Estimated non-linearities and multiple equilibria in a model of distributive-demand cycles

Abstract

We introduce the results of a non-parametric estimate of the US wage-Phillips Curve into a simplified version of the model of the wage-price spiral by Flaschel and Krolzig (2008). Making use of Okun’s law, the non-linearity in the wage inflation-employment relation translates into a non-linearity in the so-called ‘distributive curve’ of the economy. Exploiting the observed non-linearity in extending an otherwise standard demand-distribution model (Taylor 2004), we provide a dynamical analysis both in wage-led and profit-led effective demand regimes. In a profit-led scenario, shown to be the empirically relevant case for the US economy, there are two stable equilibria of Goodwin (1967) growth cycle type, identified as a stable depression and a stable boom, and a saddle-path stable equilibrium in between them. Both stable steady states are surrounded by trajectories that cycle counterclockwise around their basins of attraction. The obtained type of growth fluctuations can be verified by a long phase cycle estimation for the US economy using a method developed by Kauermann, Teuber and Flaschel (2008).

Keywords:

• profit-led demand
• profit squeeze
• long phase cycles
• multiple steady states
• booms and depressions

JEL Classifications:

• E24
• E31
• E32

Notes

1. See also Flaschel, Kauermann and Semmler (2007).

2. Bhaduri and Marglin (1990a, 1990b) reason in terms of the profit share, but being distributive shares the complement of one another there is no harm in considering the wage share instead, as we do in this paper.

3. The data for the plots used in Figure 1 are taken from the Federal Reserve Bank of St. Louis (see http://www.stls.frb.org/fred). The data are quarterly, seasonally adjusted. Except for the unemployment rate U the log of the series are used. See also Kauermann et al. (2008) for the econometric methodology that allows us to separate endogenously long phase cycles from cycles occurring at higher frequency and for empirical applications that are closely related to the ones shown in Figure 1.

4. In their model, the authors consider, in both equations, error-corrections for the deviation of the wage share from a certain level ψ ₀. For reasons of expositional simplicity, we do not analyze in this paper the consequences of this augmentation in both the money wage and the price Phillips curve.

5. Flaschel, Kauermann and Semmler (2007) provide estimates suggesting that κ, κ_p , κ_w are all positive. See also footnote 8. The growth rate of labor productivity is included in the definition of $\widehat{\psi }\equiv \widehat{w}-\widehat{p}-n_x$.

6. For basic standard treatments of Okun’s law, see a widely adopted intermediate macro textbook such as Blanchard (2010, Chapter 9), or Mankiw (2010, Chapter 9). Typical estimates of σ range from 1/2 (Abel and Bernanke 2005) to 1/3 (Prachowny 1993). Flaschel and Krolzig (2006), instead, parameterize σ = 1, an assumption that is confirmed by recent estimations by Proaño et al. (2007). Also, Foley and Michl (1999, 179) can be can be read as arguing that σ = 1.

7. The other estimated p-spline functions are not statistically different from linear ones – including the price Phillips curve – with the exception of the inflation climate which however does not matter for the law of motion of real wages.

8. Flaschel, Kauermann and Semmler (2007), specifying the inflationary climate as 12-quarter moving-average, obtain estimates of κ_w = 0.4464, β_p = 0.0026 and 1 − κ_p = 0.6859 so that, given the traditional σ = 1/3, the composite parameter $\frac{1-{\kappa }_w}{\sigma (1-{\kappa }_p)}{\beta }_p$ is roughly equal to 0.0063. Clearly, different specifications for the inflationary climate may lead to different results. It is worth to keep in mind, however, that recent estimations of the Okun’s coefficient, such as the one provided in Proaño et al. (2007), point toward σ not being significantly different from 1. In this case, the composite parameter would look closer to .002.

9. Clearly, ū, $\overline{\psi }$ are shift parameters in the model.

10. Skott (1989, Chapter 6) is an authoritative dissenting voice on such stability condition, and especially about its plausibility in the long-run.

11. We rule out as uninteresting the case of a demand regime laying entirely in the orthant in which capacity utilization takes only negative values, and therefore we impose $\frac{\overline{\psi }}{\overline{u}}>-\frac{{\beta }_{uu}}{{\beta }_{u\psi }}$ to be satisfied everywhere.

12. However, the case in which the locus $\dot{u}=0$ is so steep that there is only one intermediate equilibrium requires an intercept of the curve higher than 1, and this is a case we would like to rule out from the analysis. Clearly, a sound empirical analysis will be crucial on this respect, but we proceed here assuming that the $\dot{u}=0$-isocline is sufficiently flat.

13. Observe finally with respect to Figure 3 that there is of course a fourth steady state at the origin of the phase space, which however cannot be reached from the positive orthant.

14. Remark: If the economy is fluctuating around the extreme equilibria and comes closer to the intermediate one, a small shock may suffice to move it into one of the basins of attraction of the steady states E ₁ or E ₂ so that the business cycle will then change its course and converge either to a depressed or a boom situation. Convergence into the depressed basin may for example be the situation experienced in Germany, while the US economy seems to fluctuate outside the basins of attraction of the investigated dynamics, as Figure 1 shows.

15. Assumption 5 in Section 4 already incorporates induced technical change effects into the evolution of the wage share. In view of this argument, imposing β_ψψ > n_xψ > 0 does not seem a very stringent requirement. It must be said however that, if such an assumption is violated, the stability properties of the steady states in the model change (Rezai 2010 analyzes theoretically all the possible cases). Nevertheless, the dynamics observed in the empirical plots in Figures 1 and 6 should be enough to convince the skeptical reader who likes to engage in Jacobian analyses.

16. We thank an anonymous referee for suggesting we make such comparisons.

17. The careful reader will have observed that the this plot displays data at the annual frequency, differently from Figure 1, which plots quarterly data. The reason is that Piketty and Saez (2003) have annual data in their dataset, which is the one we used to construct our series for these plots. Obviously, having to work with annual data as in the Netherlands makes it cumbersome to estimate ‘long-phase cycles’ using the methodology we adopted in Figure 1. In other words, there is a potential ‘apples and oranges’ problem that arises from different data frequencies for different countries. This is the reason why simple HP filtered data are used in Figure 6. The same considerations apply to the plots in Figure 5.

18. The employment data for the Netherlands are taken from the WDI website: http://data.worldbank.org/datacatalog. Data for the wage share in the Netherlands are taken from EPWT 3.0, available at http://homepage.newschool.edu/~foleyd/epwt/.