# A struggle that leads to growth essay

Lotka-Volterra equations have vortex dynamics of the type illustrated in Figure 1. Thus, as is obvious, we have cyclical dynamics of u and m. To understand the logic, look only at the differential equations for employment and the wage share.

### Focus On The Lesson, Not The Problem

This is akin to "predator-prey" where employment is the prey and wage share is the predator: The off-isokine dynamics are easy to evaluate. To get at the resulting dynamic, let us set up the differential equations as a ratio to eliminate dt: Introducing dummy variable z, then the left side can be written: Similarly, for the second equation: For instance, suppose we began at m1 in Figure 2.

Thus, m cannot fall below m 1 because then the corresponding z will be above the maximum G u, c.

Similarly, m cannot be higher than m 2. Conversely, if we take the lower bound of the admissible values of u i.

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Any u below u1 would imply a z below the extrema of F mwhich is not feasible. Thus, u cannot be lower than u1 nor, by a similar argument, can it be above u2.

The boundaries m 1, m 2, u1 and u2 all define a very clear orbit in the positive orthant - which is one out of the many possible orbits, but it will be the one that is chosen. As the boundaries are determined by the shape and position of F m and G u, c respectively, then obviously which orbit is chosen depends not only on the parameters in those functions abv, qn as well as the coefficient of integration, c. If, for instance, c changes so that G u, c declines i.

## Writing a Three-Paragraph Essay

In short, given the orbit chosen by the parameters, employment and wage share will fluctuate cyclically in a deterministic manner - without the extraneous paraphenelia of exogenous shocks, ceilings, floors, etc. Thus, Goodwin's "class struggle" cycle is "endogenous" to the system. Many economists, mostly but not exclusively of Keynesian or Marxian persuasion, have made much use of Goodwin's structure and endogenous cyclical dynamics.

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