Implications of habit formation for optimal monetary policy

Habit formation had been repeatedly shown to improve the empirical fit of small business-cycle models, but almost no one had asked what it implies for optimal monetary policy in a model where households optimise over both consumption and output supply. The authors write that “despite the large number of papers that examine the desirable empirical properties of models with habit formation, only McCallum and Nelson (1999) and Fuhrer (2000) have addressed the subject of monetary policy, and neither of these authors characterizes optimal policy in a model in which agents make optimal choices about both consumption and labor supply” (Section 1). The motivation is sharpened by a further observation: the existing literature reaches different conclusions about desirable policy partly because of how endogenous dynamics are specified – optimising sticky-price models deliver highly inertial policy with low weight on output, while non-utility-based models with lagged terms deliver neither – and “it can be difficult to distinguish… whether the observed persistence in the data is derived from endogenous dynamics or exogenous shocks.”

Felicity depends on current consumption divided by lagged consumption raised to the power h, a special case of forms analysed by Abel and by Carroll et al.; the curvature parameter must exceed one for the habit to do what it is meant to do. In the limiting case h = 1 “the instantaneous utility derived from consumption depends only on the ratio of current to previous period’s consumption,” while h = 0 “is the standard case of time-separable utility,” in which the curvature parameter is the inverse of the intertemporal elasticity of substitution (Section 2.1). The authors flag the parameter restriction explicitly: “to capture the idea that an increase in past consumption raises the marginal utility of current consumption, it is necessary to assume that [the curvature parameter] > 1, so that the exponent on [lagged consumption] is positive.” This matters for the calibration, since Rotemberg and Woodford’s estimate of that parameter is 0.16 – below one, and therefore unusable here.

The IS equation now links the expected change in output to the current change, to expected changes two periods ahead, and to the real interest rate gap, so output inherits endogenous persistence. Without habit “(5) reduces to the standard intertemporal IS equation… that characterizes the demand side in the model of Woodford (1999b) and other recent studies.” With habit, “the presence of habit formation implies a positive correlation between the change in output expected from the current to the next period, and the change in output in the adjacent two periods” (Section 2.1).

Because suppliers value their expected future revenues by the marginal utility of consumption, and habit makes that marginal utility depend on more than current output, the Phillips curve acquires lagged and expected future output gaps. The authors write that “habit formation causes inflation to depend on the lagged and expected future output gap (beyond the dependence on expected future conditions captured by [expected inflation]) because expected marginal revenues are valued by the marginal utility of consumption, which does no longer depend on current output only” (Section 2.1). They immediately note the scope of the point: “this effect of habit formation on the aggregate supply side is absent from the model of McCallum and Nelson (1999) who assume inelastic labour supply. Similarly, it is absent from Fuhrer’s (2000) study, who models the supply side by using a reduced-form VAR equation.” So the reduced-form processes for output and inflation change for two reasons – the altered IS equation feeding the gap, and the altered valuation of revenues.

Quarterly: discount factor 0.99, Phillips-curve slope 0.031, curvature parameter 1.1, supply-elasticity parameter 0.6, habit at 0, 0.4 and 0.8, with serially uncorrelated shocks calibrated so the model reproduces Rotemberg and Woodford’s VAR law of motion. The slope of 0.031 is “consistent with an average lifetime of price contracts of three quarters… and an average markup in goods markets of 15%,” following Rotemberg and Woodford’s estimate (Section 2.2). The curvature parameter of 1.1 is a compromise: aggregate-consumption estimates by Hall and by Attanasio and Weber are “on the order of 3,” Rotemberg and Woodford’s impulse-response matching gives 0.16, and the habit requires a value above one. The supply-elasticity parameter of 0.6 “is consistent with a Frisch elasticity of 5 and a Cobb-Douglas production technology with a coefficient on labour of 0.75,” and is chosen so the model’s inflation response to an interest rate innovation matches Rotemberg and Woodford’s – a choice the authors check is robust, since “the impulse response of inflation is almost invariant with respect to changes in h.” The habit value of 0.8 is Fuhrer’s (2000) estimate. The historical-policy benchmark rule used for these comparisons is a simplified version of Rotemberg and Woodford’s estimated U.S. rule, with a lagged-rate coefficient of 0.69, an inflation coefficient of 0.67 and an output coefficient of 0.15.

Yes, and the effect is quantitatively larger for inflation than for the output gap. With serially uncorrelated shocks and time-separable utility, “the model generates no positive serial correlation in either inflation or the output gap” – which the authors describe as “a well-documented feature of models in which the structural equations are entirely forward-looking.” Raising h “has the effect of inducing some positive serial correlation to the output gap and, quantitatively more important, to inflation,” which they read as the increasing persistence of the gap being transmitted to inflation. They draw the methodological moral directly: “the finding that the most pronounced effects of introducing habit formation are on the serial correlation of inflation points to the importance of modelling the effects of habit formation on the supply side of the model” (Section 2.3).

Output variability itself becomes welfare-reducing, the loss depends on past and future output and gaps rather than only current ones, and the weight on the squared output gap rises fivefold – though the net weight on output gap variance barely moves. The authors identify “three distinct effects compared to the case of time-separable utility.” First, “in our model, in which consumption equals output, output variability reduces welfare because of the link from past output to the current marginal utility of consumption.” Second, the period loss “depends on past and expected future, as well as current, output and output gaps,” which alters the relative weights. Third, the coefficient on the squared output gap “is increasing in h”: with inflation in annual percent and output in percent of steady-state output, it “is 0.063 for h = 0. As h increases to 0.8, [it] increases to 0.30.” Netting these out, “the weight… on output gap stabilization decreases from 0.108 to 0.102 as h rises from 0 to 0.8,” while the output-variability terms rise from zero to a fraction 0.044 of output variance (Section 3.1).

Because the only distortion apart from imperfect competition is relative-price dispersion, which price stability would eliminate – but with a sufficiently volatile natural rate, price stability would require interest rate variability so large that a positive steady-state inflation rate is needed to stay off the zero lower bound. Following Rotemberg and Woodford, the paper assumes all interest rate realisations lie within a fixed number of standard deviations of the mean, so “only interest rate policies such that [that band] exceeds the steady-state net real interest rate… cause positive steady-state inflation rates.” With a band factor of 2.26 and a 3% annual steady-state real rate, “a tradeoff between perfect stabilization and zero steady-state inflation exists whenever the standard deviation of the natural rate of interest exceeds 1.32.” Under the paper’s calibration this bound binds “for any value of h,” so the objective is equivalently augmented with a penalty on interest rate variance whose shadow weight is solved for numerically (Section 3.1).

Output variability rises dramatically with the habit even though output gains weight in the objective, because the natural rate of interest becomes so much more volatile that policy cannot afford to follow it. Under the optimal plan the standard deviation of the natural rate rises from 22.42 (h = 0) to 25.77 (h = 0.4) to 36.29 (h = 0.8), and every variability statistic rises with it – the paper’s measure of output variance goes 13.47, 32.41, 146, and inflation variance 0.50, 0.66, 1.46. Interest rate variance, by contrast, barely moves (1.86, 1.88, 1.97), because “the small change in interest rate variability reflects the substantial increase in [the shadow weight on interest rate variability]” as the natural rate becomes more volatile. The authors summarise: “the increase in [interest rate variance] is not nearly sufficient to fully offset the effects of a more variable natural rate on the variability of inflation, output, and the output gap.” They describe the rise in natural-rate volatility as “an outward shift of the ‘optimal policy frontier’” (Section 3.2).

Two effects work against each other and the amplifying one wins. The variance of the expectational term inside the natural-rate expression falls with h, “because the polynomial in [the natural rate of output] inside the brackets is becoming more similar to a moving average.” But “the dominant effect, however, is that the factor in front of the expectations operator increases in h” (Section 3.2). Separately, the paper notes that the high base-case natural-rate volatility relative to Rotemberg and Woodford’s own value of 3.72 is a consequence of its own parameter choice: “our choice of a higher value of [the curvature parameter] in particular implies that considerably larger interest-rate movements are required to offset the effects of shocks… raising the variability of the natural rate of interest” (Section 2.2).

A three-term rule in the lagged interest rate, current inflation and current output reproduces the optimal plan closely – welfare losses about 1% higher – and is super-inertial at every habit strength, but its optimal coefficients on the lagged rate and on inflation both decline as the habit grows. The rule requires no observation of shock processes and, deliberately, no feedback from the output gap, because “Orphanides (1998) argues that rules that respond to the output gap may perform poorly because of the problems associated with accurate measurement of the natural rate of output” (Section 4). Interest rate variance, steady-state inflation, and output and output-gap variability are “almost identical” to the optimal plan; “the only marked difference in the statistics occurs for [inflation variance], the values of which under the simple rules exceed those obtained under the optimal plan by up to 8%.” The optimised coefficients are a lagged-rate response of 1.72, 1.66, 1.32 and an inflation response of 2.37, 1.97, 0.76 across h = 0, 0.4, 0.8, with the output response essentially nil throughout (0, -0.001, 0.007).

Because habit-induced inertia in output and inflation does part of the work that policy inertia would otherwise have to do, and pushing inertia further would make deviations larger and more drawn out. The authors explain that “habit formation has the effect of reducing the size of the lagged interest rate response because, given a particular path of persistently high interest rates, the additional inertia in other variables caused by habit formation forces deviations of the output gap and inflation from their steady-states to be larger and more persistent in the face of shocks. By toning-down the threat to keep interest rates high after an inflationary shock, less excessive fluctuations in all of the variables can be achieved” (Section 1). In Section 4 the same point is put as: “a higher degree of habit formation therefore acts to some extent as a substitute for interest rate inertia. Nevertheless, the coefficient [on the lagged rate] is well above 1 even when h = 0.8.”

Optimal policy responds far less to output and is far more inertial, which lets output rise more on impact but avoids the large negative output gaps that follow. Comparing the optimised rules to the historical benchmark, “the response to the lagged interest rate is much larger in the optimal rules across all values of h. Even in the case h = 0.8, this coefficient is nearly twice as large. Moreover, the response to output in [the historical rule] is at least 20 times the size of the response in any of the optimal rules” (Section 4). Tracing an impulse response to a shock that raises the marginal utility of consumption, the authors note that under optimal policy “the rise in the interest rate is smaller, but it is more drawn out,” allowing output to rise more initially while “subsequent fluctuations around the steady state are greatly reduced.” Their conclusion from the comparison is blunt: “a rule such as (11) does not respond optimally to a shock of this type.”

The qualitative conclusions do, but the welfare levels and even the direction of the habit’s welfare effect can change under different structural parameters. Setting both the curvature and supply-elasticity parameters to 2 (Case 1, with the shock variances rescaled to keep the h = 0 natural-rate volatility at 22.42) lowers every variability statistic, because “the larger value of [the curvature parameter] reduces the size of the elasticity of output with respect to the one-period expected real interest rate,” so a smaller expectational variance produces the same natural-rate variance. Under this calibration the paper reports a genuine non-monotonicity: welfare loss is 0.59 at h = 0, 0.53 at h = 0.4 and 0.84 at h = 0.8, since “the improvement in the trade-off between interest rate and output gap variability as h increases more than offsets the welfare-reducing effects of larger variation in [the natural rate].” Case 2 instead keeps those structural parameters but uses the shock variances Rotemberg and Woodford’s model implies for them, giving an h = 0 natural-rate standard deviation of 40.54 – “almost double the value in Case 1 across the range of h” – and correspondingly higher variances and welfare losses. A further observation from the alternative calibration is that combining a larger curvature parameter with a large habit makes the optimal responses “more hump-shaped, something that does not occur when h = 0.”

Commitment, an efficient steady state, serially uncorrelated shocks, and a closed economy with no capital. The authors “assume throughout that monetary policy is able to act under commitment, and benefit from the associated stabilization gains,” and evaluate policies “under the assumption that subsidies for output are in place such that the steady-state level of output is efficient despite the presence of imperfect competition. Therefore, monetary policy has no inflationary bias” (Section 3.1). The shocks are assumed serially uncorrelated, so all persistence in the simulated series is endogenous. The economy is closed with no capital accumulation, so goods market clearing forces consumption to equal output – which is exactly why output variability, and not only gap variability, enters welfare. And the paper is candid that the specific rule coefficients “are likely to be sensitive to the structure of the model as well as its calibration”; what it claims is robust is that the optimal lagged-rate coefficient exceeds one.