Indeterminacy with non-separable utility

Testo completo

(1),N+, NWV,+5WAv W5AWANA, #,+A,A 6 ,W5. ,NW `Lh!?} @Tih , L bb%

(2) e. W?_i|ih4?@?U) | L?5iT@h@M*i N|*|) -NtB,Aa w jAAj|| @?_ -N}ji . 8Bi6ji. #W 6W,5wc 5 #,W E6W.

(3) ** h}|t hitihi_ L T@h| Lu |t T@Tih 4@) Mi hiThL_Ui_ ? @?) uLh4 |L| Tih4ttL? Lu |i @|Lh. U bbb +w i??i|| @?_ +, 6@h4ih h?|i_ ? W|@*) ? U|LMih bbb ,hLTi@? N?iht|) W?t|||i @_@ 6itL*@?@ WDffS 5@? #L4i?UL E6W W|@*).

(4) IndeterminacywithN on-SeparableU tility1 R osalind L . B ennett FD IC, R oom 209 7 550-17 th Street W ashington, D C 20429 rbennett@ fdic.gov R ogerE. A . Farmer2 D epartmentofEconomics: U CL A 405 H ilgard A venue L os A ngeles CA 9 0024 rfarmer@ econ.ucla.edu N ovember19 9 7 this version M ay 19 9 8 1W. ewishtothankM ichaelW oodfordforclarifyingforustheoriginsoftheterm “Frisch demand”. Jess B enhabib, Jang-T ing G uo, Sharon H arrison, Stephanie Schmidt-G rohé, A martyaL ahiri, M artinU ribeandM arkW ederhaveallindirectly contributedtotheideasinthis paperandwethankthem withoutimplicatingthem in anymistakes thatmightremain. W ealsothanktwoanonymous referees ofthis journalfor their very thorough reading ofthe …rst draft of this paper and for providingsuggestions forthathavesigni…cantly improved the…nalversion. 2 Farmer’s researchwas supportedbyN SF grant# SB R 9 515036andbyagrant from the A cademicSenateoftheU niversity ofCaliforniaatL os A ngeles..

(5) A bstract B enhabib and Farmer[3]showed thata single sectorgrowth modelin the presenceofincreasingreturns-to-scalemaydisplayanindeterminateequilibrium ifthe demand and supply curves cross with the “wrongslopes”. W e generalize theirresultto a modelwith preferences thatare non-separable in consumption and leisure. W e provide a simple analogofthe B enhabibFarmerconditionthatworksinthenon-separablecase. O urconditioniseasy to check in practice and itallows forequilibria to be indeterminate, even when demand and supply curves have the standard slopes. W e illustrate thatequilibrium canbeindeterminatewhendemandandsupplycurves have standard slopes and the degree ofincreasingreturns-to-scale is wellwithin recentestimates byB asu and Fernald [2]forU .S. manufacturing..

(6) 1. Introduction. In this paperwe study the conditions underwhich a single sectorgrowth modelwith increasingreturns-to-scalewilldisplayanindeterminateequilibrium. Inarecentpaper, B enhabib andFarmer[3]showedthatthecondition forequilibrium tobe indeterminate in the one sectormodelwith separable preferences is thatthe laborsupply and demand curves should cross with the “wrongslopes”. O urworkgeneralizes theirmodeltothe case in which thepreferences oftherepresentativeagentarenon-separableinconsumption and leisure. T he B enhabib-Farmercondition is intuitive and can be applied in practice tocalibrate indeterminate models ortoprovide an econometrictestof indeterminacy in astructuraleconometricmodel. B utthe assumption that utility is separable in consumption and leisure is restrictive since itimplies thattheintertemporalelasticityofsubstitution mustequalone. O urgeneralization can be applied in practice tostudy the existence ofindeterminacy in a much largerclass ofmodels. A rguably, the class thatwe study is the one mostrelevanttobusiness cycleanalysis with representative agentmodels since itis the largestclass ofgrowth models thatis consistentwith the stylizedfactthathours workedintheU .S. havebeenstationaryeventhough therealwagehas grown. B enhabibandFarmerde…nedthelaborsupplycurvetobethequantityof laborsuppliedas afunctionoftherealwage, holdingconstantconsumption. W hen preferences are separable in consumption and leisure, the constantconsumptionlaborsupplycurveis identicaltoasecondwidelyusedconcept; theFrisch laborsupplycurve, de…ned as thequantityoflaborsupplied as a function ofthe realwage holdingconstantthemarginalutility ofconsumption.1 W hen preferences are non-separable, theconstant-consumption labor supply curve is di¤ erentfrom the Frisch laborsupply curve since holding constantthemarginalutilityofconsumptionis notthesameas holdingconstantthelevelofconsumption. W e…nd thatin amodelwith non-separable preferences, theappropriateconditionfortheindeterminacyofequilibrium is 1. O urusageof“Frisch” demandandsupplyfunctions follows B rowning[6]andB rowning, D eaton and Irish [7 ]who introduce the de…nition of a Frisch demand to refer to demands inwhichpreferences areintertemporallyseparableandthedemandfunctions for contemporaneous commodities areexpressedas afunctionofcurrentprices andoftheL agrange multiplierassociated with an intertemporalbudgetconstraint. B rowningD eaton and Irish citeFrisch [11]as theirsource forthe term.. 1.

(7) thattheFrisch laborsupplycurveand thelabordemand curveshould cross with the“wrongslopes”.. 2. R elated L iterature. T heB enhabib-Farmerconditionforindeterminacyhas beenwidelycriticized as beingimplausible(see, forexample, thediscussion byA iyagari [1] ), since the required degree ofreturns-to-scale is higherthan seems consistentwith recentestimates. B asu and Fernald [2] , forexample …nd thatthe returnsto-scale parameterin U .S. manufacturing is not much above unity. T his observation has led anumberofauthors tostudy alternative approaches in which indeterminacycan beobtained moreeasily. B enhabib and Farmer[4] …ndthatindeterminacyinmulti-sectormodelsdoesnotrequireahighdegree ofincreasingreturns-to-scale, andP erli [14]isabletogenerateindeterminacy forreasonableparametervalues in amodelofhomeproduction. In arecent paper, B enhabib and N ishimura [5]showthatindeterminacy can arise in a multi-sectormodelwithconstantreturns-to-scaleevenwhenexternalitiesare arbitrarilysmall. P elloni and W aldmann [13]generate indeterminacy in an endogenous growth model, with assumptions similarto ours. T heirexample is a limitingcaseofourmodelin which thesocialtechnologyis linearin capital. In this casetheequilibrium dynamics canbereduced toa…rstorderdi¤ erence equationinasinglestatevariable. Inourmodel, incontrast, thedescription ofequilibrium dynamics requires twostatevariables, as in thestandard real business cycle model. W e showthat, ifone allows fornon-separable preferences, equilibria may be indeterminate forreturns-to-scale of1.03. O ur lowerdegree ofreturns-to-scale is in agreementwith recentpointestimates by B asu and Fernald and itallows forindeterminacy to occureven when the demand curve slopes down and the constantconsumption supply curve slopes up. T he P elloni-W aldmann endogenous growth example is alsoconsistentwith standard sloped demand and supplycurves, butitrequires that returns-to-scale be equalto1.66 , when the laborelasticity is calibrated to equallabor’s shareofnationalincome. T his is unrealisticallyhigh.. 2.

(8) 3 P roduction T echnology O urtechnologyis taken directlyfrom B enhabib and Farmer[3] . W eassume alarge numberofcompetitive …rms, each ofwhich produces ahomogenous commodityusingaconstantreturns-to-scaletechnology. Y = K a L bA. (1). wherea + b = 1 andA > 0 . Each…rm takestheaggregateproductivityshock A as given. H owever, A is determined in practice by the activity ofother …rms. W emodelexternalities bytheequation A = K¹ ® ¡a L¹¯ ¡b: (2) where K¹ and L¹ denote average economy wide use ofcapitaland laborand, 1 > ® > a; ¯ > b; and ® + ¯ > 1. W elimitourselves tothecasewhen® < 1 sincewhen® = 1 thedynamicsofthemodelbecomeonedimensional. T hisis thecaseofendogenousgrowthalreadystudiedbyP elloni andW aldmann[13] . In thecaseof® > 1 growthis explosiveand werulethis outbyassumption. Substitutingfrom (2) into(1) leadstoanexpressionforthesocialproduction function: Y = K ® L¯: (3) W e assume thatfactormarkets are competitive and thatthe factors of production receive…xed shares ofnationalincome; wL b= ; (4) Y rK ; (5) a= Y where w is the wage rate, and r is the rentalrate, both measured in terms ofthe consumption good. Factorshares in nationalincome, a and b, will di¤ erfrom the socialmarginalproducts, ® and ¯ , due to the existence of externalities in production.. 4 T he Consumer’s P roblem In this section, we describe the preferences ofthe representative consumer. W eassumethatconsumers deriveutilityfrom theinstantaneous utilityfunction, [C V (L)]1¡¾ ¡1 U (C ;L) = ; (6) 1 ¡¾ 3.

(9) where ¾ > 0 ; ¾ 6 = 1; and V (L) is a non-n tive, strictly decreasingcon£ega¤ ¹ cave function, bounded above, thatmaps 0 ; L ! <: W ealsoassumethat V 0(0 ) is bounded and V (0 ) > 0 . L¹ has theinterpretation oftheconsumer’s endowmentofleisure and we allowforthe possibility (since certain simple examples have this feature) that L¹ = 1 : T he function U (C ;L) displays a constantintertemporalelasticity ofsubstitution and generalizes the utility function used byB enhabib and Farmer[3] ; U (C ;L) = lnC ¡. L 1+ ° ; 1+ °. (7 ). whilstmaintainingthepropertythatincomeandsubstitutione¤ ects exactly balanceeachotherin thelaborsupplyequation. T his propertyis important sinceitcaptures thefactthatlaborsupplyperperson is approximatelystationary in the U .S. although the realwage has grown atan average rate of 1.6 % peryearin acentury ofdata. W hen the consumerhas unitelasticity ofintertemporalsubstitution the parameter, ¾, is one. In this case, ifthe function V (L) is given by; µ ¶ L 1+ ° V (L) = exp ¡ ; 1+ ° our utility function can be shown (using L ’H ospital’s R ule) to reduce to equation (7 ). T herepresentativeconsumermaximizes thepresentvalueofutility Z1 U (C ;L)e ¡½td t 0. subjecttothebudgetconstraint: K_ = (r ¡±)K + w L ¡C ; theinitialcondition K (0 ) = K 0 ; and the“noP onzi scheme” constraint: Z1 Q (s;0 )(C (s) ¡w (s)L (s)) ·K (0 ) s= 0. 4.

(10) where½ > 0 is thediscountrate, 0 < ± < 1 is thedepreciation rate, and Zs Q (s;t) = e ¡r(v)+ ±d v v= t. is thepriceofaunitofconsumptionatdatetfordeliveryatdates. Sincethe individualproducers faceconstantreturns-to-scaletechnologies, thereareno pro…ts in this economy.. 5. SolvingtheConsumer’sP roblem andFindingaM arketEquilibrium. T osolvetheconsumer’s problem wede…nethepresentvalueH amiltonian: [C V (L)]1¡¾ ¡1 H = + ¤ [(r ¡±)K + w L ¡C ] 1 ¡¾. (8 ). ¤ = C ¡¾ V (L)1¡¾. (9 ). where¤ is theco-statevariable. T he…rstordercondition with respecttoconsumption is:. and with respecttolaborsupplyis: w ¤ = ¡C 1¡¾. V 0(L) : V (L)¾. (10). Substituting(9 ) into(10) and usingthe factthatthe wage equals the marginalproductoflaborleads tothestaticcondition: Y V 0(L) b = w = ¡C ; L V (L). (11). whichistherequirementthatthenegativeoftheratioofthemarginaldisutilityoflaborsupplytothemarginalutilityofconsumptionshouldbeequated totherealwage. A long an optimalpath, the shadowvalue ofcapital, ¤ mustobey the di¤ erentialequation: aY ¤_= ¤ (½ + ± ¡ ); (12) K 5.

(11) where we have used the …rm’s optimizingcondition (5) in equation (12) to write the rentalrate as a function ofcapitaland labor. T he transversality condition associated with this problem is represented bytheequation: lim e ¡½t¤ = 0 :. t! 1. (13). T oanalyze the dynamics ofa competitive equilibrium, the followingtransformations make the analysis easier. First, we divide the capitalformation equation bythelevelofcapital. K_ Y C = ¡± ¡ : K K K D e…ninglowercaseletters, ¸ ;l;k;candy tobelogarithms oftheirrespective uppercasecharacters, theco-stateequation becomes: ¸_= ½ + ± ¡ae y¡k;. (14). and thecapitalaccumulation equation is: _ e y¡k ¡± ¡e c¡k: k=. (15). W e would like toanalyze the stability ofthis pairofdi¤ erentialequations around a steady state. T odothis, we must…rst…nd the steady state then obtain expressions fory and cin terms ofthevariables ¸ and k.. 6 Expressions forthe SteadyState In this section weusethefactthat: V 0(L) h (L) ´¡ V (L) is monotonically increasing, to show that the model has a unique steady state. M onotonicity ofh (L) follows from the assumptions thatV 0(L) > 0 and V 00(L) < 0 since: £ 0 2 ¤ 00 V (L) ¡V (L)V (L) h 0(L) = > 0: V (L)2 6.

(12) N otice alsothatsinceV 0(0 ) is bounded and V (0 ) > 0 ; h (0 ) is positiveand …nite. W e willuse this property below to establish existence ofa unique steadystatevalueL ¤. ¤ ¤ ¤ W edenotethelogarithmsofsteadystatevariablesfY ;C ;K ;Lg asfy¤;c ;k ;l g. T oshowuniqueness, …rstchoose ¸_= 0 , and solve equation (14) to…nd an expression fory¤ ¡k¤ : µ ¶ ½+ ± ¤ ¤ y ¡k = ln (16) a _ 0 , and usingequation (16), solveequation (15) togive Similarly, settingk= ¤ an expression forc ¡k¤ : µ ¶ ½ + ±(1 ¡a) ¤ ¤ c ¡k = ln : (17 ) a ³ ´ ± ¤ Itfollows from (16 ) and(17 ) thaty¤¡c = ln ½+ ½+ > 0 canbeuniquely ±(1¡a) determined. From thelabormarketequation (11) wehave: ¤ l¤ + ln(h (L ¤)) = log(b) + (y¤ ¡c );. (18 ). L etf (L ¤) ´log(L ¤) + ln(h (L ¤)):Sinceh (0 ) is …nite f (0 ) = ¡1 :Sinceh is increasingf (L ¤) is increasingandf (L ¤) ! 1 as L ¤ ! 1 . Itfollows that thereis oneand onlyonepositivevalueofL ¤£forw hich (18) holds. SinceL ¤ ¤ is bounded, there is some L¹ forwhich L ¤ 2 0 ; L¹ and hence forany choice ofutility function there is an upperbound on laborsupply forwhich the equilibrium is interior. G iven the valueofL ¤ onecan compute l¤ and given ¤ thevalues ofy¤ ¡k¤ and y¤ ¡c onecan usetheproduction function (3) to ¤ solvefortheindividualvariables y¤;k¤; and c :. 7. D ynamicEquilibria A nequilibrium isatimepathforthestatevariablekandthecostatevariable ¸ thatsatis…es thesystem: ¸_ = _ = k. ½ + ± ¡ae y¡k e y¡k ¡± ¡e c¡k 7. (19 ) (20).

(13) with the boundary condition k(0 ) = k0 ; and the transversality condition lim e ¸¡½t = 0 ;togetherwith asetoftime paths forthe variables c ;l and y t! 1 thatsatisfythesideconditions y = c+ log(h (L)) = ¸ =. ® k+ ¯ l log(b) + y¡l; ¡¾ c+ (1 ¡¾ )log(V (L)):. (21) (22) (23). Equation(21) is theproductionfunction, (22) is thelabormarket…rstorder condition and (23) is the …rst order condition forchoice of consumption. Equations (21), (22) and (23) can bewritten as asetofapproximate linear equations byde…ningtheparameters µ ¶ ¡L ¤V 0(L ¤) Ã = ; (24) V (L ¤) L ¤h 0(L ¤) (25) ° = h (L ¤) toyield theequations: y ~ = (1 + °) ~l = ¸~ =. ~ ¯ ~l; ® k+ y ~ ¡~ c ¡¾ c ~ + Ã (¾ ¡1) ~l;. (26) (27 ) (28 ). wheretilde’s denotedeviations from thesteadystate. T he parameters Ã and °have relatively simple interpretations. From equation(11) itfollows thatÃ is theshareofwages relativetoconsumption since, ¡L ¤V 0(L ¤) w L ¤ = : V (L ¤) C Ifwecombinegovernmentandprivateconsumption, theratioofwageincome toconsumptionhas beenapproximately1 since189 0 inU .S. data. Itfollows thatÃ is approximately equalto1:T he parameter°can alsobe recovered from data. Ifone linearizes (11) around the steady state, then °would be theslopeoftheconstantconsumption laborsupplycurve.. 8.

(14) 8. L ocalD ynamics. In this section, we analyze the local dynamics of the system around the uniquesteadystate. W ehavedescribedtheeconomybyapairofdi¤ erential equations, in ¸ and k; (19 ) and (20) and by three static equations in the variables ¸ , l, y, and c . N otice …rstthatthe dynamic equations (19 ) and (20) are functions of(y¡k) and (c¡k):O ur…rsttaskis toshowthatthe three staticequations (26), (27 ) and (28) can be solved to…nd expressions for(y¡k) and (c¡k) as functions of¸ and k:W e …nd exactexpressions forthesevariables intheappendixinwhichwederiveequation(29 ) and…nd explicitparametricexpressions fortheelements ofthematrix© : · ¸ · ¸ ~ y ~ ¡k ¸~ = © ~ : (29 ) ~ c ~ ¡k k U singthenotation:. ·. Á1 Á2 © ´ Á3 Á4. ¸. we can write the dynamics of this system around the steady state as an approximatelinearsystem: · ¸ · ¸ ¸_ ¸~ = J ~ ; (30) _ k k whereweshowin theappendixthattheelements ofJ can bewritten as: ¡½+ ± ¢ ¡½+ ± ¢ · ¸ ¡aÁ ¡aÁ 1 ¢ 2 ¢ a a ¡ ¡ £ ¤ ¢ ¡ ¡ £ ¤ ¢ J´ : (31) Á 1 ½+a ± ¡Á 3 ½+a ± ¡± Á 2 ½+a ± ¡Á 4 ½+a ± ¡±. Sinceonevariableofthispairofequationsispredetermined, andtheother is free, thesystem willhavealocallyunique(determinate) equilibrium when thesteadystateis asaddle;this requires onenegativerootand onepositive rootofthe matrix J. Indeterminacy ofequilibrium occurs when both roots ofJ are negative. Since the trace ofJ is equaltothe sum ofthe roots and thedeterminantis equaltotheirproduct, determinacywould require: T r (J) 70 ; Det(J) < 0 ; and indeterminacythat T r (J) < 0 ; Det(J) > 0 : 9.

(15) T o characterize the conditions when indeterminacy occurs, as functions of theparameters ofthemodelweestablish thefollowingtworesults: P roposition1 sign (Det(J)) = sign (´) where ´ ´¾ (¯ ¡°¡1) ¡Ã (¾ ¡1): P roposition2 Tr (J) = ½ + Q ;. (32). where · µ µ ¶¶ ¸ (½ + ±) ±® Q ´¡ (¾ ¡1) ¯ ¡Ã 1 ¡ + d (1 + °)¾ ´ ½+ ± and d ´. (® ¡a) : a. P roofs ofbothresults aregivenintheappendix. Inthecasewhen¾ = 1, ourmodelcollapses to the B enhabib-Farmer[3]modeland in this case ´ collapses to¯ ¡1 ¡°andthedeterminantofJ is positivewhen¯ > 1 + °as in B enhabib and Farmer. N otice alsothatTr (J) is positivewhen thereare noexternalities and ¾ = 1 since, in this case, T r (J) = ½. Forsmallcapital externalities, however, the trace ofJ becomes negative as soon as ´ passes throughzero, from asmallnegativenumbertoasmallpositivenumbersince, when ¾ = 1, (½ + ±) Q = ¡ d (1 + °): (33) ´ If´ is small(close tozero) and positive then Q is large and negative and from proposition 2 itfollows thatthe trace condition forindeterminacy is met. H ence, when1 > ® > a and¾ = 1, anecessaryandsu¢cientcondition for indeterminacy is that there exists a value ´¤ at which the trace ofJ switches sign. In thecaseof¾ = 1 indeterminacyoccurs when 0 < ´ < ´ ¤ = (½ + ±)½¡1d (1 + °): T heconditions forindeterminacywhen ¾ is notequalto1 arethat: 10.

(16) ³ ³ ´´ 1. d (1 + °)¾ + (¾ ¡1) ¯ ¡Ã 1 ¡ ½+±®± > 0 ; and. 2. ´ > 0 :. In this caseindeterminacyoccurs forvalues of´ in therange: · µ µ ¶¶¸ ±® (½ + ±) ¤ d (1 + °)¾ + (¾ ¡1) ¯ ¡Ã 1 ¡ : 0 < ´< ´ = ½ ½+ ± T he reason forthe condition that´ should be positive is obvious since itimplies thatthe determinantofJ is positive. Condition 1 is su¢cientto imply a negative trace atthe pointwhen ´ crosses 0 from above, since at this point´ is smallandpositiveand itfollows from equation (33) thatQ is largeand negative, hencethetraceofJ is negative. Condition 1 is satis…ed when ¾ = 1 for positive capital externalities (d > 0 ) and, by continuity forvalues of¾ close to one. In computational experiments we were able togenerate examples ofindeterminate equilibria forvalues of¾ rangingfrom 0 to2 althoughvalues of¾ greaterthan1 make it harderto generate an indeterminate equilibrium, since when (¾ ¡1) is strictlypositive, ¯ mustbelargerthan would otherwisebethecasefor´ to switchsign. Inourcalibratedexamples weeasilyobtainedindeterminacyfor ¾ a little lowerthan 1 and ¯ notmuch biggerthan b. In ourcalibrations, (¯ ¡Ã (1 ¡±® = (½ + ±))) was typicallynegativeand soboth terms ofcondition1 werepositiveatthepointwhere´ changedsign. W eshowbelowthat condition ??is satis…ed when theslopes ofthelabordemand curveand the Frisch laborsupplycurvecross with the“wrongslopes”.. 9. T he Case ofEndogenous G rowth. O urresults onindeterminacyarerelatedtotheendogenous growthmodelof P elloni and W aldmann [13]whostudy the case ofaproduction function in whichthereis acapitalexternality, butnolaborexternality. T hetechnology studied byP elloni and W aldmann is ¡ ¢ Y = F K¹ L;K where K and L are private inputs ofcapitaland laborand K¹ is a capital externality. F (X ;Y ) is constantreturns-to-scaletechnology thatis linearly homogenous in X and Y . W hen F is Cobb-D ouglas, this structure is the 11.

(17) limiting case ofourmodelfor® = 1 and ¯ = b (no laborexternalities). SinceP elloni andW aldmann donotimposetheassumptionthatF is CobbD ouglas theyareabletoinvestigatetheroleoftheelasticityofsubstitution between laborand capitalin production on indeterminacy ofthe balanced growthpathas wellas consideringtheroleoftheelasticityofsubstitutionof consumption and leisurein utility. H owdoes this modeldi¤ erfrom ours?First, theequilibriaoftheP elloniW aldmann modelare balanced growth paths that can be described by a di¤ erence equation in a single state variable. B enhabib and Farmer[3]in theiroriginalpaperallowed forthis case;we have ruled itoutby assuming that® < 1.2 T heendogenous growthversionofthemodelwilltypicallyhave multiple balanced growth paths, in contrastwith ourmodelin which the steadystateequilibrium is unique. P elloni and W aldmann considerthecase with nolaborexternalities, and they are able toprove thatindeterminacy occurs around any given balanced growth path when ¾ < 1 provided the production function is concave enough. “Concave enough” means thatthe production function in intensive form has alargenegativesecond derivative and it is equivalent to the assumption that capital and labor are strong compliments.3 A lthough the balanced growth version ofthe modelis interesting, the magnitudeofthecapitalexternalitiesthatarerequiredforendogenousgrowth are extreme. Ifone calibrates the private production function usingfactor shares of1=3 to capitaland 2 =3 to laborthe aggregate technology in the P elloni W aldmann version of the modelwould have increasing returns to the socialproduction function of5=3:W e thinkthatthis is empirically implausible. T he assumption thatlaborand consumption are non-separable is however, plausible, andthereis considerableeconometricevidenceagainst theassumptionoflogarithmicutilityoverconsumption. Forthisreasonalone itis worthstudyingthemodelwithsmallcapitalandlaborexternalities.4 In 2. G eneralizingourmodeltoconsider® = 1 would lengthen ourpaperand add little. In ourmodel, with laborexternalities, indeterminacy can occureitherfor¾ < 1 or for¾ ¸ 1 although itis stilltrue thatindeterminacy is more likely forthe case oflow ¾. W e restrictourselves to a Cobb-D ouglas technology because we hope to show that indeterminacy can arise in models that are calibrated in a way that can be compared directly with standard realbusiness cycle economies, mostofwhich uses aCobb-D ouglas production technology. T he P elloni-W aldmann results suggestthatindeterminacy would be more likely iftechnology were calibrated as a CES production function with inputs thatarecompliments ratherthan substitutes. 4 In related econometric work, Farmerand O hanian [10]have estimated a structural 3. 12.

(18) related work, R obertoP erli [14]has shown thatamodelwith homeproduction can generate indeterminacy with a lowdegree ofreturns-to-scale and laborsupplycurves with“standardslopes”. P erli’s workis essentiallyatwosectormodelinwhichonesectorproduces anon-marketedgood. O urresults are generated in the standard one-sectormodeland forthis reason they are ofindependentinterest.. 10. T he L aborM arketand Indeterminacy. T he indeterminacy condition ofB enhabib and Farmer(thatlabordemand mustslopeup moresteeplythan laborsupply) has been widelycriticized as empiricallyimplausible. (Seeforexample, thediscussionbyA iyagari [1] ). In this section, we showthatthis counterintuitive resultis notnecessary for indeterminacyin themoregeneralcaseofnon-separablepreferences and we relate ourconditions forindeterminacy to the slopes oflabordemand and supply curves. O urmain resultis thatthe B enhabib-Farmer[3]condition thatlabordemandandsupplycurves cross withthe“wrongslopes” generalizes tothenon-separablecase: butthecorrectconceptoflaborsupplyis the Frisch laborsupply curve;de…ned as laborsupply as a function ofthe real wageholdingconstantthemarginalutilityofconsumption.. 10.1. Separable U tility. W hen ¾ = 1 the utility function is logarithmicand the determinantofJ is positivewhen ¯ ¡1 > °. InthiscasetheFrischlaborsupplycurveandtheconstant-consumptionlabor supplycurveareidenticaland given by: ln(w ) = c+ °l; and thelabordemand curveis ln(w ) = constant+ ® k+ (¯ ¡1)l. modeloftheU .S.economyandusedtheresultsthatwediscussinthispapertoinvestigate the hypothesis thatthatthe U .S. economy is welldescribed by a one sectormodelwith an indeterminate balanced growth path. In contrasttothe workby Farmerand G uo[9 ] , Farmerand O hanian [10]…nd evidence againstindeterminacy. T heirworkrelies in part on thegeneralized B enhabib-Farmercondition thatwe derivebelow.. 13.

(19) Since the slope ofthe laborsupply curve is °and the slope ofthe labor demand curve is ¯ ¡1; a necessary condition forindeterminacy is thatthe slope ofthelabordemand curveis largerthan theslopeofthelaborsupply curve.. 10.2. N on-Separable U tility. In the more generalcase when intertemporalsubstitution di¤ ers from one, thenecessary condition forindeterminacyis that´ > 0 which implies, rearrangingthede…nition of´, that: (¾ ¡1) ¯ ¡1 > Ã + °: ¾ InthiscasetheFrischlaborsupplycurveandtheconstant-consumptionlabor supplycurvedi¤ er. A linearapproximationtotheFrischlaborsupplycurve, in theneighborhood ofthesteadystate, is given by: µ ¶ 1 (¾ ¡1) ln(w ) = constant¡ ¸ + Ã + ° l: (34) ¾ ¾ Ifonesubstitutes for¸ from equation(28) intoequation(34) oneobtains the constant-consumption laborsupplycurve; ln(w ) = constant + c+ °l;. (35). whichisidentical(up toaconstant) totheseparablecase. T helabordemand curveis ln(w ) = constant+ ® k+ (¯ ¡1)l: (36). N oticethatingeneral, thenecessaryconditionforindeterminacythat´ > 0 , implies thatthelabordemandcurveandtheFrischlaborsupplycurvecross with the wrongslopes. Since the coe¢cientofl in the Frisch laborsupply curvedepends on thesign of(¾ ¡1); indeterminacymayoccurin themore generalmodelwhen the labordemand curve slopes down. T his may occur, forexample, if¾ < 1 and ¯ ¡1 (the slope oflabordemand) is negative but greaterthan(Ã (¾ ¡1)=¾ )+ °(theslopeofFrisch laborsupply). N otethat in this casetheFrisch laborsupplycurvewould slopedown also. Equation (35) slopes up forall°> 0 (a necessary condition forboth consumptionandleisuretobenormalgoods). Itfollowsthat, whenthemodel is generalizedtoallowfordi¤ eringdegrees ofintertemporalsubstitution, the labordemand curve and the constant-consumption laborsupply curve may returntotheirtraditionalslopesevenwhenthesteadystateisindeterminate. 14.

(20) P arameter ½ ¾ a m ° ±. V alue 0.06 5 1 0.3 1 0 0.10. D escription discountrate coe¢cientofrelativeriskaversion capitalshare returns toscale laborelasticity depreciation rate. T able1: B enchmarkCase. 11. A N umericalExample. In this section, wecomparethedynamicproperties ofthemodelforalternativeparametervalues. W ebegin with abenchmarkcasegiven in T able1, in which themodelhas separablepreferences and noexternalities. T hereturns-to-scaleparameter, m is relatedtoa and® bytheequations: ® = ¯ =. am bm. In the benchmark case, the trace ofthe Jacobian matrix is ½ which is positive, and indeterminacy cannotoccur. In T able 2, in contrast, we vary thedegreeofreturns toscalefrom 1 to1:9. W henm , reaches 1:4 3, thereis a bifurcation in thesystem from astablesaddlepointtoasink. A tthis point, themodeldisplaysanindeterminatesteadystateandiscapableofgenerating business‡uctuations drivenpurelybyanimalspiritsasintheworkofFarmer and G uo[8 ] . T oobtain complex roots (Farmerand G uoargue thatthis is required to mimic the U .S. data) the returns to scale parametermustbe increased stillfurtherto 1:4 8. B ecause recentempiricalestimates (see for exampletheworkbyB asu and Fernald[2] ) suggestthatan upperboundon the degree ofreturns toscale in U .S. manufacturing is 1:0 9, the separable case requires an implausibly high degree ofreturns-to-scale forthe data to beconsistentwith indeterminacy. In T able 3, we lookata case where utility is slightly di¤ erentfrom the separablelogcase;speci…callywelettheintertemporalsubstitutionparameterdrop from 1 to 0 :75. In this case we perform the same computational exercise and …nd thatthe system bifurcates from a stable saddlepointtoa sinkatamuchlowermagnitudeofreturns toscale, 1:0 3. T his is wellwithin 15.

(21) R eturns toScale 1 1.1 1.2 1.3 1.4 1.41 1.42 1.43 1.44 1.45 1.46 1.47 1.48 1.49 1.5 1.6 1.7 1.8 1.9. R oot1 0.45 -0.4017 -0.426 -0.4657 -0.55 -0.5649 -0.58 24 -0.6 032 -0.629 -0.6 623 -0.7 087 -0.7 8 43 -1.06 7 5+ 0.0848i -0.9 07 6+ 0.36 21i -0.7 9 25+ 0.4344i -0.38 + 0.4211i -0.27 14+ 0.3432i -0.2213+ 0.28 7 i -0.19 25+ 0.2443i. R oot2 -0.385 0.5384 0.6 97 2 1.08 07 3.9 15 5.8338 12.19 7 4 -7 0.28 18 -8 .381 -4.2227 -2.6 7 6 3 -1.8248 -1.06 7 5-0.0848 i -0.9 07 6 -0.36 21i -0.7 9 25-0.4344i -0.38-0.4211i -0.27 14-0.3432i -0.2213-0.287 i -0.19 25-0.2443i. D ynamics saddlepath saddlepath saddlepath saddlepath saddlepath saddlepath saddlepath sink sink sink sink sink sink sink sink sink sink sink sink. T able2: V aries R eturns toScale, B enchmarkCase. 16.

(22) R eturns toScale 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9. R oot1 -2.1306 -2.8655 -7 .8212 -0.2535+ 3.316 i -0.146 4+ 2.26 8 1i -0.1136+ 1.829 4i -0.09 7 7 + 1.57 29 i -0.0883+ 1.39 9 5i -0.0821+ 1.27 21i -0.07 7 6+ 1.17 33i -0.07 44+ 1.09 38i -0.06 17 + 0.7 08 7 i -0.0581+ 0.5532i -0.056 5+ 0.46 19 i -0.0555+ 0.39 9 2i -0.0548 + 0.3522i -0.0544+ 0.3148i -0.054+ 0.2839 i -0.0538 + 0.257 5i. R oot2 2.19 56 3.08 14 10.7 8 12 -0.2535-3.316i -0.1464-2.26 81i -0.1136 -1.8 29 4i -0.09 7 7 -1.57 29 i -0.088 3-1.39 9 5i -0.0821-1.27 21i -0.07 7 6 -1.17 33i -0.07 44-1.09 38 i -0.06 17 -0.7 087 i -0.058 1-0.5532i -0.0565-0.46 19 i -0.0555-0.39 9 2i -0.0548-0.3522i -0.0544-0.3148 i -0.054-0.2839 i -0.0538-0.257 5i. D ynamics saddlepath saddlepath saddlepath sink sink sink sink sink sink sink sink sink sink sink sink sink sink sink sink. T able3: V aries R eturns toScale, R iskA version (¾ = 0 :75) the empirically relevantrange accordingtothe estimates ofB asu and Fernald. W econcludethatbymodifyingtheutilityfunctiontoallowforvarying degrees ofintertemporalsubstitution, one is abletogenerate indeterminacy ata much lowermagnitude ofincreasingreturns than when the individual has logpreferences.. 12. D iscussion ofthe R esults. A lthough we have shown thatindeterminacy may be consistentwith a low degree ofreturns to scale; this does not imply thatthe one sectormodel amended in this way can be used to generate business cycles when driven purely by sunspots in the mannerdescribed by Farmerand G uo[8] . W hen labordemand and constantconsumption laborsupplycurves cross with the 17.

(23) conventionalslopes, purely sunspot driven business cycles willcause consumption and employmentto move countercyclically; in the data they are procyclical. T his is thesameissuediscussed by B enhabib and Farmer[4]in theirtwosectormodel. H owever, ourmodeldoes lead tothepossibilitythat indeterminacymayprovideanadditionaltransmissionmechanism forshocks originatingin therealsector. Itis alsoworth pointingoutthattogenerate indeterminacy with a low degree ofreturns-to-scale, we chose the curvature ofthe utility function to beonthelinearsideoflogarithmicpreferences (theparameter¾ was chosen tobesmaller than unity). T his contradicts thetypicalassumption in single sectormodels with constantlaborsupply thatthe curvature parameteris greater than unitytohelp explain the equitypremium puzzle. In defense of ourcalibration, thesemodels donotordinarilyallowlaborsupplytovary. It isnotclearhowoneshouldmap ournon-separableexampleintothefunction; U =. C 1¡¾ ; 1 ¡¾. thatis commonlystudied in this literature. Finally, recentworkbyL ahiri [12]onindeterminacyininternationalmodels …nds thatopen capitalmarkets make indeterminacy more likely. L ahiri points out thatopen capital markets break the link between savings and investment and permitindividuals to smooth consumption through international borrowing and lending, thereby making the representative agent behave in a more risk neutralmanner. O urwork on non-separabilities exploits asimilartheme;thecloseris ¾ tozero, theless averseis theconsumer to‡uctuations in consumption.. 13 Conclusion In this paper, we generalized the B enhabib-Farmercondition forindeterminacy to the case ofnon-separable preferences. O urcondition is simple to check in practice and itcovers a class ofutility functions thatis the most generalclass thatis consistentwith balanced growth. W e found that, once oneallowsfornon-separabilitiesbetweenconsumptionandleisure, indeterminacynolongerrequiresthatthedemandcurveandtheconstantconsumption supplycurve should cross with the wrongslopes. Instead, therequired condition is thatthe labordemand curve and the Frisch labor supply curve 18.

(24) should cross with non-standard slopes; a condition thatis simple to check in practice. B y means ofan example, we showed thatwhen the curvature parameterontheutilityfunctionis setat0 :75 incontrasttoavalueofunity thatwouldholdinthelogarithmiccase, indeterminacycanoccuratlevels of increasingreturns as lowas 1:0 3.. 14 A ppendix 14.1. P art1: D erivingthe Elements of©. T oderive the elements ofthe matrix © , we solve the staticequations (26 ), (27 ) and(28 ) for(y¡k) and(c¡k). W estartbyrearrangingequations (26) ~) and (~ ~) which leads and (28 ) as amatrixsystem in thevariables (~ y¡k c¡k totheexpression: · ¸· ¸ · ¸ · ¸· ¸ ~ 1 0 y ~ ¡k ¡¯ 0 1 ¡® ¸~ ~ + l+ (37 ) ~ ~ = 0 0 ¾ ¡Ã (¾ ¡1) 1 ¾ c ~ ¡k k. W ewritethelabormarketequilibrium condition (27 ) separatelyin terms of thesamelinearcombinations ofvariables: · ¸ ~ £ ¤ y ~ ¡k ~ ¡1 1 (38 ) ~ + (1 + °)l= 0 c ~ ¡k. N ow, dividethesecondrowof(37 ) by¾ and divideequation (38) by(1 + °) toobtain · ¸· ¸ · ¸ · ¸· ¸ ~ ¡¯ 1 0 y ~ ¡k 0 1 ¡® ¸~ ~l+ + (39 ) (¾ ¡1) 1 ~ ~ = 0 0 1 1 c ~ ¡k ¡ ¾ Ã k ¾ · ¸ ~ £ 1 ¤ y ~ ¡k 1 ~ ¡1+ ° 1+ ° (40) ~ + l= 0 c ~ ¡k Solveequation (40) for~l and substituteintoequation (39 ). #· ¸ · · ¸ " ¸· ¸ ¯ ~ ~ ¡1+¯ ° y ~ ¡k 0 1 ¡® ¸~ y ~ ¡k 1+ ° + + ¡Ã (¾¡1) Ã (¾ ¡1) 1 ~ ~ = 0 ~ 1 c ~ ¡k c ~ ¡k k ¾ ¾ (1+ °) ¾(1+ °) R earranging, " 1 ¡ 1+¯ ° ¡Ã (¾¡1) ¾(1+ °). 1+. ¯ 1+ ° Ã (¾ ¡1) ¾ (1+ °). #·. ¸ · ¸· ¸ ~ y ~ ¡k 0 1 ¡® ¸~ + 1 ~ ~ = 0 1 c ~ ¡k k ¾ 19.

(25) or A where. ·. A=. ¸ · ¸ ~ y ~ ¡k ¸~ + B ~ ~ = 0; c ~ ¡k k. ". 1 ¡ 1+¯ ° ¡Ã (¾¡1) ¾ (1+ °). B =. ·. 1+. ¯ 1+ ° Ã (¾¡1) ¾(1+ °). 1 ¡® 1. 0 1 ¾. ¸ ;. #. ;. (41). ~) and (~ ~) in terms of¸~ and k ~; Solvefor(~ y ¡k c¡k · ¸ · ¸ ~ y ~ ¡k ¸~ = © ~ ~ ; c ~ ¡k k where ¡1. © = ¡A B =. ·. Á1 Á2 Á3 Á4. ¸ ;. Invertingthe expression forA, (41) itfollows thatthe elements ofA¡1 are given by " # Ã (¾¡1) ¡¯ 1 + 1 ¾(1+ °) 1+ ° A¡1 = ; Ã (¾ ¡1) ¯ Det(A) 1 ¡ ¾(1+ °) 1+ ° wherethedeterminantofA is Det(A) =. ¾ (1 + °) + Ã (¾ ¡1) ¡¯ ¾ ¡´ ´ ; ¾ (1 + °) ¾ (1 + °). (42). and´ ´¾ (¯ ¡°¡1)¡Ã (¾ ¡1). N otethatthedeterminantofA is negative when ´ is positive. © is given by, " # (1¡® )Ã (¾¡1) ¯ ¯ ¡ 1 ¡® + ¡ ¾ (1 + ° ) ¾ (1+ ° ) ¾(1+ ° ) 1+ ° © = ¡A¡1B = (1¡® )Ã (¾¡1) 1+ °¡¯ ´ + 1 ¡ 1+¯ ° ¾(1+ °) ¾(1+ °) and theelements of© are. Á2 =. ¯ Á1 = ¡ ´. ¾ (1 ¡® )(1 + °) + (1 ¡® )Ã (¾ ¡1) ¡¯ ¾ ´ 20. (43) (44).

(26) 1 + °¡¯ ´ ¾ (1 + °) + (1 ¡® )(¾ ¡1)Ã ¡¯ ¾ Á4 = ´ Á3 =. 14.2. (45) (46). P art2: T he Elements ofJ. T o…nd theelements ofthematrixJ; weusethede…nitions ofÁ 1;Á 2 ;Á 3 and Á 4 , towriteequations (26) and (28) in thefollowingform: ~ Á 1¸~ + Á 2 k; ~ y ~ ¡k=. (47 ). ~ Á 3¸~ + Á 4 k ~ c ~ ¡k=. (48 ). N owsubstitutethesetwoequationsintothetwodynamicequationstoobtain theexpressions: ¸_ = _ = k. ~. ~. ½ + ± ¡ae Á 1 ¸+ Á 2 k+ µ0 ; e. ~ Á k ~ Á 1 ¸+ 2 + µ0. whereµ0 and µ1 areconstants µ0 = µ1 =. ¡± ¡e. ~ Á k ~ Á 3¸+ 4 + µ1. (49 ) ;. (50). y¤ ¡k¤; ¤ c ¡k¤;. thatdonotin‡uencethedynamics. L inearizingequations(49 ) and(50) leads tothesystem: · ¸ · ¸ ¸_ ¸~ = J (51) _ ~ : k k. where localinformation aboutthe dynamics ofthe system is contained in thematrixJ. T heelements ofJ aregiven bytheexpression: · ¸ ¡ ae µ0 Á 1 ¡ ae µ0 Á 2 ¡ ¢ ¡ ¢ J´ : Á 1 e µ0 ¡Á 3e µ1 Á 2 e µ0 ¡Á 4 e µ1 ¤ U sing the steady state solutions of µ0 = y¤ ¡k¤ and µ1 = c ¡k¤ from equations (16 ) and (17 ) wecan writethis expression as ¸ ¡½+ ± ¢ · ¡½+ ± ¢ ¡ aÁ ¡aÁ 2 1 a a ¡ ¢ ¡£ ¤ ¢ ¡ ¢ ¡£ ¤ ¢ : J´ Á 1 ½+a ± ¡Á 3 ½+a ± ¡± Á 2 ½+a ± ¡Á 4 ½+a ± ¡±. which is equation (31) in thetext.. 21.

(27) 14.3 P art3: P roofofP roposition1. W e prove, in this section, thatthe sign ofthe determinantofJ depends on the sign of´, a variable thatswitches sign when the labordemand curve and the Frisch labor demand curves cross with the “wrong slopes”. From equation(31) itfollows thatthedeterminantofJ is givenbytheexpression: ¶µ ¶ µ ½ + ± (1 ¡a) ½+ ± (52) Det(J) = a(Á 1 Á 4 ¡Á 2 Á 3) a a. T he term, (Á 1 Á 4 ¡Á 2 Á 3), is the determinantof© . Since a, (½ + ±=a) and (½ + ± (1 ¡a))=a are allpositive, the sign of the determinantofJ is the sameas thesign ofthedeterminantof© . sign (Det(J)) = sign(Á 1 Á 4 ¡Á 2 Á 3) = sign(Det(© )):. (53). W e nowshowthatthe determinantof© is related tothe slopes ofthe labordemandandsupplycurves throughtheterm ´:R ecallthat© is de…ned as: © = ¡A¡1B U singtheproperties ofthedeterminantofasquarematrix, Det(© ) = Det(¡A¡1)Det(B ); and sincethematrixA is ofdimension two: Det(© ) = (¡1)2 Det(A¡1 )Det(B ): T his implies that, Det(© ) =. Det(B ) : Det(A). T hedeterminantofB is. ® ¡1 ; ¾ andsinceweassume0 < ® < 1 and¾ > 0 , thedeterminantofB is negative. T herefore, thesign ofthedeterminantof© is theoppositeofthesignofthe determinantofA: Det(B ) =. sign (Det(J)) = sign(Det(© )) = ¡sign(DetA): 22.

(28) B utfrom de…nition ofDet(A); equation (42) itfollows that: sign(Det(A)) = ¡sign(´); thereforethesign ofthedeterminantofJ is thesameas thesign of´. sign(Det(J)) = sign(´): Q .E.D .. 14.4 P art4: P roofofP roposition2. From the de…nition ofthe elements ofJ in equation (31) we can write the traceofJ as: µ ¶ µ ¶ ½+ ± ½+ ± T r (J) = (Á 2 ¡Á 4 ) + Á 4 ± ¡aÁ 1 : (54) a a U singequations (44) and (46) notethat ¾ ® (1 + °) ; Á 2 ¡Á 4 = ¡ ´ and Á 4 = ¡1 ¡. ® Ã (¾ ¡1) : ´. U singtheseexpressions wecan rewrite(54) as follows: T r (J) =. (½+ ±)¯ ´. ¡. (½+ ±)¾® (1+ °) a´. X. ¡ ± ¡. Y. ±® (¾¡1)Ã ´. (55). Z. N owwritetheexpressions X , Y , and Z , as follows: X. =. Y. =. Z. =. ¯ (½ + ±)¾ ´ (½ + ±)¾ (1 + °) ¡ ´ Ã (¾ ¡1)(½ + ±) ¡ ´. ¡ ¡ + 23. ¯ (¾ ¡1)(½ + ±) ; ´ (® ¡a) (½ + ±)¾ (1 + °) ; a ´ µ ¶ Ã (¾ ¡1)(½ + ±) ±® 1¡ : ´ ½+ ±.

(29) Collectingtogetherthe …rstterms ofeach ofthe expressions forX , Y and Z and usingthede…nition of´ = ¾ (¯ ¡°¡1)¡Ã (¾ ¡1)wecan writethe sum oftheterms X , Y and Z as X + Y + Z = (½ + ±) + Q where. · µ µ ¶¶ ¸ (½ + ±) ±® Q ´¡ (¾ ¡1) ¯ ¡Ã 1 ¡ + d (1 + °)¾ ; ´ ½+ ±. and. (® ¡a) ; a is ameasure ofthe importanceofcapitalexternalities. SinceT r (J) = X + Y + Z ¡±, wecan writethetraceofJ as d ´. T r (J) = ½ + Q which is equation (32) in thetext. Q .E.D .. R eferences [1]A iyagari, S.R .“T heEconometrics ofIndeterminacy: anA ppliedStudy. A Comment,” Carnegie R ochesterSeries on P ublic P olicy, 43. (19 9 5), 27 3–282. [2]B asu, Susantoand John Fernald “R eturns toScalein U .S. P roduction: Estimates and Implications” JournalofP oliticalEconomy 105, no. 2 (19 9 7 ), 249 -28 3. [3]B enhabib, Jess and R ogerE.A . Farmer“Indeterminacy and Increasing R eturns” JournalofEconomicT heory 63 (19 9 4): 19 -41. [4]B enhabib, Jess andR ogerE.A . Farmer“IndeterminacyandSectorSpeci…cExternalities” JournalofM onetaryEconomics 37 (19 9 6 ) 421-443. [5]B enhabib, Jess and N ishimura, K., “Indeterminacy and Sunspots with ConstantR eturns,” C.V .StarrCenterforA ppliedEconomics, Economic research report9 6 -44, (19 9 6 ), N ewY orkU niversity. 24.

(30) [6 ]B rowning, M artin J. “P ro…t Function R epresentations for Consumer P references,” (19 82) B ristolU niversityD iscussion paperN o. 82/125. [7 ]B rowning, M artin J., A ngus D eaton and M argaretIrish. “A P ro…table A pproach to L abor Supply and Commodity D emands over the L ifeCycle”, Econometrica (19 85), V ol. 53, no. 3, 503–543. [8]Farmer, R ogerE.A .andJangT ingG uo.“R ealB usinessCycles andthe A nimalSpirits H ypothesis”, JournalofEconomic T heory, 6 3 (19 9 4), 42–7 2. [9 ]Farmer, R oger E. A . and Jang T ing G uo. “T he Econometrics of Indeterminacy: an A pplied Study”, Carnegie R ochesterSeries on P ublic P olicy, 43. (19 9 5), 225–27 2. [10]Farmer, R ogerE. A . and L ee O hanian “T he P references ofthe R epresentativeA merican” W orkingP aper, U CL A , M arch 19 9 8. [11]Frisch, R agnar. N ewM ethods ofM easuring M arginalU tility: T ubige: J.C.B . M ohr, (19 32). [12]L ahiri, A martya. “G rowthandEquilibrium Indeterminacy: T heR oleof CapitalM obility” W orkingP aper, U CL A , January19 9 8. [13]P elloni, A lessandraandR obertW aldmann, “IndeterminacyinaG rowth M odel with Elastic L abour Supply”, R ivista Internazioale di Scienze Sociali, (19 9 7 ) no. 3, pp. 202–211. [14]P erli, R oberto, “Indeterminacy, H omeP roductionandtheB usiness Cycle”, JournalofM onetaryEconomics, (19 9 8 ) vol41, no. 1, pp. 105–125.. 25.

(31)