Monday, March 26, 2012

Thomas Palley's Book On The Little Depression

Figure 1: A Figure Illustrating Data From A Table In Palley (2012)
I have been reading Thomas Palley's new book, From Financial Crisis to Stagnation: The Destruction of Shared Prosperity and the Role of Economics. He argues that the ongoing crisis is not just a downturn in the business cycle, but the manifestation of the exhaustion of the neoliberal paradigm for economic growth1. Palley points to underlying structural contradictions, such as the role of consumer debt in the United States of providing the mass-based aggregate demand for consumption no longer sustainable when the overwhelming majority of workers do not participate in income gains from improving productivity. The expanded power of the less-regulated financial sector fits nicely into this thesis. Palley also discusses flaws in how the United States has come to fit into the global economy.

How academic economists have forwarded flawed ideas in support of these unsustainable structures is another major theme of this book. Both the freshwater (also known as Chicago school) and saltwater (also known as MIT school) economists are neoliberals. Palley, as is typical of Post Keynesians, opposes the views of saltwater economists, who agree with freshwater economists that, if it were not for failures of competition, externalities, information asymmetries, and sticky and rigid prices, the economy would generally perform well. The disagreement, on the level of analysis, is on the empirical importance of such imperfections2. Both erroneously think that economics can be separated from politics. Palley names his contrasting, third view as "Structural Keynesianism".

Palley does not describe the ideas of economists as driving economic policy in the United States. Rather, the market capture of academic economics provides a challenging obstacle for enlightenment. Palley approvingly quotes3 both Keynes' final paragraph in the General Theory and Karl Marx from the The German Ideology, "The ideas of the ruling class are in every epoch the ruling ideas..."

I have yet to finish this book, but I still have some criticisms. I wish Palley had included more graphs in this book. It seems to me that some of his charts in Chapter 4 could more usefully be graphs. What figures he has are usually decompositions of ideas, like Ishikawa diagrams in another format. In his discussions, Palley drops some nuances4 from his text that are explained earlier. Having skipped from Chapter 2 to Chapter 11, I can see some redundancies. Also, I wish Palley had referenced more heterodox economists.

Finally, I want to note José Antonio Ocampo's cover blurb:

"This is an outstanding book: clear, concise, and comprehensive. It shows that the economic crisis is the result of economic policies derived from flawed ideas and flawed ideologies. Read it and recommend it to your friends. It provides a map to overcome the Great Stagnation and to return to shared prosperity."
Ocampo has been in the news lately; Brazil just nominated him for president of the World Bank.


  1. Palley dates the start of this growth model with the Reagan era. I would rather point to Nixon's abolishing of the Bretton Woods' system. I suppose one could say the last half-decade of the 1970s was a transitional period.
  2. Palley notes Post Keynesian agreements with saltwater economists on short run policy.
  3. Antonio Gramsci does not appear in the index.
  4. Such as the distinction between textbook (or bastard) Keynesianism and structural Keynesianism.

Sunday, March 18, 2012

A Nonergodic Model of the Business Cycle

Figure 1: A Fractal in the Phase Diagram for One Specification of Parameters in the Kaldor Model1

1.0 Introduction

I thought I would try to combine an ability for computers to draw fractals with an economic model that suggests practical conclusions. In this post, I merely duplicate some results in the literature. In a deterministic ergodic process, as I understand it, all trajectories pass through every state in whatever attractor may exist. Hence, the Kaldor model, like some dynamical systems arising in mathematics, is non-ergodic.

2.0 The Model

In 1940, Nicholas Kaldor proposed a model of the business cycle. It can be expressed by four equations2. National ouput evolves from the previous period as a response to aggregate demand:

Yt+1 = Yt + α(It - St),
where Yt is the value of output in year t, It is intended investment, and St is intended saving. The parameter α represents the speed of adjustment to excess aggregate demand. The evolution of the value of the capital stock depends on investment and depreciation:
Kt+1 = It + (1 - δ)Kt,
where δ is the depreciation rate of capital stock. Intended saving is directly proportional to output:
St = σYt,
where σ is the (average and marginal) propensity to save. An investment function3 is the final equation specifying the model:
It = σμ + γ(σμ/δ - Kt) + Tan-1(Yt - μ),
where μ is the expected level of output, and γ represents the costs of adjusting the capital stock. Along with some restrictions on the values of parameters, the model is now fully specified. The arc tangent function provides a s-shaped non-linear term, such that entrepreneurs increase investment when output exceeds their expectations4.

I find it convenient to define new variables normalized around a stationary state:

kt = Kt - σμ/δ
yt = Yt - μ
The model, expressed in terms of normalized capital stocks and normalized output, is:
kt + 1 = Tan-1(yt) + (1 - δ - γ)kt
yt + 1 = (1 - ασ)yt + αTan-1(yt) - αγkt
Note that the following is a solution:
For all time t, yt = kt = 0.

3.0 Some Results

The above version of the Kaldor model is a discrete-time dynamical system, defined by a map from the two-dimensional real plane (k, y) to the same space. Four5 parameters are used to define the map. Questions for the mathematician revolve around describing how the phase portrait for the system varies qualitatively with variations in the parameters. Complex and chaotic behavior can arise in the Kaldor model with appropriate choices of parameter values.

For a small enough speed of adjustment and large enough propensity to save, the dynamics is boring. All trajectories converge to the origin.

As the propensity to save decreases, the system goes through a pitchfork bifurcation, so-called because the bifurcation diagram looks like a pitchfork. The origin loses its stability, and two symmetric fixed points appear. For a small enough speed of adjustment, at least, the two symmetric fixed points exhibit local asymptotic stability. The location of these new fixed points must be found numerically. A fortiori, the computer must be used as an aid to perform a local stability analysis of these points, based on the eigenvalues of the Jacobian matrix.

As the speed of adjustment increases, the boundary between the basins of attraction becomes more complex. Figure 1 shows a case where they are entangled in a fractal-like structure, and the outer perimeter of the colored area is repelling. The limit cycle shown is a short distance outside this repelling boundary.

I have by no means exhausted the dynamics of the Kaldor model. Consider a region in which the origin is the only fixed point, and it is asymptotically stable. As the speed of adjustment increases, the system undergoes a Neimark-Sacker bifurcation, which, I gather, is the discrete-time analog to a Hopf bifurcation. Cycles exist in which the cycle is not a fixed point on the Poincaré return map, but winds around many times before repeating. And if I want my application to explore all these dynamics, I have quite a bit of programming to do. I am curious if I will be able to plot a bifurcation diagram, given that the behavior at the limit depends on the initial value.

4.0 Observations

For the parameter values illustrated in the figures, trajectories have three possible destinations:

  • A stable equilibrium with lots of capital and high output.
  • Another stable equilibrium with less capital and less output.
  • A business cycle.
Furthermore, the boundary between the basins of attraction for the stable limit points is fractal-like. These properties suggest that a random shock to the system can redirect trajectories to a very different final destination.

Although not illustrated above, the model exhibits structural instability. A perturbation of the model parameters can result in different observable behavior, of greater or less complexity.

One general way of conceptualizing business cycles is to see them as the response of a damped linear system to exogenous shocks. Their height and depth depends on the characteristics of the external impulses driving the system. The Kaldor model suggests another possibility. In this model, the properties and extent of business cycles are endogenously determined. Shocks can drive the system from one trajectory to another, but the range of possible behaviors is determined from within the system. It is my impression that the former way of understanding business cycles is dominant among mainstream macroeconomists, while the latter is closer to describing actually existing capitalist economies.


  1. Figure 1 is drawn with the parameter values specified in Figure 2(c) in Agliari et al (2007).
  2. Kaldor uses a nonlinear savings function and merely specifies the form of the investment function.
  3. The specification of investment independent of saving is an essential characteristic of Keynesian models.
  4. In this model, expectations are held constant.
  5. Notice the expected level of output does not appear in the two equations giving the normalized model.

Selected References

  • A. Agliari, R. Dieci, and L. Gardini (2007). Homoclinic Tangles in a Kaldor-like Business Cycle Model. Journal of Economic Behavior & Organization. Vol. 62: 324-347.
  • W. W. Chang and D. J. Smyth (1971). The Existence and Persistence of Cycles in a Non-linear Model: Kaldor's 1940 Model Rexamined. Review of Economic Studies. Vol. 38, No. 1: 37-44.
  • Richard M. Goodwin (1951). The Nonlinear Accelerator and the Persistence of Business Cycles. Econometrica. Vol. 19, No. 1: 1-17.
  • Nicholas Kaldor (1940). A Model of the Trade Cycle. Economic Journal. Vol. 50, No. 197: 78-92.
  • Yuri A. Kuznetsov (1998). Elements of Applied Bifurcation Theory, 2nd edition.

Sunday, March 11, 2012

The Wise On Roke

Ursula Le Guin sets her Earthsea trilogy on an imaginary world containing many islands1. Nine mages run a world-famous school for wizards on Roke.

ROKE is now also an acronym, for the Review of Keynesian Economics. It sounds like it prmises to be an interesting journal.


  1. Strangely enough, six books now comprise the Earthsea trilogy.

Friday, March 02, 2012


  • Matías Vernengo provides an introduction to the capital controversy. He concludes, like others, including myself, that the neoclassical theory of supply and demand cannot explain factor incomes in a capitalist economy.
  • Philip Pilkington interviews Yanis Varoufakis, concentrating on Varoufakis' views on the errors in mainstream economics and the sociology of economics. Update: The second part of the interview is here.
  • Dan Little reviews The End of Value-Free Economics, a book edited by Vivian Walsh and Hilary Putnam. I have only skimmed the Putnam book that the essays in this book are responding to. I have read Putnam's Reason, Truth and History, which makes the case, drawing on the later Wittgenstein, that facts and values cannot be disentangled in science. And Gram and Walsh's Classical and Neoclassical Theories of General Equilibrium: Historical Origins and Mathematical Structure is a very good textbook introduction to the Sraffian revolution.
  • Open Democracy is hosting a series called "Uneconomics". Hey, one of the articles is by Philip Mirowski. Hat tip: Rod Hill and Tony Myatt.