The Goodwin model, sometimes called Goodwin's class struggle model, is a model of endogenous economic fluctuations first proposed by the American economist Richard M. Goodwin in 1967. It combines aspects of the Harrod–Domar growth model with the Phillips curve to generate endogenous cycles in economic activity (output, unemployment and wages) unlike most modern macroeconomic models in which movements in economic aggregates are driven by exogenously assumed shocks. Since Goodwin's publication in 1967, the model has been extended and applied in various ways.
Model.
The model is derived from the following assumptions:
The model uses the variables
 "q"  is output
 "k"  is (homogeneous) capital
 "w"  is the wage rate
 "a"  is labour productivity
 "n"  is the labour force
which are all functions of time (although the time subscripts have been suppressed for convenience) and the constants
 "α"  is the rate of growth of labour productivity
 "β"  is the rate of growth of the labour force
 "γ"  is used to define the real wage change curve
 "ρ"  is also used to define the real wage change curve
 "σ"  is the capital-output ratio.
A number of derived quantities are helpful to define the model.
The amount of employed labour is given by
 formula_1,
the employment ratio is given by
 formula_2,
the workers' share in the output is given by
 formula_3,
and the share of the capitalists in the output (formula_4 for surplus) is given by
 formula_5.
The model is then defined by a set of differential equations.
Firstly, the change in labour productivity is defined by
 formula_6,
that is, steady growth, with formula_7.
The labour force changes according to
 formula_10,
again, steady growth, with formula_11.
Real wages change according to
 formula_12,
that is, the real wage change curve is modelled as linear.
Note that to correctly model the assumptions, formula_13 and formula_14
must be picked to ensure that real wages increase when formula_15 is near 1.
In other words, if the labor market is 'tight' (employment is already high) there is upward pressure on wages and vice versa in a 'lax' labor market.
Capital changes according to
 formula_16,
as the surplus is assumed to be completely invested by the capitalist.
Lastly, output changes according to
 formula_17,
that is, in proportion to the surplus invested.
Note that
 formula_18
by the assumption that k and q grow at the same rate by assumption of full utilization of capital and constant returns to scale.
Solution.
The defining equations can be solved for formula_19 and formula_20,
which gives the two differential equations
 formula_21
 formula_22.
These are the key equations of the model and in fact are the Lotka–Volterra equations, which are used in biology to model predator-prey interaction.
These equations have two fixed points. The first is when
 formula_23 and formula_24
and the second is when
 formula_25
 formula_26,
which determines the center of a family of cyclic trajectories.
Since the model cannot be solved explicitly, it is instructive to analyze the trajectory of the economy in terms of a phase diagram.
The two lines defining the center of the cycle divide the positive orthant into four regions. The figure below indicates with arrows the movement of the economy in each region. For example, the north-western region (high employment, low labor's share in output) the economy is moving north-east (employment is rising, worker's share is increasing). Once it crosses the u* line it will begin moving south-west.
The figure below illustrates the movement of potential output (output at full employment), actual output and wages over time.
As can be seen the Goodwin model can generate endogenous fluctuations in economic activity without relying on extraneous assumptions of outside shocks, whether on the demand or supply side.
Notes.
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