Goodwin Model
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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 Richard M. Goodwin (February 24, 1913 – August 13, 1996) was an American mathematician and economist. Background Goodwin was born in New Castle, Indiana. He received his BA and PhD at Harvard and taught there from 1942 until 1950. He fl ...
in 1967. It combines aspects of the Harrod–Domar growth model with the
Phillips curve The Phillips curve is an economic model, named after William Phillips hypothesizing a correlation between reduction in unemployment and increased rates of wage rises within an economy. While Phillips himself did not state a linked relationship ...
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: # there is steady growth of labour productivity (e.g. by technological improvement); # there is steady growth of the labour force (e.g. by births); # there are only two factors of production: labour and capital; # workers completely consume their wages, and capitalists completely invest their profits; # the capital-output ratio is constant (i.e. a fixed amount of output can always be turned into the same amount of capital); # real wages change according to a linearized
Phillips curve The Phillips curve is an economic model, named after William Phillips hypothesizing a correlation between reduction in unemployment and increased rates of wage rises within an economy. While Phillips himself did not state a linked relationship ...
, where wages rise when close to full employment. 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 : l = \frac, the employment ratio is given by : v = \frac, the workers' share in the output is given by : u = \frac = \frac, and the share of the capitalists in the output (s for surplus) is given by : s = 1 - u. The model is then defined by a set of differential equations. Firstly, the change in labour productivity is defined by : \frac = \alpha, that is, steady growth, with a_t = a_0 e^. (Note that \dot is the derivative over time /.) The labour force changes according to : \frac = \beta, again, steady growth, with n_t = n_0 e^. Real wages change according to : \frac = -\gamma + \rho v, that is, the real wage change curve is modelled as linear. Note that to correctly model the assumptions, \gamma and \rho must be picked to ensure that real wages increase when v is near 1. In other words, if the labor market is '
tight Tight may refer to: Clothing * Skin-tight garment, a garment that is held to the skin by elastic tension * Tights, a type of leg coverings fabric extending from the waist to feet * Tightlacing, the practice of wearing a tightly-laced corset * ...
' (employment is already high) there is upward pressure on wages and vice versa in a 'lax' labor market. Capital changes according to : \dot = q s, as the surplus is assumed to be completely invested by the capitalist. Lastly, output changes according to : \frac = \frac, that is, in proportion to the surplus invested. Note that : \frac = \frac = \frac 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 \dot and \dot, which gives the two differential equations : \dot = v(-\frac u + \frac - \alpha - \beta) : \dot = u(\rho v - \gamma - \alpha). These are the key equations of the model and in fact are the
Lotka–Volterra equations The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order nonlinear differential equations, frequently used to describe the dynamics of biological systems in which two species interact, one as a pred ...
, which are used in biology to model predator-prey interaction. These equations have two fixed points. The first is when : u = 0 and v = 0 and the second is when : u = 1 - (\alpha + \beta)\sigma : v = \frac, 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 A phase diagram in physical chemistry, engineering, mineralogy, and materials science is a type of chart used to show conditions (pressure, temperature, volume, etc.) at which thermodynamically distinct phases (such as solid, liquid or gaseous ...
. 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.


Statistics

File:Goodwin2 fredgraph.png,
Wage share In economics, the wage share or labor share is the part of national income, or the income of a particular economic sector, allocated to wages (labor). It is related to the capital or profit share, the part of income going to capital, which is also ...
(blue line) and civilian employment population ratio (red line) in the United States
According to the Goodwin model, the wage share is to be expected to lag behind the employment rate. This seems to be the case if only by a small time lag


See also

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Business cycle Business cycles are intervals of Economic expansion, expansion followed by recession in economic activity. These changes have implications for the welfare of the broad population as well as for private institutions. Typically business cycles are ...
*
Richard M. Goodwin Richard M. Goodwin (February 24, 1913 – August 13, 1996) was an American mathematician and economist. Background Goodwin was born in New Castle, Indiana. He received his BA and PhD at Harvard and taught there from 1942 until 1950. He fl ...
*
Harrod–Domar model The Harrod–Domar model is a Keynesian model of economic growth. It is used in development economics to explain an economy's growth rate in terms of the level of saving and of capital. It suggests that there is no natural reason for an economy to ...
*
Marxian economics Marxian economics, or the Marxian school of economics, is a Heterodox economics, heterodox school of political economic thought. Its foundations can be traced back to Karl Marx, Karl Marx's Critique of political economy#Marx's critique of politic ...
*
Phillips curve The Phillips curve is an economic model, named after William Phillips hypothesizing a correlation between reduction in unemployment and increased rates of wage rises within an economy. While Phillips himself did not state a linked relationship ...


Notes

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References

* Goodwin, Richard M. (1967), "A Growth Cycle", in C.H. Feinstein, editor, ''Socialism, Capitalism and Economic Growth''. Cambridge: Cambridge University Press. * Goodwin, Richard M., ''Chaotic Economic Dynamics'',
Oxford University Press Oxford University Press (OUP) is the university press of the University of Oxford. It is the largest university press in the world, and its printing history dates back to the 1480s. Having been officially granted the legal right to print books ...
, 1990. * Flaschel, Peter, ''The Macrodynamics of Capitalism'' - Elements for a Synthesis of Marx, Keynes and Schumpeter. Second edition, Springer Verlag Berlin 2010. Chapter 4.3. Economics models Economic growth Business cycle theories