Fishr s Equation: Som Mthodological Doubts Rajas Parchur Gokhal Institut of Politics and Economics, Pun 4 4, India. Th ida of a ral rat of intrst, as opposd to th nominal or mony rat of intrst, th rat at which ral world transactions tak plac, was introducd by Irving Fishr in his 93 classic, Th Thory of Intrst. Th objct of this distinction was to unvil th intrst phnomnon of its montary shroud and to invstigat th rality which was supposd to li bnath a rality in which thriftinss (tim prfrnc in Fishr s words) and productivity govrn th lnding and borrowing of ral capital to stablish a ral rat of intrst. Onc th ral rat is stablishd in this way thn, all ls rmaining th sam, montary forcs may b allowd to affct only th nominal rat. This conclusion appard to b consistnt with th doctrin of nutrality of mony as applid to financial markts and squard wll with Fishr s quantity thory of mony. In kping with this logic, Fishr first dfind th ral rat of intrst as, + R r = R p rp R p + p () with R bing th nominal rat and p th rat of inflation. Nxt, having dcidd that th ral objct r was dtrmind outsid th montary world, h chos to rwrit th idntity () in th form R r + p (2) This mannr of writing it is intndd to convy th ida that th dpndnt variabl is th nominal rat which, in som sns, must b xplaind by th ral rat and th inflation rat; that th lattr ar th building blocks that mak R what it is sn to b. Of cours th fact rmains that (2) is as much an idntity as () is. To convrt it into a bhavioural quation Fishr took th nxt stp of supposing that r would b dtrmind by ral forcs and that only th xpctd rat of inflation can hav any influnc on th nominal intrst rat. H thus obtaind th quation R r + = p (3) which has now bcom on of th most widly usd quations in conomics during th last fifty yars. 2 In Fishr s words, Th influnc of changs in th purchasing powr of mony on th mony rat of intrst will b diffrnt according to whthr or not that chang is forsn [Fishr 93, p. 45]. 2. Aftr dcads of mpirical inquiry, this quation has bcom a foundation ston of modrn macroconomic, montary and financial thoris. It is today a sacrd cow of not only th scinc of conomics but also of th art of cntral banking. I wish to
show that dspit its stablishd status, Fishr s quation stands on rathr shaky foundations and I lav it to th radrs to judg whthr I should b hangd, or only floggd, for this hrsy. Considr quation () which dfins an unobsrvabl ral intrst rat in trms of obsrvabl variabls, th nominal intrst rat and th rat of inflation. This quation has bn transformd into quation (3) making th obsrvd nominal intrst rat th sum of two unobsrvabl variabls, th ral intrst rat and th xpctd inflation rat. Thn quation (3) is rad in th following way: givn th ral rat as dtrmind by ral forcs that ar givn in th long-run, changs in th xpctd inflation rat ar simply transmittd into changs in th nominal intrst rat. Th suspicious thing about this is that th transmission mchanism is omittd; spcifically, th mannr in which th xpctd rat of inflation causs a chang in th dcisions of markt participants and thir transactions and thus causs a chang in th nominal intrst rat is nowhr articulatd. But surly that is what Fishr mans in his statmnt quotd arlir. Spcifically, it is by rprsnting th inflation rat in trms of xpctations that th idntity () is convrtd into th quation (3). Thus, considr th consqunc of allowing xpctations to affct th transactions of markt participants. Suppos popl xpct inflation to occur at a rat p in futur. Thy hav an amount of mony M which thy can invst in a dposit/bond that pays th nominal intrst rat R (no mattr whthr it contains p or not) or, altrnativly, to buy goods at thir currnt prics (in th sam wightag as thy appar in th pric indx) and sll thm latr. If thy do not hav accss to M, thy simply borrow mony and buy commoditis if R < p or sll commoditis and invst at R if R > p. This by itslf will caus a chang in th going markt prics of commoditis and bonds. Th trminal walth for th two invstmnts M ( + R) and M ( + p ) ar qualisd in quilibrium, so that R = p (4) and th ral intrst rat, which in this contxt shows th nt (arbitrag) rat of profit, bcoms zro. In othr words, Fishr s quation (3) dos not obtain if forsn inflation is allowd to affct portfolio dcisions. Incidntally, it may b rcalld that ths wr th prcis grounds on which Kyns (936, p. 42) took objction to Fishr s ida,. If it [inflation] is forsn, th prics of xisting goods will b forthwith so adjustd that th advantags of holding mony and of holding goods ar again qualizd. 3. Considring th important rol playd by Fishr s quation in modrn macroconomic thory, it may b subjctd to anothr tst. Considr a static conomy. Suppos it undrgos continuing inflation in th mony supply, wag rat and prics for a priod sufficintly long so as to mak it th xpctd inflation rat. Thn th nominal valus of th national incom for this conomy from any on yar to th nxt ar; Y = + 5(a) w L RK
Y = + 5(b) wl rk If th capital stock is rgardd as th valu of producd mans of production, Y Y w K = = = + p w K which on substitution in 5(a) and 5(b) rsults in, R = r (6) Equation (3) is not obtaind. Evn if w suppos that K rprsnts ral capital whos siz lik that of labour or of ral output dos not chang from yar to yar quation (5) stands rplacd by Y w L + RK = Y w L + rk = Y Thn substituting w = = ( + p) givs Y w R = r( + p) (7) i.., vn if th impact of inflation wr to b coaxd to pass through xclusivly into th nominal intrst rat th rsulting quation is not Fishr s quation! If, to mak on mor attmpt to find a suitabl macroconomic rol for Fishr s quation, w wr to substitut ( + r)( + p) in 5(a), th rsult is, p = which clarly contradicts th initial assumption that th conomy is undrgoing inflation. Nor dos substituting th approximation r + p for R hlp bcaus it simply rsults in r =, which is nonsnsical. Anothr way to mak th point is to writ th nominal national incom of th conomy undrgoing continuous uniform inflation for any givn yar as Y N = WL + [(+r) (+p) ] K whr W is th nominal wag rat. To obtain th ral incom dflat both sids by th pric factor so that
YN = + p Y R W + p L + = wl + rk + ( + r) = wl + rk + K + p p + p K K K + p Th last trm shows an rror in th masurmnt of th ral national incom. Th invitabl conclusion is that Fishr s quation is incompatibl with th national incom idntity! 4. Economtric studis of th inflation-intrst rlationship has bcom an industry in its own right alongsid such subjcts as tsting th CAPM, tsting for markt fficincy or tsting for th purchasing powr parity thory. Countlss paprs hav bn writtn on this subjct and although thir conclusions go this way and that, as is only to b xpctd whn th data handld diffrs across countris, conomic conditions, tim priods and dfinitions of th variabls, th consnsus sms to b that Fishr s quation passs th conomtric tsts of validity. Sinc a rviw of that litratur is byond th scop of this papr, and mor frankly, byond th ndowmnt of my conomtric capabilitis, I shall not attmpt it. I shall confin my commnt to on simpl tst of Fishr s quation, a tst whos multivarigatd gnralizations ar found in th litratur. Fishr s quation (3) is usually cast into th corrsponding conomtric form R = a + bp u (8) t t + whr Rt is a suitably chosn nominal rat, pt is xtractd from a suitably chosn pric indx by using on of many xpctations modls and th random trm ut is supposd to hav th customary proprtis. Thn, if a, th stimat of th ral rat r is found to b significantly positiv and b is found to b significantly clos to, th undrlying quation (3) is rgardd as valid. But what this procdur ignors is that quation (3) itslf is a numrical approximation with rp which is mad purly for convrsational convninc; th xact quation must includ th multiplicativ trm and ought to b rad as, = r + ( + r p 9(a) R ) with th conomtric form R = a + ( + a) p + u 9(b) t t t Sn against th backdrop of quation 9(a) th findings that a > which implis r > and b = which implis r = ar actually contradictory! And thr is no concivabl way of idntifying whthr th rgrssion of Rt on pt is on that prtains to th mathmatical form (3) or 9(a). I conclud that vn whn th data ar bst suitd to
tst Fishr s quation, it is impossibl to say from th rsults that Fishr quation is vrifid. 5. Th quation may b approachd from yt anothr viwpoint which prmits a mor dtaild xamination of th ffcts of inflation. Considr a firm that uss a stock of working capital K, flow inputs of C and labour worth W for its annual production which gnrats a sals rvnu S. If r is th rat of intrst, its annual activity is givn by rk + C + W = S If K=, C=5, W=5 and r=2 pr cnt and th firm is fully lvragd, its profit and loss account is as follows: Sals Rvnu 2 Lss Cost of Goods Sold Lss Intrst 2 Profit Th firm can continu its oprations from yar to yar simply by rnwing th loan. (Not that w ar assuming implicitly that 2 pr cnt is th normal rat of profit so th zro profit should b considrd as zro suprnormal profit) Nxt, suppos that thr is inflation at a pr cnt rat which in th first stag is confind to th output sold by th firm. Th profit and loss account will show Sals Rvnu 32 Lss Cost of Goods Sold Lss Intrst 2 Profit 2 Th xcss profit of 2 that th ownrs bgin to njoy is dirctly attributabl to th (opportunity) losss suffrd by th input supplirs and workrs which is worth and thos of lndrs which is worth two. Thr ar two notworthy points. Firstly, th rat of oprating profit (intrst plus profit) R can b xprssd as, R = ( + r)( + p) i..,.32 = (.2)(.) Kyns (936, p. 42) had mad a similar obsrvation whn criticizing Fishr, Th mistak lis in supposing that it is th rat of intrst on which changs in th valu of mony will dirctly ract, instad of th marginal fficincy of a givn stock of capital. Th rat R=.32 is not th nominal rat of intrst, it is th nominal rat of oprating profit, which has an obvious similarity with th marginal fficincy of capital. Scondly, if lndrs wr to actually charg th rat of intrst of 32 pr cnt in accordanc with Fishr s formula thy would ntirly rcovr what thy had lost to th ownrs;
Sals Rvnu 32 Lss Cost of Goods Sold Lss Intrst 32 Excss Profit This looks okay at first glanc but bgins to crat difficultis whn inflation is considrd to b conomy wid so that input prics and wags too inflat by pr cnt. Thn th position of th firm is as follows; Sals Rvnu 32 Lss Cost of Goods Sold Lss Intrst 32 Profit - And this lads to a furthr worsning whn inflation touchs th valus of stocks thmslvs whos rquirmnts would incras to with th rsult that th quantum of financing incrass to. Th pictur is, Sals Rvnu 32 Lss Cost of Goods Sold Lss Intrst 32.2 --- Profit -3.2 --- In othr words, if nominal intrst rats wr to chang according to Fishr s formula othrwis viabl firms in a zro inflationary condition ar drivn into insolvncy in a condition of inflation at a uniform rat. Not a vry convincing conclusion. Th mor likly scnarios would b as follows. Whn, in th first bout, inflation causs xcss profits to appar, two forcs will b activatd. Firstly, inflation will bgin to sprad through th systm of production considring that th inflatd sals rvnus of som firms ar costs of production for othr firms, and so on. This will caus th costs of production of all firms to ris which, in tim, will bgin to inflat th valus of th stocks and fixd capital itms K as thy com up for rplnishmnt and rplacmnt. Ths ffcts will work towards rducing th xcss profits. Scondly, th divrgnc btwn th rat of profit and th rat of intrst will caus th financial portfolios to b rshuffld away from dposits and bonds and towards quitis. Th rsult will ithr b an outflow of dposits or a slling prssur in th bond markt. If it is th formr, banks will rais th intrst rat offrd on dposits to bid thm back. Thy ar abl to do this by raising th intrst rat on loans which ntrprnurs do not mind paying du to th incrasd profitability. This causs intrst rats to ris. If it is th lattr bond prics will drop and bond yilds will ris. Ths ffcts xplain why Fishr s quation may b found to hold good in th shortrun. In tim, howvr, aftr inflation has sprad throughout th systm of production and xcss profits xprincd in th arly stags mov southwards, intrst rats too follow suit and in th nw quilibrium th profit and loss account will look as follows:
Sals Rvnu 32 Lss Cost of Goods Sold Lss Intrst 22 Profit Th intrst paymnt of 22 on an inflatd capital of onc again is a 2 pr cnt rat; th nominal and ral rat ar qual to on anothr and back at thir original lvl. Ndlss to say, continuing inflation at disuniform rats from yar to yar or at disuniform rats across classs of commoditis will kp both th production systm and th financial systm in a disturbd stat but that is outsid th scop of a discussion of Fishr s quation. 6. Fridman (956) was prhaps th first conomist to mak Fishr s quation a foundational building block of macroconomic modls. But th vrsion that Fridman advancs [Fridman (956), quation 9] of Fishr s quation is: r b = r + dp p dt whr r b is th rat of intrst on bonds and r, Fishr s ral rat, is mad quivalnt to th rat of rturn on, of all things, quity. But this is xactly th opposit of our finding in th xampl of th prvious sction viz. that it is th rat of oprating profit (including that on quitis) which incrass with inflation at th prvailing nominal rat of intrst. Morovr, th rturn on quity is th sol non-contractual pric in an conomy; all othr prics including th rat of intrst ar contractually fixd and vary by a rvision in th trms of th contract as th conomic forcs govrning thir dmands and supplis vary ovr tim. Th rturn on quity can bcom ngativ, somtims for sustaind priods of tim. Surly, it cannot stand for Fishr s ral rat, which must b strictly positiv on account of its bing dtrmind by th ral forcs of th marginal productivity of capital and a tim prfrnc for currnt ovr dfrrd consumption. So I think Fridman s vrsion of Fishr s quation is simply a misrprsntation. It is intuitivly far mor comfortabl to suppos that a variabl rat of inflation affcts, in th first plac, a variabl non-contractual rturn on quity with th contractd rat bing tmporarily fixd. To my knowldg Fridman s vrsion of Fishr s quation has nvr bn mpirically tstd. On dos not nd to b a propht to fortll that if it wr tstd with short-trm rats, th rsults would ithr b rjctd or inconclusiv and if it wr tstd with long-trm rats, th quation would b simply rjctd. 3 7. Having said all this dos not absolv m from th task of answring two important qustions. Firstly, why dos short-trm intrst rat bhaviour sm to adhr closly to Fishr s quation whn it dos so? [For xampl, s Mishkin 29] Scondly, why do markt playrs, particularly mony and bond markt tradrs, swar by Fishr s quation in thir day-to-day work? To my mind th answr to both qustions is on and th sam. Cntral bankrs us th intrst rat instrumnt to achiv thir shorttrm inflation targts. If that is so, bond tradrs must us somthing lik Fishr s quation to anticipat cntral bank actions on intrst rats in rspons to inflation rats and maximiz thir profits by slling scuritis if an intrst rat incras (consqunt upon a rising th inflation rat abov th targt) is anticipatd and buying
thm if an intrst rat dcras (th actual inflation rat falling blow th targt) sms imminnt. Anticipations of intrst rat incrass caus bonds to b sold and thir yilds to ris and vic vrsa. If th vry sam bond yild data is mappd against th inflation rat, lo and bhold, it must giv a good fit. To conclud, if Fishr s quation is found to work it could only b bcaus of th gnral imprssion that sombody as powrful as a Cntral Bank is using it! Nots. Fishr s trminology, now almost univrsally adoptd in th conomics litratur, is most mislading. Both th adjctivs ral and nominal ar usd to convy th opposits of thir convntional manings. Thus, th adjctiv nominal which usually mans unral, unimportant or insignificant has bn attachd to th actual rat of intrst, but actual convntionally mans ral. On th othr hand, th adjctiv ral has bn attachd to th unobsrvabl rat and no grounds ar givn for it to b vn concivably obsrvabl so that, in convntional trms, it has a flavour of unrality. Evn if it is undrstood that Fishr usd th trm ral rat of intrst to rhym with ral capital, I suspct that th transfr of pitht has somwhat prjudicd mattrs in such a way that only an inquiry into ral intrst rats is rgardd as a ral = tru = gnuin inquiry into intrst rat thory. 2. Similar dfinitions can b st up for othr contractually dtrmind prics as wll. For instanc, suppos a wag (or rnt) contract to rciv an amount of mony wag W. If inflation is xpctd to tak plac th xpctd rmunration rcivd in ral trms is W w = W ( p ) + p which can b cast into an appropriat conomtric form, W = A + B [ p ( + p )] + u and tstd. It is surprising that ths Fishr-lik quations for othr prics hav bn ntirly ignord in th macroconomic litratur vn though a whol school of macroconomic thought viz. rational xpctations thory, rlis on thir mpirical validity. 3. Prscott and Mhra (985) hav shown that for th US conomy ovr long timsris th rturn on quitis always xcds th rat of intrst. Th quity prmium puzzl has bn subsquntly found to b a fatur of svral financial markts. Rfrncs Fishr Irving (93), Th Thory of Intrst, Macmillan, Nw York. Fridman Milton (956), Th Quantity Thory of Mony: A Rstatmnt, in Fridman (Ed.), Studis in th Quantity Thory of Mony, Chicago Univrsity Prss, Chicago. Kyns, John Maynard (936), Th Gnral Thory of Employmnt, Intrst and Mony, Macmillan, London.
Mishkin Frdric (29), Th Economics of Mony, Banking and Financial Markts, Addison-Wsly, Nw York. Prscott E.C. and R. Mhra (985), Th Equity Prmium: A Puzzl, Journal of Montary Economics, Vol.5, pp. 45-6.