A quick, uninformed, probably wrong, and completely unoriginal resolution to the "Brenner Paradox"

In comments Jonathan points to a problem with the Domar explanation of reemergence of serfdom in Eastern Europe in the 16th/17th century. Because of real life responsibilities I've just gotten around to reading Jonathan's paper (From the part I've gotten through, it's highly recommended.) and it's gonna be a few days before Brenner's book arrives in the mail.

But steppin' back, asking' "what's the difference?", thinkin' "what's the simplest explanation?" and comin' up with something on the spur of the moment here's my answer to what I guess we can call (if we roughly buy into Domar's analysis) the "Brenner Paradox" (yes. I am aware of the fact that the preceding sentence has a lot of "s and 's which makes for a whole host of vertical lines). Note also that Krugman actually discusses this particular problem with Domar's explanation in his essay.

Ok, so what I'm calling the Brenner Paradox is the seemingly opposite effect that changes in the land/labor ratio had on the institutional development of Eastern and Western Europe.

According to Domar, an increase in the land/labor ratio raises the marginal product of labor in a competitive market, which means that, if competitive markets are the rules of the game, then labor wins and land(owner) loses. In turn, starting from a position of roughly competitive markets, or at least labor mobility - the ability of peasants to transfer between their employment between various landowners - what this means is that the incentive that landowners have to change the rules of the game, enact an institutional change and introduce some kind of a bonded-labor arrangement, increases.

16th century Polish-Lithuanian Commonwealth and late 16th/early 17th century Muscovy (Russia) both experienced a rise in the land/labor ratio, the first one through the opening up of the Ukrainian 'frontier' (and in fact, consistently with Domar's story, the Kresy region is where serfdom first began to reappear in PLC) and the second through eastward expansion (the beginning of what eventually turned into the exploration and settlement of Siberia) and a bit later also, through acquiring lands in the Ukraine. So the reappearance of serfdom in Eastern Europe is consistent with Domar's story.

The problem is that Western Europe also experienced an increase in the land/labor ratio, though a bit earlier, say 14th century, as a result of the Black Death. More land, less labor, due to the wiping out of roughly the third of (Western) Europe's population. But the trouble is that the same mechanism postulated by Domar - a rise in the land/labor ratio - has been used (by North and Thomas) to also explain the END of serfdom in Western Europe during this time.

Peoples need to make up their minds. Or listen to what I'm about to say. (Err..., see the title of the post first)

So how can we reconcile the two views while keeping the basic structure of Domar's explanation intact and without relying too much on non-robust just-so stories about exogenous institutional or cultural changes (at the moment, linking it up with the rise of Absolutist states in the West seems to be a bit ad hocish - but like I said, the book hasn't arrived yet)?

Well, I think the easiest way is to note that landowners' incomes in the pre-industrial world depend not just on the land/labor ratios but also on the relative number of landlords and peasants. Here's how (I'll lay out the 'model' first then discuss if some of the other historical facts match up with the explanation).

Suppose we have an economy with just two classes, laborers and landowners. Landowners don't work, they own the land and live off of the rents they receive. Laborers don't own land, they supply their labor in exchange for some kind of compensation from the landlord. The nature of this compensation depends on the institutional set up of the economy. Very very roughly speaking the institutional arrangement as how farm product gets split up can take three forms:
' a competitive labor (and land) market (I think one's gonna imply the other by Walras' Law in a general equilibrium set up. It also seems like Jonathan's paper relaxes or gets around this but I haven't gotten there yet). In that case workers get their marginal product and landowners get the marginal product of land without doing anything (lesson is: you can get what you're worth and still be exploited).
' a rough sharecropping (more precisely a fixed crop rent) arrangement where the product of the land gets split between the landowner and the workers according to some kind of institutionally-constrained contract agreed to under a system where there is NO labor mobility. Here basically a landowner takes a constant fraction of output and the rest goes to workers. Now, this would actually also be true with constant returns to scale and, say, a Cobb-Douglas production function, so what we assume here is that this share is greater than the share that the landowner would get under a competitive market
' a system where the workers get a fixed subsistence wage and landowners get everything else. The difference between this arrangement and the previous one is that if total output goes up, under the previous system peasants' incomes will still rise whereas here, any kind of increase is fully appropriated by the landlords.

In the back of my head I have a Malthusian view of these and if you play around with some fertility and mortality dynamics it turns out that there's some theoretical problems with analyzing the third kind of system (basically, it's either unstable, or ends up looking like the second one) and I also think it's empirically unsupported; there was a quite a bit of fluctuation in workers' wages (by pre Industrial world standards) which is inconsistent with a fixed wage rate and, as Clark notes, most societies seemed to have average incomes (which means peasants' incomes) quite above that required for bare subsistence.

So I'm going to focus on the difference between the institutional arrangement 1) and 2). In other words, I'm going to assume that under a bonded labor system landlords (in addition to getting all that stuff for free simply by virtue of owning land) can extract a portion of total output which is greater than that which they would obtain with competitive labor markets. But of course, if this was all there is to it then landlords would always prefer serfdom and then we'd have to recourse to those exogenous cultural and institutional factors that we want to avoid in order to provide an explanation. So what I'm going to assume is that there is some fixed cost to the landlords, per unit of time (though it could also be like a present discounted value cost of a one time institutional change), of keeping the peasants in bondage under serfdom. Now, this cost could be small and in fact it probably was. Jonathan quotes Moses Finley and it's worth repeating:

“[I]n the context of universal history, free labour, wage labour, is the
peculiar institution (historian Moses Finley, 1976).”

But if we were to assume otherwise we'd get a model where it's all serfdom all the time and part of what we want to explain is why in fact free labour, wage labour, emerged as an alternative to serfdom in Western Europe and why it disappeared in the East. So there's a cost to keepin' the people down. Furthermore, 'c' could represent all the monitoring cost associated with bad effort incentives under serfdom as well as the necessary expenditures required to enforce a more coercive legal environment.

I'm also going to assume that all landowners are identical to each other and the same for peasants. This is because I wish to explain things WITHOUT having to make the explanation depend on some particular distribution of land. In essence what this means is that there's constant returns to scale in land and labor (Jonathan's paper departs from this)- so we can treat a 'large' landowner as several 'small' landowners for purposes of analysis and we don't specify any kind of power dynamics between the big landowners and the small. So if T is the total amount of land in the economy and N is the number of landlords, T/N is the land per landowner. Also because landowners are identical, each one's going to hire/bondage L/N peasants where L is the total population of laborers.

Output of an individual farm is (and we will focus just on landowners' income here, in keeping with the spirit of Domar's paper):



With a competitive market the landlords have to pay the workers they hire a market wage, w, so their income is given by:



They maximize this income with respect to the number of workers they hire given the wage rate. This means that in equilibrium w=MPL, and solving for that, and plugging it back into landlords' incomes we have that it is:



Or in other words, in a competitive labor market landowners get share alpha of the output their farm (and workers) produces. Alpha, though a purely technical parameter, ends up being land's share in aggregate income. Plugging in T/N and L/N for T(i) and L(i) in the above equation we get each landlords' income as a function of TWO variables; the land labor ratio t=T/L and the landlord/peasant ratio phi=L/N:



Ok. Now, under a bonded labor system - serfdom (and ignoring some very important nuances) - we assume that the landlord just takes a share of total output, theta:




So it looks much like what happens with competitive markets but we assume that theta>alpha which captures the fact that laborers are screwed because they cannot move between farms. The difference in (gross) incomes for landlords between serfdom and the competitive is then just:



But, as mentioned above, let's assume there's also a (possibly small) fixed cost 'c' to the landlords associated with making sure the peasants stay enserfed. Hence the net difference between landlords' income is actually:



To figure out which one they prefer, or in other words, the size of the landlords' incentive to adopt one system over the other, we first look at the situation in which they are indifferent. We set the above equation equal to zero and solve for t (land/labor ratio) as a function of phi (landowner/peasant ratio):



The figure below illustrates this relationship. It's immediately obvious that if t is above the curve in the picture then, since total output is larger with a higher land/labor ratio and landlords get to appropriate more of it under serfdom, in that case they will prefer serfdom. If t is below the curve on the other hand the extra portion of output that landlords could take away from the peasants under serfdom is just not justified by the cost of keeping serfdom in place.



The key - and this is the departure from Domar that makes this reconciliation of the paradox possible - is that phi, the labor/landlord ratio, also matters. Basically, if there's a lot of landlords around, this means low land per landlord and this means low landlord income regardless of whether you're in serfdom or a competitive world. But this means that moving from competition to serfdom is not going to increase incomes much and it might just not be worth it to the landlords.

Alright, now the 'steppin' back' part alluded to in the intro. What's the basic difference in how the land/labor changed in Eastern Europe in the 16th/17th century and how it changed in Western Europe in the 14th? Well, in Eastern Europe it went up because the numerator in T/L, T, went up (the opening up of Ukraine and lands to the east of Moscow). But in Western Europe it went up mostly because the denominator in T/L, L, went down (the deaths due to the Black Plague). But if L goes down this also means that L/N, phi, goes down as well. More precisely, what we need here is that the death rate of peasants due to the Black Death was greater than the death rate of landowners. I believe that somewhere someone (perhaps on my old Malthusian posts) said that that was indeed the case (if not, then the whole explanation in this post falls on its face).

So, it is possible that in Eastern Europe the land/labor ratio went up, labor became scarce (and hence, expansive in a competitive environment) but the relative numbers of peasants and landowners stayed the same. As a result, to prevent the erosion in their incomes the Slavic (and the East German ones too) landowners managed to carry out a change in institutional structure that was the reemergence of serfdom. This is the classic Domar explanation:



At the same time, the land/labor ratio went up in Western Europe a century or two earlier but for different reasons. But the shock also profoundly changed the relative number of landowners and laborers. While the decrease in the number of laborers put downward pressure on landlords' incomes, the fact that there were now fewer workers per landowner (hence per unit of land) and that there were some enforcement costs to keeping serfdom in place made it just not worth it to stick with a system of bonded labor:



According to this, Domar was essentially, though only partly right. If you just focus on Eastern Europe, as he did, then casting everything in terms of changes in the land labor ratio made sense since that's all that was changing. But expanding the model to Western Europe makes it appear to be wrong simply because the land labor ratio was NOT the only thing that was changing. There was something else going on at the same time. Ignoring the changes in the relative number of peasants and landlords (or, workers per land per landlord) creates the paradox.

Some notes:
Like I've tried to indicate, all of this could be bunk. In particular, I'm not exactly sure that the Black Death killed proportionately more peasants than landlords (in absolute numbers it did but what matters is the rates).

Also, whether or not this explanation is feasible depends on the 'initial' levels of t and phi.

Roughly speaking I think that even before the opening up of new land in Eastern Europe it tended to be relatively more land abundant than Western Europe. This was definietly true for Poland-Lithuania (with the Polish kings encouraging immigration from abroad as much as possible, at least prior to late 17th century, since there was all this fallow land laying around) and maybe true for the lands of former Kievan Rus though there the whole Mongol Yoke throws a wrench into the spokes of an easy answer. For other parts of Eastern Europe it was the expansion of the Ottoman Empire that wields a similar wrench, though at least for a few centuries Hungary was very similar to PLC. Bohemia had that whole Hussite War thing going which pretty much trumped all the other shocks (basically those were a big upward shock in phi but that was an effect not a cause so it doesn't apply - at least not until the Hapsburgs got a good hold on it). Overall, if we accept a higher t for Eastern Europe than for Western, that fits our explanation since a transition from free labor to serfdom is more likely if t is already high.

On the other hand, a transition from free labor to serfdom is also more likely if we start with a high phi - lots of peasants, few landlords. But my understanding is that if we go by the proportion of folks with noble titles to the masses without them then Eastern Europe had a much greater share of nobility in population. In Hungary between the beginning of Ottoman invasions and/or Austrians getting in there it reached as high as 20%. Poland had somewhere a bit above 10% of everyone running around calling themselves nobles, as did Lithuania (before the Union of Lublin). Moscow might've been different due to the simple fact that the Mongols shaved off a few of those percentage points. But of course having a noble title and/or being a member of the aristocracy is not the same thing as owning land. In fact there's some historical evidence for the prevelance of the 'landless noble' class in Eastern Europe during this time - folks who had their title and their ancestry, some cloths on their back and not much else. So maybe phi wasn't as high as one would think by looking at noble/peasant ratios.

In our modern world economic factors change quickly while institutions evolve slowly and sometimes lag behind. Given the previous post on the speed of convergence, in the Malthusian world and putting it together with this Domar style analysis, suggests that the opposite was the case in the Malthusian world. Economic pressures, while ever present, moved slowly and so institutions were given plenty of time to evolve and change and determine how the world was shaped.

Anyway. This is just meant to throw out a possible explanation which would necessarily involve an unnecessarily long blog post.

Re-Elect Gaius Baltar!

The new seasons of Battlestar Galactica started. And we're all wondering; is Ellen really the Final Five? Or is she an older version of No. 6? Can Tigh be trusted as a human-wannabe-Cylon? Is Tyrol gonna go all Cylon or is he gonna be loyal to the fleet? Was that really Kara Thrace in the cockpit of that burned out raider or did she time travel?

But as an economist the one question in my mind is:

Why the hell was the MPK so low on New Caprica?

They had a year before the Cylons showed up. And after that year they were still living in freakin' tents. TENTS! After a year! Lazy ass bastards!

Now, the first explanation that comes to mind is that, you know, they just didn't have enough natural resources to work with. But that can't be. There's obviously woods around on New Caprica - where the search party from Galactica landed and the ambush took place. Cut down some of them trees and build some goddamn log cabins. You're in an virgin environment. Exploit it! You're in a world where diminishing returns have not set in. Take advantage! It took settlers in the American West less than two months to put up their cabin and that probably included plowing the land around it (I had a reference for that but lost it). What the hell where you people doing?

We are talking about a civilization that has FTL capability here! And they can't fashion tree trunks into adobes.

Also, there's obviously stone - where the execution of Roslin, Zarek and others was gonna take place. Now, if you take stones, and pile one on top of each other, at some point you make these things called *WALLS*. It's not a long shot from there to make a *CEILING*. And then you got a house. But when the Cylons showed up there was no *walls* or *ceiling* or *houses*, just freakin' tents.

And when the Cylons DID show up apparently they didn't have the same problems. They build a very large concrete building very quickly. Granted, them being Cylons, they build a prison. But they still build it. Why couldn't the New Caprica crew build a series of concrete buildings which, if they had been so inclined they could've called "apartments" (I think that word was mentioned somewhere in the Pythia prophecies). Was it just cuz they've all been feeling so sorry for themselves through out?

Alright, the obvious explanation here is the Presidency of Gaius Baltar. But as much of a shit as that guy is you just can't blame everything on him. Was Gaius Baltar really a *growth killing president*, say, like Robert Mugabe of the New Caprica?

It doesn't seem likely. From all indications Gaius just wanted pills, booze and hookers. In other words "non-distorting lump sum taxes". Or in yet in other words, do you really think that Gaius Baltar had the time to supervise the extraction of surplus from hard working New Capricans who were busy chopping down trees to build some comfy log cabins for themselves. No, in all likelihood he probably just collected enough lump sum taxes to fund the above mentioned activities and left everyone alone.

(As an aside, here I completely sympathize with the rabble rousing that Tyrol's union is about to do in the relevant episodes. Trade Unions make perfect sense - even more - when they're all anti-government in a government system. The Polish Solidarity being the prime example. The technical term for this is the *double marginalization problem*)

In fact, if you really think about it, then Laura Roslin seems like a much more of a "anti-development", "growth-killing" president than Gaius. She's got that 3rd grade autocratic marmish school teacher thing going and she knows what's best for everybody. Tom Zarek is not entirely wrong about her. Prime exhibit: the idea of randomly assigning people to tilium mining regardless of their background. I mean, even Lee Adama, who, like, plays the role of the "lovable dumbass" picked up on the fact that it's not really good to have English teachers, or cow farmers or ... people who turn logs into log cabins and stones into *walls* ... be miners.

You can't blame this one on Gaius. In fact I blame it on Roslin. She promised everyone she'd take care of them, they believed her and in the extraordinary circumstances it actually made sense for a while. People needed hope and change and then she gave it to them. And that's what she was good at. But once it settled into the nitty gritty the basic laws reasserted themselves. And ever since then she's been messing up the economy of the fleet, And they got all lazy.

They deserved it. Or if they didn't, it was Laura Roslin's fault.

Krugman and Domar go to Whitecastle

Ay, Krugman and Domar are good.

Having written the last post I've re-read both the Evsey Domar paper and Paul Krugman's old article on it and was struck by how good both were. One amazing thing was how much old arguments get repeated over and over again even if they're, well, wrong.

Paul Krugman quotes James Surowiecki on the Black Death(and JS is in turn quoted by commentators on blogs all over);
" "The Black Death helped undermine feudalism. The population decline was so severe that the individual�s labor grew more valuable, which enabled serfs to abandon their lords and become tenant farmers or urban workers. "

and then Paul:

"That sounds plausible, but it's not the way it happened. According to Domar, serfdom actually withered away before the Black Death, as European population grew close to its Malthusian limit. The puzzle is why serfdom wasn't reinstituted after the Black Death. "

And then we go from there. The Domar paper really needs to be the basis for any kind of discussion of how political factors interacted with economic and demographic ones in the Malthusian world.

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")