Goodwin model

The Goodwin model is a model for explaining the business cycle developed by Richard M. Goodwin . It uses the math of the Lotka-Volterra equations . The cyclical interplay between the employment rate and the wage rate is modeled. If the employment rate is high (denoted by v), the bargaining power of workers is high. The wage pressure and thus the wage share (u) increases. The profit rate (1-u) decreases accordingly. The companies lay off because of low profits. The employment rate then falls. When the employment quota is low, the bargaining power of the workers is low, the wage quota falls, the profit quota rises. For companies, the incentive to hire more increases and the employment rate rises again. Mathematically, the wage quota corresponds to the “predators”, the employment quota to the “prey animals” in the predator-prey relationships based on the Lotka-Volterra equations .

Mathematical representation

The output , the overall economic production, is given by

${\displaystyle q=\min \left(a\ell ,{\frac {k}{\sigma }}\right)}$

where q is the macroeconomic output, ℓ is employment , k is the stock of capital and a is labor productivity . All variables change over time, the time indices are not listed. σ is the assumed constant capital coefficient .

The capacity utilization is 100%, i.e. full utilization of the existing capacities:

${\displaystyle a\ell ={\frac {k}{\sigma }}=q}$

The employment rate is

${\displaystyle v={\frac {\ell }{n}}}$

where n is the labor supply that grows at the rate β . In addition, the labor productivity a grows at the rate α ( technical progress ). Employment grows with it

${\displaystyle {\frac {dv/dt}{v}}=g_{v}=g_{\ell }-\beta .}$

The job supply increases with it

${\displaystyle {\frac {d\ell /dt}{\ell }}=g_{\ell }=g_{q}-\alpha }$

The wages are determined from the Phillips curve :

${\displaystyle {\frac {dw/dt}{w}}=g_{w}=\rho v-\gamma .}$

The wage share u is defined as

${\displaystyle u={\frac {w\ell }{q}}={\frac {w}{a}}.}$

The growth rate of the wage share is therefore

${\displaystyle {\frac {du/dt}{u}}=g_{u}=g_{w}-\alpha }$

It is assumed that the workers spend their wages on consumption , while the capital owners save part of their profits and that capital depreciates at the rate delta ( depreciation ). The growth rate of output and capital is therefore the same (due to the assumed full utilization of capital)

${\displaystyle {\frac {dk/dt}{k}}=g_{k}=g_{q}=s(1-u)(q/k)-\delta .}$

So

${\displaystyle {\frac {dv/dt}{v}}=g_{v}={\frac {s(1-u)}{\sigma }}-(\delta +\alpha +\beta ).}$

Solving the equations

There are two differential equations for the growth rates of the wage share u and the employment rate v:

${\displaystyle {\frac {dv/dt}{v}}=g_{v}={\frac {s(1-u)}{\sigma }}-(\delta +\alpha +\beta )}$

${\displaystyle {\frac {du/dt}{u}}=g_{u}=\rho v-\gamma -\alpha }$

They correspond to the Lotka-Volterra equations . The constant quantities of the equations can be combined to form new constants a, b, c and d, each greater than zero:

${\displaystyle {\frac {dv/dt}{v}}=-{a}{u}+b}$

${\displaystyle {\frac {du/dt}{u}}=cv-d}$

It is

${\displaystyle {a}={\frac {s}{\sigma }},\quad {b}={\frac {s}{\sigma }}-(\delta +\alpha +\beta ),\quad {c}=\rho ,\quad {d}=\gamma +\alpha }$

If you set the two equations equal to zero, you get values ​​for u and v in which v and u do not change.

${\displaystyle u*={\frac {b}{a}}=1-{\frac {(\delta +\alpha +\beta )\sigma }{s}}}$

${\displaystyle v*={\frac {d}{c}}={\frac {\gamma +\alpha }{\sigma }}}$

Illustrations

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  Goodwin cycle

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  with random disturbance variable

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  The wage share (blue) and the employment rate (red) in the USA. According to the Goodwin model, the wage share would follow the employment rate with a time lag.

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  Development of the wage share and employment share in the USA from 1949 to 2015 with theoretically expected directions of movement of the cycles (thick arrows) and actual partial movements (thin arrows).

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  Development of the wage share and employment quota in the FRG 1991 to 2015 with theoretically expected directions of movement of the cycles (thick arrows) and actual partial movements (thin arrows).

literature

• RM Goodwin: A Growth Cycle. In: CH Feinstein (Ed.): Socialism, Capitalism and Economic Growth. Essays presented to Maurice Dobb . Cambridge University Press, Cambridge 1967, pp. 54-58.
• Richard M. Goodwin: Chaotic Economic Dynamics. Clarendon Press, Oxford et al. 1990, ISBN 0-19-828335-0 .
• Peter Flaschel: The Macrodynamics of Capitalism. Elements for a Synthesis of Marx, Keynes and Schumpeter. 2nd revised and enlarged edition. Springer, Berlin et al. 2009, ISBN 978-3-540-87931-2 , chapter 4.3.