Sticky Price Models of the Business Cycle: Can the Contract Multiplier Solve the Persistence Problem?

The problem is that generating persistent output movements from monetary shocks requires prices not to change much for a long time after the shock, and the unappealing way to get that is simply to assume prices are exogenously fixed for long periods (Section 1, Minneapolis Fed Staff Report 217, pp. 1-2). “A preferable way is to construct models in which small frictions lead to long periods of endogenous price rigidity and, hence, persistent output movements.” Staggering is the candidate because of the mechanism Taylor (1980) set out, which the authors quote in full: “Because of the staggering, some firms will have established their wage rates prior to the current negotiations, but others will establish their wage rates in future periods. Hence, when considering relative wages, firms and unions must look both forward and backward in time to see what other workers will be paid during their own contract period. In effect, each contract is written relative to other contracts, and this causes shocks to be passed on from one contract to another – a sort of ‘contract multiplier’.” The authors’ contribution is to embed this in a quantitative general equilibrium model with rational price-setters, where Taylor and Blanchard “both exogenously specify the rules for setting wages or prices.”

As the ratio of the half-life of output deviations under staggered price-setting to one-half the length of exogenous price stickiness (Section 4, pp. 14-15). The denominator is justified by the fact that under synchronized price-setting “on average the half-life of output is one-half the length of exogenous stickiness. The reason is that monetary shocks occur randomly between price adjustments, and the associated movements in output last only until the next price adjustment.” Thus “a multiplier of, say, 5 means that staggered price-setting generates output movements that last 5 times longer than they would with synchronized price-setting.” The authors flag an attractive property: “in our quantitative models, this multiplier is approximately invariant to the length of exogenous price stickiness.”

A multiplier of 20, derived from a 10-quarter output half-life in the data against one quarter of exogenous price stickiness (Section 4, pp. 14-15). The half-life is measured by fitting an ARMA process to quadratically detrended log real GDP, with the fitted equation having AR coefficients of 1.30 and -0.38 (standard errors 0.066 each), a Ljung-Box Q of 13.8 at significance level 0.46 indicating little evidence of residual serial correlation. The authors are explicit about the limits of the exercise: “This measure is admittedly imperfect, but provides a useful benchmark.” They also state the alternatives: “With one month of exogenous price stickiness, the needed contract multiplier is 60; with one year, 5.”

Money demand parameters from an estimated velocity regression, the markup from the productivity literature, and the remaining parameters from balanced-growth relationships matched to standard business cycle statistics (Section 4, pp. 12-14). The share parameter and interest elasticity of money demand come from regressing log consumption velocity on the interest rate term implied by the model’s bond first-order condition, using 1960:1-1995:4 Citibase data on M1, the GDP deflator, consumption of services, nondurables and durables, and the three-month Treasury bill rate; this gives a share parameter of 0.94 and an interest elasticity of 0.39 with a standard error of 0.033 – “similar to that estimated by Mankiw and Summers (1986) and Lucas (1988) and smaller than that of Stock and Watson (1993).” Based on Basu and Fernald, Basu and Kimball, and Basu, the demand parameter implies an 11 percent markup and a demand elasticity of 10. The model then predicts an annualized capital-output ratio of 2.65, an investment-output ratio of 0.23, and one-third of time allocated to the market, with a capital share of one-third under the assumption that monopoly profits are allocated proportionately to capital and labour. Adjustment costs are set so HP-filtered model data reproduce the U.S. relative volatility of investment to output (3.25), and set to zero where the no-adjustment-cost volatility is already too small. Money growth follows an AR(1) with serial correlation 0.57, estimated from quarterly M1 over 1959:3-1995:2. Most experiments set exogenous stickiness to one quarter with 13 cohorts, so prices are fixed for a quarter, a cohort resets weekly, and consumers decide weekly. The authors describe this as “only setting the midpoint of the ranges for each of the parameters,” with sensitivity ranges “set wide enough to encompass most estimates of the various parameters in the literature.”

Without capital and with a static money demand equation, the model reduces to exactly Taylor’s (1980) system – except that Taylor’s key coefficient is a free parameter while here it is pinned down by preferences and technology, and is necessarily greater than one (Section 5, pp. 15-18). Setting the capital share to zero, using two cohorts, and imposing that real balances equal consumption equal output, the log-linearised system is a money demand equation, a price index equation, and a price-setting equation. “The only difference between our price-setting equation and Taylor’s is that our value of [the coefficient on the sum of future output] depends on the underlying preferences and technology while Taylor’s … is a structural parameter.” Large output movements require small price level movements, which in turn require that coefficient – the elasticity of the equilibrium real wage with respect to consumption – to be small. But under the benchmark utility function it equals one plus a positive term, so it exceeds one, which makes the relevant root of the characteristic equation negative and hence output non-persistent. “When [the coefficient] is treated as a free parameter, Taylor’s model … can produce the needed contract multiplier … For a multiplier of 20, we need [a root of] 0.965 and hence [a coefficient of] 0.00031. In our model, [it] is not a free parameter and is necessarily greater than 1.”

A multiplier of roughly 1 in all versions, and increasing the number of cohorts makes it slightly worse rather than better (Section 5, pp. 18-19, and Table II). “In Table II we see that for all versions of our benchmark model, the contract multiplier is roughly 1. With this contract multiplier, in order to generate the half-life of output seen in the data, the exogenous length of price stickiness would have to be 5 years.” On the Taylor-Blanchard conjecture that, holding the length of price fixity constant, more staggering should raise persistence, the authors test 2, 13 and 26 cohorts and obtain multipliers of 1.06, 0.99 and 0.87: “These results show that increasing the amount of staggering does not increase the persistence of output responses.” They also confirm the multiplier does not vary with the length of exogenous stickiness – changing it from one quarter to one year leaves the benchmark multiplier unchanged.

They do without intertemporal links and fail dramatically with them (Section 6.1, pp. 19-22, and Table II). Making consumption and leisure near-perfect substitutes drives the real-wage-to-consumption elasticity toward zero: with a consumption curvature of 0.0002 and a labour supply elasticity of 9,091, the coefficient reaches 0.00031 and the multiplier reaches 20. “With N = 2, no persistence in money growth, and no intertemporal links, the multiplier is 21.97. The same economy with intertemporal links has a multiplier of only 0.50.” The authors identify capital accumulation as the key link and draw the methodological moral explicitly: “Eliminating intertemporal links makes developing analytical solutions easy; that step is, therefore, usually thought of as a useful shortcut. The results here show that this shortcut can be misleading.” The mechanism is that a money shock triggers an impact-period investment boom, which raises labour demand and wages; with utility nearly linear in both consumption and labour, that produces a large impact-period rise in consumption and labour relative to later periods, shrinking the half-life. Two further objections are recorded: with 13 cohorts, persistent money growth and both links, the relative volatility of investment is 0.42, “substantially less than the corresponding statistic in the data,” and adding adjustment costs would push it lower still; and with 2 percent annual growth in wages and consumption, these preferences imply “labor input rises 57 percent each day.” The authors also use this case to make a general point – that in this model “although the price level responds slowly, output is not affected very much, and the contract multiplier is tiny,” because interest rates fall enough for households to willingly hold higher real balances.

It raises the multiplier but only at a degree of demand convexity the authors judge empirically untenable, and intertemporal links cut it further (Section 6.2, pp. 22-24, and Table II). With constant-elasticity demand the markup is constant so a 1 percent cost rise produces a 1 percent price rise; Kimball’s alternative final-goods technology makes the elasticity of demand rise with relative price, so markups fall and prices respond less than one-for-one. Matching Kimball’s parameterization – scaled to this model’s steady-state demand elasticity of 10, so that a 1 percent rise in market share lowers the elasticity from 10 to 7 – gives a curvature parameter of -289 and a multiplier of 2.2 in the version without links. But a second-order expansion of the implied demand function shows “a 2 percent increase in relative prices results in a 78 percent reduction in demand, and a 2.3 percent increase in relative prices results in zero demand. A demand function with this extreme level of convexity is clearly inconsistent with both casual empiricism and a wide variety of demand studies for a wide range of products.” With 13 cohorts and persistent money growth the multiplier is 3.79 without links and 1.55 with them. Pushing the curvature 100 times higher still does not produce a multiplier above 5, and at that curvature “a 0.5 percent increase in relative prices results in zero demand.”

Only at demand elasticities with wildly counterfactual implications for the cross-firm output distribution, and again intertemporal links shrink the gain (Section 6.3, pp. 25-27, and Table II). Adding an inelastically supplied factor specific to each intermediate good makes marginal cost slope upward, which sets two forces against each other: an output effect (higher aggregate output raises the monopolist’s costs through decreasing returns, making prices more sensitive to output) and a wage effect (with decreasing returns, wage increases induce a smaller price increase, making prices less sensitive), with the wage effect stronger the more elastic is demand. At the calibrated parameters and a decreasing-returns parameter of 3/2, the relevant coefficient is 0.19, yielding a multiplier of only 1.35. Reaching 20 requires a demand elasticity of about 6500 – under which “a 1 percent difference in relative prices implies a difference in relative outputs” that the authors report as astronomically large. With capital and interest-sensitive money demand added, the multiplier falls from 1.82 to 1.33, “which is not much higher than the multiplier’s value in our benchmark economy.”

The multiplier stays around 2, and the three mechanisms do not reinforce one another (Section 6.4, pp. 27-29, and Figures 1-2, Table III). Varying parameters one at a time in the benchmark economy while holding the relative volatility of investment fixed: the multiplier is roughly constant in the interest elasticity of money demand, rising only below 0.05; it does not change much with the capital share; it falls as risk aversion rises above log utility, because keeping investment volatility at its data value requires reducing adjustment costs as utility becomes more concave, and above a risk aversion of 2.2 no adjustment costs are needed and the multiplier is small; and it rises with the demand elasticity, for reasons similar to the specific-factors case, since “capital does not move much, it acts like a factor in fixed supply.” Varying the interest elasticity, capital share, risk aversion, weight on leisure, demand elasticity and money growth persistence simultaneously over wide ranges, “the largest value of the multiplier we find is 3.05,” at an extreme corner (interest elasticity 0.2, capital share 0.22, leisure weight 9, log utility, demand elasticity 300, money growth serial correlation 0.96). The analogous maxima are 3.20 for the convex demand economy and 4.17 for specific factors. Combining near-perfect substitutes, convex demand and specific factors at calibrated values raises the multiplier only “modestly to 1.81 from 1.55 (convex demand only) and 1.33 (specific factors only),” with the same 4.17 maximum under simultaneous variation. The reason no interaction emerges is that “the convex demand specification gives a large multiplier when the demand elasticity is small, while the specific factors specification gives a large multiplier when the demand elasticity is large,” so combining them just reproduces whichever mechanism the chosen elasticity favours: “In no sense do these three variations tend to reinforce each other.”

That staggered price-setting alone does not generate monetary business cycles, stated as a conditional on what one finds plausible rather than as an unconditional impossibility (Section 7, p. 30). “Here we find that the staggered price-setting mechanism is not the long-sought solution.” The conditional form is worth carrying exactly: “One interpretation of our findings is that if 5 years of exogenous price stickiness is plausible, then conventional models used in the business cycle literature can generate persistent output fluctuations. With the amount of convexity in demand posited by Kimball (1995), or with extremely elastic demand and sizable departures from constant returns to scale, generating persistence requires 2½ years of exogenous price stickiness. If one quarter of exogenous price stickiness is plausible, then generating persistence requires extraordinarily convex demand or extremely elastic demand. Altogether, these findings suggest that mechanisms to solve the persistence problem must be found elsewhere.”

That endogenous price stickiness is not sufficient for persistent output – the two can come apart (Section 6.1, p. 20). In the near-perfect substitutes economy the authors “find that both with and without intertemporal links, prices change hardly at all after a monetary shock, but with intertemporal links, output is not persistent. This finding shows that endogenous price stickiness, by itself, does not necessarily imply persistent output fluctuations.” This is a distinct claim from the multiplier result: it says that even confirming that firms choose not to move prices does not settle whether output movements will last.