WILLIAM POOLE* Special Studies Section, Division of Research and Statistics, Board of Governorsof the Federal Reserve System The Role of Interest Rates and Inflation in the ConsumptionFunction MOST OF THE ANALYSIS of the effects of interestrate changes on con- sumptionhas been concernedwith (1) the relativeimportance of income and substitutioneffects in determininghow householdswill allocatetheir resourcesover time, and (2) the substitutioneffect at a moment of time determiningthe demands for durable versus nondurableconsumption goods. But two othertypes of interestrate effects on consumption-effects that have receivedlittle attentionin the literature-may be of some im- portanceand are the subjectof this report.The rate of inflationenters the analysisbecause of the wedgeit drivesbetween the nominaland real rates of interest. One of the effectsto be consideredis a consequenceof the fact that the realrate of interesthelps determine the servicesyielded by the stock of con- sumer durables.Following a common practicein econometricwork on consumption,the servicesof durablesare includedin consumptionand purchasesof consumerdurables are excluded.Services of durablesmust also be addedto disposableincome. Since the value of such servicescannot be ascertainedfrom markettransactions, it mustbe imputed.The stock of consumerdurables is firstestimated, and then multipliedby a depreciation rate and a net rate of returnto obtain the gross yield on the stock. Since * Bonnie Garretthandled the computerwork. The authoris solely responsiblefor the views expressedand any errorsof analysis. 211 212 BrookingsPapers on EconomicActivity, 1:1972 consumerscan be expectedto equatethe net rate of returnon durablesto the interestrate on financialassets, the interestrate entersinto the deter- minationof the yield on consumerdurables. Another neglectedfactor in the study of consumptionbehavior is the treatmentof the gross interestincome of households.In calculatingreal income duringan inflationaryperiod, households should take accountof the depreciationin the real value of theirfixed income assets. Sincean in- flationpremium finds its way into nominalinterest rates precisely because of this depreciation,and compensatesfor it, consumerscan allow for the depreciationby savingin entiretythat part of their gross interestincome that representsthe inflationpremium. A theoreticallycorrect definition of "income"should either exclude that partof interestincome that is an infla- tion premium,or, equivalently,include the anticipatedpart of the capital lossesin realterms on the household'sportfolio of fixedincome assets. Both of these issuesbecome important whenever nominal and real rates of interestdiverge, that is, wheneverinflationary or deflationaryanticipa- tions develop.To studythese issuesI have used the consumptionfunction of the SSRC-MIT-Penn(SMP) model. This function will be called the "standard"function and will serveas a benchmarkin makingcomparisons with formulationssuggested by the analysis.An alternativeformulation will be examinedafter an outline of the currentformulation of the SMP consumptionfunction.' The SMP ConsumptionFunction Real consumption,CON, is definedby (1) CON=ECN + YCD + WCD, whereECN is consumerexpenditures on nondurablegoods and services, and YCD and WCD are, respectively,the net yield on and depreciation of the stockof consumerdurable goods. These variables are all in realterms (1958 dollars).The quarterlyequations for YCD and WCDare (2) YCD = 0.0379 (0.125 ECD + KCD_1) 1. The SMP consumptionfunction is a modifiedversion of the life cycle consumption model. See Albert Ando and Franco Modigliani,"The 'Life Cycle' Hypothesis of Sav- ing: Aggregate Implicationsand Tests," AmericanEconomic Review, Vol. 53 (March 1963), pp. 55-84. WilliamPoole 213 and (3) WCD = 0.225 KCDJ1 + 0.45 (0.125 ECD) = 0.225 (0.25 ECD + KCDL1), where ECD is expenditureson consumerdurables at annual rates and KCD is the stock of consumerdurables at the end of the quarter. Theassumptions underlying equation (2) areeasily explained. Since ECD is measuredat annualrates, it is necessaryto divide it by 4.0 to obtain a quarterlyrate. Thus, in the absenceof depreciation,KCD = 0.25 ECD + KCD-1. However,the amount of servicesyielded by consumerdurables over the quarterdepends on the integralof the instantaneousrate of yield on the level of the stock. This amountmay be approximatedby applying the rate of yield to the averagelevel of the stock over the quarter,which is approximately1/2(KCD + KCDJ1) = 0.125 ECD + KCD-1. The rate of yield appliedto this stockis 3.79 percent.The reasonfor selectingthis yield will be explainedbelow. Equation(3) may be derivedin a similarfashion. The stock of consumer durablesis assumedto depreciateat a rate of 22.5 percentper annum,ex- cept for the firstquarter, during which the rate of depreciationis assumed to be 45 percent. The equationsfor nominalmagnitudes are as follows: (4) ECD$= PCD(ECD) (5) KCD$= PCD(KCD) (6) YCD$ = 0.01 RCB (0.125 ECD$ + KCD$_1) (7) WCD$ = 0.225 (0.25 ECD$ + KCD$_1) =PCD (WCD) (8) CON$ = ECN$ + YCD$ + WCD$ CON$ (9) PCON = CON' In theseequations variable names ending with a dollarsign arethe current- dollarequivalents of the realvariables. In equations(4), (5), and(7) current- dollar variablesare obtainedfrom real variablesby multiplyingby PCD, the price index for consumerdurable goods. RCB is the corporatebond rate, and PCON is the consumptiondeflator. One difficultywith the standardmodel arisesin equation(6). The nomi- 214 BrookingsPapers on EconomicActivity, 1:1972 nal interestrate ought to be replacedwith the real interestrate, because the presentvalue of the streamof servicesyielded by a durablegood can be calculatedeither by discountingthe futurenominal stream of servicesby the nominalinterest rate, or by discountingthe futurereal streamof services by the realinterest rate. The two formulationsare equivalent since the differ- ence betweenthe nominaland real servicesis the accumulatedamount of anticipatedinflation, and the differencebetween the nominaland real in- terest rates is the anticipatedrate of inflation.The procedurefollowed in equation (6) is equivalentto discountingthe (assumedconstant) future streamof realservices by the nominalinterest rate. Correcting equation (6), therefore,requires the substitutionof a real rate of interestfor RCB. Equation(2), determiningYCD, employsa constantyield of 3.79 per- cent, a resultstemming from the way in whichprice indexes are calculated. This resultis derivedas follows: In equilibrium, (10) St = (rt + dt)PCDt, whereSt is the current-dollarrental rate for one unit of consumerdurables, dt is the depreciationrate, and rt is the realrate of interest.An index,IS, of the rentalprice of consumerdurables may then be definedas (11) IS = _St (rt + dt)PCDt So (ro + do)PCDo It is convenientto assumethat the baseyear for the priceindex PCD is year t = 0 so that PCDo = 1. In terms of the model's notation, S = (YCD$ + WCD$)/KCD. If one ignoresthe minorcomplication raised by the assumptionof a more rapid depreciationrate in the first quarterand assumesa constantdepreciation rate, deflatingthe sum of (6) and (7) by the indexdefined in (11) yieldsthe sum of (2) and (3), where0.0379 is the valueof RCB for 1958,the baseyear for the priceindexes. Once the consumptionvariable is defined,the analysiscan move on to the consumptionfunction itself. Real consumption,CON, is a functionof real disposableincome, YD, and of real householdnet worth, VCN. Real disposableincome and real net worthare obtainedby deflatingthe corre- spondingcurrent-dollar magnitudes by PCON. With all items in current-dollarterms, the model's disposableincome, YD$, equals:(1) personalincome; plus (2) the grossyield on the stock of consumerdurables, YCD$ + WCD$; less (3) federal, state, and local WilliamPoole 215 personalincome tax liabilities;less (4) federalestate and gift taxes; less (5) interestpaid by consumers. The householdnet worthvariable, VCN$, involvesa numberof separate items, many of which are subjectto substantialmeasurement problems. The attempt is to measurethe market value of household net worth. Basically,the variableincludes the net financialassets plus tangibleassets of householdsand of noncorporatebusiness, both farm and nonfarm. Tangibleassets include the value of (1) the inventoriesof noncorporate businesses;(2) the stock of consumerdurables and of noncorporatebusi- ness plant and equipment;and (3) farm land and structures,nonfarm residentialland and structures(excluding nonfarm corporate residential structures),and nonfarmnonresidential land. Most of the cyclicalfluctua- tion in VCN$ is causedby stock marketfluctuations. The consumptionfunction is estimatedusing real per capitamagnitudes. A twelve-quarterdistributed lag is used for disposableincome, and a four- quarterdistributed lag for householdnet worth.The distributedlags are constrainedto lie on a seconddegree polynomial and the interceptto equal zero. Estimationis by ordinaryleast squaresusing a rho transformationto eliminateserial correlation of residuals.After the coefficients,the R2, the standarderror of estimate,and the Durbin-Watsonstatistic have beenesti- mated,the post-estimationpredictions are calculatedwith rho set equal to zero.The estimationand prediction periods are 1954:1through 1967:4 and 1968:1 through 1971:4, respectively. Theregression statistics and post-estimation performance of the standard consumptionequation are shown in the column labeled "Standardequa- tion" in Table 1. Onlythe sumsof the lag coefficientson disposableincome