Implications of habit formation for optimal monetary policy

Habit formation had been shown to improve how well small business-cycle models fit U.S. data, but at the time of writing only two papers had asked what it implies for monetary policy, and neither characterised optimal policy in a model where agents choose both consumption and labour supply optimally. This paper fills that gap by inserting a ratio-form habit into a model otherwise identical to Woodford’s – a closed economy, no capital, a continuum of monopolistically competitive household-producers, Calvo pricing – and tracing the consequences through three channels the authors keep carefully distinct. First, the Euler equation makes current marginal utility depend on past as well as expected future consumption, so the IS equation acquires lagged output. Second, and less obviously, because suppliers value their expected revenues using the marginal utility of consumption, the Phillips curve acquires past and expected future output gaps on top of the usual current gap and expected inflation; the authors point out that this supply-side effect is absent from McCallum and Nelson, who assume inelastic labour supply, and from Fuhrer, who models supply with a reduced-form VAR. Third, the second-order approximation to household welfare changes: variability in the level of output, not just the gap, becomes welfare-reducing, and the period loss involves lagged and led output terms. Calibrating at quarterly frequency with a discount factor of 0.99, a Phillips-curve slope of 0.031 (implied by three-quarter average price contracts and a 15% markup), a curvature parameter of 1.1, a supply-elasticity parameter of 0.6, and habit strengths of 0, 0.4 and 0.8 (the last being Fuhrer’s estimate), the paper finds the dominant quantitative force is that the habit sharply raises the volatility of the Wicksellian natural rate of interest: its standard deviation goes from 22.42 at h = 0 to 36.29 at h = 0.8. Because the zero lower bound makes interest rate variability costly, policy does not fully track that more volatile natural rate, and the result is that output variability rises dramatically – the paper’s measure of output variance goes from 13.47 to 146 between h = 0 and h = 0.8 – even though output has gained weight in the welfare function (the output-variability terms rise from zero to a fraction 0.044 of output variance). Turning to implementation, a simple rule in the lagged interest rate, current inflation and current output gets within roughly 1% of the optimal plan’s welfare at every habit strength, without requiring the central bank to observe any shock process or to measure the natural rate of output. That rule is super-inertial throughout – its lagged-rate coefficient exceeds one for every h – but both that coefficient and the inflation response decline as the habit grows, falling from 1.72 to 1.32 and from 2.37 to 0.76 respectively, because habit-induced inertia in output and inflation partly substitutes for policy inertia. The authors are explicit that the particular coefficient values “are likely to be sensitive to the structure of the model as well as its calibration,” and check robustness by doubling both the curvature and supply-elasticity parameters and by rescaling the shock variances.

Rule-of-thumb behaviour and monetary policy

Standard optimisation-based sticky-price models have no lagged variables in their structural equations, which makes them hard to square with the high serial correlation actually observed in output and inflation; this paper asks what happens to optimal monetary policy once a fraction of agents is allowed to skip the optimisation and follow a simple backward-looking rule instead. The model is otherwise identical to Woodford’s – a closed economy with no capital accumulation, a continuum of monopolistically competitive household-producers, and Calvo price setting – and the two departures are deliberately symmetric. Each period a household draws an independent optimisation cost; a fraction of households with costs above a threshold sets consumption equal to last period’s aggregate per-capita consumption rather than solving its Euler equation, and among firms offered a Calvo price-reset opportunity a fraction follows Gali and Gertler’s rule of setting its price to last period’s average newly chosen price scaled up by last period’s inflation. Both departures put a lagged endogenous variable into the structural equation – lagged output into the IS curve, lagged inflation into the Phillips curve – and both, the paper shows, also change the welfare criterion that policy should be maximising, a point not previously noted in the literature: rule-of-thumb price setting adds a penalty on the squared change in inflation, and rule-of-thumb consumption adds a penalty on the squared change in output. Rule-of-thumb behaviour works in two opposing directions. It raises endogenous persistence, which on its own would make inflation and the output gap more variable; but it also weakens transmission, reducing the sensitivity of inflation to the output gap and of output to expected real interest rates. In the paper’s calibration – Woodford’s parameter values, based on Rotemberg and Woodford’s estimates on U.S. data for 1980-95, with Calvo parameter 0.66 per quarter, discount factor 0.99, and a natural-rate-of-interest shock with standard deviation 0.93 percent per quarter – the weakening of transmission dominates, so inflation variability falls as rule-of-thumb price setting becomes more prevalent and output gap variability falls sharply as rule-of-thumb consumption becomes more prevalent. The central policy result is that highly inertial, indeed “superinertial,” interest rate policy – a sum of coefficients on lagged interest rates exceeding one – remains optimal at every fraction of rule-of-thumb behaviour examined (the paper reports results for optimising fractions of 1, 0.6 and 0.2), and survives every robustness check it runs: a lower weight on interest rate variability, logarithmic preferences, serially correlated shocks, and the introduction of inefficient supply shocks that create a genuine inflation/output-gap trade-off. Two rules stand out as robust: the four-argument rule that implements the optimal plan when all agents optimise (current inflation, the change in the output gap, and two lags of the interest rate), and a first-difference version of Taylor’s 1993 rule. By contrast, rules feeding back only from inflation and the lagged interest rate, and price-level rules, have optimal coefficients that shift dramatically with the rule-of-thumb fraction – an unattractive property given how hard that fraction is to measure. Throughout, the policymaker is assumed able to commit; the authors are explicit that further work is needed to show these particular rules of thumb are good approximations to actual decision making.

Jeffery D. Amato | Macro Paper Warehouse

Implications of habit formation for optimal monetary policy

Rule-of-thumb behaviour and monetary policy