Federal Reserve Ba nk o f Minneapolis Research Department Inter est Rates and I nflation Fernando Alvarez, Robert E. Lucas, Jr., andWarrenE.Weber ∗ Work ing Paper 609 January 200 1 ∗ Alvarez, The University of C hicago; Lucas, The Univ ersity of Chicago and Federal Reserve Bank of Minneapo- lis; Weber, Federal Reserve Bank of Minneapolis. We would like to thank Lars Svensson for his discussion, Nurlan Turdaliev for his assistance, and seminar participants at the Federal Reserve Bank of Minneapolis for their comments and suggestions. The views expressed herein are those of the authors and not n ecessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System. 1. Introduction A consensus h as emerged about the co nduct of m onetary policy t hat now serves as common ground for discussion of the specific policies called for in particular situations. The central elemen ts of this consensus are that the instrument of monetary policy ought to be the short term interest rate, that policy should be focused on the con trol of inflation, and that inflation can be reduced by increasing short term in terest rates. For monetary econo mists, participating in d iscussions wh ere these p ropositions are tak en as given would seem to entail the rejection o f the quantity theory of money, the class of theories that imply that inflatio n rates can be controlle d by controlling the rate of gro wth of the money supply. Suc h a rejection is a difficult step to take, because th e syst e m atic evidence that exists linking monetary policy, inflation, a nd in terest rates–and there is an enormous amount o f it–consists almost entirely of evidence that increases in average rates of money growth are associated with equal increases in average inflation rates and in interest rates. Under the q uantity t heory, rapid m oney g rowth i s t he de fining characteristic of monetary ease, and it is associated with high interestratesaswellaswithhighinflation. Evidence from the postwar period, from the United States and elsewhere, shows that the quan tity theory of money contin ues to provide a reasonable description of the long run av er age relationships am o n g interest r a tes, inflationrates,andmoneygrowthrates. Inpar- ticular, the U.S . inflation of the 1970s and 80s can be fully accounted for by the corresponding increase in M2 (or M1 ) grow th rates, and th e return to relatively low inflation rates in the 1990s can be explained by the correspondingly low average rate of money supply grow th in that decade. Inflation in the 90s w as about 3.5 percentage poin ts lo wer than its average in the 70s and 80s, and th e growth rate of M 2 was about 5 percentage poin ts lower. The long 1 run behavior of in terest rates, in the U.S. and elsewhere, can similarly be understood in terms of Fisherian inflation p rem ia. To lose sight of these co nnections i s to lose si gh t of the on e reliable means society has for controlling th e long run average inflation rate. These observations n eed not rule out a constructive role for the use of short term in t erest ra tes as a monetary instrume nt. One possibility is that increasing sho rt rates in t he face of increases in inflation is just an indirect w ay of reducing money gro w th: S ell bonds and take money ou t of the system . Anothe r possibility is tha t wh ile control of mon etary agg reg ates is the key to low long ru n aver age inflation rates, an in terest rate policy can impro ve the short run behavior of interest rates and prices. The short run connections between m oney g rowth and inflation and interest rates are very unreliable, so there is much room for improvem ent. Thes e possibilities are surely worth exploring, but doing so require s new theo ry: The analys is needed to reconcile in tere s t rate policies w ith the evidence on which the qu antity theory of money is g rounded cannot be found in old textbook diagrams. 2. A n Economy with Segmented Markets Many theoretical models ha ve been introduced in the last few years, designed to ratio- nalize the use of in tere st rat e policies to control inflation rates. Many of these are centered on a c lass of policies k n own as “Taylor rule s,” ru les that specify the i nterest r a t e set by the central bank as an increasing function of the inflation rate (or perhaps o f a forecast of the inflation rate). 1 Theories differ considerably in their specification of the econo my to whic h the Taylor rule is assumed to apply. One class of inflation-targeting models combines an IS-curve, relating the nominal 1 See Taylor (1993). 2 in t erest rate to expected inflation (for Fisherian reasons) and production, w ith a Phillips-like curv e relating inflation to production. 2 Given the interest rate, these t wo equations can be solved for inflation a nd production. 3 These new-Keynesian models hav e been used to analyze the design of Tay lor rules, to dete rm in e the forms that would maximize a we lfare function that depends on the variab ility of inflation a n d real growth rates. But w hatever they can tell us about high frequency mo vements in interest rates and inflation rates, these models contribute nothing to our understanding of why the 60s and the 90s w ere low inflation decades, relativ e to 70s and the 80s, or why Germany has been a low inflation coun try, relative to Mexico. For these questions, the re ally important ones from a we lfare viewpoint, one n eed s to rely on some explicit v er sion of t he quantity th eo ry of money. To be useful in thinking about the r ole of interest rates and open m arket operations in th e con trol of inflation, a model of monetary equilibrium needs to deal w ith the fact that most coherent monetary theories do not ha ve any thing lik e a downw ard sloping demand for nominal bonds: With a complete set of financial markets, i t is just not true that when the gov e rn men t buys bonds, the price of bonds increa se s. We may believe th at such a “liqu id ity effect” occurs in reality (thou gh it is hard to see it in the data ) an d ma y r egard it as a deficiency that so much of monetary theory ignores it, but the fact rem ain s that one cannot take take a Sidrauski (1967), Broc k (1974), or Lucas and Sto key (1987) model off the shelf and us e it to think about increase s in money reducing interest rates. To engage th is liquidity effect it is necessary to adopt a framework in which some agents are excluded from money 2 See Clarida, Gali, and Gertler (1999) for a helpful review. 3 Where is the LM (which is to say, money demand) curv e? It no longer exists, some say, and if it did, we wouldn’t have any use for it: Everything we care about has been determined by the two curves already discussed. 3 markets, at least some o f t he time. This idea t h at m arkets must be segmen t ed, in some sens e, for a liquidity effect to occur, is taken from the original work of Grossman an d Weiss (1983) an d Rotemberg (1984). The particular ve rsion of the idea that we use here is ada pt ed from recent papers by Alvarez and Atkeson (1997), Alvarez, A tkeson, and Kehoe (2 000) and Occhino (2000), The model we de velop is an excha n g e economy : There is no Phillips cu r ve and no effect of monetary policy c hanges on production. Segmented mark et models that hav e suc h effects in clude contributions by Christiano and Eich enbaum (1992) and Carlstrom and Fu erst (2000). Our simp ler m odel permits a discussion of inflation, but not of all o f in flation’s possible consequences. Think, then, of an exchange econom y with man y agen ts, all with the preferences ∞ X t=0 ( 1 1+ρ ) t U(c t ), where U(c t )= c 1−γ t 1 − γ , o v er sequences {c t } of a single, non-storable consumption good. All of t hese agen ts attend a goods mark e t ev ery period. A fraction λ of agents also attend a bond m arket. We call these agents “traders.” The remaining 1 − λ agen ts–w e call t hem “non-traders”–never attend the bond mark et. We assume that no one ever c hanges status between being a trader and a non-trader. Agents o f both types have the same, constant endow ment of y units of the consumption good. The economy ’s resouce constraint is th us y = λc T t +(1− λ)c N t , (1) where c T t and c N t are the co nsumptions of the tw o ag en t-t ypes. We ensure that mon ey is held in equilibrium by assuming that no one c on s umes h is own endowment. Eac h hou s eh old 4 consists of a shopper-seller pair, where the seller sells the household ’s en dow ment for cash in the goods mark et, while the shopper uses cash to buy the consumption good from others on the s ame mark et. Prior to the opening of t h is goods m arket, money and one-period go vernment bonds are traded on a another market, attended only b y traders. Purchases are subject to a cash-in-advance constrain t, modified to incorporate shocks to v elocit y. Assume, to be specific, that goods purc hases P t c t are constraine d to be less than the sum of cash brough t into goods trading by the household, and a variable fraction v t of current period sales receipts. Think of th e shopper as v isitin g the seller’s s tore at some time during the trading da y, empt ying the cash register, and returning to shop some more. Th us every non-trader carries his unspen t r eceipts from period t−1 sales, (1−v t )P t+1 y, in t o period t trading. He a dds to these balances v t P t y from period t sales, giving h im a total of (1 − v t )P t+1 y + v t P t y tospendongoodsinperiodt. In order to k eep the determination of the price level as simple as possible, we assume that ev ery househo ld spends all of its cash, ev ery period. 4 Then every non-trader spends P t c N t =(1− v t−1 )P t−1 y + v t P t y (2) in period t. Trad ers, w ho attend both bond and goods markets, hav e more options. Like the non-traders, each trader has available the a moun t on the righ t of (2) to spend on goods in period t,buteachtraderalso absorbs his share of the increase in the per capita money supply that occurs i n the open m arket operation i n t. If the per capit a increase in mo ney 4 After solving for equilibrium prices and quantities under the assumption that cash constraints alwa ys bind, one can go bac k to individual maximum problems to find the set of parameter values under whic h this provisional assumption will hold. See Alvarez, Atkeson, and Kehoe (2000), Appendix A. 5 is M t − M t−1 = µ t M t−1 then each trader leaves the date t bond market with an additional µ t M t−1 /λ dollars. 5 Consumption spending per trader is thus given by P t c T t =(1− v t−1 )P t−1 y + v t P t y +(M t − M t−1 )/λ. (3) No w using the cash flow equations (2) and (3) and the market-clearing condition (1) we obtain P t y =(1− v t−1 )P t−1 y + v t P t y + M t − M t−1 = M t−1 + v t P t y + M t − M t−1 , since M t−1 =(1− v t−1 )P t−1 y is total d ollars carried forward from t − 1.Thusaversion M t 1 1 − v t = P t y (4) of th e equation of exchange must hold in equilibrium, and the fraction v t can be in terpreted (approximately ) as the log of velocity. Introducing shoc ks t o velocity c ap tu res th e short r un in s tab ility in the emp irical rela- tionship betwe en money and pric e s. In addition, it allo ws us to study the way interest rates react to news about inflation for different specifications of mon etary policy. In the formulation of the segmented markets m odel that we use here, there are no possibilities fo r s u b stitutin g 5 If B t is the value of bonds maturing at date t and if T t is the value of lump sum tax receipts at t,the market clearing co ndition for this bond m arket becomes B t − ( 1 1+r t )B t+1 − T t = M t − M t−1 . We assu me that all taxes are paid by the traders, so Ricardian equivalence will apply and the timing of taxes will be immaterial. These taxes pla y no role in our discussion, except to give us a second way to change the m oney supply besides open market operations. W ith this flexibility, any moneta ry policy c an be made consistent with the real debt remaining bounded. The arithmetic that follows will be both monetarist and pleasant in the sense of Sargent and Wallace (1981). 6 against cash, so the interest rate does not appear in the money demand function–in (4)–and v elocit y is simply given. Gi ven the beha vior of the money supply, then, prices are entirely determ in ed by (4 ): This is the q ua ntity th e ory of money in its ver y simplest form. The exogeneity of velocity in the model is, of course, easily relaxed without altering the essen tials of the model, but at the cost of complicating the solution method. In the ve rsion we study here, the two cash flo w eq uation s (2) and (3) de scribe th e way the fixed endo wment is distributed to the tw o consumer types. The three equations (1)-(3) thus completely determine the equilibriu m resource allocation and the behavior of the price level. No maximum problem has been studied and no derivativ es hav e been tak en! But to study the related beha v ior of interest rates, we need to exam ine bond market equilibrium, and th ere the real interest rate will depend on th e cu rre nt and expected futur e consumption of the traders only. Solving (1), (2), and (4), w e derive the form ula for c T t : c T t = " 1+µ t v t + µ t (1 − v t )/λ 1+µ t # y = c(v t ,µ t )y, where the second equality defines the relative consumption function c(v t ,µ t ). Then the equi- librium nomin al in te re st rate must satisfy the fa m iliar margin al condition 1 1+r t = 1 1+ρ E t " U 0 (c(v t+1 ,µ t+1 )y) U 0 (c(v t ,µ t )y) 1 1+µ t+1 1 − v t+1 1 − v t # , (5) where E t (·) mean s an expectation con dit io nal on eve nts dat ed t and earlier. We use t wo approxima tion s to simp lify e qu ation (5). The first involves expand ing t he function log(c(v t ,µ t )) around the point (v, 0) to obtain the first-order appro ximation log(c(v t ,µ t )) ∼ = (1 − v)( 1 − λ λ )µ t . 7 (Note that the first-order effect of velocity changes on consumption is zero.) With the C RRA prefere n c es we h ave a s sumed, the m arginal utility of trad ers is then approxima ted by U 0 (c(v t ,µ t )y)=exp(−φµ t )y, where φ = γ(1 − v)( 1 − λ λ ) > 0. Taking logs o f both sides of (5), we h ave: r t = ρ − log à E t " exp{−φ(µ t+1 − µ t )} 1 1+µ t+1 1 − v t+1 1 − v t #! . We apply a second appro ximation to the r ight hand side to obtain r t =ˆρ + φ(E t [µ t+1 ] − µ t )+E t [µ t+1 ]+E t [v t+1 ] − v t , (6) where ˆρ − ρ > 0 is a risk cor rection factor. 6 From equation (6) o ne can see that the immediate effectsofanopenmarketoperation bond purchase, µ t > 0, is to reduc e inter est rates by φµ t . This is the liquidity effect that the segmented market m odels are designed to c aptu re. If we drop the segmentation and let everyone trade in bonds, then λ =1, φ =0, and the liquidity effect vanishes. In this case, open market operations can only affect interest rates through information effects on the 6 In fact, ˆρ − ρ = 1 2 Var t (z t ), where z t = −φ(µ t+1 − µ t ) − µ t+1 − v t+1 + v t . In the applications of (9) that we consider below, Var t (z t ) will not vary with t under a given monetary policy rule, though it will v ary with changes in the policy rule. 8 inflation premium. Interest rate increases can on ly reflect expected inflation: monetary ease. With φ > 0, the m odel com bines quantity-theoretic predictions for the long run behavior of money gro w th, inflation, and interest rates, with a potential role for in terest rates a s an instrum e nt of inflation c ontrol in th e short run. We explore this potential in the next s ection . 3. Inflation Control with Segme nted M arkets In this section, w e work through a series of thought experimen ts based on the equi- librium co nd ition (6) that illum inate various aspects o f mone tary policy. T h e se examples all draw on the fact, ob tain ed by differencing the equation of exc hange (4), that the inflation rate is t he sum of the money g r owth rate and the rate of chang e in velocity: π t = µ t + v t − v t−1 . (7) Example 1 : (Constan t velocit y and money grow th.) Let v t be constant at v and µ t be constan t at µ, T hen (6) becomes r = ρ + µ. We can view this equa tion in terchangeab ly as fixing money growth, giv en the interest rate, or a s fixing the interest rate given money gro wth and inflation . Th is Fisher equation must alwa y s chara cterize the long run average money grow th, inflation, and in terest rates. Example 2 : (Constant money grow th and iid shoc ks.) L et the velocity shocks be iid random variables, with mean v and variance σ 2 v .Letµ t be constant at µ. Under these conditions, (6) implies r t = b ρ + µ − (v t − v). 9 [...]... References Alvarez, Fernando and Andrew Atkeson 1997 “Money and Exchange Rates in the GrossmanWeiss-Rotemberg Model.” Journal of Monetary Economics, 40: 619-640 Alvarez, Fernando, Andrew Atkeson, and Patrick Kehoe 2000 “Money, Interest Rates, and Exchange Rates with Endogenously Segmented Asset Markets.” Federal Reserve Bank of Minneapolis working paper Brock, William A 1974 “Money and Growth: The Case... interest rates, money growth rates, or inflation rates The way it is distributed over these three variables can, in the presence of a liquidity effect, be determined by policy However this is done, the long run connections between money growth, inflation, and interest rates are entirely quantitytheoretic Our next four examples consider versions of Taylor rules Suppose, to be specific, that interest rates. .. reducing expected inflation b This induces a transient decrease in interest rates In this example, rt is iid, with mean ρ + µ and variance σ 2 ; the inflation rate has mean µ and variance 2σ 2 v v Example 3 : (Exact inflation targeting.) It is always possible to attain a target inflation rate π exactly Just set the money growth rate according to ¯ ¯ µt = π − vt + vt−1 Then interest rates will be given... increase leads in this case to a reduction in the the interest rate The variance of the inflation rate under such a policy is σ 2 = [1 + ( π 1−φ−θ 2 2 ) ]σ v φ The Taylor rule on the expected inflation rate that minimizes the inflation variance has the coefficient θ = 1 − φ 14 4 Conclusions Can a policy of increasing short term interest rates to reduce inflation be rationalized with essentially quantity-theoretic... the inflation rate is, using (7), σ 2 = V ar(µt+1 + vt+1 − vt ) = [1 + ( π φ−1 2 2 ) ]σ v φ 10 Comparing this case to Example 2, one sees that pegging the interest rate is inflationstabilizing, relative to constant money growth, if and only if φ > 1/2 In Examples 2, 3, and 4, the economy is subjected to unavoidable velocity shocks The variability of these shocks must show up somewhere, either in interest. .. model we used to generate all of our specific examples, production is a given constant, velocity is an exogenous random shock, and the equation of exchange determines the equilibrium price level, given the money supply In this theory of inflation, consistent with much of the evidence, interest rates play no role whatsoever To this simple model we have added segmented markets: Only a fraction of the agents... Examples 2 and 3, a policy at any date is set in advance of the realization of the velocity shock in that period: One can commit to a given rate of money growth, leaving interest rates free to vary with the velocity shock (Example 2), or one can commit to an interest rate, leaving money growth to be adjusted later to maintain this rate (Example 3) Neither policy can reduce the variance of inflation to... the relative effectiveness of the interest rate rule in stabilizing inflation rates about a target rate In the remaining examples we consider, policy (however specified) is permitted to 15 respond to contemporaneous velocity shocks In Example 4, we show that under this assumption an inflation target can be hit exactly by a money supply rule that is conditioned on the shock, and that this is true whatever... increase it in the next period, and return to the target growth rate thereafter This will smooth the inflationary impact of the velocity increase, whether or not there is a positive liquidity effect φ If φ > 0 and 2θ + φ > 1, (12) implies that these open market operations will raise the interest rate initially in response to a velocity increase, then b reduce it below the target, and then return it to ρ +... Economic Review, 15: 750-777 Carlstrom, Charles T., and Timothy S Fuerst 2000 “Forward-Looking Versus BackwardLooking Taylor Rules.” Federal Reserve Bank of Cleveland working paper Christiano, Larence J., and Martin Eichenbaum 1992 “Liquidity Effects and the Monetary Transmission Mechanism.” American Economic Review, 82: 346-353 Clarida, Richard, Jordi Gali, and Mark Gertler 1999 “The Science of Monetary . age relationships am o n g interest r a tes, inflationrates,andmoneygrowthrates. Inpar- ticular, the U.S . inflation of the 1970s and 80s can be fully accounted. variab ility of inflation a n d real growth rates. But w hatever they can tell us about high frequency mo vements in interest rates and inflation rates, these