The SWJ ‘textbook’ derivation, although useful from an intuitive perspective, is based upon a non-forward-looking, reduced form model rather than a forward- looking, structural model and is therefore potentially subject to the Lucas (1976) critique. We therefore proceed to derive a set of implicit interest rate rules from several structural economic models, in keeping with the ‘Modern Macro’ approach (e.g. Gillman, 2011). To begin with we consider two ways in which an implicit interest rate rule can be derived from the NK model under two alternative specifications for monetary policy – the first is where the standard interest rate rule is replaced by ‘strict inflation targeting’ and the second is where it is replaced by a money supply (growth) rule.
Consider the following two equations of the simplest version of the NK model (as presented by Woodford 2003, p.246):
̅ ̅ ( ̅ ̅ ̅ )
(1.10) ̅ ̅ ̅
(1.11) where ω (>0) is the intertemporal elasticity of substitution (the inverse of the coefficient of relative risk aversion) from the underlying utility function, κ (>0) is an amalgam of parameters which reflects, amongst other things, the assumed degree of nominal rigidity emanating from the price-setting behaviour of firms
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(i.e. the Calvo, 1983, ‘reset probability’), β (0<β<1) is the discount factor and all other terms have been defined elsewhere.20
Adopting the usual terminology, equation (1.10) represents the ‘New Keynesian IS equation (NKIS)’, which is essentially an intertemporal Euler equation expressed in terms of the output gap rather than consumption and y=c in this simple model, and equation (1.11) represents the ‘New Keynesian Phillips Curve’ (NKPC) which documents the trade-off between inflation and the output gap engendered by the nominal rigidities incorporated into the model. Unlike the original Phillips curve, the NKPC takes inflation expectations into account so that the Phillips curve shifts with expected future inflation. This two-equation system requires a representation of the monetary policy process to close the model. Usually a Taylor-type rule such as (1.2) would be added but following Woodford (2003, p.290), suppose that the central bank commits to deliver the target rate of inflation (π*) – a ‘strict inflation targeting’ regime – but still using the nominal
interest rate as its policy instrument. Woodford explains that:
“Under such a specification of the policy rule, the central bank’s explicit commitment is to the achievement of the target criterion, which need not involve any explicit reference to the desired path of the nominal interest rate, even though this is the central bank’s policy instrument.” (Woodford, 2003, p.290, emphasis added)
In this sense there is an interest rate rule implicit in the NK model under this alternative policy regime. To derive the precise form for this implicit rule, begin by substituting the NKIS equation for the output gap (1.10) into the NKPC (1.11) to give (this derivation is presented and discussed by Woodford, 2003, pp.290- 295):
̅ ( ) ̅ ̅ ( ̅ ̅ )
(1.12)
20 Equations (1.10) and (1.11) are written in deviation-from-steady-state form. Woodford (2003)
often employs this sort of approximation hence his analysis applies to situations in which variables do not ‘stray too far’ from their steady-state values. Further note that steady-state inflation is usually set equal to zero (Woodford, 2003, p.79); Coibion and Gorodnichenko (2011) generalise the model to allow for trend inflation.
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Further suppose that the central bank has full knowledge of the ‘correct’ model of the economy so that it is able to evaluate period t inflation as a function of its interest-rate decision. In this case, the nominal interest rate that would lead it to project an inflation rate consistent with the target (π*) is obtained by equating the
left-hand side of (1.12) with the inflation target and solving for the nominal interest rate. Also using the steady-state counterpart to the NKPC, ̅ ̅ (
) , we obtain:
̅ ̅ ̅ (
) ( ̅ ̅ ) ( ̅ ̅ )
(1.13) which represents the interest rate rule implicit in ‘strict inflation targeting’ according to Woodford’s (2003, p.246) model. It differs from Taylor’s original rule and equation (1.9) for SWJ’s derivation because it features forward-looking terms but is similar in that it includes terms in inflation and output gaps. Equation (1.13) shows that the coefficient on (expected future) inflation deviations from target exceeds unity since β, κ, ω > 0 and that the coefficient on the (expected future) output gap is positive and will usually be smaller than the coefficient on inflation in a manner consistent with conventional rules which typically prescribe more decisive responses to inflation than to output. As Woodford (2003, p.294) notes, this means that the coefficients of (1.13) would satisfy the ‘Taylor principle’ if it were to be interpreted as a conventional rule. For example, under a ‘standard’ calibration of β=0.99, κ=0.024 and ω-1=0.16 (Woodford, 2003, p.341), the
coefficient on inflation in (1.13) would take a value of 7.60 and the coefficient on the output gap would take a value of 0.16. Under this calibration, the inflation coefficient takes a magnitude substantially higher than the value of 1.5 that Taylor (1993) assigned to βπ in his contemporaneous rule (1.2) or the inflation
coefficient of 2.15 that Clarida et al. (2000, Table II) obtain for a ‘Volcker- Greenspan’ sample of U.S. data.21 On the other hand, the magnitude of the output
21 Clarida et al. (2000) evaluate an interest rate rule which accounts for ‘interest smoothing’ by
including a lagged dependent variables in the estimating equation. The estimate of 2.15 refers to their long-term coefficient estimate, i.e. once the protracted adjustment process has fully played out. We return to this issue in Chapter-IV when conducting similar estimation exercises to theirs.
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gap coefficient is substantially smaller than Taylor’s calibration of 0.5 or Clarida et al.’s (2000, Table II) ‘Pre-Volcker’ estimate of 0.83. Consistent with Woodford’s (2003, p.294) interpretation of (1.13) as the interest rate rule which “implements” strict inflation targeting in the NK model, policymakers react more decisively to inflation deviations from target and less decisively to the output gap than a conventional interest rate rule usually requires. It is difficult to compare equation (1.13) with SWJ’s equation (1.9) because the model used to derive the former doesn’t contain a money demand function and the model used to derive the latter doesn’t contain structural parameters such as β, κ and ω. For the time being, we simply note that the coefficients on both inflation and the output gap in (1.13) vary inversely with the intertemporal elasticity of substitution, ω.
Although Woodford’s derivation yields some intuitively appealing results, his derivation still takes place in a model in which the short-term nominal interest rate is assumed to be the instrument of monetary policy. Minford (2008), on the other hand, presents a modified NK model which frames the monetary policy process in terms of a monetary aggregate instead.22 He derives an implicit Taylor-
type rule by combining the money-supply-based approach taken in the SWJ derivation with the micro-founded nature of Woodford’s derivation. Just like the standard NK framework, his model consists of a NKIS equation and a NK Phillips Curve but a money growth rule is then added to represent monetary policy and a money demand function is also specified in order to form equilibrium in the money market. The modified NK model is as follows:
̅ (1.14) ̅ ̅ ̅ ( ) ̅ (1.15) ̅ (1.16)
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̅
(1.17) ̅ ̅ ̅
(1.18) where (1.14), the NKIS equation, is similar to (1.10) above but it features the level of output rather than the output gap and it does not restrict the parameter on expected future output to take a value of unity. The NKPC (1.15) is similar to (1.11) except that it allows for some degree of ‘backward indexation’. The two equations relating to money supply and money demand (1.16) and (1.17) do not feature in Woodford’s derivation and the latter now contains a constant interest elasticity of money (λR) as opposed to the constant semi-elasticity (ηR) which
featured in equation (1.3), and the Fisher relation (1.18) provides the link between real and nominal variables.
In order to derive the model’s implicit interest rate rule, Minford (2008) sets the first difference of the money demand function (1.17) equal to the exogenous process for the money supply (1.16), where the term in the first difference of output is eliminated using the first difference of the NKIS equation (1.14) to give:
̅ ̅ ̅ ( ̅ )
(1.19) Using the first difference of the Fisher relation (1.18) to eliminate ∆ ̅ and adding steady-state values in first-difference form (there is no trend growth in any of the variables so the latter take a value of zero) gives:
( ) ( ̅ ̅ ) ̅ ̅ ( )
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Rewriting this expression by expanding the Δ ̅ term and applying the steady- state Fisher relation ( ̅ = ̅ ̅ ) to one of the two ̅ terms contained in ̅ gives: ̅ [( ̅ ̅ ) ( ̅ ̅) ( )] {( ̅ ̅ ) [ ̅ ( )] ( )} (1.20) where χ≡αλR–ωηy. The expression in square brackets takes the same form as a
conventional interest rate rule such as (1.2) if the steady-state condition ̅ ̅ is applied. The remaining terms are collected in braces and would form a ‘composite error term’ if one was to try and estimate a conventional interest rate rule from model-simulated data. Using the definition of χ provided above, the coefficient on inflation in (1.20) is:
(1.21) The magnitude of this coefficient depends inversely on the interest elasticity of money demand and positively on the income elasticity of money demand whereas the coefficient associated with inflation in (1.9) for the SWJ model depended only on the former. The coefficient on inflation also depends positively on the intertemporal elasticity of substitution, ω; this differs from equation (1.13) derived from the NK model under strict inflation targeting. The coefficient on inflation in Woodford’s (2003) implicit rule also depends upon the parameters of the NKPC but these do not feature in (1.21). Applying the Rotemberg and Woodford (1997) calibration to (1.21), the ratio of the structural parameters of the IS equation (α/ω) is found to be equal to 0.16.23 Therefore, under a ‘velocity
specification’ (ηy=1), (1.21) exceeds unity if λR is smaller than 0.86
23 Specifically, α=1 and ω=6.25. However, this calibration may not be entirely appropriate because
Rotemberg and Woodford (1997) and Woodford (2003) use Calvo contracts to generate nominal rigidities whereas as Minford (2008) uses Fischer contracting instead – i.e. ‘sticky wages’ (Fischer) rather than ‘sticky prices’ (Calvo).
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(approximately). As discussed in the context of the SWJ derivation, a money supply rule generates an interest rate rule which mimics a ‘desirable’ interest rate rule if money demand is not too sensitive to the nominal interest rate – i.e. if the elasticity λR is ‘small’.
Using the definition of χ provided above, the coefficient on the output gap in (1.20) is:
(1.22) This is found to depend upon the ratio of income and interest elasticities in the same way that the corresponding coefficient in (1.9) did for the QTM-based model but, again, this coefficient can now be linked back to structural parameters of the intertemporal IS equation. The intertemporal elasticity of substitution is found to enter (1.22) with the opposite sign to Woodford’s (1.13), where the coefficients on both inflation and the output gap depended negatively upon ω.
The composition of the term in braces in (1.20) is also particularly interesting because it features a lagged dependent variable with a coefficient of unity, as found in interest rate rule written in first differences. It also features a lagged output gap term, as found in ‘speed-limit’ interest rate rules (e.g. Walsh, 2003b) and a term in expected future inflation, as one would find in a forward-looking interest rate rule (e.g. Clarida et al., 2000). Equation (1.20) could be rewritten so that any of these terms are extracted from the error term and placed into the systematic component in square brackets. The complex nature of the error term also suggests that estimating (1.20) would be challenging from an econometric perspective. In summary, this derivation shows that either a contemporaneous (Taylor, 1993) rule or one of the several prominent descendants of the original rule can be generated as an implicit rule derived from a NK model in which the central bank uses the money supply as the instrument of monetary policy.
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