Optimal monetary policy with staggered wage and price contracts

It asks whether the monetary policy tradeoff between inflation variability and output-gap variability survives in a fully specified optimizing-agent model, given that recent dynamic general equilibrium models with staggered price setting and completely flexible wages had appeared to eliminate it (Section 1, IFDP 640, p. 1). The authors note that in those models “monetary policy rules that keep the inflation rate constant also minimize output gap variability,” so that the monetary authorities “can achieve the Pareto-optimal welfare level … through the remarkably simple policy regime of strict price inflation targeting, irrespective of the parameter values or other specific features of these models” (p. 1). Their move is to add staggered wage setting alongside staggered price setting, on the reasoning that with staggered wage contracts “aggregate wage inflation induces inefficiencies in the distribution of employment across households,” just as price inflation induces inefficient dispersion of output across firms — so achieving the Pareto-optimal equilibrium “would require not only a zero output gap and complete price stability, but also complete stabilization of nominal wages” (p. 1).

It is a dynamic general equilibrium model with monopolistically competitive firms setting prices in Calvo contracts and monopolistically competitive households setting wages in Calvo contracts, with a fixed aggregate capital stock and Cobb-Douglas production (Section 2, pp. 2-8). A representative “output aggregator” combines differentiated goods into an output index with a Dixit-Stiglitz technology; each good is produced by one firm hiring capital services and a labour index, with the aggregate capital stock fixed and factors perfectly mobile across firms. Prices are reset with a fixed per-period probability, and a firm that cannot reset has its price automatically raised at the unconditional mean gross inflation rate. In contrast to most recent contributions of the period, households — not firms — set wages, and in staggered contracts of the same Calvo form. Because monopolistic competition would otherwise leave output and labour supply below their Pareto-optimal levels even with flexible wages and prices, the authors follow recent studies and assume output and labour are each subsidised at fixed rates, so the flexible-wage, flexible-price equilibrium is Pareto-optimal: “the central task of monetary policy is to mitigate the effects of nominal inertia, while fiscal policy assumes responsibility for offsetting distortions associated with imperfect competition” (p. 2).

The key equations are a consumption Euler equation (“goods demand”), a marginal-product-of-labour schedule, a marginal-rate-of-substitution schedule, a price-setting equation, a wage-setting equation, and an identity linking the change in the real wage to the difference between wage and price inflation (Table 1, p. 9). Price inflation depends positively on expected price inflation and on the percentage by which the real wage exceeds the marginal product of labour; wage inflation depends on expected wage inflation and on the percentage by which households’ average marginal rate of substitution exceeds the real wage. The marginal product of labour falls, and the marginal rate of substitution rises, with the output gap, so the output gap is zero exactly where the two schedules intersect — at the Pareto-optimal real wage. The authors observe a “formal similarity to the earlier work on ‘disequilibrium’ models,” in that wages and prices are subject to nominal inertia and adjust partially toward the Pareto-optimal equilibrium, but stress that here the wage and price equations “are derived from optimizing behavior, and thus depend on the underlying structure of preferences and technology as well as exogenously-specified mean contract duration parameters” (p. 10).

The policymaker is assumed to maximise the unconditional expectation of the unweighted average of household utility; a second-order approximation reduces this to a negatively weighted sum of the unconditional variances of the output gap, price inflation and wage inflation, scaled as a fraction of steady-state consumption (Section 3, pp. 10-14, following “essentially the same methods employed by Rotemberg and Woodford”). The mechanism is that price dispersion across firms creates inefficient cross-sectional variation in output, and wage dispersion across households creates inefficient variation in hours. The welfare cost of price inflation volatility rises with the substitutability across differentiated goods (inversely related to the price markup rate) and with the mean duration of price contracts; the welfare cost of wage inflation volatility rises symmetrically with substitutability across differentiated labour inputs and with mean wage contract duration. The authors also note an asymmetry: if markup rates and contract durations are identical for wages and prices, “the costs of wage inflation volatility exceed those of price inflation volatility for reasonable values” of the capital share, because the wage adjustment coefficient is smaller than the price adjustment coefficient (p. 14). Finally, the welfare cost of output-gap volatility does not depend on either contract-duration parameter, so “the relative weight on output gap volatility declines with the mean duration of price contracts and the mean duration of wage contracts” (p. 14). Welfare losses from fluctuations in real balances are assumed small enough to ignore.

Proposition 1 states that with staggered wage and price setting it is impossible for more than one of the output gap, price inflation and wage inflation to have zero variance, and hence that monetary policy cannot achieve the Pareto-optimal level of social welfare (Section 4.1, pp. 14-15). The proof of part (A) works from the supply relations: price inflation remains constant if and only if all firms are continuously on their labour demand schedules (real wage equal to the marginal product of labour); wage inflation remains constant if and only if all households are continuously on their labour supply schedules (real wage equal to the marginal rate of substitution); and the Pareto-optimal output and real wage occur where the two schedules intersect, so any two of the three zero-variance conditions imply the third. But the Pareto-optimal real wage moves in response to each of the exogenous shocks, whereas the actual real wage “would always remain at its steady-state value if neither prices nor wages ever adjust.” Given this contradiction, no more than one of the three variables can have zero variance when the shocks have non-zero variance. Part (B) then follows because all three variables enter the welfare function with strictly negative weights.

Proposition 2 establishes that with completely flexible wages it is possible to stabilise both price inflation and the output gap, and with completely flexible prices it is possible to stabilise both wage inflation and the output gap; in either case monetary policy can achieve Pareto-optimal welfare (Section 4.2, pp. 15-17). With flexible wages, the condition that households are on their labour supply schedule no longer pins wage inflation, so combining it with the other relations yields a familiar expectational Phillips curve with no error term, in which stabilising price inflation also delivers a zero output gap. With flexible prices there is a formally symmetric wage-setting relation: stabilising wage inflation stabilises the output gap while prices adjust freely to keep the real wage at its Pareto-optimal value. In each case the variance that cannot be controlled is precisely the one carrying zero weight in the welfare function.

In the authors’ model the price-inflation/output-gap variance tradeoff arises endogenously, from the term in the real wage’s deviation from its Pareto-optimal value, rather than from an exogenous residual appended to the Phillips curve (Section 4.3, pp. 16-17). One existing approach adds a shock to the flexible-wage Phillips curve, so that under complete inflation stabilisation the variance of the output gap is proportional to the variance of that exogenous shock; the authors note such a shock “does in fact arise as an estimation residual, and has been viewed as representing aggregate pricing mistakes or other unexplained deviations from the optimality condition.” Substituting the marginal-product-of-labour schedule into the price-setting equation instead yields a relation in which the output gap’s variance under complete price stabilisation is proportional to the variance of the real wage’s deviation from its equilibrium value – so the tradeoff “depends on the preference and technology parameters of the model as well as the exogenous disturbances,” not on an ad hoc residual. They also note that the other common route – staggered wages with completely flexible prices – does produce a tradeoff, but one with no welfare consequences, since price inflation variability does not enter the welfare function in that case.

Quarterly calibration: discount factor 0.99; near-logarithmic utility in consumption and leisure; Cobb-Douglas capital share 0.3 (labour elasticity of output 0.7); wage and price markup rates both 1/6; Calvo parameters for both wage and price contracts of 0.75, implying a mean contract duration of four quarters; and an AR(1) productivity shock with first-order autocorrelation 0.95, scaled so that Pareto-optimal output at an annual rate has a standard deviation of about 3.1 percentage points (Section 5.1, pp. 17-18). The analysis focuses exclusively on volatility from productivity shocks, the authors noting that consumption and leisure shocks imply qualitatively similar properties of the policy frontier. The optimal rule takes the form of an interest rate rule with a coefficient on current price inflation large enough to ensure determinacy plus optimally chosen coefficients on lagged interest rates, lagged real wages, and current and lagged shocks; the model is solved with the Anderson-Moore algorithm and the rule parameters chosen by hill-climbing. Under this calibration “the expected welfare loss compared with the Pareto optimum is about 0.0024 percent of steady-state consumption, and is about one-quarter as large as the baseline estimate of Lucas (1987) for the potential welfare gains from eliminating aggregate consumption fluctuations” (p. 19).

Sweeping both Calvo parameters from 0 to 0.9 (mean durations from one to ten quarters), the authors report three findings (Section 5.3, pp. 19-20). First, when only one nominal variable is sticky it is possible to smooth that variable completely and reach Pareto-optimal welfare: along the axis where prices are sticky but wages flexible, the optimal variances of price inflation and the output gap are both zero and welfare losses are zero even though wage-inflation variance is at its maximum, and symmetrically along the other axis. Second, “it is optimal ceteris paribus for the more flexible nominal variable (that is, the variable with the shorter contract duration) to bear a relatively larger share of the burden of achieving real wage adjustment” — optimal price-inflation variance rises as wage contracts lengthen or as price contracts shorten. Third, output-gap volatility is quite low under the optimal rule for every combination of contract durations, not just along the axes, which the authors describe as “somewhat surprising” given the output gap’s relatively low welfare weight under their parameters and which “suggests that strict output gap targeting is nearly optimal for a wide range of values of the structural parameters.”

It is clearly suboptimal, and in some parameter regions catastrophically so (Section 6, pp. 20-23, and Table 2). Under complete price stabilisation the real wage must lie on the marginal-product-of-labour schedule, so all real-wage adjustment has to come through nominal wage movements; the resulting equilibrium is where the zero-price-inflation schedule crosses the labour demand schedule, and the costs depend on the slopes of the two schedules and on the sensitivity of nominal wages to the current marginal-rate-of-substitution gap. “If the mean wage contract duration is long (that is, only a small fraction of households revise their wage contracts in any given period), then strict inflation targeting is associated with relatively large movements in the output gap and in substantial cross-sectional dispersion in wages and hours worked. Large output and wage fluctuations also occur if the [marginal product of labour] schedule is relatively flat (corresponding to highly elastic demand for labour)” (p. 22). In Table 2 the pattern is stark: at the baseline four-quarter mean wage contract duration, the welfare loss under strict price inflation targeting is roughly eight times that under the optimal rule, and it grows further as wage contracts lengthen to eight quarters. As the labour elasticity of output falls toward zero the loss under strict price inflation targeting rises by more than two orders of magnitude relative to the optimal rule.

Nearly, but not always: it matches the optimal rule closely across wage contract durations and labour elasticities, yet imposes substantial costs when the price markup rate is much smaller than the wage markup rate (Section 6, pp. 22-23, and Table 2). When the output gap is held at zero, the marginal product of labour and the marginal rate of substitution both equal the Pareto-optimal real wage, and the implied ratio of price-inflation variance to wage-inflation variance is simply the ratio of the two adjustment coefficients — neither of which depends on the price markup rate, while the welfare cost of price inflation volatility rises non-linearly with it. So “if the degree of substitutability across differentiated goods were much greater than the degree of substitutability across differentiated labor inputs … then strict output gap targeting would tend to induce excessive price inflation variability compared with the optimal policy rule” (p. 23), and Table 2 confirms a noticeable gap at the smallest price markup rate examined.

Both hybrid rules — one responding to price inflation and the output gap, the other to price inflation and wage inflation — perform nearly as well as the optimal rule in all cases, despite containing no lagged state variables (Section 6, p. 23, and Table 2). The authors point out that the wage-inflation hybrid “is a natural one to consider in a model with two sources of nominal inertia, and has the advantage that the policymaker need not know the Pareto-optimal level of output.” The two free parameters of each hybrid rule are re-optimised for every structural-parameter configuration examined, as are the optimal rule’s parameters.

They argue the tradeoff turns on two features that are not specific to staggered wage and price setting: several input and/or output prices must be set in nominal contracts that are not completely synchronised, and the relative prices of some of those inputs and outputs must have to move in the Pareto-optimal equilibrium (Section 7, pp. 23-24). In the special case of staggered price setting with flexible wages, the second condition fails because the Pareto-optimal relative prices of the differentiated goods are invariant to aggregate shocks. They sketch other structures that would satisfy both conditions: differentiated goods at multiple stages of production with both intermediate and final prices staggered; homogeneous inputs with one-period input price contracts and staggered output price contracts; and even staggered output prices with flexible input prices if producers face idiosyncratic productivity shocks or relative demand shifts.

They acknowledge that recent contributions emphasised sticky prices over sticky wages partly because state-contingent employment contracts could in principle prevent any misallocation of labour from nominal wage contracts (Section 7, pp. 23-24). Their answer is symmetry: “one can also imagine state-contingent output contracts which ensure that sticky prices have no allocative effects; such state-contingent contracts are neither more nor less plausible than the analogous employment contracts,” so it “seems reasonable to assume that both wage and price contracts have significant allocative effects, at least until further guidance is provided by empirical research.” They flag four extensions: richer dynamics from capital accumulation, adjustment costs and habit persistence; relaxing the strong assumption that all current variables and shocks are observable to agents and the policymaker, which would introduce persistent output-gap measurement error; endogenous determination of contract duration; and incomplete asset markets, since the welfare costs of output-gap deviations are likely sensitive to the assumption of complete consumption risk-sharing across households.