Goodwin model (economics)

The Goodwin model is a model for the explanation of the business cycle, the Richard M. Goodwin has developed. It uses the mathematics of the Lotka -Volterra equations. It is modeled the economic interplay between employment rate and wage rate. With high employment rate ( denoted by v) the bargaining power of workers is high. The wage pressure and thus the wage rate (u ) is increasing. The profit ratio ( 1 -u) decreases accordingly. Due to low profits dismiss the company. The employment rate then decreases. At low employment rates, the bargaining power of workers is low, it decreases the wage rate, which increases profit share. For companies the incentive to set more increases, the employment rate is rising again. Mathematically, the wage rate the " robbers ", the employment rate of the " prey " in the predator-prey relationships based on the Lotka -Volterra equations.

Mathematical representation

The output, aggregate output is given by

While q is the total economic output, ℓ is employment, K is the stock of capital and a is labor productivity. All variables change with time, the time indices are not listed. σ is the capital coefficient assumed to be constant.

The capacity utilization rate is 100 %, that full utilization of existing capacity:

The employment rate is

Where n is the labor supply, the β grows with the rate. In addition, labor productivity grows with a rate α (technical progress ). The employment grows so with

The supply of labor increases with

Wages are determined by the Phillips curve:

The wage share u is defined as

The growth rate of the wage rate is therefore

It is believed that the workers spend their wages on consumption, while capital owners save a portion of their profits, and that capital at the rate delta loses value ( depreciation). The growth rate of output and capital is therefore (because, assuming full utilization of capital equal to)

So

Solution of the equations

This results in two differential equations for the rates of growth of wage rate and employment rate u v:

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

It is

Substituting the two equations equal to zero, one obtains values ​​for u and v, in which v and u is not change.

With Zufallsstörgröße