Complexity

Lotka’s model was easily derived, simply by acknowledging that sharks eat fish, and by using the simplest possible mathematical operation to link the two species together.[68] It’s also easily analyzed, since with just two dimensions, its dynamic properties depend on a simple quadratic, as I’ll explain later in Chapter 11. The next model, which is the first simulated model in the history of complex systems analysis, is an entirely different … kettle of fish.

9.1 Lorenz model

The derivation of Lorenz’s model of turbulent flow[69] required mathematical skills well in advance of those possessed by the vast majority of economists—including me—so don’t let the simplicity of the equations in (59) fool you. Superficially, they are only slightly more complicated than the Lotka predator prey model: rather than 2 variables and 4 parameters (Lotka), there are 3 variables and 3 parameters:

However, the behavior of the model is from another planet: Planet Complexity—see Figure 200. Unfortunately for mainstream economists, this happens to be the planet on which we actually live.

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69 https://www.scribd.com/document/395983652/lorenzderivation-pdf

Figure 203: Lorenz's model with its chaotic behavior and "strange attractors"

I won’t repeat the step-by-step explanation of deriving this model that I gave for Lokta’s model— there is no simple “step by step” process to explain, and you should be able to pretty rapidly design this model in Minsky yourself by converting the differential equations in equation (59) to integral equations, and then coding them in Minsky . But one feature of this model is worth noting: the “sliding window” plot that shows a ten time-unit slice of the plots for y and z . This uses the “range” inputs for the plots—the angled inputs on each of the X, Y1 and Y2 axes. Normally these are constants, but they can take variable inputs, and the variable inputs for this plot are the input for the system’s simulation time at the far right of the plot, and on the far left. What is worth repeating is the exercise of deriving the equilibrium of the model, by setting all the A−10

What is worth repeating is the exercise of deriving the equilibrium of the model, by setting all the differential equations in (59) to zero:

One obvious solution here is where . The non-zero solutions to (60) give us these three conditions for the equilibrium values, which I identify using the subscript E : T= E= z= 0

A bit of algebraic manipulation yields:

There are thus 3 equilibria for this model:

One reason I love this model, as a non-mainstream economist, is that it makes a mockery of the Neoclassical obsession with equilibrium modelling, because it has three equilibria, all of which are unstable . The equilibria are the colored dots on the phase plots of z against x & y, and y against x. The simulation starts at the values (0.1,0.1,0.1) , just a slight displacement from the (0,0,0) equilibrium. Because the simulation starts so close to this equilibrium, the system is rapidly pushed away from it: this equilibrium is stable on two of its three eigenvalues, but unstable on one.

The system is then attracted towards one of the other two equilibria, but they are “strange attractors”: they attract the system from a distance but repel it—in a cyclical fashion—when it gets closer to them. We’ll get into the detail of how to analyze this instability in Chapter 11, but for now its primary characteristics are that the system will never converge to any of its equilibria, and yet the system will also never return values that are unrealistic. Its dynamics are therefore necessarily farfrom-equilibrium dynamics: the very idea of “equilibrium dynamics”—as ensconced in Neoclassical “ Dynamic Stochastic General Equilibrium ” modeling—is an oxymoron.

Figure 204: Lorenz model with equilibria. Simulation starting from (0.1,0.1,0.1)

My model of Minsky’s Financial Instability Hypothesis, which we’ll develop in the next section, has related, though not quite so complex, far-from-equilibrium dynamics.

9.2 A complex systems model of economic instability

On page 82, I introduced Richard Goodwin’s model of cyclical growth, which reduced to the following two equations for the employment rate and the wages share of output . d   1 − ω λ   ω[70]

The model generated everlasting cycles, like those generated by Lotka’s Predator-Prey model:

Figure 205: Reduced Form Goodwin Model

Since that derivation was some time back(!), and I don’t have to worry about my publisher’s constraints on word length, I’ll repeat the derivation here, along with the third factor I introduced when I constructed my model of Minsky’s Financial Instability Hypothesis in 1992 (Keen 1995): the private debt to GDP ratio ⁄ . I’ll derive the model using the basic rules of calculus that: D  1 dx S ≡DY

The three definitions that we will turn into a simple model that extends Goodwin’s model to include the role of finance in capitalism, are the employment rate, the wages share of GDP, and the private debt to GDP ratio:

r Y Two ancillary definitions are needed: the output to labor ratio , and the capital to output ratio : T v

The first step is to apply the rules in (65) to the definitions in (66)—and to save time I’ll use the definitions in (67) to extend the equations as far as possible without introducing any assumptions:

To proceed any further, we need to introduce some simplifying assumptions. Most of the assumptions Goodwin made are shown in Equation (69): exponential growth of the output to labor ratio and of population, a constant capital to output ratio, and a uniform wage:

At this point, we have three mathematical statements that are easily interpreted verbally: the employment rate will rise if economic growth exceeds the sum of the growth rates of the output to labor ratio and population; the wages share of GDP will rise if wage demands exceed the growth rate

of the output to labor ratio; and the debt ratio will rise if debt rises faster than the rate of economic growth. We now need to define the rate of economic growth and the rate of growth of private debt to arrive at a final model. Given the assumption that the capital to output ratio is a constant, the rate of economic growth is the same as the rate of change in the capital stock: v

Now we need to define the rate of change of the capital stock.[71] The obvious starting point is that the rate of change of capital equals investment minus depreciation, which is normally assumed to be a linear function of the amount of capital. Using for gross investment , for the depreciation rate, and ⁄ for the ratio of gross investment to output, this yields: G K I δ LG = IG Y dK

This leaves just the rate of change of private debt to be defined:

dr  D  gr To determine , we need a realistic simplifying assumption. The simplest—which is far too kind to the finance sector, since it omits the modern finance sector’s main business model of Ponzi D� finance[72] — is that capitalists invest more than profits during a boom, and less than profits during a slump—and that they have to borrow money[73] to enable this. Borrowing thus finances investment, so that the rate of change of private debt was equal to gross investment minus profits: d D

This appears to be an impasse, since the denominator is D, rather than something we can work with like Y. But there’s a handy trick for situations like this, taught to me by the wonderful (if irascible) mathematics lecturer Professor Williams when I was studying first year undergraduate mathematics at Sydney University in 1971. To quote Williams:

There are 3 rules of mathematics: (1) what have you got that you don’t want? Get rid of it; (2) what haven’t you got that you do want? Put it in (3) Keep it balanced. We can bring in Y by multiplying the right hand side of , and then rearranging terms:  Y IG −Π Y/Y

d s r Here stands for the profit share of income: = Π⁄ . This now gives us three fairly simple equations: t t π π Y

d r To proceed, we need to add functional forms for the rate of change of wages and the investment share of output . For the former, I use the same linear “Phillips curve” function used by Goodwin. w� For the latter, though a common practice in Post Keynesian economics is to model investment as G L driven by a target level of capacity utilisation, I base investment on the rate of profit, using exactly the same form as the wage change function, with the rate of profit taking the place of the employment rate:

Profit in the original Goodwin model was just output minus wages. The introduction of private debt means that profit now equals output minus wages minus interest payments on outstanding debt:

This gives us a 3-dimensional model which, as I explain in Manifesto , reproduces the essence of Minsky’s Financial Instability Hypothesis:

Where I’ve used the abbreviations:

— — To simulate and analyze this model, we need to express it in terms of differential equations 1 ∙ rather than rates of change— . This just involves multiplying both sides of Equation (81) by dt dx the relevant variables : T�= x dt D λ, ω, S d

These equations are easily entered into Minsky—see Figure 203.

Figure 206: Reduced form version of Minsky's Financial Instability model

The peculiar dynamics in this model—the initially falling and then rising cycles in the growth rate and employment, the rising private debt to GDP ratio, and the decline in the workers’ share of GDP, even though they do no borrowing in this model—turn out to be a particular subset of the dynamics of the Lorenz model, known as the “Intermittent Route to Chaos” (Pomeau and Manneville 1980).[74]

We’ll check this behavior out analytically in Chapter 11—and the linear form of the model, shown here, is essential to that task. But I’ve also heard comments from Neoclassicals that this model generates unrealistic cycles—look at how large the fluctuations are in wages share, employment, and the growth rate! Therefore, this is a useful point at which to discuss the proper role of nonlinear functions in complex systems models.

9.3 Nonlinear Functions in Nonlinear Models

Figure 204 adds two nonlinear functions to the previous model, one for the “Phillips Curve” and the other for the investment function. You will note that the cycles in this model are much smaller in scale than for the previous model with linear behavioral functions.

This is because the nonlinear functions themselves curtail a model’s behavior to realistic bounds, in a way that linear functions do not. They do not generate the cycles themselves in the model shown here, since those cycles are intrinsic to the model itself via the nonlinear interaction of system states.

The actual cyclical behavior of the model is due to the several points in it in which one variable (say, the debt ratio) is multiplied by another (say, the wages share of GDP).[75] For example, as the employment rate rises, it increases the rate of growth of the wages share of GDP because the variables are multiplied together in the equation for the rate of growth of the wages share.

With a linear “Phillips Curve” function, this intrinsic nonlinearity is multiplied by a constant slope of the Phillips Curve, whether the model economy is close to an equilibrium or far away from it. This applies whether the employment rate is well below or well above its equilibrium value, and a constant slope means that it shows wages fall with low employment as easily as they rise with high employment.

But with a nonlinear Phillips Curve, the intrinsic nonlinearity is increased much more when it is a significant distance above the equilibrium than it is when closer to it, while an employment rate well below the equilibrium leads to only a small fall in wages, rather than a very large one.

Nonlinear functions also let you use much more subdued assumptions about the magnitude of a system’s response to an imbalance. The functions used here are both exponentials, and are entered using a generalized formula that takes an (x,y) coordinate, the slope at that coordinate, and a minimum value as inputs:[76]

The slope of both these functions at their (x,y) points—the employment level where wage change equals zero for the wages function, the profit level at which all profits are invested for the investment function—is 2, versus a slope of 10 for both of their linear counterparts. An extreme slope is needed for the linear functions because, with a more gradual slope, nonsense values could be returned—such as an employment rate of more than 100%, for example. With a nonlinear function, the slope near the equilibrium can be quite modest, while the nonlinearity of the function itself makes this slope steeper further away from the equilibrium. This is realistic, and serves the purpose of constraining system outcomes to realistic bounds.

Figure 207: Nonlinear behavioral functions generate more realistic cycles

9.4 The switch function

Figure 204 makes use of Minsky’s switch function to enable the model to switch from using linear to nonlinear functions easily.

Figure 208: The switch functions used in Figure 204

The switch function takes an input which can be zero or non-zero, and returns the first input if its value is zero, or the second if it is non-zero. The parameter L has an initial value of 0, a maximum of 1, and a step-size of 1, so it acts as the on-off switch. To make it easier to see what is SATg happening, the indicator wire within the switch block will switch from one input to the other when the input condition is altered.

Now let’s turn our attention to the essential missing ingredient in economic models of production: energy.

Footnotes

68 Things get far more complicated when 3 or more species are considered: with 3 dimensions, as I explain in Manifesto, you enter the realm of chaotic dynamics—which we’ll explore using Lorenz’s model.

70 The model’s parameters are , where is the rate of technical progress, the rate of population growth, the depreciation rate, the capital to output ratio, the slope of a linear “Phillips K λ λ Curve”, and the employment rate at which the rate of change of wages is zero. α, β, δ , v, S , Z α β K λ δ v S λ Z

71 The very issue of an aggregate measure of capital is a fraught one, given the Cambridge Controversies. However the import of this was much greater for Neoclassical economics than Post Keynesian, because in Neoclassical Economics, the rate of return on capital is its marginal product. This leads to the circularity in the theory that Sraffa exposed. In Post Keynesian economics, this link does not exist: the rate of return is a function of class struggle over the distribution of income. So the concept of an aggregate quantity of capital is less problematic for Post Keynesian modelling, even though this issue of how one aggregates disparate types of capital equipment still exists. In the next chapter on energy, I make a novel suggestion as to how to do this, though at a highly abstract level.

72 Ponzi finance can easily be added by including debt that doesn’t create new productive capacity. See (Giraud and Grasselli 2019).

73 “Money” at this point in developing the model is effectively “real”—there is no inflationary mechanism.

74 The unrealistic values of some variables—in particular, the debt ratio—are largely a consequence of the use of linear behavioural functions. More realistic nonlinear functions result in more realistic variable values, which indicate that the role of nonlinear functions is not to generate the cyclical behaviour (which results from the intrinsic nonlinearities in the model itself) but to constrain the behaviour within realistic bounds.

75 This isn’t apparent in the equations in this section, since they use abbreviations to make the equations more compact. The full nonlinear interactions in this model are shown in Equation (214) in section 11.4, which starts on page 237.

76 This function is, I hope, much easier to read than the flowchart renditions of it on the canvas. One thing we hope to enable one day is direct entry of equations onto the canvas, rather than requiring the use of a flowchart format. But as with all things computing, implementing this will require funding.