Speed of Convergence in the Malthusian World

"Hola amigos. I know it's been a long time since I rapped at ya."

Alright, alright, might try getting back into this blogging thing despite lots of real life work, but let me start with something easy (yeah right); speed of convergence in the Malthusian world.

There's a review of Greg Clark's 'Farewell to Alms' in the new issue of Journal of Economic Literature by Robert Allen. Allen lists the the six claims made by FtA and argues that the data does not support any of the six claims:

1) the preindustrial world was in a Malthusian preventive check equilibrium
2) living standards were unchanging and above subsistence for the last 100,000 years
3) bad institutions were not the cause of economic backwardness
4) successful economic growth was due to the spread of "middle class" values from the elite to the rest of society for "biological" reasons
5) workers were the big gainers in the British Industrial Revolution
6) the absence of middle class values, for biological reasons, explains why most of the world is poor


I'm going to focus on the first two here since this is where, contra Allen, I think the data is actually the strongest in support of the claims (I also think claims 3 and 5 don't do too bad while evidence for the other two is at the moment at best circumstantial).

In fact the first portion of Allen's critical review reads like he's either actually providing evidence for the thesis of a Malthusian equilibrium and stagnation (g.e. standards of living after invention of agriculture were lower than before it) or nit picking details and statements not essential to the main thesis (in the book Clark says that post-agricultural incomes were "about the same" whereas Allen insists they were lower. He also mentions fertility was higher. But that's very much inline with the Malthusian model. Or Allen stresses frequently that even if there was some relationship between fertility and income other, social and historical factors had much more influence on the birth rate. But in fact, a fertility which is completely independent of income works even better in the Malthusian model (and that's what I'll have below) and Clark repeatedly stresses the various social custom, such as postponing marriage which were the main determinants of fertility in the Malthusian world).

But a frequent criticism of the Malthusian model of world history, which also shows up in Allen, that at various points in time income did seem to be significantly higher than at others. In other words that the claim #2 above is wrong. But the way the claim is presented is a straw man. FtA does not argue that there was never any change in incomes - in fact both the evidence on English laborers' wages (which doubled between roughly 1300 and 1450) and the historical comparison between wages in the ancient world in wheat pounds (Ancient Babylonia had about 2/3 the wages of Classical Athens, two times the wages of Roman Egypt and only slightly higher wages than 18th century England) shows exactly that kind of variation. The book simply argues that there is no long run trend, from 100,000 BC until the Industrial Revolution. Sometimes up, sometimes, down, but at the end of the day (or the millennia) all technological progress simply shows up in higher populations, not higher living standards.

Still, how does one reconcile the idea of "Malthusian stagnation" with the observation that incomes were significantly (though not by modern standards) higher in some places and at some times? One recourse would be to rely on differences in exogenous fertility and mortality rates which are not related to income to explain it. But as the book argues, and I think Allen would mostly agree, pretty much all pre industrial societies limited fertility in some way through social custom (be it getting folks to marry later, space the births within marriage more or simple infanticide). Basically, there just isn't enough variance in fertility rates (more specifically, birth rates) between regions to account for the variation in living standards. The other possibility is mortality rates but here the same problem arises. Could it really be true that Classical Athens had really high standards of living (unprecedented for the pre industrial world) simply because it had a ... really really high mortality rate?

Once you throw out the "Malthusian" explanations of fertility and mortality rates, how do you explain the dispersion of income in the pre industrial world? Well, there's technology, or land. But in the simple Malthusian story those factors are not supposed to matter - they get overwhelmed by the population pressures. To misquote Malthus, population grows geometrically while technology (and land) grow arithmetically. In fact, for example Kremer in a pretty famous paper (better link out there somewhere)assumes that the adjustment to the Malthusian equilibrium is instantaneous which is pretty essential for his empirical analysis.

So the question is, in the absence of large enough differences across regions and time in fertility and (exogenous) mortality, how does one explain persistent fairly large scale (by pre industrial standards) deviations from the Malthusian equilibrium? In other words, how can the world from 100,000 BC and the Industrial Revolution be both Malthusian and at the same time not be Malthusian at many points in time.

The answer to this lies in how fast the pre industrial economies adjust to their Malthusian steady states. Assuming roughly equal fertility and (exogenous) mortality rates for all pre Industrial Revoultion regions one can still obtain a dispersion of incomes at any point in time simply by assuming that the Malthusian adjustment take a long time. But in order for this explanation to work, we have to establish that in fact, these Malthusian pressures are in fact pretty slow.

The rest of this post tries to argue exactly that: the pre Industrial World was Malthusians but the Malthusian mechanisms took a long time to work. Over the millenniums, Malthusian pressures always dragged living standards down to those determined solely by demographics (fertility and mortality) and not by the availability of technology or land. But at the frequency of decades or even centuries technology (and land) could still play a very significant role in determining living standards.

Or in other words, both Clark and some of the more traditional historians who emphasize the riches of Ancient Rome or Classical Athens or 14th century China can be correct.

Ok, here's how the argument is going to work. First let's take a simple model of the Malthusian world in which there is SOME technological progress (or institutional improvements or land acquisition). From that we derive a simple condition which tells us whether or not we have Malthusian stagnation or not. Given that productivity growth in the pre industrial era was fairly low this condition is not of interest of itself, rather, it highlights an important phenomenon:

In the Malthusian world with some technological progress there is always a race between technology and demographic pressures. But this means we need to know the speed of demographic pressures. And this is the rate of convergence to the Malthusian stagnation. If the economy adjusts very quickly to the Malthusian equilibrium, then any kind of technological progress is going to get eaten up by more people rather than higher standards of living. If the economy adjusts slowly then technological progress can come into its own as an important (although not a consistent) determinant of living standards. Somewhere in between, we got somewhere in between.

Here's a Malthusian model with technological progress. Output per capita (living standards) is produced with technology and labor, there are diminishing returns to labor, and land is constant (which here means it's included in A):







while population grows according to:







(the mortality rate function m/y is not perfect as I've mentioned in my previous posts on the Malthusian model but for all relevant purposes it'll do here).

Let's suppose that productivity grows at a constant rate g or







Taking logs and time derivatives of the equation for living standards y we get the growth rate of per capita income:







If there is some level of income at which population growth is zero then it is:







A bunch of simple math can verify that in fact if the growth rate of productivity, g, is less than alpha*f, then the economy will converge to this level of per capita income. In that sense, the Malthusian model WITH technological growth is QUALITATIVELY NO different than the one WITHOUT technological growth in terms of its main predictions; long run stagnating incomes (quantitatively economies with higher g will have a higher LEVEL of income). But if somehow the rate of productivity growth is very high (relative to the birth rate and alpha) then output will grow without bound and we are no longer in the Malthusian trap. As it turns out for all intents and purposes growth rate of productivity (though not zero and even non trivial) in the Malthusian world was much lower than birth rate * alpha.

This means that the world before the Industrial Revolution really WAS in a Malthusian trap. But another question is how *tight* was that trap? How long did it take for a Malthusian economy to adjust to its Malthusian level, if it deviated from it?

To give away the punch line, the Malthusian economy took some time to adjust and so deviations from some kind of fertility/mortality determined subsistence living standard could persist for quite some time.

How can we get this? Well, one way to ask the question is to think about the HALF LIFE of a income deviation of a Malthusian economy: IF a Malthusian economy deviated from its long run equilibrium how long did it take to close HALF the gap between its current level and its ultimate, equilibrium, level?

Mathematically, if a variable X(t) grows at a constant rate b, then its half life is given by:







where t_HL is the number of years it takes to half the % distance to steady state and b is the (negative) growth rate. This is essentially the famous "Rule of 70" (since ln(2)=.69)) in reverse (shrinking in half rather than doubling).

So what we want is to write the growth rate of the living standards as a constant fraction of the % gap from its long run value:







The trouble is that in fact







which is a nonlinear function of the % gap. I.e., the convergence rate parameter which determines the half life "b" is NOT constant. BUT, following standard procedures (like those used to estimate the parallel parameter in the Solow model) we can approximate "b" by log-linearizing the actual g_y around its steady state to obtain a "b" that's gonna give us something pretty close to the half life.

Here's a graph of growth of y as a function of y:



Here's a linear approximation of growth of y by a tangent function around y*:



So, close to the steady state, the rate of convergence to the Malthusian eqiulibrium can be found by the slope of the tangent line. We get this from a Talor Expansion approximation:







(the "=" sign above is a "approximately equal" sign actually).

Now g_y evaluated at y* is obviously zero (i.e. in Malthusian equilibrium there is no growth in living standards). And we have that







So our approximation is:







I'm gonna leave it up to you to do all the basic calculus and algebra, but what you should get is:







Or, the "b" that we need for half life is exactly equal to alpha*f:







and in fact that is the rate of convergence to the Malthusian equilibrium.

This shouldn't be surprising. It's the reason why I considered the 'Malthusian model with technological progress' above. Again - in the Malthusian world there's a race between the rate of technological progress and the speed of Malthusian pressures. Above, we've assumed that technology changes at the rate g. Now, we've calculated the speed of the Malthusian pressures as alpha*f. And our condition for the world to be still Malthusian even in the presence of technological progress was







which is just another (fancy math) way of saying that Malthusian demographic pressures are faster than the growth of technology. (Generally economic "math talk" corresponds very well to intuitive verbal concepts with much less ambiguity).

Ok, but what does this mean? Well, we have that b=alpha*f and b give us the half life as t(HL)=.69/(alpha*f). If we got alpha and f then we can calculate the half life of a Malthusian economy and say something about how fast the adjustment takes place.

Alpha.
At the most basic level alpha measures the rate at which diminishing returns to land set in. If you got a competitive market in land (or by Walras' law) in labor) then alpha will equal the share of land in income - it will be the portion of total output that landlords appropriate. Now, of course, very few pre industrial (and a good number of ones today) economies had competitive markets in either land or labor. But some did have something approximating it (for example post serfdom England) and there's also other ways to estimate it. Without quoting a bunch of literature a not-unreasonable estimate for alpha is 1/4. It could be as low as 1/5 and as high as 1/3, depending on the institutional and land specific factors. But for the sake of a general description we'll take 1/4 here.

f
Strictly speaking, f, is not the fertility rate (number of births and average woman has over her lifetime) but the crude birth rate (number of births as a fraction of population). Fertility rate is actually easier to obtain and given the age structure of population and other demographic factors it's not actually straight forward to get a crude birth rate from a fertility rate. But what we're interested in here is an approximation. And one way to get it is to note that in a steady population, life expectancy equals the reciprocal of the birth rate (I'm leaving out why this is so). So we can get a rough birth rate by taking 1/Life Expectancy for a pre industrial economy. For pre industrial England life expectancy at birth was about 37 years and this seems to be roughly the mean/median for a lot of pre industrial economies (as mentioned above, there isn't THAT much variance in the fertility/birth rate). This would imply a crude birth rate of f=.027.

Combining the two we have the half life:







In other words:
If there was a shock to incomes in the Malthusian world, it would take about 103 years (or roughly 3 generations) for about half the effect of it to disappear. After about 6 generations, a quarter of the initial shock would still be present.

If these shocks occurred often enough (but not often enough to generate a long term trend - i.e. the g was still less than alpha*f) then in fact you can get a centuries long deviation from what should be a Malthusian equilibrium observed in historical data.

So the at the frequency of millenniums (or even several hundreds of years) the world was Malthusian. But on the order of a couple of generations or a century, innovations (broadly understood) mattered. We're reconciled the two seemingly contradictory views of history.

Some extrapolations.
Historians, by the nature of their subject, tend to focus on dramatic events (and that's what makes history interesting. It's also what makes the History of England the most boring kind of history one can study.) In other words they tend to focus on time periods when the pre industrial economies, almost by assumption, were deviating from their long run state, when they were getting shocked. For example, much has been written on the effect of Black Death in the 14th century (killing 1/3 or more of Europe's population) but very few make links between incomes in 16th century and the original (i.e. ignoring the subsequent "Little Black Deaths") shock - but this analysis suggests hundreds of years later it still mattered. For a historian what's interesting is wars, plagues, revolutions; times when history was deviating from its trend. The long, but ever present times in between where history took its inevitable march back to where it began are boring, but they are there.In that way, Clark's analysis reminds us to pay attention to long run effects and long run phenomenon - like the lack of trend in long run living standards between the emergence of modern man and the Industrial Revolution.

At the same time, acknowledging that the adjustments took place over multiple generations also allows us to reconcile two conflicting views of human history - that of stagnation and that of occasional periods of prosperity.

More importantly perhaps it makes economic sense. After all, if you're living in a Malthusian world, in which living standards are always determined by the fertility and (exogenous - i.e. socially determined) mortality what incentive is there to ever adopt new more productive technologies? You get the new technology, you have more children, diminishing returns mean that they will be just as poor as you.

So let's posit a form of "Malthusian Rational Expectations" - the people in the Malthusian world KNEW that they were living in a Malthusian World in which ultimately, over the long run, standards of living were determined by fertility and mortality. What point would they have in working harder, in adopting new technologies, in striving for better institutions?

They wouldn't, unless the transition period after the adoption was long enough.

So think about it this way. Say, you're a hunter gather that just stumbled upon this new technology called "agriculture" which will give you food security, increase the amount of food you can produce and generally increase your "standard of living". The only catch is you got to work a bit harder for it. Now, if you know the world is Malthusian, you know that these gains that this new technology "agriculture" promises are only going to be temporary.

But how temporary?

Higher standards of living means everybody will have more surviving children which means more people, which with diminishing returns ever present, means that everyone including your offspring will wind up right back where they started perhaps working harder for it.

But humans did switch to agriculture and people did continue to adopt advanced technologies, despite the fact that these produced no long run trend in living standards. Assuming some minimal cost to technology adoption, why did they do this?

If the demographic adjustment took place quickly there would have been no reason to do so (and in fact, in some societies and at many points in history people did resist new technologies even where these productivity enhancing - why bother when uncle Malthus will undo it anyway?). But if people's planning horizons stretched only to the next ... three?, four? generations (come on, how far is YOUR planning horizon?) then the above analysis implies that people adopted new technologies simply because the Malthusian equilibrium was too far away to matter for them.

Some quick afterthoughs:
(there's a testable prediction here - economies with higher land share or higher birth rates would adopt new technologies at a lower rate since in that case the half life would be lower and so any advantages of new technologies would be more short lived, so why bother?)

(there's some model of optimal technology adoption in the Malthusian world with a given birth rate and death rate here . Given how fast the adjustment takes place and how much extra work effort new technology requires whether or not new technology is adopted is going to depend on how much present generations value the well being of future generations (the income of their children). For a given rate of time preference the slower the adjustment the longer it takes for Malthus to disintegrate the benefits, the longer will the benefits of a better technology last the more incentive is there to adopt it.
Now, the interesting part is that this intersects with Clark's story about the spread of 'middle class values' - roughly speaking here, people becoming more patient over time. This means that over time the rate of time preference could go up which given the length of the adjustment process and all that other stuff, could make it more likely for folks to adopt a new technology. Basically more patient people would be willing to accept a faster adjustment time. Chicken and egg type of thing)

(Thinking about all this stuff it seems like there's a WEALTH of analysis yet to be done on the political economy of these questions. The one parameter which keeps popping up in all kinds of questions about the Malthusian economy is ALPHA - which, roughly is the share of land in income. But surely, in a feudal world this share is subject to all kinds of political, institutional and historical pressures. I readily admit that estimating it by land's share when markets are competitive is a bit of a cop out. But then we need to have some kind of a political model of how income shares get determined, which go beyond the standard Classical Ricardian analysis of land rents. I think the starting point for thinking about it is the very excellent article by Evsey Domar: ""The causes of slavery or serfdom: a hypothesis."" which Paul Krugman himself tried to get people interested in way back when (as far as I can tell, the response was "Oh, that's interesting but slightly weird, let's think about something else")