We compare two ways of modeling Calvo-type wage stickiness. One in which each household is the monopolistic supplier of a differentiated type of labor input (as in Erceg, et al., 2000) and one in which households supply a homogenous labor input that is transformed by monopolistically competitive labor unions into a differentiated labor input (as in Schmitt-Grohe and Uribe, 2006a,b). We show that up to a log-linear approximation the two variants yield identical equilibrium dynamics, provided the wage stickiness parameter is in each case calibrated to be consistent with empirical estimates of the wage Phillips curve. It follows that econometric estimates of New Keynesian models that rely on log-linearizations of the equilibrium dynamics are mute about which type of wage stickiness fits the data better. In the context of a medium-scale macroeconomic model, we show that the two variants of the sticky-wage formulation give rise to the same Ramsey-optimal dynamics, which call for low volatility of price inflation. Furthermore, under both specifications the optimized operational interest-rate feedback rule features a large coefficient on price inflation and a mute response to wage inflation and output.
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