Analyzing a Model
This should not be is a difficult chapter, because it is telling you how to do something that you should already know how to do : to work out the qualitative properties of a dynamic system.
However, if you have done a degree in economics—even a PhD—you probably don’t know how to do that. This is regardless of whether you’re a Neoclassical, Austrian, Sraffian, Marxist or Post Keynesian, because most (not all!) modelers in these disparate traditions have one thing in common: they model the economy using equilibrium-oriented methods.[94] This implicitly assumes that the equilibrium of their model—and by implication, the economy itself—is necessarily stable. It’s not, as this chapter will explain.
If you’ve got this far into this book, I am assuming that you know the basics of linear algebra— specifically, what a determinant is and how to work it out. I also assume that you don’t know how they’re applied in dynamic analysis—basically, in working out the stability or otherwise of a dynamic system using “eigenvalues” and “eigenvectors”.
It was also difficult for me to write this chapter, since, though this material used to be second nature to me, after dedicating most of the last two decades to debunking Neoclassical economics (Keen 2001, 2002, 2003e, 2003c, 2003a, 2003d, 2003b, 2004b, 2004a; Lee and Keen 2004; Standish and Keen 2004; Keen 2005; Keen and Standish 2005; Gallegati et al. 2006; Keen and Standish 2006; Keen and Ormerod 2007; Keen 2009; Keen and Standish 2010; Keen 2011a, 2011b, 2011c, 2015; Keen and Standish 2015), I’ve barely used these techniques myself this century. With mathematics, like sports, if you don’t use it, you lose it.
Finally, large scale dynamic systems— and that means anything with more than two dimensions —are extremely hard to analyze qualitatively, and there is a hard limit : the mathematical prodigy Galois proved in 1830 that almost all 5th order and above polynomials do not have a symbolic solution.[95] This matters for analyzing dynamic systems because, as you’ll see below, the qualitative properties of a dynamic model can be reduced to the properties of a polynomial of the same order.
Why is this a problem? Here, what economists do know has led to delusions about what they don’t know.
Virtually all schoolchildren learn the solution to a quadratic, a “second order polynomial”. Figure 219 shows the formulas, using the symbolic engine of the mathematics program Mathcad.
Figure 222: The well-known solutions to a quadratic equation
Pretty simple, right? Most of us learn this by rote at school: “the roots of a + b ∙x + c ∙x[2] = 0 are minus b, plus and minus the square root of b squared minus 4ac, all divided by 2 times c”. If you don’t study mathematics to an advanced level (say, 2nd year undergraduate), it’s not unusual to think there must be equivalent formulas for higher order polynomials—and that there’s no limit to how high you can go.
At the next level, a quartic, the equation for even one of the four formulas wouldn’t fit on a single page, and there is no formula—there cannot be a general formula, as Galois proved almost two centuries ago—for a quintic or above.
That’s the bad news. The good news is twofold. Firstly, advances in computing power have meant that the numerical analysis of the properties of a dynamic model are possible. Secondly, the actual number of fundamental dimensions to a model is often below five, even for a very complicated model—for example, even the government-based extension of my model of Minsky, which has six equations (Keen 1995, pp. 625-632), is actually a 3-dimensional model, because its fundamental dimensions are the wages share of GDP, the employment rate, and the private debt to GDP ratio.
However, as is often the case with mathematical analysis, the good news comes with bad news (and so on ad infinitum).
In numerical analysis, the number of dimensions is based not on the fundamental variables in a model, but on its parameters: so a model with 3 variables and ten parameters is ten-dimensional when you wish to work out how the model’s behavior changes with different parameters. If each parameter can take fifteen different values, you have one million billion possible combinations. That’s just too many even for a modern computer to analyze, so mathematicians and computer programmers have worked out ways to explore the parameter space—genetic algorithms, simulated annealing, etc.—without having to check every possible combination of parameters.
In symbolic analysis, while the dimensionality depends on the number of fundamental variables, the task of converting a model into a form where its equations are strictly in terms of the fundamental variables can be extremely difficult. Using my model of Minsky’s Financial Instability Hypothesis as an example again, it had three fundamental dimensions—the wages share of GDP, the employment
rate and the private debt to GDP ratio. The mathematician Bernardo Costa-Lima’s devoted his entire PhD thesis to analyzing its properties.[96]
Despite those discouraging remarks, its worth knowing at least the basics of the qualitative interpretation of complex dynamical models. It will, for a start, disavow you of the notion that equilibrium modelling is sufficient. And it will allow you to appreciate the processes that give rise to the complex dynamics of systems like Lorenz’s butterfly effect, and my own models of financial instability.
It will also help you appreciate that the instability of input-output dynamics—something that I think played a large role in Neoclassical economists finally abandoning CGE (“Computable General Equilibrium”) modeling—wasn’t at all due to the fixed proportions involved in an input-output matrix. Even with variable input proportions , these equilibrium models would still have been unstable, because ironically, the linear components of a model determine the stability of its equilibrium, while the nonlinear bits only come into play far from equilibrium. Neoclassicals should not have abandoned Computable General Equilibrium modelling for Dynamic Stochastic General Equilibrium modelling, which was a backward step (Romer 2016): instead, they should have abandoned their fetish for equilibrium, and embraced far from equilibrium dynamics.
11.1 Of linearity and nonlinearity
One of the quirks of nonlinear models is that the stability of their equilibrium points is determined by their linear components —so that if the equilibrium of a nonlinear model is unstable, it’s because of its linear bits, not the nonlinear ones. Therefore, if the equilibrium of some dynamic process is unstable—as is the case for Computable General Equilibrium models —then adding nonlinear bits to it (say, replacing a Leontief input-output matrix with a different production model that allows for substitution between inputs) won’t help one zot: the equilibrium will remain unstable. The easiest way to illustrate this is with a Taylor series expansion for a periodic function, like . Imagine a model where the equilibrium is given by . Then, if you’re a long way from this ALS(T) equilibrium—say at —the behavior of the model is dominated by its nonlinear bits, and the linear component is effectively irrelevant: see Figure 221. The linear approximation to ALS(T) is , or about 6.3 , which is hopelessly wrong, since the actual value is zero. The full 17 2πth order polynomial approximation is 0.011 , which is only wrong by 1%. ALS(2π) 2π
96http://cms.dm.uba.ar/actividades/seminarios/sanlsd/PhD_Thesis_Bernardo_R_C_Costa_Lima_Final_Submiss ion.pdf.
Figure 224: sin(x) and its approximations out to 10
Figure 222 shows the same functions, but for values less than 1. Here, the nonlinear terms add very little to the accuracy of the approximation: the linear approximation to sin(0.1) is 0.1; the actual value of . The higher order terms improve the accuracy of the linear approximation by less than 0.02% . Near the equilibrium, the linear term rules. ALS(0.1) = 0.099833
Figure 225: sin(x) and its approximations out to 1
sin(x) and its first nine Taylor series approximations
This is because the nonlinear components are the divergence from equilibrium, raised to a power of 2 or more. When you’re a large distance from the equilibrium, these numbers dominate the linear component, since . But when you’re close to the equilibrium, the order is reversed: . T[5] > T[4] > T[3] > T[2] > T[1] fRg T> 1 This enables a simple way of analyzing the stability of a nonlinear dynamic model: convert it into a T[1] > T[2] > T[3] > T[4] > T[5] fRg T< 1
This enables a simple way of analyzing the stability of a nonlinear dynamic model: convert it into a polynomial approximation; drop all but the linear terms to generate a linearized model; and work out whether its rates of change are negative near the equilibrium—which means the system will converge to the equilibrium—or positive—which means the system will diverge. It’s slightly more complicated than this, but that’s the gist.
11.2 The absolute basics of stability analysis
Let’s give it a try with the simplest complex model we consider in this book, Lotka’s predator-prey model—see Figure 223 (which reproduces Figure 85 on page 66). This generated, to Lotka’s surprise, everlasting cycles, rather than the convergence to equilibrium that he expected:
It was, therefore, with considerable surprise that the writer, on applying his method to certain special cases, found these to lead to undamped, and hence indefinitely continued, oscillations. (Lotka 1920,p. 410)
Figure 226: Lotka's predator-prey model: everlasting cycles
Equation (171) expresses this as a differential equation, in a way that illustrates that Lotka’s model is the simplest possible extension of a model of exponential growth. If there were no Sharks, the number of fish would grow exponentially, while if there were no Fish, Shark numbers would fall exponentially. But the existence of sharks linearly decreases the growth rate of fish, while also linearly decreasing the death rate of sharks:
The first step in working out why this model generates everlasting cycles is to express it as a vector equation:
Next we create a matrix from this vector, where the entries are the differentials of the equations with respect to F and S. This is known as the Jacobian matrix (I’ll explain why this is needed later):
d Then we replace F and S with their equilibrium values. From Equation (171), = 0 if and = a dS 0 if :[97] dt S= b dt c F= d a c b c
This is now the linear component of the predator-prey model, which, in the vicinity of the equilibrium, dominates the nonlinear components. So, to work out whether the model converges to the equilibrium or diverges from it, we have to analyze Equation (175)—where I’ve used the subscript L to indicate that this is a linearized model:
Before we do, it’s easy to add this to the Minsky model of the full nonlinear model, so that we can see how it behaves:
This is a linear model of the deviation of the full system from its equilibrium values (hence the positive and negative values that it generates), and you can see that it reproduces the same closed cycle as the full nonlinear model (though it’s circular in shape, as opposed to the egg-like shape of the full nonlinear model). So the model neither converges to the equilibrium, nor moves away from it.
Figure 227: The Predator-Prey model and its linearized deviation from equilibrium__98
We can now use some algebra to show why this model generates the same magnitude cycles forever. The logic starts from the nature of a single ordinary differential equation, and simply works out how to apply that to a system of equations.
A linear Ordinary Differential Equation is an equation of the form:
dt Here is a constant and is a variable, and what we’re trying to find is the correct functional form for . This might be a model of population growth, or radioactive decay. So the general solution for as a function of time is that T E must be some function whose rate of change equals itself times a constant E . The exponential function is the only candidate, since the differential of an exponential E E(A) function is the coefficient for the exponent times the function. A exponential is of the form:
98 Notice that the text on the y-axis for the second graph spills outside the graph? This is obviously a bug. It has been there for some time as we’ve focused on improving other aspects of Minsky. This one, I hope, will be eliminated with the remaining funds from the Friends Provident grant.
Its differential is the exponent times the function itself:
We then feed this “guess”[99] into Equation (177)
y t a y t It’s obvious that our “guess” answer is correct if , but I’m going to labor the point a bit here by rearranging the last line of Equation (180): T= λ
Equation (181) is only true for non-zero values of if . Therefore the solution to Equation (177) is: E(A) λ= T
y t c e (182)[100] The value of tells you whether a system tends towards zero over time—if —or explodes—if —or doesn’t change—if . T T< 0 Have I bored you with this exposition? I hope so, because the process for working out the same T> 0 T= 0 results for a system of equations like Equation (175) is much more demanding, but it is essentially the same process—only following the rules of matrix mathematics, since we’re working with a pair of equations rather than a single equation.
We start with the linear component of the predator-prey model in Equation (175)—reproduced here:
We equate this to the matrix equation in (183):
Next we rearrange the equation so that the zero vector is on the right hand side:
To process this according to the rules of matrix mathematics, we need to multiply the first term by the identity matrix:
We subtract the second matrix from the first to yield:
b For this equation to allow for non-zero values of and over time, the matrix in (189) must somehow be like in Equation (181): it must have a magnitude of zero. Then the equation can L L be solved for non-zero values of and . This will be the case if the determinant of the matrix is F S zero. The determinant is a quadratic in (λ−T) : L L F S λ b c
So the roots of this polynomial give us the values of that both solve Equation (189), and tell us whether this linear system will converge to the equilibrium—if the roots are both negative—or λ diverge from it—if they are both positive:
However, the roots of this equation are pure complex numbers—numbers including the square root of −1 , symbolized by the letter : 2 L a c
They therefore they describe purely cyclical behavior, as we’ve already seen in the simulation.
In general, there are 4 major classes of behavior:
- a stable fixed point: both roots—called eigenvalues—are real and negative;
- an unstable fixed point: both eigenvalues are real and positive;
- a saddle, which is also unstable: one positive real eigenvalue and one negative; and
- a cyclical system, where both are complex numbers—which can have positive, negative, or zero real parts.
In a way, this is all rather ho-hum: the behavior is relatively simple, and with computers, we can see the model’s behavior in the simulation anyway. But the behavior isn’t simple when we get to what are called complex systems: systems of three or more nonlinear ordinary differential equations.[101]
11.3 Analyzing a complex model
As a warmup to analyzing my model of Minsky’s Financial Instability Hypothesis, we’ll analyze the first such system to be numerically simulated: Lorenz’s “butterfly” model of turbulent flow (Lorenz 1963). We simulated—but did not analyze—this in section 9.1 on 182. Its deceptively simple equations in 3 variables with three parameters are:[102] dx T, E, z T, L, c
The first step is to work out the equilibria of this model, which is relatively easy to do. We set the differentials in Equation (193) to zero, and then solve for these specific values of : T, E, z
The obvious solution is that all three are zero.[103] One obvious element of the non-zero solutions is that , which is easily derived from the first equation. Feed this into the second and third equations, and you get: E= T
We calculate the “Jacobian” matrix—so called because it was first developed by Carl Gustav Jacob Jacobi—which is a matrix formed by differentiating each function with respect to each variable in the system (I’ll explain why it’s used later):
We make the same “ansatz” that the solution is of the form:
Substituting this into (199) gives us an algebraic equation to solve:
Rearrange it using the rules of matrix algebra:
0 0 0 0 c zL 0 We’re now looking for values of that allow the variabes to take non-zero values. This is revealed by the roots of the polynomial in generated by this determinant: L L L λ T , E , z a a 0 λ
There’s a lot more grunt work to express this fully, but the key point for stability is that, for the equilibrium to be stable, the biggest “real” part (that is, the part that doesn’t involve the square root of minus one) of these numbers—known as the “dominant eigenvalue”—must be negative. We know that is negative, since are all positive numbers; but the value of the other two depends on the magnitude of the square root term. Here we have to plug in numerical values for −c T, L, c , which are parameters in fluid mechanics. The realistic values that Lorenz first used were: T, L, c
When we plug those into Equation (204) we get:
What this means that this equilibrium is stable along two of its three “eigenvectors”, but unstable along the third. So the zero equilibrium is a “saddle node repeller”: it attracts the system along two axes, but repels along the third. The other two equilibria are symmetric, so we can just consider the second. We feed the values for 2 2 2 into the Jacobian: E E E T , E , z
The linear component of the dynamic system near this second equilibrium is thus:
We assume that this is equivalent to:
Substitution yields:
Grouping via the rules of linear algebra yields:
This is true if the determinant of the matrix is zero, which yields this third order polynomial:
Remember the formula for the roots of a cubic equation in Figure 220? Feed this into that, and you get a nightmare expression that would fill several pages. It’s much easier to fill in the values for the parameters—see Equation (205). Feed these into Equation (212) and you get one real root (the negative number on its own) and a pair of “imaginary” numbers—numbers involving √ −1 : 13.855, L=
The first root shows that this equilibrium is a strong attractor along one of its dimensions (eigenvectors). But the next two show that it is cyclical (the two complex numbers generate cyclical behavior) and that it is a repeller: the real part of the pair of complex roots is greater than zero. This means that as the system approaches this equilibrium, it is repelled from it in a cyclical fashion— which is what we saw in the simulation.
The whole model has one equilibrium which attracts along two dimensions and repels along a third, and two equilibria which attract along one dimension and repel cyclically on a plane. Figure 225 shows its three equilibria, and the dynamics of the Lorenz model, projected onto the and planes. This behavior had never been seen in a mathematical model before Lorenz, and the T, E z equilibria were dubbed “strange attractors” as a result.
104 Since it is symmetric, the other non-zero equilibrium has the same roots.
Figure 228: The dynamic behavior of the Lorenz model projected onto the x,y and z planes
This also illustrates why equilibrium modelling is a waste of time if the underlying system is complex—and that is true when the system in question has three or more dimensions that interact with each other in nonlinear ways (Li and Yorke 1975). A complex system will almost certainly never be in any of its feasible equilibrium states: if you want to model it, you have to use modelling techniques that can handle far-from-equilibrium behavior.
This discovery convinced meteorologists of Lorenz’s point, that weather forecasting should not use equilibrium techniques. If his simple model, which was derived from the complicated equations that describe fluid flow, generated chaotic behavior, then the weather—where the fluid is air—must also be chaotic. Equilibrium models of the weather were therefore useless.
This was not the only factor involved, but there then ensued a revolution in meteorology which has led to the advanced capacity meteorologists have to predict the weather today—within limits also determined by the properties of complex systems. This is why the disaster of Cyclone Sandy had such a tiny impact on human life: meteorologists were able to predict where it would make landfall to a far higher degree of accuracy than was possible before Lorenz, and therefore people could be advised to evacuate before the disaster hit.[105]
But in economics? In 2007, economists advised that politicians that the economy would sail smoothly into 2008, since their equilibrium-based economic models predicted it would be a year of tranquil economic growth.
11.4 Analyzing the Keen model of Minsky’s Financial Instability Hypothesis
I predicted something different—a financial crisis—because I was informed by the non-equilibrium model I had built of Minsky’s Financial Instability Hypothesis, way back in August of 1992 (Keen 1995). That model used a nonlinear Phillips curve function devised by John Blatt (Blatt 1983, p. 213) for the express purpose of avoiding nonsense outcomes like an employment rate of more than 1; in this book, I have used a linear Phillips curve, not because I believe it is linear—anything but, as I explain in Section 5.2.1, starting on page 51—but because it’s easier to mathematically analyze a model with linear rather than nonlinear functions.
It also means, as Carl Chiarella emphasized (Chiarella and Flaschel 2000; Chiarella 2005; Asada et al. 2006), that if you use linear behavioral functions in an otherwise nonlinear model, the nonlinearities in the model itself arise not from the functions (which inevitably involve assumptions by the modeler) but from the structure of the model itself: they are intrinsic. Once these are identified and analyzed, nonlinear functions can be added at a later stage when you are attempting to fit your model to data.
My solution to the problem Blatt identified with linear functions—that they can give you a employment rate of more than 100%--was to use the employment to population ratio (which is about 60%) rather than the employment to workforce ratio (which is about 90-95%). That way, the countervailing intrinsic nonlinearities in the model will kick in well before the model reaches an employment to population ratio of 100%, even with an unrealistic linear Phillips curve.
The model—which as I show in Section 9.2 on page 185, can be derived directly from the macroeconomic definitions for the employment rate, the wages share of GDP, and the private debt ratio—is reproduced in Equation (214), with the intrinsic nonlinearities, generated when one variable is multiplied by another, highlighted in red.
The full analysis of this model’s stability properties is in (Grasselli and Costa Lima 2012). I won’t attempt that level of detail here, because it would just be too lengthy, and also because examining just one equilibrium is hard enough on its own—so hard in fact, that I debated whether or not I should include this section at all.
In the end, I decided to keep it, because it reveals the role of symbolic analysis in explaining why some phenomena that can be seen in a simulation actually occur. The two specific features of my model that cannot be explained by the equations themselves, nor understood simply by looking at a simulation, are that:
- The crisis is preceded by a period of diminishing volatility in the rate of economic growth—a “Great Moderation”; and
- Before it collapses in a final crisis, the capitalist share of income fluctuates around a constant level, while the workers’ share of GDP falls as the debt ratio rises.
Both these phenomena are apparent in Figure 226. But neither were either predictions by Minsky in his Financial Instability Hypothesis, nor assumptions built into the model itself.
Minsky did say “stability … is destabilizing”, but this was with reference to a single cycle—that a period of tranquil growth would lead to rising and eventually euphoric expectations, leading to a boom which changed the distribution of income, and caused a bust. However, he did not predict that the scale of cycles would get smaller before they got larger: that was something that was first seen in the simulations for my 1995 paper. The phenomenon was so striking that I ended the paper with a rhetorical flourish about it:
The chaotic dynamics explored in this paper should warn us against accepting a period of relative tranquility in a capitalist economy as anything other than a lull before the storm. (Keen 1995, p. 634)
Figure 229: The Keen-Minsky model (same equations as (214) but with compact functions to save space)
The fact that the profit share was stable (before it collapsed at the end of the simulation), while the wages share fell as the debt level rose, was also an enigma: how come workers’ share of GDP falls as debt rises, even though—in this model—workers did not borrow?
The solution to this enigma became obvious when I first tried to work out the model’s equilibria in terms of its three system states: the wages share , the employment rate , and the debt ratio . An equilibrium will be defined in terms of the model’s parameters, which was easy to do for the D ω λ S employment rate. But the “equilibrium” wages share included the “equilibrium” debt ratio as one of its arguments: try as I might, I either couldn’t eliminate a variable from the solution, or I got something too complicated to work with. So I decided to work with the profit share instead:[106]
That results in a much more compact set of equations:
Given that and are both positive, we can simplify finding the equilibrium to solving the following equations: λ ω
The first two are relatively easy to work out, while the third, for the debt ratio, is quite involved. Equation (218) shows the equations using the substitution of the profit share, which allows us to express both the equilibrium profit share and equilibrium employment rate in terms of the model’s parameters:
We have to go further to express the debt ratio in terms of the model’s parameters only though, since the profit share itself is part of the debt ratio. Making this substitution yields the unholy mess in Equation (219):
This complicated expression simplifies drastically once you realize that part of it is the expression for the equilibrium rate of profit from Equation (218) . Using superscripted (as in ) to emphasize that these are equilibrium values, we have: DE E S
Even this isn’t the end of the process, because the differential equations in Equation (216) are in terms of the wages share, not the profit share. We either have to derive the equilibrium wages share, or convert the wages share differential equation into a profit share equation. The easier route is the former, and it also reveals something interesting. Firstly, we define the equilibrium wages share in terms of the equilibrium profit rate and debt ratio:
Remember that this model was derived from strictly true macroeconomic definitions? Since the equilibrium profit share is a constant, then a higher rate of interest means a lower wages share of GDP. In the dynamic model as well as in this equilibrium calculation, the higher the debt level, the lower the workers’ share of GDP. This explains the phenomenon we can see in the dynamic path of the model as well: as the debt level rises, the wages share falls. A higher interest rate also leads to a lower wages share. The real class struggle in capitalism is not between workers and capitalists, but between workers and bankers.
That fun observation aside, we still need to substitute the expressions for the equilibrium profit share and debt ratio from Equation (220) into (221) to generate equilibrium values for the wages share, employment ratio and debt ratio. Using the expression for yields this relatively compact statement of the values of the “good” equilibrium: tE π
These equilibrium values have to be substituted into the Jacobian—which is just too big to show in full form here, so I’ll represent what it is instead: the partial derivatives of each of the functions in the system (Equation (216)) with respect to each of the variables D ∂ d ∂ d ∂ λ, ω, S d
Fortunately a fair degree of cancellation occurs, so that the Jacobian isn’t quite as horrific as it could have been—but it’s still pretty horrific:
We’re not home and hosed yet: this is the linear component of the model in the vicinity of this equilibrium, and to know whether this linear model will converge to the equilibrium, we need to calculate the roots of the polynomial that results from the same process as shown for the Lorenz model. These roots are just too complicated to calculate symbolically, so—as is often the case in complex systems analysis—we are forced to numerical means: we calculate the polynomial for given parameter values.
The key parameters that shape the system’s behavior are the interest rate, the slope of the investment function , and the point at which investment equals profits (and therefore, firms don’t borrow) . The key behavior of the model—a flip from a stable to an unstable equilibrium—can be π S seen in the equations below: where the slope of the function , is 8.4, 8.5 and 8.6 respectively:[107] π Z
Each set of eigenvalues includes a negative real value, which attracts the system towards it. Each also contains a pair of complex eigenvalues, which makes the system cycle. The key difference is that, in the first case, the “real part of the complex eigenvalues” is negative—which means the system will converge to the equilibrium; in the second, the real part is zero—so it will repeat the same cycle indefinitely, neither converging on the equilibrium, nor diverging from it; in the third however, we get “strange attractor” behavior, because the real part is positive—it repels the system from the equilibrium. So the system is attracted along one axis and cyclically repelled along another.
This doesn’t mean that the value of 8.5 is a critical one for the system—the behavior of this linearized model only properly characterizes the full nonlinear model in the immediate vicinity of the equilibrium, so if you start much further away, as do the simulations I’ve done of it in this book, then instability can apply at a lower value for the slope of the investment function. But it does indicate that there are conditions under which the model can be stable, and others under which it can be unstable, so the model “bifurcates” when the parameters change across this critical value. Figure 227 shows a simulation with a low value ( = 5 ) for the slope of the investment function, and you can see the model converging to the equilibrium values over time. S π
Figure 230: Linear Keen-Minsky model with a low value for the slope of the investment function
Figure 228 shows what happens with a higher value (7.5), and starting from a long way from the equilibrium position (the debt ratio I start the simulation with is zero, while the equilibrium debt ratio is 252%). The model starts to converge on the equilibrium, but then diverges—and ultimately it will collapse onto the “bad” equilibrium of zero employment, zero wages share, and an infinite private debt to GDP ratio. This is the stylized equivalent of a debt-deflation, which implies a total breakdown of society—and it’s why “Big Government” is needed, according to Hyman Minsky, because the government’s counter-cyclical spending can prevent this collapse from occurring.
Figure 231: Linear Keen-Minsky model with a high value for the slope of the investment function
This simple model emphasizes a key point from a genuine dynamic model, as opposed to fake dynamics of “Dynamic Stochastic General Equilibrium” modelling: you can’t reduce the behavior of a complex system to the properties of its equilibrium.
11.5 Why the Jacobian?
The easiest way to explain why the Jacobian matrix—a matrix formed by differentiating each function with respect to each variable—is needed as part of this process is by analogy to what happens when you try to approximate a function like using a polynomial. What you are doing in that instance is declaring that the numerical value of ALS(A) is equal to the numerical value of an infinite sum of polynomials, which is an infinite sum of terms of a constant, ALS(A) plus another constant times t, plus another constant times t squared, and so on:
>
108 https://en.wikipedia.org/wiki/Brook_Taylor
This is already “too much information” for what you need to know here (were you feeling that way? Sorry!), which is that the linear term in this whole process is the first differential of the function. This rule for a “scalar function”—a function of just one variable, in this case, t—is the first differential of the function with respect to . The Jacobian just generalizes this to a “vector function”—a function of more than one variable, in A
The Jacobian just generalizes this to a “vector function”—a function of more than one variable, in the case of the Keen-Minsky model, the variables of . The Jacobian is the first differential of every function with respect to every variable. So by deriving it, we build a linear approximation of D λ, ω, S
the system, and the properties of this linear approximation dominate the system’s behavior around its equilibrium.
If this linear system is stable—if it converges to the equilibrium—then so will be the full nonlinear system, but only if it starts “close enough” to the equilibrium that the linear forces can “do their thing”. It is possible to have a system whose equilibria are stable, but only if you start close to those equilibria where the linear components dominate. If you start further away, then the nonlinear terms dominate—which can be seen in the simulation shown in Figure 228, since the value of is below the critical level for stability of the equilibrium, and yet the simulation still diverges from it. π S
This behavior of a complex system leads to the concept of a “basin of attraction”: a region around an equilibrium where the system will converge to the equilibrium—or remain within a finite distance of it that is less than the overall phase space of the system—if the initial conditions lie within this basin. Other concepts that turn up at this level of analysis include the Lyapunov exponent, to determine whether a system is chaotic or not (a system can generate aperiodic cycles, but not actually be chaotic). Were I twenty years younger, and that much closer to my own mathematical education, I’d attempt an explanation here. But having wasted so much time fighting Neoclassical economists, I’ve forgotten much of what would be needed to give a decent explanation. So, if you want to understand these concepts—and they are worth understanding if you wish to really contribute to nonequilibrium economics—then I suggest either doing a course in complex systems at your local University mathematics department, or undertaking self-tuition via online resources like The Chaos Book (https://chaosbook.org/).
Footnotes
94 To evolutionary and complex systems economists, this chapter should be old hat.
95 Here I had to pause in my writing to look up the name of the mathematician who proved this—which is a sign of how much I’ve forgotten. I thought it was Galois (it was, though he had predecessors), but I was no longer sure.
97 There’s another equilibrium, when . The same technique used here shows that this equilibrium is unstable. F= S= 0
99 This is called the “ansatz” in mathematics (see https://en.wikipedia.org/wiki/Ansatz). https://en.wikipedia.org/wiki/Ansatz). ). .
99 This is called the “ansatz” in mathematics (see https://en.wikipedia.org/wiki/Ansatz). https://en.wikipedia.org/wiki/Ansatz). ). . 100 This defines not just one solution to the equation, but a whole family of solutions, each with a different initial condition . c
101 This can be confusing, because the word “complex” is also used to describe numbers involving the square root of minus one: “Complex numbers”. Complex systems analysis uses complex numbers, but the two areas are different topics.
102 The treatment here is only the very first step in analyzing this model, whose properties are still an active field in mathematics research today—see for example (Chen 2018) and (Kudryashov 2015); for a teaching-level exposition on this model, see https://core.ac.uk/download/pdf/236407976.pdf.
103 is also an equilibrium of the predator prey model, and it is unstable. F= S= 0
105 This wasn’t the case for Cyclone Ida in August 2021, because, with the additional heat in the Caribbean from global warming, Ida grew too quickly for most people to be able to evacuate in time. It was just luck that Ida didn’t hit a major population centre, as did Sandy.
106 Substitutions like this are often necessary with complex systems models. Grasselli and Costa-Lima found that they had to substitute the debt ratio with the inverse of the debt ratio to analyze what the called the “bad” equilibrium, with zero wages share and zero employment but an infinite debt ratio. (Grasselli and Costa Lima 2012, p. 199)
107 The other parameters are the same: = = 3% . K λ λ α= 1.5%, β= 1%, δ = 6%, v= 3, g= 4%, S = 10, Z π π 60%, S = 10, Z