1 I

iImfl~rm’4ai’diuL liii’ c’tinm.li’J umtedIicnt~t~pni’t’l in

theory is accepted.

magnitude with regard to perceived price as with re-gard to the amount of goods and services that per-ceived nominal income can purchase. Therefore, thetheoretical formulation was in terms of perceivednominal income. This assertion implies in EquationsA and B that for each lag the coefficients for the

Tables I and II, If the estimated coefficients are con- rates of change in price, in the amount of goods andsistent with the implied coefficients, at the five per- services that nominal income can purchase, and incent level of statistical significance, this part of the nominal income are equal. This implication was tested

The model as a whole constitutes the theory of

by the ability of the empirical model to forecastnominal income.12 Relatively small root mean

nominal income determination. This theory is tested this test, estimnates of the coefficients of Equation B

and could not be rejected at the five percent levelof statistical significance (Appendix).” As a result of

determination can be accepted.

are used for the balance of this study (Table II).

squared errors in forecasting nominal income are by households and business firms implies that all oftaken as evidence that the theory of nominal income the coefficients in Equation B are statistically signifi-

The theory of determination of nominal spending

Testing Implications of Theory Regarding the and b, is negative.tm4 All of the estimated coefficientsResponses of Households and Business Firms in Table II are statistically significant from zero at

cant from zero at the five percent level and that theyhave the following signs: b3, b,, and b, are positive

It was asserted in the theoretical section that theresponse of holders of money balances is of the same

12

A more appropriate test would be to solve for the reduced-fonn of the model, and then to test whether the estimatedcoefficients of this equation are consistent with those impliedby the theory. The reduced-form is a nonlinear equationwith variable coefficients, which poses some vesy difficultproblems of estimation. The forecasting test was thereforeselected.

the five percent level and have the implied signs. The

~~The proposition that the elasticity of desired money bal-ances with respect to the perceived level of nominal incomeequals unity was tested and rejected (Appendix). Theproposition that this elasticity is greater than unity wasaccepted.

“These implied signs are derived by applying the postulatedsigns for Equations 1’ and 2’ to Equation 3’. The magni-tude of b

3is derived by adding the coefficients on the

lagged income terms,

Page 15


empirical evidence is consistent with the implicationsof this part of the theory, which is, therefore, accepted.

Testing Theory of NominalIncome Determination

A test of the theory of nominal income determina-tion is the ability of the empirical form of the modelto forecast nominal income with relatively smallroot mean squared errors. This forecasting ability ofthe model is determined by dynamic simulations. Ina dynamic simulation actual values of the quarterlyrates of change in money balances, in governmentspending plus exports, and in the nominal short-terminterest rate are used. At the start of the simulation,actual lagged rates of change in nominal income areused, but, subsequently, simulated rates of change areincorporated. Two types of simulations are performed— ex post for the sample period and a number ofcx ante simulations beyond various sample periods.

The simulation model consists of Equation B (TableII), Equation (4’), and the definition of the weights.Using the discrete-time linear approximation to ratesof change in Equation (4’):

AlnYt w~tin Y~+ (1-We) tin Z5

.d

W1~

’(1-8) .!_i±Yt-t

Z government spending plus exports.

8 = ratio of imports to total spending on product. The ratiois held constant at its average value in the sampleperiod.

Lx post simulations are used to ascertain the abilityof the model to capture movements in nominal in-come over the sample period. The root.mean-squared-error (RMSE) for the quarterly rates of change ofnominal income (expressed as annual rates) is 2.85percentage points using M1 and 2.95 using M2. Al-though the errors in the quarterly rates of change(expressed as annual rates) are quite large, the errorstend to be offsetting over the sample period. In thefourth quarter of 1973, the error in the level of nomi-nal income was 0.36 percent using M1 and 0.33 per-cent using M,. Over the sample period, the RMSEfor the quarterly levels of nominal income is 1.57percent using M1 and 2.07 percent using M,.

Lx ante simulations are used to determine the abil-ity of the model to forecast nominal income beyonda sample period, with known values of the exogenousvariables. The coefficients of Equation B are estimatedfor the sample period from first quarter 1955 to fourthquarter 1961. The coefficients are then re-estimatedfor the sample period from first quarter 1955 to fourthquarter 1962. This procedure of lengthening the sam-

ple period by four quarters is continued until fourthquarter 1973, the terminal date.

Dynamic simulations are then performed for theeight quarters following each sample period. Thesimulated annual rate of change in nominal incomeover the first four quarters and over the entire eightquarters are calculated. For the set of first fourquarters, using all the post sample periods, theRMSE’s of the annual rate of change in nominal in-come are 1.59 percentage points using M1 and 1.40using M,. For the set of eight quarter periods, theRMSE’s are 0.99 percentage points using M1 and0.82 using M,.

For each post sample period, the simulated level ofnominal income is calculated for the fourth quarterand the eighth quarter. The RMSE’s are then cal-culated for the set of all post sample periods. UsingM,, the RMSE in the fourth quarter level of nominalincome is 1.59 percent and using M, is 1.40 percent.For the eighth quarter level, the RMSE is 1.98 per-cent using M1 and 1.64 percent using M2.

The cx ante simulation results indicate that theempirical model forecasts the level of nominal incomewith relatively small RMSE’s, compared with fore-casts from nine major econometric models of the U.S.economy. Carl F. Christ, in reviewing the cx anteforecasting performance of these models, concluded,“All have RMSE’s for real and nominal GNP that are1 percent or less for one quarter ahead, and 3 percentor less for five or six quarters ahead.”15 The modelpresented in this study, which is based on a markedlydifferent theory than eight of these nine models, fore-casts nominal GNP with smaller RMSE’s than severalof the large scale econometric models of the U.S. Thisforecasting evidence thus leads to acceptance of thetheoretical model of nominal income determination.

CONCLUSIONSA theoretical model of nominal income determina-

tion was developed, based on a set of postulates re-garding the behavior of households and business firms.The central postulate is that the rate of change innominal spending by households and business firmsfor newly produced final goods and services respondsto a discrepancy between the rates of change in actualand desired nominal money balances.

On the basis of the empirical tests, the theoreticalmodel was accepted as representing the determination

iSCarl F, Christ, “Judging the Performance of EconometricModels of the U.S. Economy,” International Economic Re-view (February 1975), p. 64.

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of nominal income in the period 1955 - 1973. Theestimated coefficients of the empirical model have thesigns implied by the theory. In addition, cx postand cx ante dynamic simulations indicate that it fore-casts nominal income with relatively small errors.

The empirical results mentioned in the introductionof the St. Louis reduced-form equation relating changesin GNP to changes in money and government spend-ing are consistent with the theoretical properties ofthe model presented in this study. These theoreticalproperties are developed in the section which dis-cusses the changes in the dynamic equilibrium state.

A theoretical property of the model is that a changein the trend growth of nominal money balanceschanges the trend growth of nominal income. An-other theoretical property is that short-run changesin the growth of nominal balances result in short.run

changes in the growth of nominal income. A thirdtheoretical property (based on partial analysis) is thatchanges in the growth of government spending exerta short-run, but not a long-run, influence on growthof nominal income. Thus, there is “crowding-out” ofspending by households and business firms, but onlyin partial equilibrium analysis. Since the desire tohold money balances is negatively related to theshort-term interest rate, the financing of the growthof government expenditures by issuing debt may havea positive influence on growth on nominal income.The estimated interest elasticity of desired’ moneybalances is very small (about .03); therefore, for debtfinancing to have much of an influence on growth ofnominal income, the elasticity of the interest rate withrespect to government debt must be very large or theincrease in debt outstanding must be exceedinglylarge.

APPENDIX

Data ProblemsThe linear approximation of rates of change in dis-

crete thne is given by the following:dInY 1 dY_____ — In Y, — In It-i.

dt Y dt

The technical efficiency of the payments system isassumed, on average, to increase at a constant rate givenby the following exponential growth function:

E(t) = a&e; or In E(t) = In a + la.

e = the base of natural logarithms.

I) = constant rate of growth.

a = beginning level of E.

= an index of time.

Differentiating with regard to time yields:

tlI,m E —

dt

A variation of Phillip Cagan’s procedure is used for ap-proximating the rate of change irk the perceived level ofnominal income. It is assumed that changes in the rate ofchange in the perceived level of nominal income areproportional to the discrepancy between the actual rate

of change in nominal income and the rate ofthe perceived level of nominal income.

d2InY

5’ =

1dInY dlnY~

dt2 i3~ dt dt

change in

d2

In yP= the change in the rate of change in the perceived

dt2

level of nominal income.p = the adjustment coefficient. 0 < $ ‘v The larger

is A the faster is the speed of adjustment.

According to Cagan’s procedure, the rate of change inthe perceived level of nominal income can be approxi-mated in discrete time by the following:

tin ~F=(l-e0

) , ~ tin Ys.te~t

(l-e~) e~’ = 1.

In the discrete time form of the model, it is assertedthat the rate of change in perceived nominal income isthat at the beginning of the period. The index (i), there-fore, runs from 1 to T. Instead of assuming, as Cagan did,various values of (3 and then using their implied valuesfor Aln Y~’direcfly in the regression equation and se-lecting the one which has the largest R2, the coefficientsfor each lagged income term were estimated directly.Four lagged changes in nominal income (the lag struc-ture used in this study) implies that 3 = 1.15, with the

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implied weights( 0.68, 0.22, 0.07, and 0.02) summing to0.99. The estimated coefficients for the four lagged incometenns, given their standard errors of estimate, are approxi-mately consistent with those implied by these weights.

Test for Structural ChangeA test of the hypothesis of a structural change in the

coefficients of the model is performed by introducing azero-one dummy variable into equation (A) of the text.This equation is used rather than Equation B, whichincorporates a special assumption regarding the param-eters on perceived price and the amount of goods andservices that perceived nominal income can purchase.The dummy variable has the value of zero in the M

1equation from first quarter 1955 to fourth quarter 1968,and then a value of one to fourth quarter 1973. It has azero value in the M2 equation from first quarter 1955 tofourth quarter 1961, and then a value of one to fourthquarter 1973.

Two forms of equation (A) are estimated. One is theoriginal specification. The second one has all the originalvariables, but adds the dummy variable itself (to meas-ure a change in the constant) and the product of thedummy and each of the original variables (to measurea change in the regression coefficients).’

If the estimated coefficients for these added varia-bles are not statistically significant from zero, the struc-tural change hypothesis is rejected. An F test is con-ducted to test the null hypothesis that all the regressioncoefficients of the added set of variables are equal tozero. If the calculated F value is less than the critical Fvalue, at the five percent level of significance, the nullhypothesis is not rejected.

Since the number of lagged MnQ and AInP termshave not been specified at this point, the test forstructural change was conducted for lag specificationsfrom one quarter to ten quarters (Table A). For everylength of lag, the calculated F value is less than thecritical value, As a result, the hypothesis of a structuralchange in both the M and M equations is rejected foreach of the lag structures exam ned.

Table A

Test For Structural ChangeEquation (Al

Number Calculated ~ CrmticaI

of Lag Mj M Ff05)I 1.64 167 2362 1,45 134 2173 158 154 2054 LOS 1.42 1 985 .99 117 1.936 87 91 1907 82 93 1.898 75 .98 1.899 .75 1 21 1.91

10 .83 1 12 1.91

‘Except D, and I)a

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Determination of Lag StructureA statistical procedure is used to determine the ap-

propriate number of lagged AlnQ and AInP terms inequation (A). Each equation (M, and M

2) is estimated

eleven times, first with no lags and then increasing thelength of lag for A1nQ and AlnP by oue quarter until tensets of the two lagged tenns are included. The F test isused to test the null hypothesis that the two estimatedcoefficients for each added lag are zero. If the cal-culated F value is greater than the critical value, at thefive percent level of significance, the null hypothesis isrejected. Table B shows that the null hypothesis is re-jected when lags one and four are added, but not whenany of the other lags are added.

Table K

Test of Lag StructureEquation (A)

Each Added Calculated F Cnt,cot

Lag Mj M2

F( 0S~

1 16.88 1692 4.002 119 .89 4.003 AS 16 4004 411 483 4,005 .20 .20 400

6 102 1.00 4007 24 .26 4.008 124 1.41 4.009 39 .30 4.04

10 2.30 278 4S4LagsI 4 6.37 &09 2.535 10 91 1.01 2.29

The F test is ne t used to test the null hypothe is thatas a set all the coefficients for lags one to four are equalto zero. This test, at the five percent level, rejects thenull hypothesis (Table 13). A similar test is conductedfor the coefficients as a set for lags five to ten, whenlags one to four are included in the regression. In thissecond test the null hypothesis cannot be rejected (TableB). On the basis 0f these results, four lagged AlnQ and~lnP terms are considered to be appropriate. Table I inthe text presents the estimated coefficients for this speci-fication, equation (A).

Test of Equal Response HypothesisThe hypothesis that the response of desired nominal

money balances is the same with regard to both per-ceived price and the amount of goods and services thatnominal income can purchase is tested by imposing alinear constraint on equation (A). An F test is used totest this constraint. If the calculated F value for the con-straint is less than the critical value, at the five percentlevel of significance, the proposition is accepted.

Assuming that the weights in forming Mn Q° andMn P’ are equal for each lag, the equal response hypo-thesis implies that for each lag the estimated coefficientsfor A1nQ and MnP are equal. Since by definition


Table C

Test of Unitary Elasticity HypothesisRegression Results

2.035 2.0223.864) I 3.708 I

0 2 486 2.09514 424) (3.550)

AIr .015 .025V 672) (2A38)

Airt V Al, fr. - .696 — 6454./23) (.4.276)

AmY’ ;—AlrM - 730 - .1487971 I 1.1 29)

Am1 V. ‘, — Am M .264 .240(2.1 I8~ Il 859~

Aln V Al,, itt .213 ‘ .1842.~.07) 1—1.6801

car.tani .571 .040063) 1 356)

12 .526SEE .897 .918

2.045 2027

Mn Qm ±AlnP~= Mn Y., the coefficients on each set oflagged AlnQ and LinT terms are constrained to be equal

by specifying AlnY terms (Equation B). Table II of thetext presents the estimated coefficients for equation B.Regression results for this equation are tested againstthose for equation (A). The calculated F value are .58for M, and .64 for M,. The critical F value is 2.53, there-fore, the equal response hypothesis is accepted in bothcases. These results also lead to the acceptance of theassumptions made regarding the equality of weights informing Mn QP and Mn ~u,

A frequent hypothesis in monetary economics is thatthe elasticity of desired money balances with regard tonominal income is unity. Since the weights sum to unity,the hypothesis implies that the coefficient on money inequation (B) is equal to, but opposite in sign, the sumof the coefficients on the lagged Mn? terms. This con-straint is imposed on equation (B) by dropping theMnM, term and subtracting AlnM~ from each of thelagged tInY terms. The regression results are reported inTable C and are tested against the results for equation(B). The calculated F value for M1 is 4.57 and for M, is7,99, The critical value of F is 4.00; therefore, the propo-sition that the elasticity equals unity is rejected in bothcases. The estimated elasticity of desired money balanceswith regard to perceived nominal income .— 1.62 for M,and 1.97 for M2, — is derived by dividing the sum of thecoefficients on the lagged income terms by the coefficienton money and changing the sign.

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A Monetary Model of Nominal Income Determination