The methodology for assessing interest-rate policy rules: some comments
Martin WattsEmeritus Professor of Economics and Research Associate, Centre of Full Employment and Equity, The University of Newcastle, Newcastle, NSW, Australia

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Category:
Research Article
Published:
Dec 2021
Page Range:
275–285
DOI:
https://doi.org/10.4337/ejeep.2021.03.02
Keywords:
zero interest-rate policy (ZIRP); zero real policy rate (ZRRP); inflation models
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This paper is critical of the conceptual foundations and methodology adopted by in his exploration of the impact of different interest-rate policy rules on inflation. His modelling framework is too narrow to adequately discriminate between different interest-rate rules in terms of their broader macroeconomic impacts.

1 INTRODUCTION

Since the monetarist revolution of the late 1960s, monetary policy has been the dominant arm of macroeconomic policy in developed economies, initially in the form of unsuccessful attempts to target the money supply and since the 1980s the targeting of an overnight (interbank) rate. The Taylor rule formalised the belief that monetary policy should be responsive to both inflation and deficient aggregate demand (see Taylor 1993), despite the strictures of Tinbergen (1952) about the number of policy instruments equalling the number of targets.1

The aftermath of the global financial crisis saw the perpetuation of this dominance (Sharpe/Watts 2012), albeit with increasing reliance on unconventional monetary policy in the form of both quantitative easing and latterly negative interest rates in some countries, despite a brief period of stimulatory fiscal policy in 2009–2010. It is only since the onset of the COVID-19 pandemic in early 2020 and the ensuing macroeconomic and social crisis that the need for a major, but many economists would argue temporary, fiscal stimulus has been acknowledged.

However, a number of scholars have been debating the merits or otherwise of different fixed interest-rate (‘park it’) rules (see for example Aspromourgos 2011; Forstater/Mosler 2005; Lavoie/Seccareccia 2019; Mitchell 2009; Pasinetti 1981; Rochon/Setterfield 2007; 2012; Smithin 2007; 2016a; 2016b; 2018; 2020; Wray 2007).

In a recent paper, Smithin (2020) examines the properties of a model of inflation, which is grounded in his alternative monetary model (AMM) (Smithin 2018) and is designed to assess the consequences of different interest-rate policy rules.

By design, the implementation of a rule in the form of a constant or zero nominal or real interest-rate policy must be complemented by active fiscal policy in the pursuit of public purpose. Thus, Smithin’s assessment of the desirability of parking the interest rate must also take into account the capacity of fiscal policy to fill the macroeconomic policy void and the repercussions for financial stability.2 Smithin’s paper, however, is designed to make a much narrower and quite specific point, namely that through an examination of a series of inflation models, it can be demonstrated that policymakers have the capacity to implement a constant real policy interest rate.

The objective of this response is to argue that Smithin’s use of difference equations to model the impact of interest-rate rules on the inflation rate is problematic from both a methodological and conceptual perspective and has major algebraic flaws, some of which were exposed in earlier exchanges (see Smithin 2016b; Watts 2016; 2018). Also, the monetary formulation for his theory of inflation is unpersuasive. Thus, he does not make a strong case for any particular interest-rate rule on the basis of his analysis.

Formal modelling is just one way of conveying economic ideas and certainly important insights can be gained. But the use of a narrowly focused model to ‘prove’ that one interest-rate policy is superior to another is highly problematic.3 This modelling framework does not adequately discriminate between the different interest-rate rules in terms of their broader macroeconomic impacts.

We first present the three-equation model and assess its appropriateness to the exploration of the inflationary consequences of the different interest-rate rules. This is followed by a critical analysis of the theoretical foundations of the inflation model. Concluding comments complete the paper.

2 INFLATION MODEL

2.1 Introduction

To explore the impact on inflation of different interest-rate policies, Smithin (2020: 382) develops a three-equation model which takes the following form:
p=p0−λ(r−r−1)+w−a=z−λ(r−r−1)  0<λ<1,
where p is the inflation rate expressed in logs, r is the real rate of interest in financial markets, and w and a are the natural logs of average real wages and average labour productivity, respectively.4 p0 is defined as ‘the inverse measure of the state of bearishness in the money market, whereas λ is effectively the interest elasticity of money demand in a speculative theory of the demand for money, such as that found in The General Theory’ (Smithin 2020: 382).5 We replace p0+w−a with z.

Equation (1) shows that an increase in the real interest rate reduces the rate of inflation, but that the continuation of a high constant real rate of interest does not further decrease the inflation rate.6

The relationship between the nominal policy rate and the nominal interest rate in financial markets, that is, ‘pass through’, is represented by:
i=m0+m1i0   m0>0; m1<1,
where i is the nominal market interest rate and i0 is the nominal policy rate set by the central bank, m1 is the pass-through coefficient and m0 is defined as the average commercial bank mark-up between deposit and lending rates.
Equation (3) is claimed to be the usual definition of the real interest rate on money, where p+1 is the future rate of inflation. Then
r≡i−p+1=m0+m1i0−p+1.

2.2 Nominal interest-rate policies

We first consider a fixed nominal policy rate, i0, which of course includes the case of zero interest-rate policy (ZIRP). Substituting into the inflation equation (1) for the real rate of interest from (3) yields
p=z−λ(r−r−1)=z+λ(p+1−p).
After lagging the equation one period and collecting terms, Smithin (2020: 383, eq. 7) shows that this representation of ZIRP, or any constant nominal policy rate, yields an unsustainable time path for inflation, since the coefficient on the lagged rate of price inflation exceeds unity (see also Smithin 2016a).
p=[(1+λ)λ]p−1+(1λ)z
Thus, ZIRP is rejected and constant real interest-rate policies are considered. However, closer inspection reveals an error in equation (5), with the correct formulation being:
p=[(1+λ)λ]p−1−(1λ)z.
Thus, inflation rises ceteris paribus if the wage share declines, which indicates a form of misspecification, as opposed to a policy rule which generates instability of the inflation rate.

This observation highlights an underlying problem with this specification of the inflationary process by Smithin, with the current real rate of interest (which impacts on the current inflation rate) being defined in terms of the future rate of inflation.

There are two issues here. First, equation (6), which has ostensibly been solved for the current rate of inflation, is actually solved for the future rate of inflation. This is the source of misspecification which is a consequence of the real rate of interest prevailing in the current period being affected by the rate of inflation prevailing one period ahead, which is hard to justify. The real rate of interest should be redefined as
r≡i−p,
since it is the current rate of inflation which directly affects the current real interest rate and in Smithin’s view the prevailing rate of inflation, ceteris paribus.7

Smithin (2020: 384) acknowledges in his footnote 9 that the implementation of the policy rule would be driven by expectations of the future rate of inflation, p+1. However, in recent correspondence he acknowledges that he assumes perfect foresight on the part of economic agents about the rate of inflation that will prevail, and argues that his three-equation monetary model of inflation is designed to demonstrate a specific technical point, namely that real interest-rate stability can be achieved. Further, with the assumption of perfect foresight, solving a difference equation model is not required to demonstrate his claim about the achievement of the desired real market rate of interest by the central bank through its setting of the nominal policy rate.

In other words, if the inflation rate resulting from a given interest-rate policy is known with certainty, formal modelling is redundant. The policymaker can ensure that the real market rate of interest remains constant given correct knowledge about the pass-through relationship, in addition to perfect foresight. This will immediately stabilise the inflation rate at its steady-state value, z.

However, the respecification of the inflation equation based on (3′) is insufficient, because the current inflation rate which appears on the right-hand side (RHS) is unknown.8 If perfect foresight is dropped, then the real interest-rate term on the RHS must incorporate an expectations-based formulation for the inflation rate such as
pe=p−1e+γ(p−1−p−1e),
which represents adaptive expectations.9 Thus, the expected inflation rate in the current period is based on the expected inflation rate in the previous period adjusted by the difference between this rate and the actual inflation rate in the previous period. If the lagged form of expectations generation is used (that is, pe=p−1), as mentioned by Smithin, then γ=1.
r=i−pe=m0+m1i0−pe
Then, from (7), the inflation equation based on a constant nominal policy rate, i0, can be written as
p=z−λ(r−r−1)=z−λ(p−1e−pe)=z+λγ(p−1−p−1e)
p−1e=z+λγp−1−pλγ.
The expectations terms in the penultimate expression in (9) can be replaced by their solution in (10) (see for example Chiang, 1984: 592). This yields the following equation:
p−(1−γ(1−λ))p−1+λγp−2=zγ.
The properties of the solution will depend on the respective values of the parameters, λ,γ. The complementary function can be written as
pt+2+a1pt+1+a2pt=0.
This second order difference equation for a constant nominal policy rate has a quadratic characteristic equation, which can be written as
b2+a1b+a2=0,
and the solution takes the form pt=A1b1t+A2b2t, in the absence of repeated roots.
If a12>4a2, then the solution is real and
b1,b2=(−a12∓a12−4a22),
and the inflation rate fails to converge to its particular integral z, if the absolute value of one or both roots exceeds unity.
On the other hand, if a12<4a2, then the roots are imaginary and the solution takes the form
pt=A1b1t+A2b2t=A1(h+vi)t+A2(h−vi)t=Rt[(A1+A2)cos(θt)+(A1−A2)sin(θt)],
where h=−a12; v=4a2−a122; R=a2; cos θ=hR; sin θ=vR (Chiang 1984: 576–581).

To investigate the properties of equation (11), a grid of values for the parameters λ,γ lying between 0 and 1 in increments of 0.01 was used to identify those values corresponding to real and imaginary solutions.

For values of γ above 0.25, and an increasing number of values of λ, of 0.99 and below, the solution to equation (11) is associated with imaginary roots. For γ>0.82, the roots are imaginary irrespective of the value of λ. However, the stability of the inflation equation, when the roots of the characteristic equation are imaginary, is determined solely by the value of R, that is, λγ. Empirical estimates suggests that λ≪1, whereas γ takes the value of unity under Smithin’s representation of inflationary expectations based on the lagged inflation rate. Since R < 1, the inflationary process is stable, when the roots of the characteristic equation are imaginary.

Turning to the case of real roots and noting that a1<0, then the stability of the inflation equation will be determined by the size of the larger root, b1=(−a12+a12−4a22). For all values of λ,γ corresponding to real roots, it can be shown that the larger real root is less than unity.

Thus the incorporation of adaptive inflationary expectations into Smithin’s modelling of inflation yields a stable inflation outcome for a constant nominal policy rate target. This in turn means that the steady-state real policy rate is constant and is given by:
r0=i0−p*=i0−z.

2.3 Real interest-rate policies

Smithin considers a real policy rate setting regime by the central bank to examine the stability of the inflationary process. ‘It will also promote financial stability, inflation stability, high growth, full employment and higher real wages’ (Smithin 2020: 381).

It takes the form
i0=r0+pe,
where r0 denotes the fixed target real policy rate.
From equations (1), (2) and (3′), the following relationship between the real rate of interest and the expected inflation rate (see Smithin 2020: 382) can be derived:
r=m0+m1i0−pe=m0+m1r0−(1−m1)pe.
Then, for a fixed real policy rate target, r0, the inflation equation can be written:
p=z−λ(r−r−1)=z−λ*(p−1e−pe)=z+λ*γ(p−1−p−1e),
where λ*=λ(1−m1).
It can be readily demonstrated that
p+1−(1−γ(1−λ*))p+λ*γp−1=zγ.
Then equation (19) has the same functional form as the model based on the fixed nominal interest rate (equation (11)), except that λ has been replaced by λ*. Given that λ*<λ, the critical value of γ at which there are some values of λ, which yield imaginary roots, is higher than 0.25, whereas the critical value of γ, at which all values of λ yield imaginary roots, is unchanged.

Thus, the imposition of pe=p−1, that is, γ=1, again leads to a complementary function with imaginary roots. Again, all solutions associated with imaginary roots yield a stable steady-state rate of inflation since λ*γ<λγ<1.

Turning to values of λ*,γ for which there are real roots, again necessarily the larger root will be less than unity and again the inflationary process will be stable.

In private correspondence in 2017, Smithin develops a new real policy rate rule of the form:
r0=r′0+[(1−m1)/m1]p+1,
so the real policy rate is based on the future rate of inflation, rather than being fixed (Smithin 2020: 384, eq. 10). Again, it can be argued that the expected current inflation rate is appropriate, so the rule should be written as:
r0=r′0+[(1−m1)/m1]pe,
so that the nominal policy rate is:
i0=r′0+[(1−m1)/m1]pe+pe=r′0+pe/m1,
which means that the nominal interest rate, i, is
i=m0+m1i0=m0+m1r′0+ pe.
Then the expected real interest rate is
r*=m0+m1r′0.
Thus, Smithin’s real policy rate rule (that is, equation (20)) is effectively a constant real interest-rate rule, but it remains non-zero, when r′0 is set to zero. So, he fails in his objective of designing an interest-rate policy rule which achieves a zero-real rate with its consequences for income distribution. It can be readily shown that the nominal policy rate should be set at
i0=( pe−m0)/m1.
As noted at the outset, stable inflation of z is achieved by imposing a constant real interest rate of arbitrary magnitude, but, as Smithin readily concedes, his model of inflation provides no means of achieving say a higher or lower rate of inflation than z, which would require the use of agile fiscal policy and possibly incomes policy.

Later Smithin (2020: 392–393) returns to the consideration of setting a target real policy rate, which he considers to be near optimal. Here he correctly defines the real policy rate as i0 – p, that is, the nominal policy rate and the rate of inflation are contemporaneous.

3 A CRITIQUE OF THE INFLATION EQUATION

In response to Palley’s (2015) critique of modern money theory (MMT), which includes ZIRP, Smithin (2016a: 73–74) builds on his earlier work (see Smithin 2007; 2013; 2015), and specifies the demand for and supply of endogenous money as follows:
MD=μPY  0<μ<1
MS=φW−1N−1  φ>1,
where money demand is a fixed multiple of nominal GDP, given by the Cambridge coefficient. He refers to it as ‘the old-fashioned Marshallian “cash balances” approach’ (Smithin 2015: 14), where these balances are held in the form of deposits: a stock.

According to Smithin, however, money supply depends on the lagged wage bill (given a one-period production lag) which is derived from circuit theory and the work of Graziani (2003: 7).10 The coefficient φ is greater than one, which is justified because it covers ‘all other types of borrowing over and above what is needed to finance the aggregate wage bill (Smithin 2013, p. 228–230)’, quoted in Smithin (2016a: 73).11

From (26) and (27), solving for equilibrium in the money market yields a model of price determination (see Smithin 2016a: 73, eq. 12):
P=(φμ)W−1A,
where labour productivity, A is defined as Y/N–1. Based on assumptions about the determinants of \phi and \mu, Smithin (2016a: 73–74) derives the inflation model as shown in equation (1).

There are serious issues with the above specifications. This is a monetary theory of inflation, rather than one based on the real economy in the form of excessive expenditure relative to the economy’s production capacity (demand pull) and/or sellers attempting to raise their incomes via higher prices or money wages (cost push). Smithin is equating two conceptually different entities, with money demand being a stock and money supply a flow. Also, the narrow definition of the money stock is not confined to the demand deposits of the non-bank private sector since transactions using notes and coins still occur.

The demand for new loans, a flow, is initiated by the non-bank private sector, and enables payments to be made for goods, services, and financial and physical assets. The provision of credit to a borrower is subject to the judgment of the banks about her creditworthiness and the profitability of making the loan, so credit rationing may occur. After a loan application is approved, the bank account of the borrower is credited (so money is created out of thin air) and the borrower can then engage in transactions which may be in the form of cash or through transfers between the accounts of the loanee and seller(s).

The role of the central bank is to ensure that there are sufficient notes and coins in circulation for banks to meet the needs of the non-bank private sector, and that banks have adequate reserves in the form of vault cash and deposits at the central bank. Otherwise, the payments system will malfunction and cause uncertainty in the financial system. This highlights the endogeneity of the money supply, which means that equilibration of the money market will not drive changes in the price level, in contrast to equation (28).

Also, this equation appears to resemble a mark-up pricing equation, but in earlier correspondence Smithin argues that equation (28) and a mark-up pricing equation12 of the form
P=(1+K)(1+i)[W−1/A]
must be reconciled by quantity (employment) adjustments but does not explain how this adjustment mechanism operates.

The Phillips curve, the expectations-augmented Phillips curve (EAPC) devised by Friedman and Phelps, and contemporary models of inflation use a range of proxies for the level of aggregate demand, such as unemployment, capacity utilisation (Watts/Mitchell 1990) and underemployment (Mitchell et al. 2013), and incorporate the role of hysteresis (Cross 1987; Mitchell 1987), which can be based on distinguishing between short- and long-term unemployment as an explanation.

Moreover, Smithin’s inflation model does not account for the state of the real economy by incorporating a proxy for aggregate demand, which could be linked to the more active use of fiscal policy given the non-discretionary operation of monetary policy under a ‘park it’ rule.

In fact, drawing on the AMM model, Smithin (2020) argues that the economy operates at full employment, with the real wage at its equilibrium level, but such a priori assumptions cannot be the basis for claims about any of the park-it interest-rate rules.

The steady-state solution of the inflation model is p = z, with z incorporating the logarithm of the wage share and the ‘inverse measure of the state of bearishness in the money market’, which would appear to be incompatible concepts. However, the growth of money wages which impacts on the wage share cannot be treated as a given and equal to the growth in labour productivity. In fact, it is an integral part of the inflationary process and should be modelled as such. This is often captured through an inflationary expectations variable, but in Smithin’s models, the role of expectations is limited to its impact on the real interest rate outcome.13 Also, as noted above, a constant high real rate of interest has no ongoing impact on the inflation rate since the interest-rate terms in (1) would cancel out.

Finally, while the distributional issues relating to interest-rate policy are important and a zero real rate makes sense on equity grounds, I am not convinced that the formal distributional analysis presented by Smithin (2020: 385–392) adds any analytical support to this policy, particularly the derivation of log shares and the use of unrealistic real rates of interest.

4 CONCLUSION

The debate as to whether the setting of a constant real or nominal (policy) interest rate is appropriate must be articulated in the context of the appropriate role for fiscal policy and the consequences for financial, as well as inflation, stability. Smithin (2020) has adopted a limited model of price inflation, which has problematic theoretical foundations, and examined its properties in the light of different interest-rate rules, but the application of the model has fundamental flaws, which include the definition of the real rate of interest and the assumption of perfect foresight, which leads to the solution procedure initially being based on the future, rather than prevailing, rate of inflation. Once appropriate corrections are made, both fixed nominal and fixed real target policy rates yield convergent inflationary outcomes, which highlights the inability of Smithin’s specification of inflation to discriminate between different fixed interest-rate policies. Also, effective demand and the role of fiscal policy are neglected along with the impact of money wage growth and the role of inflationary expectations more broadly.

We have no issue with the examination of different fixed interest-rate policies, particularly given the poor macroeconomic outcomes across developed economies which have relied primarily on monetary policy, but more sophisticated modelling than is provided by Smithin, such as stock–flow consistent (SFC) modelling, is required to make a convincing case for a particular fixed rate policy. Indeed, carefully argued non-mathematical analysis can also play an important role in casting light on this important issue.

ACKNOWLEDGMENTS

I would like to thank George Pantelopoulos and an anonymous referee for helpful comments and suggestions on earlier drafts of this paper. In addition, I have appreciated the long discussions (and disagreements!) with John Smithin over the past four years. I am responsible for any omissions and remaining errors.

  • 1↑

    Bernanke (2015) argued that the weight on the output gap should be 1.0, rather than the 0.5 for both targets which was recommended by Taylor (1993).

  • 2↑

    Fiscal policy is not mentioned until the conclusion of his paper (Smithin 2020: 394).

  • 3↑

    For example, Franke (2019) shows that a model represented by the integration of two stable sub-models may be unstable.

  • 4↑

    The steady-state solution is represented by z. Rather than assuming constancy of both w and a, it is more satisfactory to consider the log of the wage share (w – a) as constant.

  • 5↑

    Thus, the interest-rate term λ(r−r−1) represents the product of the interest elasticity of money demand and the change in the real interest rate which is the product of the rate of change of money demand and the interest rate. This is hard to interpret, unless λ is treated as constant. The impact of (r−r−1) on inflation is being transmitted via the demand for money, which is at odds with monetary endogeneity. Also, neither effective demand nor inflationary expectations impact on the rate of inflation.

  • 6↑

    His deterministic inflation model represents a generous interpretation of the macroeconomic impact of policy induced changes in interest rates. With debt to GDP ratios of about 100 per cent in some countries, the rise in spending out of interest income by bondholders following an interest-rate increase may well dwarf any effect of the differing consumption propensities of borrowers and lenders (Wray 2007: 133). Also a systematic relationship between interest rates and economic activity and hence inflation is questionable given that the relationship is long, variable and uncertain (Bank of England 1999; Batini/Nelson 2002).

  • 7↑

    An algebraic advantage of using my equation (3) to depict the real interest rate is that, irrespective of whether a real or nominal rate is targeted, the resulting difference equation is first order. Smithin (2020) is inconsistent in his use of this equation. The claim that r = 0 for the specification of r′0 in his equation (40) is based on equation (3′). See also his equations (52) and (53).

  • 8↑

    While the algebraic solution of a difference equation does not reflect the nuances of post-Keynesian thought, the specification should represent an inflationary process occurring in historical time, in which all variables on the RHS of the equation are predetermined, rather than one, namely the current inflation rate being simultaneously determined with the dependent variable.

  • 9↑

    When using the incorrect definition of the real interest rate (3), Smithin (2020: 383) acknowledges that p+1 is an expected inflation rate. Also, in his footnote 9 he suggests that ‘[t]he actual implementation of such a rule would likely involve using the currently observed inflation rate, or the lagged inflation rate, as a “proxy for” the expected rate of inflation, as in the original Taylor rule (Taylor 1993: 202)’.

  • 10↑

    These wage costs were incurred in the previous period and must be paid for then, so it is incorrect to argue that the current money supply is related to borrowing to finance labour costs in the past period. The lag in production is irrelevant. The production lag does not apply to the provision of services.

  • 11↑

    Also, credit will be required to pay for other production inputs, such as raw materials.

  • 12↑

    Ironically, Smithin’s inclusion of the interest rate in his mark-up pricing specification highlights why monetary policy should not be adopted as an instrument to control inflation.

  • 13↑

    Also, labour productivity growth could possibly be modelled via Verdoorn’s law (see Rochon/Setterfield 2007; 2012).

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