The model has just four equations that describe the evolution of the populations of Elites and Commoners, Nature, and accumulated Wealth. Mechanisms leading to collapse are discussed and the measure "Carrying Capacity" is developed and defined, The model suggests that the estimation of Carrying Capacity is a practical means for early detection of a collapse. Collapse can be avoided, and population can reach a steady state at the maximum carrying capacity, if the rate of depletion of nature is reduced to a sustainable level, and if resources are distributed equally.The problem, dear reader, is that the model builds on the simple Lotke-Volterra predator-prey equations. I couldn't copy and paste them from the .pdf of the paper, so, with some abuse of notation, here goes. Let W be the population of Workers, B the population of Bosses, X the stock of natural resources, and Y the accumulated wealth.
The four equations of motion of the model are
(1) dW/dt = (b - a)W, where b is the birth rate and a the death rate;
(2) dB/dt = (c - d)B, where c is the birth rate and d the death rate;
(3) dX/dt = gX(L - X) - uWX, where L is the exogenous carrying capacity of the environment, g a regeneration factor, u is the constant marginal product of workers, whose productive activities deplete Nature;
(4) dY/dt = uWX - S - A, where S is the consumption of Workers, assumed to be at a subsistence level, and A the consumption of Bosses, who extract surplus value.
The dynamics of this relatively simple model are not radically improved on those of Hamurabi, an ancient computer game I probably wasted too much time on years ago, and which you can play online at your leisure.
HAMURABI: I BEG TO REPORT TO YOU,The simulator gives you some information, and you try to keep your subjects alive for ten years.
IN YEAR 1, 0 PEOPLE STARVED, 5 CAME TO THE CITY,
POPULATION IS NOW 100
THE CITY NOW OWNS 1000 ACRES.
YOU HARVESTED 3 BUSHELS PER ACRE.
RATS ATE 200 BUSHELS.
In the working paper, the structure of production has a Marxian feel to it, with the Bosses appropriating some of the production for themselves without depleting Nature, and the Workers getting only subsistence pay. Not surprisingly, a gripe session about the modelling, and the folly of using NASA money to fund the research, ensues at Insta Pundit.
But there are opportunities for further research, some of which have been anticipated by the authors (See Motesharrei, Rivas, and Kalnay p. 7.)
[The initial model] models the Depletion side of the equation as if it includes the reduction in Nature due to Pollution. Future versions will differentiate Depletion from Pollution. The depletion term includes a rate of depletion per worker, [u], and is proportional to both Nature and the number of workers. However, the economic activity of [Bosses] is modeled to represent executive, management, and supervisory functions, but not engagement in the direct extraction of resources, which is done by [Workers]. Thus, only [Workers] produce.The paper notes that the source code for the model is available online at http://www.atmos.umd.edu/~ekalnay/handy-ver1.mdi, on the Maryland site of Professor Kalnay. I received a 404 error checking that link earlier this evening. My hope is that we're dealing with a temporary server hiccup, perhaps account high casual interest coming in from the public intellectuals.
Technological change can raise the efficiency of resource use, but it also tends to raise both per capita resource consumption and the scale of resource of [c.q.] extraction, such that, absent policy effects, the increases in consumption often compensate for the increased efficiency of resource use. ... The extent of these effects varies, but in this initial model, we assume that the effects of these trends tend to cancel each other out. In future versions, the rates of these trends could be adjusted in either direction.
One of my early interests in economics was playing around with simple Keynesian Cross models, with the help of a Commodore calculator and pencil and paper. There are a number of tweaks that occur to me in the simple model, including making L endogenous, having improvements to u arrive randomly, having depletion or pollution surprises arrive randomly (the Newcomen engine improves u, depletion of England's coal begins immediately, and carbon concentrations in the atmosphere increase). As a further complication, I want to have u affect the compensation of both Bosses and Workers. Perhaps u attaches to a Worker who then contracts with other Workers to implement the idea, raising the living standard of what becomes a Boss and his spawn, as well as the living standards of some Workers.
Yes, that begins to get messy, but in my second childhood, I hope to have some thinking time.