Labor Markets and Monetary Policy: A New Keynesian Model with Unemployment

Because standard versions “do not generate movements in unemployment, only voluntary movements in hours of work or employment,” which sits awkwardly with the model’s adoption as the backbone of central bank models (Section 1, NBER WP 13897, p. 3). The authors add in a footnote that this feature was “paradoxically … viewed as one of the main weaknesses of the RBC model, but was then exported to the NK model.” Their extension introduces frictions similar to those of the Diamond-Mortensen-Pissarides model, with the specific aim of asking how optimal monetary policy should differ depending on whether the labour market is fluid, as in the United States, or sclerotic, as in continental Europe.

Hiring costs are paid per hire and rise with labour market tightness; vacancies are filled immediately rather than after a random delay (Section 2.1, pp. 5-8). Employment evolves with an exogenous separation rate and new hires start working in the period they are hired, so beginning-of-period unemployment is one minus the surviving stock of last period’s employment. Tightness is hires over that pool, lies in the unit interval, and doubles as the job-finding rate. On the contrast with DMP: “in the DMP model, the hiring cost is uncertain, with its expected value corresponding to the (per period) cost of posting a vacancy times the expected time to fill it. This expected time is an increasing function of the ratio of vacancies to unemployment, which can be expressed in turn as a function of labor market tightness. Thus, while the formalism used to capture the presence of hiring costs is different, both approaches share the basic characteristic that the cost of hiring is increasing in labor market tightness.” The authors also flag that they deliberately scale hiring costs by productivity “to avoid effects of productivity shocks on the cost of hiring relative to the cost of producing, an effect we believe is best left out of the model.”

Because with log consumption and balanced-growth-consistent preferences the income and substitution effects on labour supply exactly offset, so employment does not move – and the authors explicitly label this special (Section 2.2, pp. 8-11). Without frictions the planner’s condition collapses to a constant employment level. With frictions, the planner’s optimality condition (equating the marginal rate of substitution to a marginal rate of transformation that nets out current hiring costs and the savings in next period’s hiring costs) admits a constant job-finding rate, and therefore a constant unemployment rate, with consumption and output proportional to productivity. The authors state the caveat directly: “This invariance result is obviously a special one (e.g. it would no longer hold if we introduced capital accumulation). It is, however, very convenient for our purposes, since it establishes a simple benchmark. And it contains a more general lesson. Even in a model with labor market frictions, the behavior of the marginal rate of substitution remains central to the outcome.”

The equilibrium unemployment rate generally differs from the constrained-efficient rate, but shares its invariance to productivity shocks (Section 3.2, pp. 12-15). The household’s surplus from an employment relationship is the wage less the marginal rate of substitution plus a continuation term; the firm’s surplus is simply the hiring cost, “since any current worker can be immediately replaced with someone who is unemployed by paying the hiring cost.” The bargained wage equals the marginal rate of substitution plus a term rising in current tightness and falling in expected future hiring costs and in the probability of not finding a job next period. The two unemployment rates coincide only if the gross markup is one and worker bargaining power equals the elasticity of hiring costs with respect to tightness – “a Hosios-like condition, familiar from DMP models.”

They are different results with different sources, and the authors argue theirs makes the puzzle worse, not better (Section 3.2, pp. 14-15). “Shimer’s result was derived under the assumption that the first term – the marginal rate of substitution – was constant,” and he argued that under reasonable friction parameters the second term implied large wage movements and hence small movements in profit, job creation and unemployment. “In contrast, our neutrality result follows entirely from movements in the marginal rate of substitution. Under our assumptions, the marginal rate of substitution moves one-for-one with productivity, so employment does not change, and labor market frictions have no role to play.” Because in richer settings the marginal rate of substitution is still likely to rise with productivity, “the wage response will be stronger than in the DMP model. Put another way, the Shimer puzzle will be even stronger than in the original Shimer set-up.” This is the stated motivation for imposing real wage rigidity: “This large response of the wage to productivity movements appears counterfactual.”

Through a schedule in which the real wage is proportional to productivity raised to the power one minus a rigidity index, justified by the wage band that frictions create (Section 3.3, pp. 15-16). Following Hall, the surplus from existing relationships means many wage paths are consistent with private efficiency – any path keeping both the household’s and the firm’s surplus non-negative, which here amounts to the wage lying between the marginal rate of substitution evaluated at full employment and productivity divided by the markup. The authors note the Nash-bargained wage satisfies this, assume the economy fluctuates near the Nash steady state, and restrict attention to shocks that are not too large. They acknowledge the modelling choice is provisional: “How to formalize real wage rigidity is still very much an open research question. To keep the analysis as simple as possible, we assume a wage schedule of the form…” At index zero the schedule reproduces the Nash wage exactly; at index one it is Hall’s canonical rigid wage. The formulation “is meaningful only if technology is stationary, an assumption we shall maintain here.”

Inflation depends on expected inflation and on current, lagged and expected unemployment, plus productivity – with the split between the level and the change in unemployment governed by labour market fluidity (Sections 4.2 and 5, pp. 19-24). Under two simplifying approximations (hiring costs small relative to output, separation rate small), inflation depends positively on tightness and negatively on productivity when the rigidity index is positive, with the productivity effect larger the greater the rigidity or the more persistent the productivity process. Substituting the relation between tightness and unemployment gives an equation in the level and the change in unemployment: “the more sclerotic the labor market, the weaker the effect of the level of unemployment, and the stronger the effect of the change in unemployment.” The intuition: “In a fluid labor market, average flows are high and, given the constant separation rate, depend on the level of employment … Changes in employment … lead to small relative changes in the flows, thus to small relative changes in labor market tightness. In a sclerotic labor market, average flows are low. Changes in employment … lead to large relative changes in the flows.” Without the approximations, the exact log-linearised Phillips curve carries expected inflation and lagged, current and expected unemployment.

Because productivity shocks move the wedge between the natural unemployment rate and the constrained-efficient rate whenever wages do not adjust fully (Section 4.2, pp. 21-22). “Stabilizing inflation, which is equivalent to stabilizing unemployment at its natural rate, does not deliver constant unemployment. Symmetrically, stabilizing unemployment does not deliver constant inflation.” The authors tie this explicitly back to their 2007 paper: “The reason is the same as in our earlier paper, the fact that productivity shocks affect the wedge between the natural rate … and the constrained-efficient unemployment rate.”

Unemployment stabilisation makes inflation move with productivity; strict inflation targeting makes unemployment move, with intrinsic persistence set by labour market fluidity (Section 5.1, pp. 23-24). Holding unemployment at its efficient level fixes hiring costs, so real marginal cost varies negatively with productivity and inflation fluctuates; the amplitude rises with the rigidity index and with the persistence of the productivity process, and falls with the degree of nominal rigidity. Setting inflation to zero requires real marginal cost to be fully stabilised, which forces unemployment and hence hiring costs to vary negatively with productivity. Two properties follow: “the volatility of unemployment under that policy regime is proportional to [the rigidity index], since the coefficients … are independent of that parameter,” and unemployment displays intrinsic persistence – serial correlation beyond that inherited from productivity – with the degree of persistence depending critically on the separation rate and the steady-state job-finding rate. “In a ‘sclerotic’ labor market, that is, a market with low [job-finding rate] and low [separation rate], and under strict inflation targeting, unemployment will display strong persistence, well beyond that inherited from productivity.”

A second-order approximation to household welfare around a steady state assumed to coincide with the constrained-efficient one gives a discounted sum of squared inflation and squared unemployment deviations, with the weight on unemployment derived from the model’s parameters (Section 5.2, pp. 25, and Appendix A). The authors state the simplifying assumption openly: they assume unemployment fluctuates around a steady state corresponding to the constrained-efficient allocation “to simplify the analysis and avoid well understood but peripheral issues.” Optimal policy minimises that loss subject to the exact Phillips curve; the first-order conditions plus the Phillips curve and the productivity process form a linear system solved by standard methods. The implied weight on unemployment is 0.0237 in the U.S. calibration and 0.0283 in Europe – values the authors describe as “seemingly low” but “of the same order of magnitude as the weight on the output gap in calibrated loss functions found in the literature.”

Quarterly; preferences from the literature, the labour market parameters chosen to hit fluid and sclerotic targets (Section 6, pp. 26-27). Discount factor 0.99, unit Frisch elasticity, elasticity of substitution 6 (gross markup 1.2); Calvo slope 1/12, “consistent with an average duration of prices between three and four quarters, in accordance with much of the micro and macro evidence on price setting.” The hiring cost elasticity is set to 1 by mapping to the DMP matching function – the elasticity corresponds to the matching elasticity divided by one minus itself, and “since estimates … are typically close to 1/2, we assume [elasticity] = 1.” The rigidity index is set to 0.5, “the midpoint of the admissible range,” with the authors “having no hard evidence”; a footnote reports that a regression of real wage growth on productivity growth in postwar U.S. data would suggest 0.6 to 0.7 under “an overly strict interpretation of our model,” and adds “obvious caveats apply, from the measurement of productivity growth, to the direction of causality.” The U.S. calibration targets 5 percent unemployment and a quarterly job-finding rate of 0.7; Europe targets 10 percent and 0.25. These imply separation rates of 0.12 and 0.04. The hiring cost scale is set so hiring costs are 1 percent of U.S. GDP – “which seems a plausible upper bound,” and lacking direct evidence – and the same value is used for both calibrations.

Modest under a transitory shock, large under a persistent one, with strict inflation targeting the costly extreme (Section 6.1, pp. 27-29). All responses are to a one-percent decline in productivity, reported in percentage points with inflation annualised. With a purely transitory shock, stabilising unemployment implies a one-period rise in inflation of less than 20 basis points, nearly identical across calibrations; stabilising inflation raises unemployment by about 65 basis points on impact in the U.S. calibration and 50 in the European one, with unemployment remaining elevated well after the shock has vanished and significantly more so in Europe. With an AR(1) coefficient of 0.9, stabilising unemployment implies about a 150 basis point rise in inflation on impact under both calibrations – an amplification the authors attribute to “the forward looking nature of inflation and the persistent anticipated effects on real marginal costs generated by the interaction of the shock and real wage rigidities.” Strict inflation targeting raises unemployment by about 3 percentage points on impact under both calibrations, with a prominent hump-shaped path reaching about 8 percentage points in Europe. The authors annotate that last figure themselves: “While the size of this response may be viewed as unrealistically large, it is important to keep in mind that the policy assumed is also unrealistically extreme.”

Optimal policy sharply reduces unemployment volatility at modest inflation cost, and an optimised interest rate rule captures much of the gain (Section 6.1, pp. 29-30, and Table 1). Under optimal policy unemployment rises 50 basis points in the U.S. calibration and about half that in Europe, “several times smaller than under the strict inflation targeting policy,” with inflation persistently higher by roughly 1 percentage point in the U.S. and 1.4 in Europe. The authors note that optimal policy is “tougher on inflation” (more hawkish) in the U.S. than in Europe, because inflation-stabilising policies impose a larger persistent unemployment cost under the European calibration. Table 1 reports welfare losses relative to optimal policy: “The welfare losses associated with a strict inflation targeting policy appear to be very large relative to the optimal policy, especially so under the European calibration, which yields losses that are 25 times larger than under the optimal policy.” An optimised simple rule responding to inflation and unemployment, with coefficients searched over a grid, gives 5 and -0.8 for the U.S. and 2 and -0.6 for Europe; it “reduces considerably the losses relative to the extreme policies under both calibrations and, at least under the European one, comes close to replicating the welfare outcome obtained under the optimal policy.”

As the tractable, analytically solvable member of a family of richer but simulation-only models (Section 7, pp. 30-32). They review the integration of standard preferences with frictions (Merz, Andolfatto), with frictions and Calvo pricing (Chéron and Langot, Walsh, Trigari), with frictions and real wage rigidity (Shimer, Hall, Gertler and Trigari), and the three papers closest to theirs that combine all four ingredients (Krause and Lubik, Christoffel and Linzert, Faia) – noting repeatedly that those models “are substantially richer than ours, and are solved through simulations.” Their claim is narrow and comparative: “we see the comparative advantage of our paper as being in its simplicity, its analytical characterization of the effects of productivity shocks and optimal monetary policy in relation to labor market characteristics. We think that our analytical model is a needed step in the development and full understanding of these richer but more complex models.” They also single out a live disagreement: Thomas (2007) studies a closely related structure with staggered nominal wage setting instead of real wage rigidity and reaches “implications substantially different from ours – which suggests that a more thorough exploration of the different implications of the two alternative assumptions is needed.”