1.0 IntroductionThis 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 nonlinear 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 TechnologyAssume 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 capitaloutput 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 = a_{0} exp( α t )
The labor force also grows at a constant rate:
n = n_{0} 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, InvestmentLet
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 ModelThe 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 capitaloutput 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
Hence,
(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
LotkaVolterra predatorprey model. It is a canonical nonlinear dynamical system used to model, say, lynx and rabbits.
5.0 Solution of the ModelFor what its worth, a trajectory in phase space has the equation:
(u^{ν1}) exp( θ_{1} u) = H(v^{ ν2}) exp(θ_{2} v)
where
θ_{1} = s/σ
θ_{2} = ρ
ν_{1} = s/σ  (α + β + δ)
ν_{2} = (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* = ν_{1}/θ_{1}
v* = ν_{2}/θ_{2} = (α + γ)/ρ
(The origin in phase space is also a limit point; the origin has the stability of a saddlepoint.)

Figure 1: Phase Space 

Figure 2: A Trajectory 
6.0 DiscussionI 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.
ReferencesThere 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.