Hysteresis in a Three-Equation Model


This paper introduces two post-Keynesian hysteresis mechanisms into a standard textbook three-equation model. The mechanisms work through wage bargaining and price setting. Workers are assumed to change their wage aspirations when the actual wage differs from their target wage, and firms are assumed to change their mark-up norm when the actual profit share differs from their target share. These mechanisms do not themselves guarantee hysteresis. A pure inflation shock will create hysteresis even if expectations are anchored to the central bank’s inflation target. After a demand shock, if inflation expectations are not anchored, these mechanisms generate persistence but not true hysteresis. But if expectations are partially (as they seem to be) or fully anchored, a demand shock will have a permanent effect on output, employment, and the real wage, because in this case the central bank is not obligated to reflate as aggressively in order to manage expectations. Hysteresis effects may explain the absence of disinflation and the fall in the wage share in the aftermath of Global Financial Crisis.

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  1. 1.

    Post-Keynesian in the broad sense refers to economists who have taken their cue from Keynes’s inner circle, including Joan Robinson, Nicholas Kaldor, and Roy Harrod, rather than from the North American interpreters of Keynes such as Paul Samuelson or Franco Modigliani. But there are many variants. Lavoie [2014] and Taylor [2004] provide handy guides to this terrain.

  2. 2.

    It may actually be less confusing to formalize this point. Recalling the unit labor productivity assumption and assuming labor is the only cost of production for simplicity, the actual real wage will be \(w = 1/(1+\mu )\) where \(\mu \) is the actual mark-up. The real wage reflecting the mark-up norm, \(\mu _N\), will be \(w^{PS}=1/(1+\mu _N)\). Clearly, if \(w<w^{PS}\) then \(\mu >\mu _N\).

  3. 3.

    To be clear, \(\Delta z = z_{+1} - z\).

  4. 4.

    An alternative is to minimize a discounted loss function over an extended time horizon. This approach, called optimal control, has played a role in the internal discussions at the FOMC because Janet Yellen has alluded to it in public speeches. See Brayton et al. [2014].

  5. 5.

    This is often erroneously described as a measure of how “hawkish” or inflation-phobic the central bank is. In fact, after a negative demand shock, a large \(\beta \) would be dovish since it would lead to aggressive job-creating reflation.

  6. 6.

    The Taylor Rule is normally implemented with both the inflation gap as it is here and the output gap, \(y-y_e\). As explained by Carlin and Soskice [2006, pp. 153–157], this form emerges when there is a time lag between output and inflation in the model specification. The model here includes no lag between output and inflation.

  7. 7.

    To be precise define \(R_y=\Sigma _1^\infty (y_{+1}-{y_e}_{+1})/(y_1-{y_e}_0)\).

  8. 8.

    Note that \(\Sigma _1^\infty \Delta \pi =\pi ^T-\pi _1\).

  9. 9.

    This is due to the behavior of \(R_y\) which takes the value \(-\alpha \theta /(1+\theta )\) when \(\chi =1\).

  10. 10.

    An alternative strategy would use the Phillips curve, the IS equation, and the Taylor Rule to solve for inflation, the interest rate, output and equilibrium output.

  11. 11.

    For more details on the formal properties of this type of system, see Elaydi (2005, Ch. 3)orGandolfo (1997, Ch. 18).

  12. 12.

    Another feature of the characteristic equation is that its discriminant is strictly positive which rules out complex roots that would generate cyclic behavior.

  13. 13.

    Formally, we can solve for the steady state where \(\mathbf {y=y_{-1}=y^*=(I-A)^{-1} b}\). This is sometimes called the particular solution to the system. In the presence of a unit root, \(\mathbf {(I-A)}\) is singular and its inverse is not defined.

  14. 14.

    For further discussion of the role played by unit roots (or zero roots in continuous time models) see Amable et al. [1993] or Dutt [1997].

  15. 15.

    Both cases pose an initial condition problem. In practice, distinguishing between permanent and temporary shocks raises hard questions about how the central bank knows what the structure of demand and sustainable rate are in the first place. I have chosen to avoid this complication in order to focus attention on the pure interaction of hysteresis and policy, not because permanent shocks are unimportant.

  16. 16.

    The inflation rate will deviate by \(\pi -\pi ^T=(\tilde{r}-r_s)/h\) where \(\tilde{r}\) is the perceived stabilizing rate of interest.

  17. 17.

    Fully anchored inflation expectations, as we showed, are reflected in a policy of setting the interest rate at its stabilizing level. But the rate of interest acts with a lag, so until it converges on the steady state the central bank will find itself missing its desired inflation rate because it does not perceive the forward change in equilibrium output.

  18. 18.

    In the simulation in Figure 3 with \(\chi =0.5\) the equilibrium level of output falls from 100 to 99.3. With full anchoring, it falls to 99.0.

  19. 19.

    This is true for two reasons. First, this ratio is larger than the pure sacrifice ratio, \(R_\pi \), which can be seen by substituting for h to make the two expressions comparable. The myopic \(R_\pi =\alpha \beta (1-\chi )/((\chi +\alpha ^2\beta )+\theta \chi (1+\alpha ^2\beta ))\). Second, in the myopia model, the output effect, \(R_y\), reduces the overall sacrifice ratio in the myopic model.


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Correspondence to Thomas R. Michl.

Additional information

This paper is based on a conversation with Engelbert Stockhammer at Kingston University in Spring 2014 where I was an EPOG (European Project on Globalization) Scholar. I’ve also benefited from comments by Wendy Carlin, Duncan Foley, and Amitava Dutt and two referees of the Eastern Economic Journal. I take full responsibility for views expressed and of course for any errors.



This appendix provides some additional mathematical detail about the two models and the numerical values used in the simulations.

Stability in the Basic Model

The eigenvalues or characteristic roots of the basic system, equation 4, are \((1, \lambda _2, \lambda _3)\) where

$$\begin{aligned} \lambda _{2,3} = \frac{1}{2}\left( -(\chi +\alpha a h - (1-\theta ) \pm \sqrt{\Delta } \right) \end{aligned}$$

and \(\Delta = (\chi +\alpha a h -(1-\theta ))^2 + 4 \theta (1-\chi )\) is the discriminant. Note that with the parameter restrictions in the model, \(\Delta > 0\) so we know the roots are all real numbers. To characterize stability, we replace h with its definition in the paper, solve for \(\vert \lambda _{2,3} \vert < 1\) and arrive at this stability condition:

$$\begin{aligned} \theta < \frac{2 - \chi + \alpha ^2 \beta }{(2-\chi )(1+\alpha ^2 \beta )} . \end{aligned}$$

For example, with the numerical examples in the paper and no anchoring, stability requires that \(\theta < 3/4\). With full anchoring (\(\chi =1\)) stability is guaranteed since \(\theta <1\). Higher values of \(\beta \) (the central banks relative preference for closing its inflation target) require a smaller value of \(\theta \) to maintain stability.

Perfect Foresight Model

The system with central bank foresight of the one-period ahead equilibrium level of output has the same form with slightly different coefficients. Let us call the matrix and vector \(\mathbf {B}\) and \(\mathbf {c}\) to avoid confusion. We have

$$\begin{aligned} \mathbf {B} = \left( \begin{array}{ccc} (a h)/(\alpha \beta ) &{} 0 &{} 0 \\ -a h &{} \theta &{} (1-\theta )\\ 0&{}\theta &{}(1-\theta ) \end{array}\right) \end{aligned}$$
$$\begin{aligned} \mathbf {c} = \left( \begin{array}{c} \frac{(\chi +\alpha ^2 \beta )\pi ^T}{1+\alpha ^2 \beta } \\ a h \pi ^T \\ 0 \end{array}\right) \end{aligned}$$

There is an obvious linear dependence between the last two columns indicating that the matrix is less than full rank and has only two non-zero roots. The roots are \((1, (a h)/(\alpha \beta ), 0)\). Expanding \((a h)/(\alpha \beta ) < 1\) gives the stability condition

$$\begin{aligned} \frac{\alpha \beta (1-\chi )}{1+\alpha ^2\beta } < 1 \end{aligned}$$

which is satisfied for any permissable parameter values.


$$\begin{aligned} b_0=0.5 \qquad \qquad&b_1 = 0.001 \\ c_0=0.7 \qquad \qquad&c_1 = -0.001\\ \phi = 0.5 \qquad \qquad&\pi ^T=5\\ \psi =0.2 \qquad \qquad&\sigma =0.3 \\ \alpha =1\qquad \qquad&\beta =1\\ A=105\qquad \qquad&a=1 \\ y_{e0}=100\qquad \qquad&r_{s0}=5 \end{aligned}$$

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Michl, T.R. Hysteresis in a Three-Equation Model. Eastern Econ J 44, 305–322 (2018). https://doi.org/10.1057/s41302-016-0083-9

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  • Hysteresis
  • Three-equation model
  • Path dependence
  • Inflation-expectations anchoring

Mathematics Subject Classification

  • E11
  • E12
  • O42