Sunday, October 05, 2008

For Whatever Can Walk - It Must Walk Once More

1.0 Introduction
This post presents a simple macroeconomic model that combines trend and cycle. It presents some possible aspects of economic growth and business cycles. This model has some features that I find objectionable, but I find it interesting nonetheless. It is a non-linear model of dynamics presenting a formalization of some ideas to be found in Marx's Capital. And it is a model that does not impose equilibrium, but allows for the stability of equilibrium to be analyzed.

2.0 Technology
Assume a Leontief (fixed coefficients) production function:
q = min(a l, k/σ)
where q is gross output, l is the labor employed, k is the value of capital, a is labor productivity, and σ is the capital-output ratio. Both constraints in the production function are always met with equality:
l = q/a
σ = k/q
The capital stock is always employed, but sometimes employment can fall short of the entire labor force, as explained below.

The capital stock depreciates at a rate of 100 δ percent. That is, output can either be consumed or added to a capital stock that experiences a force of mortality of δ. Technical progress is disembodied, and labor productivity increases at a constant rate:
a = a0 exp( α t )
The labor force also grows at a constant rate:
n = n0 exp( β t)
where n is the labor supply. Hence, v is the employment rate, where the employment rate is defined as follow:
v = l/n
When v is unity, the labor force is fully employed. v ranges from (a subinterval of) zero to unity in this model.

3.0 Wages, Profits, Investment
Let w be the wage rate. Then (w l) or (w q/a) are total wages. Define u to be the workers share of the gross product:
u = w/a
Then (1 - u) is the capitalists' share of the product. Assume that wages are entirely consumed and that a fixed proportion of profits are saved and invested:
dk/dt = s (1 - u) q - δ k
where s is the savings rate out of profits.

Finally, assume that the rate of growth of wages is a (linear) increasing function of the employment rate:
(1/w) dw/dt = - γ + ρ v
The above equation could also be written as:
(1/w) dw/dt = ρ [v - (γ/ρ) ]
In words, wages grow faster in a tight labor market. The marginal productivity of labor is beside the point in this model.

4.0 Derivation of the Model
The rate of growth of the employment rate is the difference between the rate of growth of employment and the rate of growth of the labor force:
(1/v) dv/dt = (1/l) dl/dt - β
By similar manipulations, one can show that the rate of growth of employment is the difference between the rate of growth of output and the rate of growth of productivity:
(1/l) dl/dt = (1/q) dq/dt - α
Combining these two equations yields an equation relating the rate of growth of the employment rate to the rate of growth in output:
(1/v) dv/dt = (1/q) dq/dt - (α + β)
The derivation of the following equation from the definition of the capital-output ratio and the equation for the rate of change in the value of capital is simpler:
(1/q) dq/dt = (1 - u) s/σ - δ
(1/v) dv/dt = (1 - u)(s/σ) - (α + β + δ)
The rate of growth of workers' share in gross ouput is the difference between the rate of growth of wages and the rate of growth of productivity:
(1/u) du/dt = (1/w) dw/dt - α
Substitute from the postulated relation between the rate of growth in wages and the employment rate:
(1/u) du/dt = ρ v - (α + γ)

The following pair of equations, the fundamental equations of this model, restate equations derived above:
dv/dt = (s/σ - α - β - δ) v - (s/σ) u v
du/dt = -(α + γ) u + ρ u v
Here's the cool part - this is the Lotka-Volterra predator-prey model. It is a canonical non-linear dynamical system used to model, say, lynx and rabbits.

5.0 Solution of the Model
For what its worth, a trajectory in phase space has the equation:
(uν1) exp(- θ1 u) = H(v- ν2) exp(θ2 v)
θ1 = s
θ2 = ρ
ν1 = s/σ - (α + β + δ)
ν2 = (α + γ)
and H is an arbitrary integrating constant. All these trajectories consist of cycles, as illustrated in Figure 1. The limit point around which these trajectories cycle is given by:
u* = ν11
v* = ν22 = (α + γ)/ρ
(The origin in phase space is also a limit point; the origin has the stability of a saddle-point.)
Figure 1: Phase Space

Figure 2: A Trajectory

6.0 Discussion
I think it interesting to note that the rate of growth of wages at the limit point is positive due to growth in productivity; in fact, the rate of growth of wages at the limit point is equal to the rate of growth in productivity. If productivity did not grow, if the labor force were stationary, and if there were no depreciation, wages would consume the entire product at the limit point; the capitalists would receive no profits.

A single cycle can easily be described in intutitive terms. Start with low unemployment. Wages will increase as a share in output. Consequently, saving and investment will decline. The growth of output will slow. Eventually, the "reserve army of the unemployed" will be recreated. Wages will decrease as a share in output, although they may still be increasing in absolute terms. Eventually, investment will pick back up. When the growth of output exceeds the growth in productivity by more than the growth of the labor force, the employment rate will increase.

Clearly, this model can reproduce a qualitative resemblance to some empirical properties of some economic time series. If one plots empirical data in the illustrated phase space, one may see a suggestion of motion in the indicated directions, but one will not find a single cycle. Perhaps shocks change the parameters of the model on some occasions. Or perhaps important considerations are not embodied in the model. This model is Classical in important respects, where I mean by "Classical" to refer to the economics of Smith and Ricardo. Richard Goodwin, the inventor of this model, has done important work attempting to integrate this model with Keynesian and Schumpeterian themes.

There was a conference in Sienna a number of years back devoted to this model. There's also discussion of this model in a Festschrift volume devoted to Richard Goodwin.
  • Richard Goodwin, "A Growth Cycle," in Socialism, Capitalism, & Economic Growth: Essays Presented to Maurice Dobb, (edited by C. H. Feinstein), Cambridge University Press, 1967.
  • Richard Goodwin, Chaotic Economic Dynamics, Oxford University Press, 1990.
  • Paul Ormerod, The Death of Economics, St. Martins, 1994.

Update: Serena Sordi has a recent generalization of this model to four dimensions, presented at a sort of festschrift for Barkley Rosser, Jr.


YouNotSneaky! said...

This is a very cool model and you've got a very nice presentation there. But you leave it a bit under analyzed.
For example, there's some parameter restrictions for that non-origin limit point to exist. Specifically, you need (d+a+b)(o/s)<1 (where d is delta, a is alpha, b is beta, o is sigma). Otherwise the only stable limit point is the crash state (0,0). While it is intuitive that higher depreciation and pop growth, as well as lower saving rate on the part of the capitalist would make that more likely, it is a bit strange that a higher rate of productivity growth (a) also makes the crash state more likely. What's going on in intuitive terms?
Also, I'm having trouble figuring out if there's a "golden rule" of s here, say, with alpha and beta = 0.
I might draw up some phase diagrams for this.

Robert Vienneau said...

There are probably conditions to ensure that neither component of that limit points exceeds unity. And maybe some conditions are needed to ensure that limit point is dynamically unstable, with periodic orbits around it.

Anyways, think of the special case where the rate of growth of productivity is zero (α = 0), the rate of growth of the labor force is zero (β = 0), and all capital is circulating capital (δ = 1). Then that condition is that σ < s. In words, it must be possible for savings to at least replace the capital used up in production. If some of the capital is fixed capital - so δ is less - not as much savings is needed. But if population or productivity grows, more savings is needed to maintain the natural rate of growth. In the sort of growth models I like, the rate of population growth and the rate of growth in productivity sum together to comprise the natural rate of growth.

YouNotSneaky! said...

Yes, that's clear. But what is the intuitive explanation for why a really high rate of productivity growth rate causes convergence to the crash state (share of labor income, as well as the employment rate going to 0)?

(we can find the (dv/dt)/v=0 line by setting the relevant equation to zero. It's a vertical line whose position depends negatively on alpha. So a high enough alpha pushes the line past zero, insuring that (0,0) is the only steady state (no cycle in this case))

One may be tempted to say "machines replacing capital" but with a Leontief PF there's no substitution. So I'm on clear on the reasons.

Also, it might also be worth noting that the 'pump' that is doing the work here of creating the fluctuations is not the particular form of the production function but rather the Phillips curve equation. I think a very similar result can be obtained with a version of the Solow model with unemployment, in which wages are given (by the PC as here) but marginal product is equal to them (hence resulting in unemployment if they go too high). Of course there the K/Y and K/L ratios aren't constant so it's substitution between K and L that is doing the work of changes in labor's share (which is constant).

Also, I assume that when you say "This model has some features that I find objectionable, but I find it interesting nonetheless." you are referring to the homogeneous capital of the model. But then why not take the same open minded approach to other models that commit the same sin?

YouNotSneaky! said...

Quick question, what parameter values did you use for Figure 1?

YouNotSneaky! said...

I was right that you can do this in a neoclassical model (I actually got excited for a second thinking it hadn't been done before):

In fact it's pretty interesting as that paper considers both the possibility of substitution between capital and labor as well as increasing returns to scale.

Basically what you get(from my own playing around with it):
With CRS
- with Leontief you get the limit cycle as in your post
- with substitution (even just a little bit of it) you get fluctuations but it gets to the steady state rather than orbiting around it. With Cobb-Douglas you get constant shares of course but v fluctuates as does K/L (which is assumed constant in your post)
- with a lot of substitution you get wild swings in employment even as capital keeps growing (as in AK model) simply because it's either all or nothing; wages are either crazy high with zero employment or they're really low with full (slavery) employment. Since in this case K - and adjustments in it - doesn't affect labor productivity v never settles down.

With IRS (I haven't played with this version as much - it's from the paper) you get more more "destabilizing effects" which here I take it to mean you're more likely to get the limit cycle rather than the darned thing making its way to where some two curves cross (either v and u, or v and k). But apparently the effect of allowing some substitutability dominates the IRS of it - you need only a bit of substitution to make up for a lot of IRS to get to where the two curves cross.

Anyway, it's pretty interesting. But it's pretty much a neoclassical model. Homogeneous capital and all.

Robert Vienneau said...

The literature on this and related models is apparently immense and growing.

I drew the curves for α = 0.05, β = 0.1, δ = -0.1 (that's bad), γ = 0.95, ρ = 1, σ = 0.2, and s = 0.25. I haven't done much in the way of numeric experimentation.

Think of that limit point in the special case where the rate of growth of productivity is zero. Then growth will be smooth. Now introduce productivity growth, keeping the rate of growth of output unchanged. Less labor will be needed over time to make the same stuff. Unemployment will rise and wages will fall. But if the rate of growth of output were increased by the rate of growth of productivity, the level of employment would remain unchanged. "Substitution" has nothing to do with it.

An objection to my formulation of the model is that savings drives investment. In a sense, the cyclical growth curve in the model is a warranted growth curve. Sordi (2008) addresses this objection, and introduces an investment function. That generalization has a varying capital-output ratio and relates the model to the Kaldor-Pasinetti-Robinson theory of income distribution.

One would like to tell tales that generalize, at least qualitatively, to multi-sector models. One cannot do that with neoclassical aggregate models in which the interest rate equaling the marginal product of capital is a central mechanism.

I resist the urge to describe continuously differentiable production functions and discrete-coefficient production functions as "neoclassical" and "non-neoclassical", respectively.

YouNotSneaky! said...

""Substitution" has nothing to do with it."

Right, which is exactly what I said. But still the fact that higher labor productivity can lead to a situation where unemployment goes 100% AND wages grow to zero definitely falls in the "bug not a feature of the model" category. Step back away from the model and ask if this makes sense. And why or why not? As workers become more productive not only are they gotten rid of but they get lower wages as well.

The problem arises simply because v is the main "jump" variable in the model so it always has to absorb any kind of exogenous shocks that occur. If these shocks occur "often enough" - continuously, at a high enough rate alpha, v is always absorbing those shocks (u also falls simply because, with fixed wages it's = constant*a). With a growing fast enough all the other adjustment dynamics don't matter and the whole thing has to go to (0,0).

In particular, what does it is the assumption that at all points in time q=al and q=k/o (hence k/l=ao). The model, unlike the traditional Harrod-Domar version, does not allow for capital under utilization. So v's is forced to go to zero to make sure those two equalities hold. It's not a good assumption - but I think it's pretty much how you got to close this model with the fixed coefficients technology.

Or think about it this way. Labor productivity can be thought of as including a certain level of "effort". It's not to much of a stretch to assume that workers can coordinate on their level of effort, since they can coordinate on the level of wages (as embodied in the Phillips curve, presumably embodying some kind of bargaining process and a wage/employment trade-off). But in this case, workers would have an incentive to reduce their effort - since lower a means higher v and u - so as to get to (1,1). That's crazy enough. But obviously that's not a steady state, since we got a limit cycle here. So even with the lower level of effort (a) the economy would start moving towards a lower v and u. But then the workers would have an incentive to lower their effort even further. In the end workers' effort is zero (well, epsilon) and they get (almost) full employment and (almost) full share of output. All this is without any kind of disembodied technological progress.
The case with such progress works in exactly opposite way (so even if you don't buy the 'effort' story, it's a useful way to think through how the model operates) except to the benefit of capitalists (workers, in the limit, infinitive productive (as opposed to zero effort)) but none of them (well, epsilon) employed and getting none of the output.

There's also other interesting parts to this;

Sticking with the Leontief function you can get stability (i.e. it circles inward) with a non-linear saving function.

Ignoring the weird effects noted above, it starts to matter WHEN technological progress occurs - if it's in hi-v, hi-u times, it's stabilizing (smaller cycles, at least for awhile), if it's during lo-v, lo-u times it's destabilizing and in other cases it destabilizes one but not the other. And some other stuff...