The B.E. Journal of Macroeconomics

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1 The B.E. Journal of Maroeonomis Contributions Volume 7, Issue 2007 Artile 9 Nominal Debt Dynamis, Credit Constraints and Monetary Poliy Liam Graham Stephen Wright University College London, liam.graham@ul.a.uk Birkbek College, London, swright@eon.bbk.a.uk Reommended Citation Liam Graham and Stephen Wright (2007) Nominal Debt Dynamis, Credit Constraints and Monetary Poliy, The B.E. Journal of Maroeonomis: Vol. 7: Iss. (Contributions), Artile 9. Available at: Copyright 2007 The Berkeley Eletroni Press. All rights reserved.

2 Nominal Debt Dynamis, Credit Constraints and Monetary Poliy Liam Graham and Stephen Wright Abstrat We onstrut a dynami general equilibrium model in whih household debt is stiky in nominal terms and debtor households are redit onstrained. Interest payments on debt ontrats may be at floating rates or fixed for the duration of the ontrat. A key result is that a simple stati Taylor Rule an result in a prolonged period in whih real interest rates are ut rather than raised in response to an inflationary shok. We show how the proportion of fixed rate ontrats affets the monetary transmission mehanism and its impliations for the distributional effets of an inflationary shok. KEYWORDS: nominal debt, dynami general equilibrium, monetary poliy Corresponding author: Liam Graham, Department of Eonomis, University College London, Gower Street, London WCE 6BT, UK. L.Graham@ul.a.uk. We are grateful for useful omments from Roland Meeks, Andrew Oswald, and Fabrizio Zampolli and from seminar partiipants at the Bank of England; Birkbek College, London, University College, London, the Institut für Weltwirtshaft, Kiel and the Universities of Birmingham, Exeter and Warwik. The editor and two anonymous referees provided omments whih greatly improved the paper.

3 Graham and Wright: Nominal Debt Dynamis, Credit Constraints and Monetary Poliy Introdution Collateralised household debt in EU ountries ranges from only 6% of GDP in Greee, up to 65% in the Netherlands and 80% in the United Kingdom, with an average for the EU-5 of 36%. The assoiated debt ontrats are almost always written in nominal terms, have quite signi ant assoiated transations osts and as a onsequene are renegotiated relatively infrequently (for example, the UK Counil of Mortgage Lenders estimates the average length of a mortgage before re naning to be 5 years). In many ountries a high proportion of the interest payments on this debt are at a xed rate, with the proportion ranging from 80% in Frane to 30% in the UK down to zero in Portugal. Some of these features of the data have been investigated in the literature. A number of papers have examined " nanial" models of the business yle and the role of ollateralised debt (for example Bernanke et al., 999) and some reent papers investigate the role of nominal debt (for example Aoki et al., 2002, Iaoviello, 2005)). However the stikiness of debt ontrats, and the observation that xed rate ontrats are ommon, has still to be addressed. Yet nominal debt stikiness is arguably easier to understand than produt prie stikiness. A well-known ritiism of the standard model of produt prie stikiness is that the menu osts that ultimately must generate stikiness are unlikely to be large. In the ase of debt ontrats, in ontrast, the osts of adjustment are larger, sine typially they will involve re-assessment of ollateral or other features of reditworthiness. In this paper we present a standard new Keynesian dynami general equilibrium model modi ed in two key respets to re et the data features noted above. Firstly, to model stiky debt in a tratable manner, we assume that debt ontrats adjust in a proess very lose to the widely used Calvo (983) model of produt pries. This Calvo adjustment proess also allows us to model oating or xed interest rates on the debt. Seondly these features of debt ontrats matter in our model beause some onsumers fae binding redit onstraints. To simplify the analysis, we swith o the nanial aelerator hannel by assuming that the real value of ollateral is xed. We use this model to examine the impat of these features on the response of an eonomy to an in ationary shok. Sine debt is denominated in nominal terms, an in ationary shok hanges its real value. The response of monetary poliy to this shok hanges the interest payment on the debt. The intera- Data on stoks of mortgage debt and proportion of xed rates from Malennan et al. (999); data on total UK household debt from Brierley et al. (2002); data on the duration of debt from the Counil of Mortage Lenders (2004). Debt renegotiation osts are desribed in detail for the UK in Miles (2004) Published by The Berkeley Eletroni Press, 2007

4 The B.E. Journal of Maroeonomis, Vol. 7 [2007], Iss. (Contributions), Art. 9 tion of these e ets, and the response of nanial institutions and onsumers, determines the behaviour of the eonomy following the shok. Our main results are:. Sine the diret e et of an in ationary shok on nominal interest rates is strongly ontrationary, the response of real interest rates neessary to ahieve a given ontrationary response is initially small or negative in eonomies where a signi ant proportion of debt ontrats are at oating rates. Many European ountries fall into this ategory 2. The presene of sluggishly adjusting nominal debt ontrats introdues additional, long-lived dynami responses, as optimising nanial institutions bring real debt levels bak into line with real ollateral, boosting debtors onsumption. As a result real interest rates need ultimately to rise, and stay above steady state for a prolonged period. 3. These new features thus introdue an explanation for a sluggish response of nominal interest rates to in ationary shoks that does not rely on any assumption of interest rate smoothing on the part of the entral bank. 4. Fixed rate debt redues the responsiveness of the eonomy to monetary poliy and shifts the burden of adjustment to a shok onto those with oating rate debt, as well as unonstrained optimising onsumers. It therefore requires a more aggressive response of real interest rates to ahieve a given ontrationary response to in ation. Both of the novel features of the model are ruial to these results. If the level of indebtedness, or the nature of debt ontrats, is to matter at all for the monetary transmission mehanism, some households must fae binding redit onstraints. Were this not the ase, in ationary shoks would simply ause small distributional wealth e ets and the only impat of di erent debt systems would then be at most a seond-order one due to the di ering risk harateristis of di erent debt ontrats (Campbell and Coo, 2003). These would ompletely net out in the perfet risk-sharing framework that underpins the standard representative agent model. But given our maintained assumption that redit onstraints are binding, the nature of debt ontrats does matter for the transmission mehanism. 2 We assume that debt is ollateralised but that there are osts in measuring ollateral, so that ontrats only adjust infrequently. This a ets the response 2 Sine the issue of redit onstraints has been widely addressed in the literature (see for example Mankiw, 2000, and Gali et al., 2004) we do not examine it further here. 2

5 Graham and Wright: Nominal Debt Dynamis, Credit Constraints and Monetary Poliy of the eonomy in two ways: the response of optimising nanial institutions in o ering new debt to onstrained onsumers after an in ationary shok; and the impat of hanges in the real value of nominal interest payments on existing debt. The rst of these will arise in any system where debt is stiky in nominal terms; the nature of the seond e et will depend ruially on whether debt interest payments are made on a oating or xed rate basis. Some evidene suggestive of a ombined role for nominal debt ontrats and redit onstraints is the well-known and long-standing empirial orrelation between onsumption and nominal (as opposed to real) interest rates (e.g., Blinder and Deaton, 985, Fuhrer and Moore, 995). Fair (2005) ites this as one of the primary reasons why strutural maroeonometri models and unrestrited VAR models (Giordani, 2003) imply that in ationary shoks have e ets that are ontrationary ex ante, rather than expansionary as implied by standard models. Our model helps to provide a theoretial rationale for these features. There is muh evidene that the monetary transmission mehanism di ers between ountries (for example Angeloni et al., 2003). Our model implies that institutional features ould aount for a signi ant part of these di erenes. Finding diret empirial support for this is ompliated by the onstant strutural hange in European mortgage markets over the past 20 years. However Calza et al. (2006) present some preliminary eonometri evidene that supports this hypothesis. In what follows, setion 2 presents the model and desribes how it is solved and alibrated. Setion 3 desribes our results, Setion 4 disusses impliations for monetary poliy, and Setion 5 onludes. Appendix A shows the linearised system, and Appendix B ontains derivations. 2 The Model The model is, in most respets, a simple version of the standard dynami new Keynesian model ommon in the literature (e.g. Goodfriend and King, 997). Households onsume nal goods, supply labour and hold nanial assets. Intermediate-goods rms produe di erentiated goods whih are imperfet substitutes in the prodution funtion of nal-goods rms. Calvo priing on the part of intermediate-goods rms gives rise to a new Keynesian Phillips urve. A monetary authority sets the real interest rate as in Clarida et al. (999). The non-standard features are the presene of redit-onstrained households, and of nanial institutions who lend to these households by means of Published by The Berkeley Eletroni Press,

6 The B.E. Journal of Maroeonomis, Vol. 7 [2007], Iss. (Contributions), Art. 9 stiky nominal debt ontrats. Note that upper ase letters refer to levels, lower ase to their log-linearised deviations from non-stohasti steady state values. Symbols without time subsripts refer to steady-state values. Full derivations an be found in Appendix B. 2. Households Households onsume, supply labour, lend or borrow and are endowed with a single physial asset whose real value is exogenously given. All households have in nite horizons and rational expetations. Following Iaoviello (2005), we divide households up into two types: type having a higher disount fator than type 2, > 2. A household of type j maximises its utility given by: U t = E t X i=0 i ju (C jt+i ; N jt+i ) () where the instantaneous utility funtion is u (C jt ; N jt ) = C jt n + ( N jt) (2) n where and n are the elastiities of intertemporal substitution of onsumption and of leisure. We follow the growing onvention in the literature of building a monetary model without money (see MCallum, 200). The maximisation is subjet to the real budget onstraint 3 A jt+ = + R jt + t+ (A jt + Y jt C jt ) + Div jt+ (3) where C jt is onsumption, Y jt = W jt N jt + T jt is after-tax labour inome (W jt is the real wage, N jt labour supplied and T jt a transfer from government), A jt the household s stok of nanial assets at the start of period t. Div jt+ are dividends paid by intermediate rms and nanial institutions. t+ = P t+ =P t is the rate of in ation between periods t and t +, and R jt the 3 This an be derived from an asset evolution equation in nominal terms P t+ A jt+ = ( + R t )P t (Y jt C jt + A jt ) + P t+ Div jt+ 4

7 Graham and Wright: Nominal Debt Dynamis, Credit Constraints and Monetary Poliy nominal interest rate, set in period t; and payable in period t + ; given by R jt = R t if A jt > 0 (4) R jt = R D t if A jt < 0 (5) where R t is the short-term interest rate set by the entral bank, Rt D is the rate payable on borrowings whih we onsider in detail later. While nanial assets pay an interest rate that is safe in nominal terms sine it is set in period t; the real return is unertain due to in ation. Households fae a further onstraint A jt D t (6) where D t is the level of debt whih nanial institutions are prepared to lend to households (the proess determining this level of debt is desribed in Setion 2.2). Given the di erene in subjetive disount fators, the existene of a redit onstraint in some form is a neessary ondition for the existene of a nonstohasti steady state in whih type 2s have non-zero onsumption. To see this, note that the Euler equation for a household of type j is C jt = j E t C jt+ + R jt + + jt (7) t+ where jt is the Lagrange multiplier on the redit onstraint. In the steady state we assume below that R = R 2 = R due to ompetition in redit markets, and if onsumption is non-zero for both types of household, (7) beomes + R = j + + j (8) whih, given > 2 annot hold for both types if redit onstraints bind for neither type, sine this would imply both and 2 are zero. 4 However a steady state an exist in whih type s hold positive assets, so that (6) annot bind and = 0: This implies R = = and hene 2 = 2 > 0, so redit onstraints must bind for type 2s in steady state. We assume that deviations from steady state are su iently small that redit onstraints bind 4 Equivalently, any notional steady state in whih C was onstant and redit onstraints did not bind would imply ontinuously falling C 2 and hene ever-inreasing indebtedness whih ontradits the assumption of a steady state. In ation and/or real growth would not a et this result. Published by The Berkeley Eletroni Press,

8 The B.E. Journal of Maroeonomis, Vol. 7 [2007], Iss. (Contributions), Art. 9 for type 2s in all time periods, and never bind for type s. Thus for type (unonstrained) onsumers, setting t = 0; equation (7) an be linearised in standard fashion: t = E t t+ (r t E t t+ ) (9) while the binding redit onstraint means that type 2 (onstrained) households are at a orner solution and are not full optimisers. Their onsumption is then given by the budget onstraint (3). Linearising (3) gives: R D 2t = y 2t rt D + (d t+ + t+ ) + + R y D 2t d t rt D (0) where = D C 2 is the steady-state debt to onsumption ratio of type 2 households. The sum of the rst two terms on the right-hand side is linearised disposable inome after interest payments. The third term represents the hange in nominal debt over the period: if banks o er new debt onstrained households aept it to fund onsumption. The fourth term is the hange in interest payments due to debt deviating from its steady state value. Although there is some evidene to support it (for example Lawrane, 99), the assumption of di ering disount rates is prinipally a devie to motivate redit onstraints. Another way of thinking about redit onstraints is within a life yle model of onsumption. For some households the optimal level of onsumption, out of urrent nanial wealth and future lifetime earnings, would imply a urrent level of debt greater than that whih a nanial institution would be prepared to lend, given that household s ollateral. This might apply, for example, for households relatively early in the life yle, for whom future labour inome (that annot be ollateralised) makes up a signi ant fration of total wealth. Eah type of household supplies a di erent type of labour. The linearised rst-order onditions for labour supply are standard: Nj n jt = n w jt jt () 2.2 Finanial institutions N j Finanial institutions make loans to households. Following Kiyotaki and Moore (997) we assume that lenders annot fore borrowers to repay their debts unless they are seured. The optimal value of debt is then given by some onstant fration (whih we normalise to unity) of households ollateral 6

9 Graham and Wright: Nominal Debt Dynamis, Credit Constraints and Monetary Poliy K; whih we assume onstant aross time and aross households. 5 A nanial institution faes osts in deviating from this level. If it lends more than this level, part of the debt is unseured and there is default risk on this unseured portion. If it lends less, the ost arises from foregone pro t opportunities. Debt ontrats are stiky in nominal terms. To apture this stikiness in a tratable way, we progress by analogy to Calvo s model of the aggregate prie level. We assume a onstant probability that any given debt ontrat will be adjusted in the next period, with omplete adjustment towards its optimal value if adjustment does take plae. 6 This simple model impliitly assumes that banks fae osts, analogous to the "menu osts" disussed in the prie adjustment literature, in measuring the reditworthiness of individuals. As a result, households only re-mortgage infrequently. When deiding the nominal value of a new ontrat Z t+ at time t the nanial institution s problem is max Z t+ E t X t;t+i ( i= ) i R z t+i R t+i Z t+ P t+i t+i (2) where t;t+i is the stohasti disount fator of the owners of the nanial intermediaries (the unonstrained households): t;t+i = i Ct+i C t (3) and nanial institutions harge a rate R z t on a new ontrat, and raise oatingrate funds from unonstrained households at the entral bank s target rate. t is a ost, assumed to be quadrati, of deviating from the optimal value of debt Zt+ =P t+i t+i = $ K 2 K K (4) where $ is a saling parameter. Sine all households have idential ollateral this aggregate ost will simply be a saling of the ost of debt deviating from 5 This swithes o the " nanial aelerator". The value of the ollateral ould be modelled as time-varying by assuming that it provides a onstant ow of servies and its value is therefore the present value of these servies. We investigated this spei ation and found our key results were unhanged. 6 As in the original Calvo model, this probability will be treated as exogenous. In reality it will be in part endogenous, but is likely to be onstant for any given stable monetary regime. See Graham and Snower (2004) for an example of nominal ontrats whose frequeny of adjustment varies with the steady state value of in ation. Published by The Berkeley Eletroni Press,

10 The B.E. Journal of Maroeonomis, Vol. 7 [2007], Iss. (Contributions), Art. 9 ollateral for any individual household. Deriving the rst-order ondition, then linearising gives [ B ()] F z t+ E t p t+ = E t t+ B (F ) (5) where F is the forward shift-operator (F i x t = x t+i ) and B (F ) = ( ) F. This ondition gives the expeted value of a new real debt ontrat at time t+ in terms of the expeted path of in ation. Note that the saling parameter in (4), $, has no impat on the dynamis of the system. If $ inreases, a nanial institution s osts and marginal ost inrease in the same proportion, but the zero pro t ondition for nanial institutions implies that revenue and hene marginal revenue do too. So $ anels out of the equilibrium ondition Aggregate debt At time t a proportion ( ) i of nanial institutions have reset their ontrats i periods earlier and have not had the opportunity to reset them sine. So we an sum over all ontrats and all nanial institutions to obtain the real value of aggregate debt: D t = P t X i=0 ( ) i Z t i (6) Linearising this gives an expression for the evolution of the nominal value of aggregate debt in terms of the value of individual debt ontrats: d t + p t = A () A (L) z t (7) where A (L) = ( ) L and L is the lag operator (L i x t = x t i ) Floating and xed rates The interest rate payable on debt an be either oating or xed. We do not attempt to model households deision on whether to hold xed or oating rate debt, though we will have something to say about the fators in uening the deision. On oating rate ontrats nanial institutions harge the nominal interest rate plus a onstant spread, whih has no impat on the linearised dynamis, so we set it to zero. For xed rate debt, nanial institutions hoose the 8

11 Graham and Wright: Nominal Debt Dynamis, Credit Constraints and Monetary Poliy (fairly pried) xed rate on a partiular debt ontrat, R z t, aording to a no-arbitrage ondition: E t X X t;t+i ( ) i Rt z Z t = E t t;t+i ( ) i Rt+iZ D t (8) i= i= whih when linearised gives 7 r z t = E t rt+ + ( ) r z t+ (9) The average xed rate R F t payable on xed rate debt will then be R F t X = (( ) L) i Rt z (20) i=0 Linearising gives r F t = r z t + ( ) r F t (2) If there is some proportion of borrowers in xed shemes the average rate payable on aggregate debt will be r D t = r F t + ( ) r t (22) where we linearise around a steady state where R is onstant so R D = R Ownership of nanial institutions Finanial intermediaries are owned by unonstrained households. Sine these households have aess to omplete markets any idiosynrati risk arising from the resetting of the Calvo ontrats will be eliminated at the level of the representative household. Competition among nanial intermediaries drives expeted pro ts over the life of the ontrat to zero so, if all debt is oating rate, nanial institutions will never earn pro ts. With some xed rate debt, pro ts will be earned on xed rate ontrats after unexpeted shoks. This pro t is remitted in full to unonstrained households. 7 Note that in both this expression and (5) the stohasti disount fator anels out in the linearization, just as in the standard derivation of Calvo prie setting. Published by The Berkeley Eletroni Press,

12 The B.E. Journal of Maroeonomis, Vol. 7 [2007], Iss. (Contributions), Art Firms In standard fashion we model two types of rms. Intermediate-goods rms with market power produe Dixit-Stiglitz di erentiated goods that are inputs to the prodution proess of nal-goods rms who ostlessly aggregate them to produe a single onsumption good. Intermediate-goods rms produe di erentiated goods by means of a tehnology in whih the labour of the two types of households are imperfet substitutes. The ost minimisation problem is then: subjet to the prodution funtion giving rst-order onditions min W t N t + W 2t N 2t (23) Y t = NtN 2t (24) W t = MC t Y t N t (25) W 2t = MC t ( ) Y t N 2t (26) where MC t is real marginal ost. The output of intermediate-goods rms are imperfet substitutes in the prodution funtion of nal goods rms whih produe a single homogenous onsumption good using no other fators of prodution. Calvo priing by intermediate-goods rms allows us to derive a New Keynesian Phillips urve as the solution to an intertemporal pro t maximisation problem. But it is well known that the New Keynesian Phillips urve annot of itself generate observable degrees of in ation persistene, whih (ating via the nominal interest rate) plays an important role in our results. To generate in ation persistene we follow Clarida et al. (999) by adding ad ho a term in lagged in ation to obtain t = t + ( ) E t t+ + m t + u t (27) where the parameter aptures the stikiness of in ation,, a funtion of the underlying parameters, measures the sensitivity of in ation to deviations in real marginal ost m t and u t is a white noise "ost-push" shok. 8 This 8 Koziki and Tinsley (2002) generate a New Keynesian Phillips Curve with a bakward- 0

13 Graham and Wright: Nominal Debt Dynamis, Credit Constraints and Monetary Poliy partiular form form for the Phillips Curve is not ruial to our results, but our spei ation has the attrative feature as disussed in Clarida et al. (999), that in redued form in ation persistene is endogenous to monetary poliy. 9 Firms, like nanial institutions, are owned by unonstrained households. To simplify the model, we assume the existene of a government whose only role is to tax away monopoly pro ts and remit the proeeds to households lump sum in proportion to their (onstant) shares of labour inome. Given the Cobb-Douglas tehnology this has the onvenient property that the e etive shares of eah type in total inome are onstant and equal to their respetive labour shares. We disuss the impat of relaxing this assumption, and allowing pro ts to ow diretly to the rms owners in Setion Monetary poliy We haraterise monetary poliy by a simple rule for output y t = t (28) Under standard assumptions, < 0 and the entral bank leans against the wind, hoosing its poliy instrument to ontrat demand when in ation is above target. The transmission mehanism of the eonomy then gives a rule for the poliy instrument, in this ase the real interest rate, of the form: r t E t t+ = (L) t (29) where (L) is a polynomial in the lag operator. In the standard model (as in e.g. Clarida et al., 999) the transmission mehanism is the eonomy s optimising IS urve and (L) is a onstant, so that there is a diret equivalene between a stati output rule and a stati real interest rate rule. This in turn an be reparameterised as a Taylor Rule for the nominal interest rate under other reasonable assumptions. 0 In our model, the nature of debt ontrats determines the transmission mehanism and hene (L) implies additional long-lived dynamis in both real and nominal interest rates. In the short run however, with appropriate alibration of, the resulting looking omponent from an optimising framework with higher order lag polynomial adjustment osts of hanging pries. 9 Exogenous in ation persistene ould also be introdued straightforwardly with a standard New Keynesian Phillips Curve by setting = 0 but allowing u t to be serially orrelated, without signi ant hanges to our results. 0 Clarida et al. (999) show that demand shoks and in ation persistene an be inorporated straightforwardly into this framework. Published by The Berkeley Eletroni Press, 2007

14 The B.E. Journal of Maroeonomis, Vol. 7 [2007], Iss. (Contributions), Art. 9 rule for nominal rates very losely resembles the stati Taylor Rule. Our results are not signi antly altered if we allow monetary poliy to follow a Taylor Rule diretly (see setion 4.); but our approah has the advantage of allowing us to fous on distributional onsequenes of in ationary shoks by making outomes for output and hene aggregate onsumption invariant to the transmission mehanism. 2.5 Identities Total output is given by Y t = C t + C 2t (30) and inomes of both types of household exhaust total output so Y t = Y t + Y 2t (3) While the eonomy also inludes a entral bank and rms the model implies that they have zero net nanial assets in eah period. We an ombine (3) with the budget onstraints for both types of households (3) and the binding redit onstraint (6) to give A t = D t (32) The assets of type households will thus equal the liabilities of type 2 households at all times. 2.6 Solution method and alibration The system of linearised equations desribing the eonomy is shown in Appendix A. We solve this system by the method set out in MCallum (998). We alibrate our model on quarterly data with the values shown in table whih orrespond to those for the UK eonomy. With the exeption of the monetary poliy parameter ; and ; the proportion of xed rate ontrats, the qualitative nature of our results is insensitive to a wide degree of variation in the assumed parameters. There is onsiderable unertainty as to the quantitative signi ane of redit onstraints: Campbell and Mankiw (99) nd the onsumption share of redit onstrained onsumers to be between 0.2 and For simpliity we assume equal labour inome shares of one half for both types of onsumer. Along with other assumptions on debt and steady state interest rates, this in turn implies that the onsumption share of onstrained onsumers is just under 2

15 Graham and Wright: Nominal Debt Dynamis, Credit Constraints and Monetary Poliy one half. Brierley et al. (2002), using data from the British Household Panel Survey, give debt to annual inome ratios ranging from 4 among the lowest inome households to among the highest. We take, the steady state debt to quarterly onsumption ratio of type 2 households, to be 8. 2 We assume that the expeted life of debt ontrats is given by the average frequeny with whih individuals re-mortgage. For the debt reset probability,, we hoose 0.05 implying the expeted life of a debt ontrat is = 20 quarters or 0:05 5 years. This is shorter than the notional maturity of most mortgage debt, but re ets the fat that suh debt is generally renegotiated on a number of oasions before maturity, most notably on moving house. We desribe monetary poliy by an output rule rather than a Taylor rule to make outomes for output (and aggregate onsumption) invariant to the transmission mehanism so allowing us to fous on distributional onsequenes. To alibrate, the measure of how strongly monetary poliy responds to in ation, we rst hoose the proportion of xed rate ontrats to math that found in the UK ( = 0:3). Then, given this transmission mehanism, we hoose to give the same impat response of the nominal interest rate as would be implied by a simple Taylor rule of the form r t = :5 t + :5y t (in setion 4. we brie y disuss the e et of diretly implementing a Taylor Rule). We hoose, the oe ient on lagged in ation in the Phillips urve, to give a realisti degree of in ation persistene given the other parameters. As to the preferenes of the households, the disount rate of unonstrained households is hosen to give an annual real interest rate of approximately 4%. For 2 we follow Iaoviello (2005) in hoosing a value of We set the elastiity of intertemporal substitution of onsumption, to and the 2 elastiity of intertemporal substitution of labour, n to 2. We hoose steady state labour supply to math observed hours as in King and Rebelo (999). Given the prodution funtion and the impat of the lump-sum tax, the onsumption share of unonstrained onsumers is given by = + R +R ( ) : On our alibration = 0:537 2 This mathes well with UK aggregate data. In 2000 total annual onsumption (ABPB) in billions of GBP was (blue book table 6.2). Finanial Statistis Table 3.G gives total M4 debt for the household setor (sum of lending on property (AVHG), onsumer debt (95.) and lending to uninorporated businesses (AVHI) as 48:8 + 95: + 34 = 60:9 while total M4 deposits of the personal setor (VSCL) were very similar at (ie the household setor is a modest net debtor). Thus annual onsumption is very lose to total debt, whih with a onsumption share of onstrained onsumer of just under one half on rms our hoie of = 8 Published by The Berkeley Eletroni Press,

16 The B.E. Journal of Maroeonomis, Vol. 7 [2007], Iss. (Contributions), Art. 9 Table : Calibration Desription Value labour inome share of unonstrained households 0.5 steady state debt to onsumption ratio of onstrained households 8.0 debt reset probability 0.05 oe ient in monetary poliy rule -.76 oe ient on lagged in ation in Phillips urve 0.75 oe ient on marginal osts in Phillips urve (implying goods pries are xed on average for one year) 0.75 disount fator of unonstrained households disount fator of onstrained households 0.95 intertemporal elastiity of substitution for onsumption 0.5 n intertemporal elastiity of substitution for labour 2.0 N steady state labour supply Results In this setion we analyze the response to a unit "ost push" shok to the Phillips urve (27) of four model eonomies: two where all debt is either oating or xed, and two mixed ases. While we onsider the rst two ases prinipally for heuristi purposes, data reported in Malennan et al. (2000) suggests the purely oating ase orresponds losely to Portugal and Finland. In our mixed ases, we onsider two eonomies, one with a relatively low (30%) share of xed rate debt, whih orresponds roughly to the ase of the UK, the other with a relatively high (80%) share orresponding roughly to Frane. To fous on the distributional onsequenes of hanging the transmission mehanism, we hold all other parameters onstant aross ases. In partiular, the spei ation of monetary poliy is held onstant aross these ases so that the response of output (and hene, from (30), aggregate onsumption) is the same in all ases: a.76% redution on impat (sine = :76) then a gradually rise bak to the steady state as in ation deays. The dynamis of debt are independent of the proportion of xed rate debt, and the dynamis of in ation largely so. Before turning to our individual model eonomies we disuss the proesses for debt and in ation that underlie all of them. 4

17 Graham and Wright: Nominal Debt Dynamis, Credit Constraints and Monetary Poliy 3. The dynamis of in ation and debt Inspetion of the Phillips urve (27) shows there are potentially two soures for dynamis in in ation: endogenous dynamis given by the parameter and the dynamis of marginal ost. To illustrated, onsider a simpli ed version of the model in whih the steady state real interest rate is zero. With this assumption, and given our baseline alibration, marginal ost is proportional to output, in deviations m t = + # y t so the Phillips urve an be rewritten as where t = t + ( ) E t t+ + ~y t + u t (33) ~ = + # (34) Then using the poliy rule (28) it is straightforward to show that in this speial ase in ation follows a rst-order autoregressive proess. t = (; ~; ) t + " t (35) where " t is a saling of the "ost-push" shok, u t, > 0: the monetary poliy leans against the wind, the more persistent will be in ation (as in Clarida et al., 999). When in ation follows this autoregressive proess, the proess for the value of a new debt ontrat (5) an be rearranged to give z t+ E t p t+ = ( ) 2 ( ) t (36) where the oe ient on in ation is inreasing in. If in ation is above its steady state value, the more persistent is in ation, the higher the real value of the ontrat hosen by nanial institutions when they reset the ontrat s value in nominal terms sine the faster it will be eroded. In the atual model, without the simplifying assumption of a zero steady state real interest rate, extra dynamis are introdued into the system due to the dynamis of onsumption a eting those of marginal ost. However these e ets are small (sine the steady state real interest rate is small) and the proess for in ation remains very lose to AR() in all our alibrations. As a result an (appropriately alibrated) output rule is always lose to a stati Taylor Rule. Figure shows the response of real debt to a unit in ationary shok. On Published by The Berkeley Eletroni Press,

18 The B.E. Journal of Maroeonomis, Vol. 7 [2007], Iss. (Contributions), Art. 9 impat, real debt falls, ontinues to fall for 5 quarters, then slowly rises bak to its steady state value. This path is the onsequene of the interation of two e ets. If the nominal value of debt were onstant ( = 0), the real value of debt would mirror the prie level, jumping down by % on impat, then falling gradually to a permanently lower long-run value (in the AR() ase this would be % below its initial value). This is shown by the dotted line in gure. But nanial institutions reset the nominal value of debt aording to the Calvo proess desribed above, whih ultimately brings the real value of debt bak to its steady state value. As an be seen from gure, on impat the % in ationary shok auses a % fall in the real value of debt sine nominal debt is set one period in advane. At rst, the in ationary shok s erosion of the real value of debt dominates the debt adjustment proess and real debt initially falls. As the in ationary shok deays, the debt adjustment proess dominates and the real value of debt gradually returns to its steady state. Figure : The dynamis of in ation and debt Inflation real debt with adjustment real debt, no adjustment From (32) the assets of unonstrained onsumers always equal the debt of onstrained onsumers. Given that the real value of debt returns to its steady state after a shok (whih it must do sine real ollateral is xed) the real value of assets must do so too so there are no long run e ets. 3.2 Floating rate debt Figure 2 shows the response to an in ationary shok of an eonomy in whih all debt is oating rate ( = 0). The burden of adjustment to the in ationary shok falls largely on onstrained onsumers (indeed on impat almost entirely 3 Quarters after the shok on the x-axis, deviations from steady state values on the y-axis. Note the x-axis here is longer than in subsequent gures. 6

19 Graham and Wright: Nominal Debt Dynamis, Credit Constraints and Monetary Poliy so). But the fall in aggregate onsumption (in this simple eonomy idential to the fall in output) is brought about despite an initial fall in real interest rates that is not reversed for nearly two years. To understand the features of the eonomy that lead to these responses it is useful to start by onsidering onstrained onsumption. Figure 2: All oating rate debt Inflation Unonstrained onsumption Constrained onsumption Output Inflation Nominal interest rate Real interest rate 0.4 The onsumption of onstrained onsumers is given by their linearised budget onstraint (0) whih, sine the steady state interest rate is small ompared to, an be written approximately as: 2t y 2t rt D (d t+ + t+ ) (37) ( ) t r D t (d t+ + t+ ) where the seond line follows from our alibration and the log linearisation of (30), whih implies that y 2t = y t = t +( ) 2t, where is the steady-state 4 Quarters after the shok on the x-axis, deviations from steady state values on the y-axis. Published by The Berkeley Eletroni Press,

20 The B.E. Journal of Maroeonomis, Vol. 7 [2007], Iss. (Contributions), Art. 9 share of unonstrained onsumption. So there are three e ets that determine the path of onstrained onsumption. The output response. Sine the entral bank sets the real interest rate to ahieve a fall in y t ; this redues onstrained onsumption in all periods. 2. The response of the debtor rate, rt D. With all oating rate debt, the rate paid on nominal debt is (in deviations) equal to the nominal interest rate, whih is above its steady state value in all periods after an in ationary shok even when the real rate is below steady state. So this redues onstrained onsumption in all periods. 3. The hange in nominal debt between periods t and t+ (given by d t+ + t+ ). The behaviour of nanial institutions means that in the period after the in ationary shok and thereafter, they issue new nominal debt to households to bring the real value of debt bak in line with ollateral. Sine onstrained households immediately onsume the additional real resoures from this new debt, this inreases their onsumption in all periods. In the simplest possible ase, if onstrained onsumers had no debt ( = 0); only the rst e et would operate, and their onsumption would simply trak the output response and hene (from the seond line of (37)) that of unonstrained optimising onsumers. Sine these in turn respond only to real interest rates the eonomy would in this restrited ase have the standard feature that a negative output response to in ation would require a positive response of real interest rates. With non-zero debt ( > 0) the response of onstrained onsumption will di er from that of unonstrained onsumption and output, the sign of the di erene depending on the relative magnitudes of e et (2) and e et (3), on whether the nominal interest rate rises by more than the rate of inrease of nominal debt. While most of the impat of the shok on nominal interest rates deays with in ation the adjustment of real debt is muh more prolonged. As a result, as an be seen from the impulse response funtions in Figure 2, in the pure oating rate ase initially e et (2) dominates; but as the nominal interest rate falls bak, e et (3) inreasingly dominates, with onstrained onsumption rising above its steady state after nearly two years before gradually falling bak towards it. This longer-term positive response to an in ationary shok is not a onventional wealth e et due to the fall in the real value of debt shown in Figure, 8

21 Graham and Wright: Nominal Debt Dynamis, Credit Constraints and Monetary Poliy sine onstrained onsumers are not intertemporal optimisers. The response is instead driven by the optimising response of nanial institutions. The fall in the real value of debt auses them losses in the short term, that they simply pass on to their owners, the unonstrained onsumers. Then they optimise by inreased lending to bring the real value of debt gradually bak in line with ollateral. Constrained onsumers respond by simply spending these additional funds. Given the time pro le of the response of onstrained onsumers, the resoure onstraint implies that unonstrained onsumption must respond on impat by less than onstrained onsumption (indeed it barely falls at all), but, with the real interest rate subsequently rising above steady state, unonstrained onsumption falls, and stays below its steady state for long after the impat on in ation itself has disappeared. However, the overall burden of adjustment borne by unonstrained onsumers is unambiguously redued, ompared to the ases where there were no redit onstraints or onstrained onsumers had no debt. 5 The pattern of falling, then rising real interest rates is onsistent with the nding (Clarida et al., 998, among others) that, exept in relatively reent years, the histori response of real interest rates to in ation in some ountries has been lose to zero or even negative. Rather than suh behaviour being due either to interest rate smoothing, or breahes of the Taylor Priniple, our model suggests that an initial fall in real interest rates may easily arise in eonomies in whih a signi ant proportion of households are redit onstrained and hold oating rate debt. 6 We disuss this issue further below, in Setion Fixed rate debt Figure 3 shows the response of the eonomy to an in ationary shok in an eonomy with only xed rate debt ( = ). While the aggregate output response is (by onstrution) idential to the ase with all oating rate debt, the distributional e ets on impat are almost preisely reversed. In this ase onstrained onsumption barely falls, so that the impat e et of the fall in output is almost entirely borne by unonstrained onsumers. 5 For a given desired output response to in ation on the part of the entral bank, this follows diretly from (37) and the identity for output. 6 It may also be optimal on a utility-based welfare riterion (see Wright, 2004). Published by The Berkeley Eletroni Press,

22 The B.E. Journal of Maroeonomis, Vol. 7 [2007], Iss. (Contributions), Art. 9 Figure 3 : All xed rate debt Inflation Unonstrained onsumption Constrained onsumption Output Inflation Nominal interest rate Real interest rate To understand this response, again onsider the three determinants of onstrained onsumption desribed above. E ets () and (3) are lose to idential to the oating rate ase sine the output response and debt responses are determined only by the proess for in ation whih (for reasons disussed in setion 3.) is barely a eted by the transmission mehanism. However e et (2) is very di erent. Now the rate on debt is equal to the xed rate rt F, given by equation (2) whih is very insensitive to hanges in nominal rates 8. This means that e et (2) is dominated by e et (3) in all periods after the shok, so that onstrained onsumption falls by muh less than unonstrained onsumption and output on impat, and responds positively after only two quarters. As in the previous ase there is a prolonged positive response as nanial institutions restore real debt levels in line with ollateral. 7 Quarters after the shok on the x-axis, deviations from steady state values on the y-axis. 8 It is doubly insensitive beause the average xed rate is a bakward moving average of the xed rate on any given ontrat, whih is itself a forward moving average of short rates. 20

23 Graham and Wright: Nominal Debt Dynamis, Credit Constraints and Monetary Poliy To ahieve a given response of output to in ation, the resoure onstraint means that unonstrained onsumption must follow a steeply rising path, requiring a large impat response of real interest rates Two mixed ases Figures 4 and 5 show the responses of the eonomy to an in ationary shok in two eonomies in whih there is both oating and xed rate debt. Figure 4 shows responses with a relatively low proportion of xed rate debt ( = 0:3, roughly orresponding to the UK); Figure 5 when the proportion is relatively high ( = 0:8; roughly orresponding to Frane). Now households an be divided into three types: unonstrained households; onstrained households with oating rate debt; and onstrained households with xed rate debt. Sine the aggregate onsumption response is idential in both ases, and the in ation response almost so these are not shown in Figures 4 and 5. Sine the aggregate output and debt responses are again determined by the path for in ation, and the interest rate on xed rate ontrats is very unresponsive to short-term interest rates, the onsumption response of onsumers with xed rate ontrats in both mixed ases is virtually idential to the ase in whih all ontrats were assumed to be xed. The big di erenes between the two ases therefore arise from the responses of onstrained onsumers with oating rate ontrats, and unonstrained onsumers. With a relatively low share of xed rate ontrats, as in the UK ase, Figure 4 shows that the overall response of onstrained vs. unonstrained onsumers is quite similar to that in the pure oating rate ase. As a result this ase displays the same pattern of an initial fall in the real interest rate that is only reversed after 5 quarters. The only di erene is that, with 30% of onstrained onsumers largely insulated from the impat of the shok, both onstrained onsumers on oating rate ontrats and unonstrained onsumers have to bear a larger burden of the aggregate adjustment. 9 Note that in this ase, for a given output response, and hene a given in ation and debt response, the approximation in (37) implies that the relative burden of adjustment on onstrained and unonstrained onsumers is almost preisely pinned down by ; the ratio of debt to onstrained onsumption. An inrease in, for example, ; will raise the responsiveness of unonstrained onsumers to real interest rates. But this will translate almost wholly into lesser rises in real rates in order to ahieve a given output response. It will have no other impat on the eonomy sine onstrained onsumers are almost entirely insulated from the impat of interest rates given the sluggish response of rates on xed ontrats. Published by The Berkeley Eletroni Press,

24 The B.E. Journal of Maroeonomis, Vol. 7 [2007], Iss. (Contributions), Art. 9 Figure 4 : 30% xed rate debt, 70% oating rate debt Unonstrained onsumption Total onstrained onsumption 4 5 Constrained ons fixed rate 6 7 Constrained ons floating rate 8 This latter feature is greatly aentuated in the Frenh ase shown in Figure 5, in whih 80% of onstrained onsumers are on xed rate ontrats. As a result the burden of adjustment has to fall on the remaining small proportion of onstrained onsumers on oating rate ontrats, and on unonstrained onsumers. This in turn requires a distintly more aggressive response of monetary poliy in terms of real interest rates. For oating rate onsumers the impat of this response on nominal rates implies over twie as large a fall in their onsumption as when all debt ontrats are at oating rates. Figure 5 : 80% xed rate debt, 20% oating rate debt Unonstrained onsumption Total onstrained onsumption 4 5 Constrained ons fixed rate 6 7 Constrained ons floating rate 8 This suggests the presene of a network externality e et: as more households join xed mortgage shemes, so the insurane bene t of a xed rate sheme inreases, in omparison to oating rate shemes. The rise in real interest rates required to satisfy the output rule (28) beomes larger, and as a 20 Quarters after the shok on the x-axis, deviations from steady state values on the y-axis. 22

25 Graham and Wright: Nominal Debt Dynamis, Credit Constraints and Monetary Poliy result more of the burden of adjustment is passed on to onstrained onsumers with oating rate debt. Other things being equal, it might be expeted that the existene of suh a strong network externality e et would drive oating rate debt out of existene. The fat that we do not observe markets in whih xed rates are ompletely dominant therefore requires other things to be happening. The most obvious explanation is the existene of a term premium whih raises the ost of xed rate mortgages. 4 Monetary poliy 4. Taylor Rules vs. Output Rules We have modelled monetary poliy as an output rule so that aggregate e ets are held onstant as we varied the transmission mehanism in the above setions. How would the results disussed above vary if we modelled monetary poliy diretly as a Taylor Rule rather than by our assumed output rule? We noted in setion 3. that the dynamis of in ation and output mean that the output rule we assume is very lose to mimiking the impat of a simple stati Taylor Rule. But we also showed that as the proportion of xed rate ontrats rose real interest rates needed to respond more aggressively to the in ationary shok in order to ahieve a given fall in output. By impliation, if instead we held the Taylor Rule oe ients onstant the impat on output would be redued, and hene there would be less of a stabilising impat of monetary poliy. The e ets are non-trivial: the implied output response on impat in the ase of 80% xed rate debt with xed Taylor Rule oe ients would be only just over half that in the ase of 30% xed rate debt. 2 This implies that a monetary authority in a ountry with a high proportion of oating rate debt (e.g. the UK) should be able to adopt a less aggressive monetary poliy in terms of real interest rates than one with a high proportion of xed rate debt (e.g. the US). However, empirial estimates for Taylor rule oe ients for reent years are roughly the same in these two ountries. Our model suggests therefore that the Bank of England is more averse to in ation than the Fed. This in ation aversion has signi ant distributional onsequenes, with muh of the burden of in ation stabilisation borne by onstrained households with oating rate debt. 2 The entire pro le with Taylor Rule oe ients of.5 and 0.5 in this ase is extremely lose to the pro le with an output rule as in (28), but with = -0.94, rather than the gure of -.76 we assume in our baseline alibration. Published by The Berkeley Eletroni Press,