Mortgages and monetary policy Liam Graham∗ and Stephen Wright† 24 February 2004
Abstract How is monetary policy transmitted through its impact on mortgage borrowings? How does this transmission mechanism depend on whether mortgages are fixed or floating rate? To address these questions, we add a model of financial institutions’ behaviour to a dynamic general equilibrium model in which some households face credit constraints. We use this model to analyse the real effects of inflationary shocks, and derive implications for monetary policy. We question the conventional wisdom that inflationary shocks necessarily benefit debtors and provide a rationale for interest rate smoothing on the part of central banks.
1
Introduction
One mechanism by which monetary policy affects the economy is through its impact on household balance sheets. An inflationary shock changes the real value of debt. The response of monetary policy changes the interest payment on the debt. The interaction of these effects determines the behaviour of the economy following the shock and the response of monetary policy need to achieve a particular target. This paper provides a theoretical framework within which this part of the transmission mechanism can be analyzed. The most significant component of household debt is borrowing to purchase a residential property: mortgages. We identify how the burden of adjustment to an inflationary shock is shared between the credit constrained and the ∗
Corresponding author: Department of Economics, University of Warwick, Coventry, CV4 7AL, UK; tel: +44 24 7652 8418; email:
[email protected] † Department of Economics, Birkbeck College, University of London, Malet Street, London W1E 7HX, UK; tel: +44 20 7631 6448; email:
[email protected]
1
credit unconstrained, and between those with fixed rate debt and those with floating rate debt. This has immediate policy implications. How should policy vary between countries with different mortgage systems? Would the UK (with predominately floating rates) incur costs or benefits from joining the euro-zone (with predominantly fixed rates)? Conventional wisdom on this topic can be summarized by three statements: 1. The nature of mortgage debt affects the transmission of monetary policy 2. Monetary policy "leans against the wind" 3. Inflationary shocks benefit debtors We find that these three statements are mutually inconsistent. The intuition behind this results is as follows. If the level of indebtedness, or the nature of mortgage finance is to matter, some households must face, or behave as if they face, binding credit constraints. Were this not the case, households could freely smooth their consumption in response to an inflationary shock. Then the level of debt would have no effect on their consumption, and the only impact of different mortgage systems would be a second-order one resulting from any change in lifetime wealth. Therefore a basic assumption of this paper is that credit constraints are binding for some households. This issue has been widely addressed in the literature and we do not propose to examine it further here1 . Inflationary shocks are held to benefit debtors by reducing the real value of their debt. However if monetary policy leans against the wind, the real interest rate will rise after the shock, and so the nominal rate will rise by more, and we show that for any conventional calibration of the model, the immediate benefit of reduced real debt is easily outweighed by higher nominal interest payments and the impact on output, so consumption of both constrained and unconstrained households is cut sharply. Therefore the average debtor does not benefit from the inflationary shock on impact. 1 For a detailed discussion of the presence of credit constraints see Mankiw (2000) and references therein. An additional piece of evidence suggestive of a combined role for nominal debt contracts and credit constraints is the well-known empirical correlation between consumption and nominal (as opposed to real) interest rates (e.g. Fuhrer and Moore, 1995)
2
What about in the long-run? Financial institutions lend to households, with contracts in nominal terms, and a particular real level of debt maximizes their profits. After an inflationary shock, banks attempt to adjust the real level of debt back to this optimal value so offer new debt to households. If there are costs to adjusting the nominal value of debt, this will be a sluggish process which we represent using a version of Calvo’s (1983) model. Households facing a binding credit constraint will accept this new debt. Hence the real level of debt in the economy returns to its steady state. So too do nominal interest rates and real debt and therefore consumption, when the inflationary shock has passed: the shock has no affect in the long run. Does the size and nature of mortgage debt matter? Central banks certainly think so as the following extract, taken from the Bank of England’s Quarterly Inflation Report for May 2002 shows: "Given the sustained strength of consumption to date, does this high level of debt constitute a serious risk to future consumption growth. One way in which debt can affect consumption is when households, having insufficient liquidity to meet their debt payment commitments,must adjust their spending" We analyze the monetary policy rule that results from a central bank minimizing a conventional loss function in the presence of this "mortgage debt" channel and compare it with the policy rule derived in the absence of this channel. A number of authors have pointed out central bank’s tendency to smooth interest rates. We show that persistence in real interest rates is a direct consequence of the sluggish adjustment of nominal debt contracts. We discuss the ways in which this transmission mechanism changes according to whether mortgages are predominately floating rate or fixed rate. We confirm the intuition that fixed rate mortgages make the economy less responsive to monetary policy. Indeed, they are often seen as insurance against nominal shocks. We show that although such an insurance effect of fixed rate mortgages exists, the insurance is only partial These questions will be of interest to central bankers and policy makers. The recent review of the UK housing market commissioned by the Treasury, Miles (2004), stated
3
"The nature of the transmission mechanism of monetary policy reflects the structure of housing finance. A move to much greater use of fixed-rate mortgages would alter that mechanism" The diversity of arrangements for mortgage finance within the European Union2 is described in Maclennan, Muellbauer and Stephens (1999). In a number of countries - most notably the UK - interest payments are either directly related to a floating rate, or are fixed for relatively short periods. In others, interest payments are often fixed for much longer terms. We analyze how different arrangements change the policy transmission mechanism, and how the burden of adjustment to the shock is shared between different groups. Our model nests all possible cases along a continuum ranging from an economy with only floating rate mortgages to one with only fixed rate mortgages. Within this framework we analyze "optimal" policy, in the sense of Clarida et al (1999)3 . Our paper will also be of interest to theoretical macroeconomists. The reaction of financial institutions to an inflationary shock and the role of mortgage debt has been largely neglected in the literature. We construct a simple version of the dynamic general equilibrium models common in the literature, for example Goodfriend and King (1997), modified to take into account the following empirical observations: 1. Nominal debt contracts (almost invariably backed up by some form of collateral - of which residential housing is the quantitatively most important element) are the norm in most countries with reasonably stable inflation 2. Adjustments to such contracts are relatively costly, and hence relatively infrequent - hence nominal debt contracts are “sticky” 3. A significant fraction of households face binding or near-binding credit constraints in the short-term, even when they may have significant positive net worth 2
In the US mortgage interest payments are usually fixed for the notional term of the contract; but borrowers have the right to repay at any point without penalty, thus giving the contract some of the features of a floating rate contract. 3 This is optimal policy in the sense that the central bank minimises an "outside" objective. Wright (2004) considers a central bank with an "inside" objective: that sets monetary policy to maximise the utility of a representative household.
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It is perhaps surprising that the combined impact of the first two features on nominal inertia has thus far played little or no role in mainstream macro analysis, despite the enormous literature dealing with various aspects of product price stickiness. One well-known criticism of the standard sticky price model is that the “menu costs” that ultimately must generate product price stickiness are unlikely to be large. In the case of debt contracts, in contrast, the costs of adjustment may well be distinctly larger, since typically this will involve re-assessment of collateral or other features of creditworthiness - neither of which is a low-cost activity. This cost differential appears to be reflected in average contract lives, implying that debt contracts are distinctly “stickier” in nominal terms than are product prices. The combination of all these features implies a potentially important impact of nominal inertia in the transmission mechanism of monetary policy, with significant effects the case of inflationary shocks, and the response of policy thereto. In what follows, section 2 presents the model and describes how it is solved and calibrated. Section 3 describes our results and section 4 concludes.
2
The Model
The model is, in most respect, an off-the-shelf dynamic equilibrium model. Households consume final goods, supply labour and hold financial assets. Intermediate firms produce differentiated goods which are imperfect substitutes in the production function of final goods firms thus motivating price setting behaviour and the presence of a new Keynesian Phillips curve with a lagged term in inflation. A monetary authority sets the real interest rate to achieve a given policy objective à la Clarida et al (1999). We only sketch these parts of the model, for a more complete exposition see Bernanke et al (1999). We modify this standard model by splitting households into two types according to whether they face binding credit constraints or not, and adding financial institutions who make collateralized loans to households. In what follows upper case letters refer to levels, lower case to their log-linearizations. Symbols without time subscripts refer to steady-state values. 5
2.1
Households
Households consume, supply labour hold financial assets4 and are endowed with a single physical asset, which can be thought of as a residential property. Throughout what follows, we assume the price of the residential property is constant. Relaxing this assumption would strengthen our results. As Miles (2004) notes, in a large-scale simulation allowing house prices to vary approximately doubles the strength of the "mortgage debt" channel. We divide households into two types: type 1 are credit unconstrained and type 2 face binding credit constraints, and we analyze a single average household of each type. All households have infinite horizons and rational expectations. Households maximize a lifetime utility function given by: max
{Cjt+i ,Njt+i }∞ i=0
U = Et
∞ X
β i u (Cjt+i , Njt+i )
(1)
i=0
subject to series of real budget constraints which equate real income (from labour and last period’s assets) with real expenditure on consumption and next period’s assets5 . Yjt+i + Ajt+i (1 + Rjt+i ) = Cjt+i + (1 + Πt+i+1 ) Ajt+i+1
(2)
where Cjt is consumption, Yjt = Wjt Njt labour income, Njt is labour supplied, Wjt the wage earned by type j labour and Ajt the household’s stock of financial assets at the start of period t. Πt+i is the rate of inflation between periods t + i − 1 and t + i, and Rjt+i the nominal interest rate payable on assets given by: Rjt+i = Rt+i if Ajt+i > 0 Rjt+i =
D Rt+i
if Ajt+i < 0
(3) (4)
where Rt is the short-term interest rate set by the central bank, RtD the rate payable on borrowings which we consider in detail later. Households face a 4
We follow the convention in the literature of building a monetary model without money. See McCallum (2001). 5 This can be derived from an asset evolution equation in nominal terms N AN jt+1 = Pt Yjt − Pt Cjt + (1 + Rt )Ajt
where nominal assets AN t = At Pt
6
further constraint Ajt+i > −Djt+i
(5)
where Djt+i is the level of debt which financial institutions are prepared to lend households. For credit unconstrained consumers, the optimal value of Ajt+i always satisfies this constraint as a strict inequality. We assume that, in the absence of this constraint, type 2 households would always choose a value of Ajt+i which would violate this constraint: they always want to borrow more than financial institutions are prepared to lend them6 . This can be most easily thought of in a life cycle model of consumption. For some consumers the optimal level of consumption, out of current financial wealth and future lifetime earnings, would imply a current level of debt greater than that which a financial institution would be prepared to lend, given that consumer’s collateral. This might apply, for example, for consumers relatively early in the life cycle, for whom future labour income (that cannot be collateralized) makes up a significant fraction of total wealth. Though households later in the life-cycle may hold significant financial assets (e.g. pensions funds) which cannot neither be realized nor used as collateral. With the following form for the instantaneous utility function 1− σ1
U (Cj , Lj ) =
Cj
1−
1 σ
+
(1 − Nj )1+ϕ 1+ϕ
(6)
it is straightforward to derive a standard linearized Euler equation for type 1 households. c1t = Et c1t+1 − σ (rt − Et π t+1 ) (7) For type 2 households, the existence of a binding credit constraint means that they are at a corner solution and are not full optimizers. Their consumption is then given by the budget constraint (2): for a fuller discussion of the effects of credit constraints on consumption see Muellbauer and Murphy (1997). Linearizing this budget constraint gives: ¡ ¢ c2t = y2t − δrtD + δ (∆dt+1 + π t+1 ) + δRD y2t − dt − rtD
(8)
6 Implicit is the assumption that the shocks we analyse are never big enough to reduce a credit constrained households optimal level of assets sufficiently to free it from the constraint.
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where δ = D C is the steady-state debt to consumption ratio of type 2 households and y2t is labour income. The sum of the first two terms on the right-hand side is the disposable income after debt repayments holding debt constant. The third term represents the change in nominal debt over the period: if banks offer new debt they constrained households accept it and consume it. The first-order conditions for labour give a linearized labour supply function: N 1 −ϕ (9) njt+i − cjt+i = wjt+i 1−N σ
2.2
Financial institutions
Financial institutions make loans to households based on nominal contracts. We assume that debt contracts are sticky in nominal terms. To capture this in a tractable way, we progress by analogy to Calvo’s (1983) model of the aggregate price level. We assume a constant probability φ that any given debt contract will be adjusted in the next period, with complete adjustment towards its optimal value if adjustment does take place7 . Within this simple model we combine two processes. Banks face costs, analogous to the "menu costs" discussed in the price adjustment literature, in measuring the creditworthiness of individuals. Households only re-mortgage infrequently. Following Kiyotaki and Moore (1997) we assume that lenders cannot force borrowers to repay their debts unless they are secured. The optimal value of debt is then given by some constant fraction (which we normalize to unity) of the level of the households collateral K which we assume is constant over time and across households. Further, we assume that the cost, Ωt , of debt deviating from its optimal level is quadratic: Ωt+i =
µ
Zt+1 /Pt+i − K K
¶2
K
(10)
where Zt+1 is the nominal value of a new debt contract set at time t which will be in force from period t + 1. When a financial institution is able to reset the value of its outstanding contracts, it does so to minimize the 7
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 constant for any given stable monetary regime. See Graham and Snower (2004) for an example of nominal contracts whose frequency of adjustment varies with the steady state value of inflation.
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cost of the debt deviating from its optimal value over the expected contract period: ∞ X minEt [β (1 − φ)]i Ωt+i (11) Zt+1
i=1
Deriving first-order conditions and linearizing gives zt+1 − Et pt+1 = Et
½
[1 − B (1)] F π t+1 B (F )
¾
(12)
where F is the forward shift-operator (F i xt = xt+i ) and B (F ) = 1 − β (1 − φ) F . This condition gives the expected value of a new real debt contract at time t + 1 in terms of the expected path of inflation.
2.3
Aggregate debt
A proportion φ (1 − φ)i of financial institutions will have reset their contracts at time at time t − i and have not had the opportunity to reset them since. So we can sum over all contracts and all financial institutions to obtain the real value of aggregate debt Dt Dt+1 =
∞ φ X
Pt+1
i=0
(1 − φ)i Zt+1−i
(13)
Linearizing this gives an expression for the evolution of aggregate debt in terms of the value of individual debt contracts dt+1 + pt+1 =
A (1) zt+1 A (L)
(14)
where A (L) = 1 − (1 − φ) L and L is the lag operator (Li xt = xt−i ).
2.4
Floating and fixed rates
The interest rate payable on this debt can be either floating or fixed. We do not attempt to model the households decision on whether to hold fixed or floating rate debt, though we will have something to say about the factors influencing the decision. See Campbell and Cocco (2003) for such a model, and Miles (2004) for an interesting discussion about the way UK households make this decision.
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If there is some proportion Ψ of borrowers in fixed schemes the average rate payable by credit-constrained borrowers on outstanding debt will be RtD = ΨRFt + (1 − Ψ) Rt
(15)
where Rt is the the short-term nominal interest rate8 and RtF the nominal interest rate payable on fixed debt. Financial institution will choose the (fairly priced) fixed rate on a particular debt contract, Rtz , as the average of expected nominal rates over the duration of the contract: Rtz
= Et = Et
P∞
i i=1 (1 − φ) Rt+i P∞ i i=1 (1 − φ)
A (1) Rt+1 A (F )
(16) (17)
The average fixed rate Rtf payable by borrowers will then be RtF
z = φRtz + φ (1 − φ) Rt−1 + φ (1 − φ) Rzt−2 + .... ∞ X ((1 − φ) L)i Rtz = φ
(18)
=
(20)
(19)
i=0
A (1) z R A (L) t
Linearizing around a steady state where R is constant so RD = RF gives rtD = ΨrtF + (1 − Ψ) rt
2.5
(21)
Firms
We model firms split between intermediate and final goods producers in the manner common in the dynamic general equilibrium literature. Here we only sketch the model, for a more complete exposition see Bernanke et al (1999). 8 We assume the spread between the central bank’s target rate and the debt rate is zero and constant.
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2.5.1
Labour demand
For convenience, we take the production function of a typical intermediate firm to be: 1−κ κ Yt = N1t N2t (22) where N1 and N2 , labour of type 1 and 2 households, are imperfect substitutes. The firm’s first-order conditions for the two types of labour are w1t N1t = (1 − κ) Yt
(23)
w2t N2t = κYt
(24)
i.e. households are paid the marginal product of their labour. Given these labour demand curves, we can use the household’s labour supply curve (9) to derive a direct relation between the labour and consumption of the two types: n2t − n1t = −µ(c2t − c1t )
(25)
where µ is a constant depending on the properties of the underlying production and utility functions. There will thus be offsetting movements in consumption and labour supplies, since, as consumption rises, the marginal utility of consumption falls, reducing the marginal incentive to supply labour. Volatility of consumption thus “spills over” into volatility of labour supply. 2.5.2
Price setting
The output of intermediate goods firms are imperfect substitutes in the production function of final goods firms who produce a single homogenous consumption good. This motivates price setting behavior on the part of intermediate goods firms and, with the additional assumption of Calvo pricing allows us to derive a new Keynesian Phillips curve. As is well known, the microfounded new Keynesian Phillips curve has difficultly in generating a realistic degree of inflation persistence so we follow the convention of adding, ad hoc, a term in lagged inflation to obtain: π t = ηπ t−1 + β (1 − η) Et π t+1 + γyt + ut 11
(26)
where the parameter η captures the stickiness of inflation, γ, a function of the underlying parameters, measures the sensitivity of inflation to deviations in output and ut is an white noise "cost-push" shock. See Kozicki and Tinsley (2002) for an attempt to provide microfoundations for a close relative of such a Phillips curve.
2.6
Monetary policy
We follow the standard “new Keynesian” approach summarized by Clarida et al (1999) in which the analysis of monetary policy without commitment can be thought of in two stages. If the central bank faces a loss function over output and inflation, they show that the optimal policy rule can be written as: yt = ζπ t (27) where ζ is a parameter related to the relative weights on output and inflation in the loss function. For ζ < 0 the bank "leans against the wind" contracting demand when inflation is above target. The second stage is to obtain a rule for the policy instrument, in this case the real interest rate, of the form rt − Et π t+1 = τ (L)π t
(28)
where τ (L) is (potentially) polynomial in the lag operator which will be determined by the transmission mechanism. In the case considered in Clarida et al (1999) the transmission mechanism is the economy’s optimizing IS curve and τ (L) is a constant When the rule for output (27) is combined with the new Keynesian Phillips curve (26)the process for inflation is given by π t = ρ (η, γ, ζ) π t−1 + εt
(29)
where εt is a mean-zero process of serially uncorrelated errors related to the Phillips curve shock ut and ∂ρ ∂ζ > 0. The more monetary policy leans against the wind, the less persistent will be inflation.
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2.7
Resource constraints
The economy’s linearized resource constraint is given by yt = λc1t + (1 − λ) c2t
(30)
where λ is the proportion of unconstrained households. The constraint for financial assets and liabilities A1t + A2t + ACBt = 0
(31)
where DCBt are the assets or liabilities of the central bank. Since there is no capital, and firms and financial institutions make zero profits, their value is zero. The intertemporal budget constraint of the central bank, using (31), is d ) + A1t (1 + Rt+1 ) −ACBt+1 = A2t (1 + Rt+1 ´ ³ d − Rt+1 A2t = (A2t − A1t ) (1 + Rt+1 ) + Rt+1
(32)
which tells us the quantity of assets or liabilities creates by the central bank in each period, determined by the behaviour of the private sector. Adherence to an interest rate rule implies that the assets of the central bank are determined endogenously. We usually think of a central bank as having net liabilities, but in the absence of outside money this will not in general be the case in our model. Our steady state assumption is that the net assets of the bank are zero.
2.8
Solution method and calibration
The economy is described by a set of ten equations (7), (8), (12), (14), (17), (20), (21), (26), (27) and (30) in ten unknowns, yt , c1t , c2t , π t , zt , dt , rt , rtD , rtF , rtZ . In general this system can be solved by standard matrix methods. However in the special cases of all floating rate debt (Ψ = 0) or all fixed rate debt (Ψ = 1) we can solve the system analytically. We do this by proposing a form for the policy rule, τ (L) =
τ 0 + τ 1L 1 − τ 2L
13
(33)
verify that it is consistent through the transmission mechanism and proceed by the method of undetermined coefficients. Our key analytical result is that, in case where all debt is floating rate τ2 = 1 − φ
(34)
and in all other cases, τ 2 will be very close to this value. This shows how the dynamics of the debt adjustment process have immediate implications for monetary policy. With this additional source of inertia a simple policy rule is no longer optimal. The slower debt adjusts (the smaller φ), the more the central bank will smooth interest rates. We return to this point below. We calibrate our model on quarterly data with the values shown in table 1. We choose the proportion of credit constrained consumers to be one half, after Mankiw (2000). Cox et al (2002), using data from the British Household Panel Survey, give debt to income ratios ranging from 4 among the lowest income households to 1 among the highest. We take the steady state debt to consumption ratio of type 2 households to be 2. An upper bound on the duration of debt contracts is given by average frequency with which individuals re-mortgage. For the debt reset probability, φ, we choose 1 0.1 implying the expected life of a debt contract is 0.1 = 10 quarters or 2.5 years. This is distinctly shorter than the notional maturity of most mortgage debt, but reflects the fact that mortgages are generally renegotiated on a number of occasions before maturity - most notably on moving house. We choose ζ, the measure of how strongly monetary policy "leans against the wind" to give an impact response of the real interest rate to roughly match the value in Clarida et al (1999) for the US economy with a given proportion of floating rate debtors. The discount rate β is equivalent to an annual real interest rate of approximately 4%. We choose the coefficient on the lagged term in the Phillips curve to give a realistic degree of persistence to inflation given the other parameters. For the coefficient of relative risk aversion we take a value of 2.
14
λ δ φ β ζ η γ σ
3
Table 1: Calibration Description proportion of unconstrained households steady state debt to consumption ratio of constrained households debt reset probability discount factor coefficient in monetary policy rule coefficient on lagged term in Phillips curve coefficient on output in Phillips curve intertemporal elasticity of substitution
Value 0.5 2.0 0.1 0.99 -2 0.75 0.08 0.5
Results
In this section we analyze the response of three model economies: (1) where all debt is floating rate (2) where all debt is fixed rate (3) a mixed case where some debt is floating and some fixed. The transmission mechanism is different in each of these cases, hence the optimal policy rule for output (27) implies different path for the real interest rate. In the floating rate case, real interest rates respond weakly to an inflationary shock. In the fixed rate case the response is much stronger. Given the monetary policy rule the response of output is the same in all cases, a 2% reduction on impact (since ζ = −2) then a gradually rise back to the steady state as inflation decays. By the resource constraint (30) aggregate consumption mirrors this. In what follows we discuss how this change in aggregate consumption is allocated between different types of household. An important component underlying each of these cases is the evolution of the real debt of constrained households, given by (14). Since this depends only on the path of inflation, we can consider it in isolation from the details of monetary policy or consumption. We then use a simple example to develop intuition for the evolution of consumption given this debt process.
3.1
The dynamics of debt
Figure 1 shows the response of real debt to a unit "cost push" shock to the Phillips curve. On impact, real debt falls, continues to fall for 5 quarters, then slowly rises back to its steady state value. This path is the consequence of the interaction of two effects. The inflationary shock on impact causes 15
the price level to increase by 1%, then to rise towards a new steady state of 1 1−ρ % above its initial value. Were the nominal value of debt fixed, the real value of debt would jump down by 1% on impact, than fall gradually to a 1 long-run value of 1−ρ % below its initial value as shown in the figure. But financial institutions reset the nominal value of the debt according to the Calvo process described above, which tends to bring the real value of debt back to its steady state value. As can be seen from figure 1, on impact the 1% inflationary shock causes a 1% fall in the real value of debt since nominal debt is set one period in advance. At first, the inflationary shock’s erosion of the real value of debt dominates the debt adjustment process and real debt initially falls. As the inflationary shock decays, the debt adjustment process dominates and the real value of debt gradually returns to its steady state.
3.2
The response of consumption
We can partition the impact of aggregate consumption into the effect were the real interest rate to remain constant; and the effect of interest rates changing to satisfy the monetary policy rule (27). Figure 2 shows the path of output in both cases. If interest rates are constant, the consumption of unconstrained households is then also constant by the Euler equation (7). For constrained households there are two effects. Firstly, the inflationary shock means that the nominal interest rate (the rate which is paid on nominal debt) rises. If the nominal value of debt were constant, this would mean repayments on debt would rise so consumption would fall. Secondly, the nominal value of debt follows the process described above, always staying below its steady state value which at a constant nominal interest rate would reduce payments on debt so increase consumption. To see the relative magnitude of these effects, use the resource constraint (30) to re-arrange the process for constrained consumption (8) to give9 : c2t =
1 [−δrt + δ (∆dt+1 + π t+1 )] 1−λ
(35)
The first term shows the effect of changes in the nominal interest rate, the second the change in the nominal value of debt both of which are exoge9
For this simple example we are assuming all debt is floating rate and that the steady state real interest rate, R, is zero.
16
nous to the household. On impact, the nominal value of debt is constant (since debt is set one period in advance) so the real value of debt falls. This means the second term is exactly zero so initially the effect of high nominal interest rates dominates and consumption falls below its steady state value. As the inflationary shock and hence the nominal interest rate, decays back towards the steady state, the debt adjustment effect takes over and constrained households benefit from newly issued debt by increasing their consumption. Of course, given the specification of monetary policy in our model, there is no monetary policy rule that will keep the real interest rate constant. However splitting the overall effect into a stage with interest rates constant neatly illustrates one of the central results of our analysis. Conventional wisdom has it that an inflationary shock benefits those with debt by eroding the real value of the debt. We show that this may not be the case in the short run, because the effect of increased nominal debt payments resulting from the inflationary shock outweighs the erosion of the debt. Nor is it the case in the long-run since financial institutions adjust the real value of debt back to its steady state value. In fact, the central bank raises the real interest rate on impact then lets it slowly decline back to its steady state value. The effect on unconstrained households is given by the Euler equation (7). On impact, unconstrained consumption will jump down, then, for as long as the real interest rate is above its steady state value, will follow a rising path back towards the steady state. The effect on constrained households is equally straightforward. Other things (specifically the process for inflation) being equal, an increase in the real interest rate raises the nominal interest rate, so increasing debt repayments and decreasing consumption. So the overall affect of the interest rate increase is to reduce the consumption of both groups. But as figure 2 shows, the dominant element is not that arising from monetary policy.
3.3
Floating rate debt
Figure 3 shows the response of the economy to an inflationary shock in an economy with all floating rate debt (Ψ = 0). The central bank leans against the wind, raising the real interest rate by a maximum of 7 basis points above its steady state value. Note how little the real interest rate has to rise to achieve the policy objective. With floating rate debt the economy is very 17
sensitive to the short-term interest rate so the central bank can lean against the wind in terms of output with only small changes in the real interest rate. Unconstrained consumption falls in response to the shock, then gradually rises back to its steady state value. However the impact on constrained consumption is much greater. On impact the nominal value of debt stays constant, so constrained consumption is given approximately by c2,0 = y0 − δr0
(36)
Since output falls, and the nominal interest rate rt rises the effect on constrained consumption will be greater than that of output. Constrained consumption falls on impact by over four times the fall in unconstrained consumption. After impact, constrained consumption rises as the nominal interest rate falls, and real debt declines. After 10 quarters it rises above its steady state value as constrained households benefit from financial institutions issuing them with new debt. This reverses the conventional wisdom regarding the impact of an inflationary shock on credit constrained individuals. With a monetary policy that leans against the wind, the positive effect on debtor consumption of reducing real values of debt is offset by a rise in the nominal interest rate and the reduction in income. Whereas unconstrained households can smooth these changes away, credit constrained households are, by definition, unable to do so hence suffer a large drop in consumption.
3.4
Fixed rate debt
How does this change if constrained households’ debt is all at a fixed interest rate? Figure 4 shows the response of an economy in which all debt is fixed rate (Ψ = 1). On impact, since nominal debt is constant and the fixed rate being sluggish, the response of constrained consumption is half that of output. Now the central bank raises real interest rates by almost 100 basis points in order to force unconstrained consumption down by three times the reduction in constrained consumption to satisfy the resource constraint and achieve the policy objective. The inflationary shock represents a net transfer of consumption from unconstrained households to constrained households10 . 10
Note that although the constrained appear to do better than the unconstrained, this is only true in deviations. Constrained households are still consuming less then they would optimally choose.
18
This reverses the relative costs from the floating rate case but the inflationary shock still imposes costs in terms of reduced consumption on both types of household. Fixed contracts are often described as providing against insurance against adverse shocks. Our model shows that this insurance can only be partial since while a fixed rate contract all but eliminates the effect of the nominal interest rate rise, the fall in income still reduces consumption.
3.5
Why take on floating rate debt?
A case with both floating and fixed rate debt is not a simple linear combination of the previous cases. Figure 5 shows a case with 50% of households borrowing on a floating rate basis and 50% on a fixed rate basis. Now we are interested in the responses of three different types of households: the credit unconstrained and credit constrained with floating or with fixed rate debt. The impact of the shock is shared unevenly between these two groups. The impact effects of the shock are -1% for unconstrained consumption, -1.4% for constrained households on fixed rate mortgages and -4.3% for constrained households with floating rate debt. The intuition behind this result is straightforward. There is an overall output cost from the central bank’s contraction of demand in response to the inflationary shock and a cost to the households of increased nominal debt payments resulting from the increase in nominal interest rates. Unconstrained households can smooth the cost of both these changes by taking on more debt. Constrained households on fixed rate debt are insulated against the second effect whereas those on floating rate debt suffer a reduction in their consumption from both effects. As the proportion of households in fixed rate schemes rises, so too does this effect. If 80% of households are in floating rate schemes, their consumption falls by 3.5% on impact whereas for those in fixed rate schemes the fall is 1.4%. If the proportion changes to 20%, the impact effect for those in floating rate schemes is 4.6%, and for those in fixed rate schemes remains at 1.4%. This suggests the presence of a network externality effect: as more households take fixed mortgage schemes, so the insurance benefit of a fixed rate scheme increases. The fact that we do not observe markets in which fixed rates are completely dominant requires other things to be happening. The most obvious explanation is the existence of a term premium which 19
raises the cost of fixed rate mortgages.
3.6
Can inflationary shocks benefit debtors?
The only way we can salvage the conventional wisdom that inflationary shocks benefit debtors only by relaxing the assumption that the central bank "leans against the wind". What if the central bank followed a (conventionally) destabilizing rule and kept output constant in the face of the inflationary shock? We can model this by setting ζ, the parameter of the monetary policy rule (27) to zero. Figure 6 shows the response of the model economy following a 1% white noise shock to the autoregressive process for inflation. We have shown above that were interest rates constant, consumption of unconstrained households would be constant and that of constrained households would vary, so output would follow the consumption of constrained households and violate the monetary policy rule. To offset the contractionary effect of constrained consumption and keep output constant, the central bank cuts the real interest rate on impact, then lets it rise to a maximum of just above its steady state value after 11 quarters, and subsequently adjust gradually back to its value. This makes unconstrained consumption jump up on impact, then fall to below its steady state value and then gradually rise back. The cut in the real interest rate shifts the response of constrained consumption up (since nominal debt repayments are everywhere lower). By the resource constraint, if the central bank sets the real interest rate to hold output constant, the path of constrained and unconstrained must be mirror images, and the inflationary shock can be seen as a transfer of resources from unconstrained households to constrained households. This seems to conform to the conventional wisdom that an inflationary shock benefits debtors. However this is only true in the short-run: an important difference is that the shock has no long run effect on constrained consumption since financial institutions gradually adjust the value of nominal debt so that real debt returns to its equilibrium value.
3.7
Monetary policy
The sluggish adjustment of debt in this model when combined with a simple rule for monetary policy gives rise to slow dynamics in the real interest rate.
20
The implied form for the policy rule (28) is given by (33) which can be rewritten as Rt = τ 2 Rt−1 + τ 0 π t + τ 1 π t−1 (37) By solving the model as described above we find that in the case of all floating rates τ2 = 1 − φ
(38)
and in the other cases we can show that τ 2 will be within a very small range of this value so the process for the interest rate will in all cases be close to Rt = (1 − φ)Rt−1 + τ 0 π t + τ 1 π t−1
(39)
This shows that interest rate smoothing emerges naturally from our model. The intuition for this result is straightforward: monetary policy must reflect the long lags in the transmission mechanisms. How does the policy rule implied by our model compare with empirically estimated Taylor rules? Clarida et al (1999) estimate rules which imply a contemporaneous response of the real interest rate to an inflationary shock of around 40 basis points. For the mixed case of 50% of households in fixed rate schemes (which corresponds roughly to the US) our model gives a coefficient τ 0 of 0.45, implying an interest rate increase of 45 basis points in response to a 1% inflationary shock. How does this response vary with the predominant mortgage system? Since the rate on fixed debt responds sluggishly to changes in the short term real interest rate, we would expect that as the proportion of fixed rate debt increases the central bank would have to vary the real interest rate by more to achieve a given monetary policy objective. Figure 7 shows the impulse response functions of the real interest rate in response to a 1% shock to the persistent process for inflation when the central bank leans against the wind (ζ = 2). With all floating rate debt, real interest rates only need to increase by a maximum of 5 basis points to achieve this policy. As the proportion of fixed rate debtors increases, so too does the impact effect of the interest rate: 15 basis points with 20% fixed rate; 43 with 60% and 110 with 100%. The weak response of interest rates in the case of floating rate debt is
21
discussed in Wright (2004). The central bank is able to lean against the wind in terms of output by a relatively small rise, or even a fall, in interest rates.
4
Conclusion
We began this paper by reference to three statements: mortgage debt affects the transmission of monetary policy; monetary policy "leans against the wind" and inflationary shocks benefit debtors. We have shown that these statements are mutually inconsistent, and that the only case in which inflationary shocks bring a net benefit to debtors is when monetary policy is neutral in output terms, so destabilizing in the conventional sense. The presence of slow lags in the transmission mechanism means that it is optimal for the central bank to smooth interest rates. Of course we are not suggesting this is the only reason a central bank would choose to smooth interest rates, but if there is sluggishness in debt contracts, there will be an "optimal" degree of smoothing arising from just this aspect of the transmission mechanism. When the central bank reduces output in response to the inflationary shock, both the magnitude of the policy response needed to achieve this and the way in which the resulting costs are shared amongst households depends on the predominant mortgage system. With all floating rates, constrained consumers incur more of the cost, with all fixed rates, it is unconstrained consumers who suffer most. In a model with both types, the highest cost is paid by constrained households with floating rate debt. This analysis has wide-ranging implications for policy. Our analysis of a mixed case consisting of households having both fixed and floating rate debt suggests that those the cost of adjustment to an inflationary shock is shared very unequally between different types of households. This implies that were a country with predominately floating rate debt (the UK) to join a monetary union consisting of countries with a higher proportion of fixed rate debt (France, Germany, Italy), it would subsequently bear a large proportion of the cost of adjustment to inflationary shocks. The model we have presented is highly stylized and focusses on a single aspect of the transmission mechanism. A fuller treatment would endogenize house prices, allowing another channel for monetary policy to be transmitted 22
to the economy, and examine how these channels interact with others.
23
Bibliography Bernanke, Ben, Mark Gertler and Simon Gilchrist, (1999), "The financial accelerator in a quantitative business cycle framework", in Taylor, John and Michael Woodford (eds.), "Handbook of macroeconomics", Amsterdam. Oxford. Elsevier, 1999. Brierley, Peter, Pru Cox and John Whitley, (2002), "Financial pressures in the UK household sector: evidence from the British Household Panel Survey", Bank of England Quarterly Bulletin. Calvo, Guillermo A, (1983), "Staggered prices in a utility maximising framework", Journal of Monetary Economics 12, pp.383-398. Campbell, John Y and Joao F Cocco, (2003), "Household risk management and optimal mortgage choice", Quarterly Journal of Economics 118 (4), pp.1449-1494. Clarida, R, Jordi Gali and Mark Gertler, (1999), "The Science of Monetary Policy: A New Keynesian Perspective", Journal of Economic Literature 37 (4), pp.1661-1707. Fuhrer, Jeffery C and George R Moore, (1995), "Monetary Policy TradeOffs and the Correlation Between Nominal Interest Rates and Real Output", American Economic Review 85 (1), pp.219-239. Goodfriend, Marvin and Robert G King, (1997), "The new neo-classical synthesis and the role of monetary policy", NBER Macroeconomics Annual. Kiyotaki, Nobuhiro and John Moore, (1997), "Credit Cycles", Journal of Political Economy 105, pp.211-248. Kozicki, Sharon and Peter A Tinsley, (2002), "Dynamic specifications in optimizing trend-deviation macro models", Journal of Economic Dynamics and Control 26, pp.1585-1611. Maclennan, Duncan, John Muellbauer and Mark Stephens, (1999), "Asymmetries in Housing and Financial Market Institutions and EMU", CEPR Discussion Paper 2062. Mankiw, Gregory N, (2000), "The Savers-Spenders Theory of Fiscal Policy", American Economic Review 90 (2), pp.120-125. McCallum, Bennett T, (2001), "Monetary Policy Analysis in Models Without Money", Federal Reserve Bank of St Louis Quarterly Review 83 (4), pp.145-160. Miles, David, (2003), "The UK Mortgage Market: Taking a Longer-Term
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View. Interim Report", HMSO Other. Muellbauer, John and A Murphy, (1997), "Booms and Busts in the UK Housing Market", Economic Journal 107 (445), pp.1701-1727. Taylor, John and Michael Woodford (eds.), (1999), "Handbook of macroeconomics", Amsterdam. Oxford. Elsevier. Wright, Stephen, (2004), "Monetary Stabilisation with nominal asymmetries", Economic Journal 114 (492), pp.196-222.
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Figure 1: The dynamics of debt 0.0 1
6
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Debt, no adjustment
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Figure 2: Output (aggregate consumption) 0.5
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-1.0 Output, constant real interest rate Output, monetary policy rule -1.5
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x-axes show time in quarters. y-axes percentage deviations from the steady state
A.1
Figure 3: Floating rate debt 0.5
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41 46 Unconstrained consumption Constrained consumption
-0.5
Output
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0.9 Nominal interest rate
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x-axes show time in quarters. Note scale on interest rate chart is shorter. y-axes percentage deviations from the steady state
A.2
Figure 4 Fixed rate debt 0.5
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41 46 Unconstrained consumption Constrained consumption Output
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A.3
Figure 5: 50% floating rate debt, 50% fixed rate debt 0.5
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A.4
Figure 6: Floating rates, neutral monetary policy 0.3
0.2 Unconstrained consumption Constrained consumption 0.1
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Figure 7: Real interest rate response with different debt mixes 1.20 1.00 0% 20% 40% 60% 80% 100%
0.80 0.60 0.40 0.20 0.00 1
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x-axes show time in quarters. Note scale on interest rate chart is shorter. y-axes percentage deviations from the steady state
A.5
