# (p.158) Appendix 3 A Goodwinian Model

# (p.158) Appendix 3 A Goodwinian Model

In this appendix, we present a model which emphasizes changes in the income distribution as the driving force behind cycles. The model is loosely based on Goodwin’s (1967) model of growth cycles. In Goodwin’s (1967) model, cycles are the outcome of the interplay between capacity effects induced by changes in the income distribution and a real wage Phillips curve. Since changes in the income distribution are likely to affect savings and, hence, capacities only very slowly, Solow (1990: 38) contends that ‘[P]erhaps the Goodwin growth cycle is not a model of the business cycle at all’, but rather a model of longer term fluctuations. The model below does not stress capacity effects. Rather, it is assumed *ad hoc* that firms choose to expand output when profits are high, and vice versa. Another caveat is in order. Probably Goodwin’s (1967) most important accomplishment was to introduce Lotka-Volterra differential equations to the economics literature. The model presented below, by contrast, uses the stochastic second-order difference equation approach to cycles. So it does not do justice to Goodwin’s achievements in nonlinear dynamic economics.

# Model

A closed economy is considered. The model has two essential ingredients. First, real wage pressure tends to rise as the economy expands (there is a real wage Phillips curve). Second, aggregate production tends to expand when profits are high, that is, when the real wage rate is low. Formally,

(p.159)
where *χ*, *γ*, and *ι* are positive constants and *ξ* is white noise. Variations in *ξ* indicate autonomous changes in the degree of real wage pressure.

# Equilibrium

From equation (A3.1), real wages grow or fall, depending on whether aggregate output is above or below *y* = 0, respectively (given that *ξ* = 0). From equations (A3.2) and (A3.3), *γ*Δ^{2} *y* =−*ι*Δ(*w* − *p*). That is, output growth decreases or increases, depending on whether the real wage rate rises or falls, respectively. Taken together, it follows that

Since *γ*/(*γ* + *ιχ*) 〈 1 and

business cycles occur for all *χ*, *γ*, and *ι*.

What is the economic explanation for the cyclical movements of aggregate production? Suppose the economy is in its steady state *y* = 0. Then a wage push shock *ξ* _{0} 〉 0 leads to an output expansion: *y* _{0} 〉 0. Thereafter, no wage push shocks *ξ* _{t} ≠ 0 occur for a while (*t* = 1, 2,…). *y* 〉 0 implies rising real wages, which in turn imply decreasing rates of output growth (Δ^{2} *y* 〈 0). So the expansion comes to a halt and output starts to fall. Since *y* is still positive, the real wage rate does not yet cease to rise. Hence, the recession gains momentum (Δ^{2} *y* 〈 0), and the economy does not settle down at the steady state *y* = 0, but overshoots into the *y* 〈 0 region. There, real wages start to decline, so that the rate of output growth, though still negative, increases (Δ^{2} *y* 〉 0). This explains the lower turning point. During the initial phase of the recovery, real wages continue to fall, since *y* 〈 0, so output growth accelerates (Δ^{2} *y* 〉 0). That is why the economy overshoots into the *y* 〉 0 region, rather than settling down at the steady state. This process repeats with declining amplitudes.

The model is un-Keynesian in that aggregate production is determined on the economy’s supply side alone. That is why it is expounded separately in this appendix, although chronologically it would fit in the chapter about Keynesian economics. In order to determine the aggregate price level and the interest (p.160) rate, the model can be closed with the standard demand-side equations:

The former equation is the IS curve, the latter the LM curve. Given an exogenous sequence of money stocks *m* _{t} and given equilibrium aggregate output *y* _{t} (determined in (A3.4)), these equations can be solved for *i* _{t} and *p* _{t}:

Increases in the stock of money *m* _{t} *ceteris paribus* raise prices *p* _{t} one-for-one. Increases in government expenditure *g* _{t} raise the interest rate *i* _{t} and prices *p* _{t}. Both monetary policy and fiscal policy are ineffective.

# Further reading

A very detailed exposition of Goodwin’s (1967) model with many modifications and extensions is in Flaschel (1993).

# References

Flaschel, P. (1993). *Macrodynamics: Income Distribution, Effective Demand and Cyclical Growth*. Frankfurt: Peter Lang.

Goodwin, R. M. (1967). ‘A Growth Cycle’, in C. H. Feinstein (ed.), *Socialism, Capitalism and Economic Growth*. Cambridge: Cambridge University Press, pp. 54–8.

Solow, R. M. (1990). ‘Goodwin’s Growth Cycle: Reminiscence and Rumination’, in K. Velupillai (ed.), *Nonlinear and Multisectoral Macrodynamics: Essays in Honour of Richard Goodwin*. London: Macmillan Press, pp. 31–41.