Some Macroeconomics for the 22nd Century

I was recently going through some old papers and came across a copy of Robert E. Lucas, Jr.’s “Some Macroeconomics for the 21st Century” (published in the Journal of Economic Perspectives, Winter 2000). Dr. Lucas, who won the Nobel Prize in Economics in 1995, proposed a model to help explain and project global growth rates and the level of inequality between nations.

Lucas’s model is quite simple: Imagine a collection of countries (or regions) all of which have a per-capita income of about $600. In the year 1800, one of those countries (I call that one the ‘alpha economy’) begins growing at a rate of 2% per year. Each year after that, any country that is not growing has some chance of starting to. A country grows at the rate of 2% per year compounded by an amount that depends on the difference between the alpha economy’s income and its own income.

Mathematically, this is written:

That is, the growth rate (GR) of economy n is 2% (the 1.02 term) multiplied by a factor that depends on the income (I) of the alpha economy divided by n’s Income to the beta power. Beta is a parameter which Lucas sets to 0.025. The income of alpha will always be greater than the income of economy n, so the term in parentheses will always be greater than 1.

So, the alpha economy will always grow at 2% per year, and all economies (once they start growing), will tend to catch up with the alpha economy. The only remaining piece is how to determine the chance that an economy will start growing.

If we simply say that, each year, x% of remaining economies start growing, then far too many economies start growing in year 1800 (when there is the largest number of non-growing economies). So, Lucas has the probability an economy will start growing go from a 0.1% to a 3% chance per year, depending on difference between economy’s income and world average income. (I won’t give the exact equation because it’s not important. But the resulting dynamic is an S-shaped curve of the number of economies that are growing.)

Here’s what the per-capita incomes of some selected economies look like in this model:

Essentially, Lucas’s model is not really one of economic growth, but rather more of the diffusion of innovations or spread of economic growth between nations. In his model, there are two ways economies influence each other: economically advancing (1) increases the chance non-growing economies will start growing, and (2) increases the rate that other economies grow.

One nice feature of this model is that, for a relatively simple model, it demonstrates an S-shaped uptake in the number of growing economies (Personally, I would have used a different approach, but Dr. Lucas won the Nobel Prize, so for this discussion, I’ll stick with his approach). Also, it generates a peak in the world economy’s annual growth rate in the post-War period and, interestingly, suggests that in the long future all the world’s economies should tend to converge into perpetually rising prosperity.

Inequality shown here is simply the standard deviation in the incomes of individual countries. (In Dr. Lucas’s article, he uses the log standard deviation as the measure of inequality, but the general concept is the same.)

With Dr. Lucas’s choice of parameters, the behaviors are even quantitatively reasonable; the US and Europe have been averaging about 2-3% growth over the past decades. In the long run, the global growth rate asymptotically approaches 2% as all the world’s economies catch up to the alpha economy.

Of course, this model leaves out lots of stuff too: no Kondratieff wave, no population growth, no demographic transition. I don’t really care about those things so much. It’s a model and every model leaves out far more than what it includes – If they didn’t, it would be impossible to understand.

But there are a couple of points I find disturbing:

  • One is that Lucas’s model has world income perpetually rising, exponentially. There’s nothing wrong with a model of this sort when used over very short timeframes, but Dr. Lucas is considering centuries here. I’m no Malthusian, but if this model is correct, economic activity will be about 10 times larger in the year 2100 than in 2010 and 10 times higher in 2200 than in 2100. That’s a growth in income of about 100-fold. Is that even possible?
  • The second is the implication that inequality will inexorably drop (a dynamic Dr. Lucas calls “conditional convergence”).

If a 100-fold increase in economic activity is possible, even with a 10-fold reduction in energy intensity, there will still be a 10-fold increase in energy demand.

For the sake of argument, let’s assume that there’s some maximal possible level of per-capita economic activity and, further, that any increase economic activity that occurs only makes further increases more difficult. (The rationale is the same one behind the concept of ‘diminishing returns‘ – since the easiest gains are generally realized first, each additional gain becomes more difficult.) I refer to this as ‘The Limits to Growth‘ scenario.

Mathematically, my only change to Dr. Lucas’ model is to make the growth equation be:

where ‘w‘ is the fraction of the way that the Gross World Product (G) has advanced from its starting level (Ginit) to a maximum amount (Gmax):

For this calculation, I’ve set Gmax = 100 x Ginit. I’ve also extended the timeframe out to the year 2200, to better see the long-term behavior.

The results here have similarities to Lucas’s case. For example, the alpha economy still generates a per capita income of about $30,000 in the year 2010. However, the long future is changed dramatically. In the long run, the growth rate of all the world’s economies slows to zero. Disturbingly, their standard deviation (the measure I’m using for ‘Inequality’) tends toward a non-zero value. That is, a steady-state discrepancy sets in.

So, in Dr. Lucas’s model, in the long run the global growth rate goes to 2% and the inequality between economies goes to zero. In my modification, the growth rate goes to 0% and the inequality stabilizes at a non-zero amount. Completely opposite results from just one minor modification.

I’m not capable of determining which scenario laid out above – Dr. Lucas’s case of perpetual growth, or my ‘Limits to Growth’ scenario – is correct. The big concern this investigation raises for me is: If adding a fundamental economic concept like ‘diminishing returns’ to a simple model like this one can completely reverse the main conclusion (from “conditional convergence” to “steady-state discrepancy“), how does a macroeconomic modeler know when to stop?


If you liked this post, you might also like my post ‘The Long Futures‘, about how physical quantities (and abstract quantities) of all kinds can grow over long periods of time.


  1. 1 Quora

    Where can I find a good review/critique of Robert Lucas’ 2000 paper “Some Macroeconomics for the 21st Century”?…

    Never got an answer to my question, so I wrote an answer myself and posted it here:

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