Real Wage Rigidities and the New Keynesian Model

It is the property that stabilising inflation is equivalent to stabilising the welfare-relevant output gap, and it rests on a chain of two equivalences (Sections 1.2-1.3, NBER WP 11806, pp. 6-8). First, the New Keynesian Phillips curve makes inflation depend on expected inflation and the output gap – the log distance of output from its natural (flexible-price, second-best) level – with no shock entering directly; supply and preference shocks appear only through their effect on the natural level of output. So stabilising inflation is equivalent to stabilising that output gap. Second, in the baseline model the log distance between the first-best (efficient) and second-best (natural) levels of output is a constant, determined by the price markup, the capital share and the Frisch elasticity. Putting the two steps together, stabilising inflation is equivalent to stabilising the welfare-relevant distance of output from first best. The authors emphasise that this constancy “is common to the vast majority of optimizing models with nominal rigidities found in the literature,” and that because an adverse supply shock does not alter it, “it does not create any incentives for the monetary authority to deviate from a policy of constant inflation.”

Because it conflicts with a broad practitioner consensus and with the medium-term orientation that inflation-targeting central banks actually adopt (Section 1, p. 3). The property “contrasts with a wide-spread consensus on the undesirability of policies that seek to fully stabilize inflation at all times and at any cost in terms of output. That consensus underlies the medium-term orientation adopted by most inflation targeting central banks.” The paper’s diagnosis is that the standard model’s real distortions are “either constant or [vary] over time in response to exogenous shocks to the distorting variables themselves (e.g. tax rates),” whereas in reality “distortions are likely to interact with shocks, leading to different policy prescriptions.”

As a partial adjustment equation in which the real wage is a weighted average of its own lagged value and the marginal rate of substitution, with the weight an index of real rigidities (Section 2, pp. 9-10). The authors are explicit that this is “an admittedly ad-hoc but parsimonious way of modeling the slow adjustment of wages to labor market conditions, as found in a variety of models of real wage rigidities, without taking a stand on what the ‘right’ model is.” They report an alternative formalisation derived from staggering of real wage decisions in an appendix, noting the algebra is more complex but the conclusions are the same. The load-bearing restriction is stated plainly: “the important assumption underlying equation (15) is that the slow adjustment be the result of distortions rather than preferences, so the first-best equilibrium is unaffected” – and they flag in a footnote that a model (Danthine and Kurmann) in which a similar rigidity arises from non-standard preferences would have clearly different normative implications.

The gap between first- and second-best output stops being constant and starts fluctuating with both preference and supply shocks, and its size and persistence rise with the degree of real rigidity (Section 2.1, pp. 10-11). An adverse supply shock decreases second-best output by more than first-best output, and the gap returns only gradually to its steady-state level as the wage adjusts. Symmetrically, a preference shock that lowers first-best output lowers second-best output by less, because the rigidity prevents the wage from adjusting upward enough to support the lower efficient level of employment. Crucially, the first link in the chain survives: stabilising inflation is still consistent with stabilising the distance of output from second best. What fails is the second link – “stabilizing the output gap … is no longer desirable … because what matters for welfare is the distance of output not from its second-best level, but from its first-best level” (Section 2.2, pp. 11-12). The resulting inflation equation contains a distributed lag of the welfare-relevant gap and a distributed lag of the supply and preference shocks, so “there is no longer an exact relation, however complex, between inflation and the welfare-relevant output gap.”

It runs through the factor price frontier: with Cobb-Douglas technology, real marginal cost is a weighted sum of the real wage and the real price of the non-produced input, so any shock that raises the real price of oil must produce either a fall in real wages or a rise in real marginal cost (Section 2.2, pp. 12-13). “Stabilizing inflation requires a proportional decline in the real wage; given real wage rigidity, this can only be delivered by a decrease in output relative to first best. Stabilizing instead the distance of output from first best will lead to higher inflation.” The authors note that the real price of the non-produced input is endogenous in their framework, so an increase in it “may come from either a negative supply shock … or a preference shock that brings down the marginal disutility of labor.”

Large at high rigidity and small at moderate rigidity; the expressions are, in the authors’ own words, “highly non-linear” (Section 3, pp. 13-15). Calibration: the discount factor taken approximately to one, an average price duration of six months following Bils and Klenow (2003) giving a Calvo slope of approximately 0.5, the share of the non-produced input in production set to 0.025, and a benchmark rigidity index of 0.9, which implies a six-quarter half-life for real wage adjustment. Under the policy that fully stabilises the welfare-relevant gap, a 10 percent oil price rise implies annualised inflation “slightly above 4 percent on impact”; at a rigidity index of 0.8 that falls to 2 percent, and at 0.5 to below 0.5 percent. Under the policy that fully stabilises inflation, and assuming a unit Frisch labour supply elasticity, the same shock implies a 1.1 percent first-quarter reduction in the welfare-relevant output gap with relatively small persistence (0.10); at 0.8 the figures are 0.5 percent and 0.05, and at 0.5 they are 0.1 percent and 0.01. The authors’ own calibration caveat is worth carrying: “Empirical estimates of [the Calvo slope] are often much smaller … For the purposes of the present exercise we take our model literally and use the implied value,” and they describe the resulting numbers as “rough and purely illustrative,” concluding only that they “suggest a far from quantitatively trivial policy trade-off.”

It multiplies the short-run output loss by a factor that is increasing, and highly non-linear, in the rigidity index – a factor of ten at the benchmark value of 0.9 (Section 4, pp. 15-17). In the standard New Keynesian model a permanent unexpected reduction in inflation produces a permanent decline in output and real wages, but a small one: with the discount factor at 0.99, the Calvo parameter at 0.75, a unit Frisch elasticity and an oil share of 0.025, cutting inflation permanently by 5 percentage points lowers output at all horizons by 0.25 percentage points and the real wage by 0.1 percentage points. (The authors explain the mechanism in a footnote: Calvo price setting implies average markups fall as inflation rises, so disinflation raises average markups, lowers real wages and hence labour supply.) With real wage rigidities, output and real wages return to their natural levels in the period immediately after a successful move to price stability, but in the period when the disinflation actually takes place “the central bank needs to engineer a much larger recession” to push the real wage down to the level consistent with price stability – at a rigidity index of 0.9, moving from 5 percent inflation to zero “now implies a short run output loss of 2.5 percentage points, a non negligible value.”

They reject the cost-push shock as incomplete and argue that the Erceg-Henderson-Levin structure only relocates the divine coincidence rather than removing it (Section 5, pp. 17-20). On appending an exogenous “cost-push” disturbance: “Taken at face value, this is a fix, not an acceptable solution: One needs to know where this additional disturbance comes from, and why it belongs to the equation.” They accept that exogenous “distortion shocks” – tax changes, changes in desired markups – provide a justification, and that with respect to those shocks the coincidence indeed fails, but object that “it still holds with respect to standard supply shocks, such as movements in the price of oil or technology shocks,” so that the extended model “still implies that keeping inflation constant in the face of increases in the price of oil is the right policy – a proposition which, again, seems implausible.”

That with staggered wage and price setting the divine coincidence “emerges again, though in a different guise”: a composite of wage and price inflation, weighted by the two Calvo slopes, satisfies a Phillips curve in the welfare-relevant output gap alone (Section 5.2, pp. 18-20). Neither wage nor price inflation on its own depends only on the output gap – both also depend on the distance of the real wage from its natural level – so in that sense the divine coincidence does not apply to either. But the composite index does satisfy the exact relation, and since the first-best/second-best gap remains constant in that model (now involving the sum of the wage and price markups), stabilising the distance of output from first best is equivalent to stabilising that composite. The authors then note the policy implication: they acknowledge that in the Erceg-Henderson-Levin welfare function “strict price inflation targeting is generally suboptimal, and often involves welfare losses that are several times larger than other, better designed policies,” but observe that strict output gap stabilisation – equivalently, stabilisation of the composite inflation index – while exactly optimal only for a specific parameter configuration, is found by Erceg, Henderson and Levin to be “nearly optimal for a large range of parameter values.” Their conclusion is therefore narrow and specific: “Hence, and for all practical purposes, a meaningful policy tradeoff is also missing in the EHL model.”

Because a change in the output gap shifts workers’ reservation wage, and that shift feeds into the real wage – and hence real marginal cost – only gradually, outliving the return of output to its natural level (Section 6.1, pp. 20-22). In the baseline New Keynesian Phillips curve “inflation will not outlive any variation in the output gap.” With rigidity, even a purely white-noise output gap produces inflation that carries a geometrically declining tail in the lagged shocks, with the weights governed by the rigidity index. Rearranging gives a representation with both lagged and expected inflation whose coefficients sum to one; as the rigidity index rises from 0 to 1, the coefficient on past inflation rises from 0 to slightly above one-half and the coefficient on expected inflation falls from the discount factor to slightly below one-half. The authors add a caveat about the empirical literature this appears to rationalise: the representation “is not directly estimable since the natural level of output, and by implication the output gap, is not observable,” and they “view the ad-hoc measures of the output gaps used in the literature … with some suspicion.” They also flag, crediting Rotemberg for the correction, that the claim that inflation persistence equals output-gap persistence absent real rigidities “is not a general proposition”; what is true in general is that persistence increases with the rigidity index.

Desired labour supply is defined implicitly from the wage and the marginal utility of income, and involuntary unemployment is the log deviation between desired supply and actual employment (Section 6.2, pp. 22-23). Absent wage rigidity there is no involuntary unemployment, since the wage always equals the marginal rate of substitution. With rigidity, the change in the real wage is proportional to the negative of unemployment, with the response inversely related to the rigidity index and positively related to the slope of labour supply. Rewriting the inflation equation in terms of unemployment and the real price of the non-produced input yields a specification in lagged inflation, expected inflation, unemployment and the change in that real price – which, apart from the forward-looking term, “is indeed quite close to traditional specifications of the Phillips curve, which typically include changes in the price of oil and other supply side factors in addition to unemployment on the right hand side.”

All coefficients carry the right sign and are statistically significant in the unrestricted specification; the theoretical sum restriction is not rejected at the 5 percent level, “though not by much” (Section 6.2, p. 23). Estimation is by instrumental variables on annual U.S. data for 1960-2004, using inflation measured by the percent change in the GDP deflator, the civilian unemployment rate, and the percent change in the PPI raw materials index relative to the GDP deflator, with four lags of the three variables as instruments. Unrestricted, the coefficients are 0.66 on lagged inflation, 0.42 on expected inflation, -0.20 on unemployment and 0.018 on the change in the real raw materials price. Imposing the sum restriction gives 0.52, 0.48, -0.08 and 0.014, and here the authors are careful: “the coefficients on unemployment and raw materials prices are only significant at the 10 percent level.” The restricted coefficients imply a discount factor of 0.92, and the authors note explicitly that “the other structural coefficients are not identified and cannot be recovered (they would be if we estimated the full model, something we have not done).”

They reject all three, on empirical rather than theoretical grounds (Section 6.3, pp. 24-25). On the hybrid Phillips curve derived from automatic indexation of non-reoptimising prices to past inflation: “We also see this as an unconvincing fix, with little basis in fact: none of the existing micro-studies of price setting uncover any form of mechanical indexation to past inflation.” On Fuhrer and Moore’s relative wage concerns: citing Holden and Driscoll, they argue the persistence there “comes in fact from the assumption that workers care about the real wages of workers in the previous period. In this sense, the results from Fuhrer and Moore come from an assumption similar to ours.” On sticky information: “Firms do not seem to adjust their prices continuously according to a pre-specified plan,” and recent survey evidence suggests “firms review their prices more often than they change them, exactly the opposite to what is assumed by sticky information models.”

They present real wage rigidity as one instance of a general proposition about the interaction of real imperfections with shocks, not as the unique fix (Sections 1 and 7, pp. 3-4, 25-26). “The optimal design of macroeconomic policy depends very much on the interaction between real imperfections and shocks”; in their model that interaction “works through endogenous variations in wage markups, resulting from the sluggish adjustment of real wages,” but “a similar interaction could work, for example, through the endogenous response of desired price markups to shocks, as in Rotemberg and Woodford (1996).” Their forward agenda is correspondingly cautious: to find microfoundations for real wage rigidity, noting that foundations based on shirking, rule-of-thumb wage setters, or bargaining “may lead to more substantial departures from the NK benchmark than we have allowed for here” – Hall’s formalisation, for instance, would modify labour demand as well as the wage equation – and to estimate a joint model of wage and price inflation and conduct a quantitative analysis of the optimal response to an oil price change.