Rule-of-thumb behaviour and monetary policy

The purely forward-looking dynamic New Keynesian model cannot reproduce the serial correlation seen in output and inflation data without assuming heavily autocorrelated structural disturbances, and the paper’s rules of thumb are offered as one tractable way of putting lagged endogenous variables into the structural equations from optimising foundations. The authors note that “this class of models has been criticised as being unable to replicate the high serial correlation found in both output and inflation data of many industrialised economies, unless one is willing to assume a substantial degree of serial correlation in the structural disturbances of the model,” citing Fuhrer (Section 1). They position their contribution as “a bridge between the studies assessing optimal monetary policy… with purely forward-looking dynamics… and models with lagged dynamics imposed,” with the advantage that “we can characterise precisely how lagged endogenous variables enter the model equations” rather than imposing them.

Rule-of-thumb agents mimic last period’s behaviour of all agents: consumers set current consumption equal to last period’s aggregate per-capita consumption, and price setters follow Gali and Gertler’s rule of setting their price to the previous period’s average newly chosen price scaled by the previous period’s inflation rate. The premise is “that it is costly to reoptimise every period”; optimisation costs are independent random draws, and “those whose cost exceeds a certain threshold use instead a rule of thumb” (Section 2). The authors give three reasons for finding these rules appealing: “they involve virtually no computational burden: all that is needed is for agents to observe last period’s consumption or price setting decisions”; “they involve passive learning of the behaviour of optimising agents”; and “all agents behave identically in the steady state,” so individual choices converge once shocks are eliminated. Crucially, because the draws are independent of history and households insure against all idiosyncratic risk, “we can still analyze price setting using a representative agent” – the wealth distribution never becomes a state variable.

The Euler equation becomes an IS relation in which the current output gap depends on its own lag as well as on its expected future value and the expected real interest rate, and smaller fractions of optimisers both raise the weight on the lag and damp the response to the real rate. Their equation (7) gives the output gap as a weighted combination of its lag and its expected future value less a term in the gap between the real and natural rates of interest. The paper notes that as the optimising fraction falls, the coefficient on the lagged gap converges to 0.5 from above “as [the optimising fraction] goes to 0,” while the elasticity of output with respect to the expected real interest rate falls. The authors are explicit that this specification is close to, but not identical with, external habit formation: log-linearising a habit Euler equation gives “the same form as (5),” and for intertemporal substitution elasticity parameters below 6 the habit coefficients “respond less to changes in h” than the rule-of-thumb coefficients respond to changes in the optimising fraction.

It converts the New Keynesian Phillips curve into a hybrid curve with lagged inflation, and simultaneously reduces the sensitivity of inflation to the output gap. Their equation (12) makes current inflation a function of the output gap, lagged inflation and expected future inflation, where “depending on the value of [the optimising fraction], the coefficients… sum to between [beta] (for [fraction] = 1) and 1 (for [fraction] = 0)” – so with the discount factor close to one they read naturally as relative weights on past and expected inflation. The authors note that “despite our slightly different interpretation of rule-of-thumb behaviour, (12) is identical to the log-linear approximation derived by Gali and Gertler (1999).” They also emphasise the second channel: “smaller values of [the optimising fraction] reduce the sensitivity of current inflation to fluctuations in the current output gap.”

The weakening of transmission dominates the increase in persistence, so inflation variability falls as rule-of-thumb price setting spreads, and output gap variability falls sharply as rule-of-thumb consumption spreads. For price setting: “the degree of endogenous inflation persistence increases as [the optimising fraction] decreases, which, ceteris paribus, would imply higher inflation variability. On the other hand, the coefficient [on the output gap] declines… As shown in Table 2, the latter effect dominates, as inflation variability decreases with [the optimising fraction] despite the slight increase in [output gap variance]” (Section 3.2). For consumption, the paper reports “the dramatic decline in output gap variability reported in Table 3,” attributing it to the same logic: “the increasingly weaker transmission of shocks as [the optimising fraction] declines dominates the effect of stronger endogenous output (gap) persistence” (Section 3.3).

A second-order approximation to household welfare gains an extra penalty term when rules of thumb are present – on the squared change in inflation with rule-of-thumb price setters, and on the squared change in output with rule-of-thumb consumers – and the two behave quite differently. With rule-of-thumb price setters the additional term on the change in inflation “does not generate any new tradeoffs between stabilisation of the various variables, since price stability still achieves the minimum of all three terms”; its “only effect… is to increase the weight on inflation stabilisation relative to output gap stabilisation” (Section 2.3). With rule-of-thumb consumers, by contrast, fluctuations in output itself – not just the gap – create welfare losses “because changes in output cause larger departures of rule-of-thumb from optimal consumption,” and this “does give rise to an additional tension among the various stabilisation goals”: “even without the penalty on interest rate variability, the output variability that is necessary to keep output at its natural rate implies welfare losses.”

Because the paper adds a penalty on interest rate variability to the welfare criterion; without it, the model’s structure would let policy stabilise inflation and the output gap completely. The authors observe that “the model presented above has the property that fluctuations in the output gap are the only source of inflation variability… It is therefore possible to completely stabilise both inflation and the output gap in response to fluctuations in [the natural rate of interest], and hence no tradeoff exists.” The interest-rate penalty is justified by appeal to Rotemberg and Woodford’s argument that stabilising the gap fully would require interest rate variability high enough that a positive steady-state inflation rate becomes necessary to avoid the zero lower bound, and such steady-state inflation is itself welfare reducing through relative price dispersion (Section 2.3). The baseline weight is 0.236; Section 4 rechecks the results at 0.077, “roughly 1/3 of its previous value.”

Highly inertial – “superinertial” – interest rate policy is optimal regardless of what fraction of agents follows a rule of thumb. In the context of simple rules this means “a coefficient greater than one on the lagged interest rate”; for the four-argument rule that implements the optimal plan, “the optimality of superinertia is shown by the sum of coefficients on the two lags of the interest rate being greater than 1” (Section 3.2). The authors state that “the key result is that superinertial behaviour of the interest rate remains optimal as [the optimising fraction] becomes small,” and in the conclusion that “our most striking finding, across all model specifications that we consider, is that highly inertial policy is desirable.” They place this explicitly against the alternative view: it “stands in direct contrast to the suggestion (e.g. Taylor 1999a) that such an inertial interest rate policy ceases to be optimal, or even feasible, once backward-looking behaviour is incorporated into the structure of the model.”

Two rules perform well across the whole range: the rule implementing the optimal plan in the fully optimising case, and a first-difference version of Taylor’s 1993 rule. Rules feeding back only from inflation and the lagged rate, and price-level rules, achieve similar welfare but with coefficients that swing wildly with the rule-of-thumb fraction. The first robust rule sets the current interest rate in response to “its level in the past two periods, the current inflation rate and the change in the output gap” (Conclusions); the paper finds it “continues to approximate the optimal plan very closely even when [the optimising fraction] is as low as 0.2,” and that “the optimal parameters of this rule change only moderately.” The Taylor first-difference rule “combines the advantages of history dependence when [the optimising fraction] is reasonably large with robustness in terms of performance across different values.” Against these, “an unappealing feature” of the inflation-plus-lagged-rate rules “is the extreme sensitivity of the optimal coefficients,” and the price-level rule “shares with the ‘a,c’ rules the drawback of extreme sensitivity of the coefficients.”

Not always: in the restricted rule that feeds back only from inflation and the lagged interest rate, with a very large rule-of-thumb fraction the optimal lagged-rate coefficient falls below one. The paper reports that when the optimising fraction of price setters is 0.2, “in fact, we find that the optimal value of c is .65, i.e. for a rule of this form a response greater than one to the lagged interest rate is no longer optimal” (Section 3.2). This is a scope condition on the headline claim rather than a contradiction of it: superinertia remains optimal in the richer rule that includes the output gap change and two interest rate lags, and in the rule-of-thumb consumption case “rules of the ‘a,c’ type also show that feedback from the lagged interest rate greater than 1 is optimal for any value” of the optimising fraction.

The paper re-runs the analysis under a lower interest-rate weight, logarithmic preferences, serially correlated shocks, and inefficient supply shocks, and superinertia survives all four. With the interest-rate weight at 0.077, “superinertial policy is still optimal, despite putting much lower weight on interest rate stabilisation.” With logarithmic utility (intertemporal substitution and labour supply elasticity parameters both set to 1, replacing the baseline’s comparatively high elasticities), “again, superinertial policy is desirable,” and the striking change is that “the coefficients of all of the rules are much more stable” across rule-of-thumb fractions. With serial correlation of 0.35 in the shock processes, “similar conclusions are reached about optimal policy rules.” Adding inefficient supply shocks – a cost-push term in the Phillips curve that makes a genuine inflation/output-gap trade-off – makes the economy harder to stabilise, but “superinertial policy remains desirable,” and here the coefficients of the simple rules become “fairly stable” with respect to the rule-of-thumb fraction, while the rule that is optimal under full optimisation becomes less attractive than the “a,b,c” and Taylor first-difference rules at low optimising fractions.

They present their specification as a reasonably general one that can be given different interpretations, since habit formation and Christiano-Eichenbaum-Evans-style indexation deliver “remarkably similar equations.” Exogenous habit formation with perfectly elastic labour supply yields an output equation of the same form as theirs (Section 2.4). The indexation alternative – where firms that do not reoptimise update their price by last period’s aggregate inflation – yields a Phillips curve of the same form, and the paper works out a directly comparable exercise: generalising indexation so that only a fraction of non-reoptimisers index, letting that fraction fall from 1 to 0.1 moves the (lagged, expected) inflation weights from (0, beta) to (0.48, 0.52) with a discount factor of 0.99, while in their own specification with Calvo parameter 0.66 and the same discount factor the corresponding weights move from (0, beta) to (0.58, 0.42). They flag one substantive objection to the indexation route: in its limiting case “the traditional view of sticky prices as being caused (at least in part) by costs to price adjustment no longer applies, as all producers change their price every period whenever last period’s inflation rate is different from zero.”

It does not claim that these rules of thumb describe actual behaviour, and it assumes throughout that the policymaker can commit. The conclusion states plainly that “further work is required to show that the type of rules of thumb we consider are good approximations to actual decision making behaviour on the part of consumers and firms.” On commitment: “we do not claim that the behaviour of policymakers is always best described by commitment, but neither do we believe… that monetary policy in the real world is clearly characterised by discretion”; the choice is justified by the focus on interest rate feedback rules “which we interpret as a commitment device” (Section 3). The model is also closed, has no capital accumulation, and – in the baseline – has serially uncorrelated shocks chosen to isolate the model’s endogenous dynamics.