There was too much fluctuation in incomes in the Malthusian world.
Updates:
In this discussion of Greg Clark's book a similar point comes up (among many others) but without the maths.
Another, um, blogger, links to this paper by Ron Lee (which is also discussed in the above ssha link). This is a JSTOR version but I think I recall seeing an ungated version somewhere which I'll try to find.
This particular graph from Clark's "Farewell to Alms" has been bothering me for awhile:
To see why, let's recap some of the basic ideas behind the pre-industrial Malthusian economy. According to Clark the Malthusian economy was characterized by:
1. Low growth rate of technology.
2. Per capita income at any point in time being a negative function of population size due to diminishing returns to labor which combined with standard "Malthusian pressures" in turn meant that over the "long run",
3. Fertility and mortality rates were the major, if not only, determinants of per capita income.
More specifically, in the context of the above graph we also have that:
1. There was little or no change in real wages between 1200 and 1800 in England.
2. The "hump", or the increase in real wages in England between roughly 1350 and 1600 was due to the Black Death (and the "little Black Deaths" that followed it).
In a qualitative sense (the sign and direction of changes) the Clark story actually matches up pretty well with the data, which is a good portion of the reason why the Malthusian model is a compelling description of the pre-industrial world. But once you start thinking about it, the quantitative implications (the magnitude of these changes) of the Malthusian model are quite a bit at odds with the data.
Specifically, the above graph of the real wages in England has two problematic features (and I'm gonna keep going with the numbered lists here):
1. A doubling of real wages as seen in England between roughly 1350 and, I don't know, 1480, as seen in the graph, is pretty much impossible in the Malthusian world, given some plausible parameter values. To get that order of magnitude would require either a very high rate of technological growth (ruled out by Malthusian Assumption 1) or a HUGE drop in population. Now, of course Black Death, which wiped out perhaps a third of Europe's/England's population at a stroke, may seem to the casual observer like a HUGE shock. But I mean really HUGE. In the Malthusian model (again, assuming some plausible parameter values) something like 7/8 of England's population would have to disappear at a stroke to double incomes. Even with an initial shock which kills 1/3 of population and recurring "after shocks" this pretty much couldn't happen.
The reason for this is that the very logic of Malthusian economy which relies on diminishing returns to labor (Malthusian Assumption 2) basically precludes this kind of an increase (given a reasonable estimate for labor's share in output).
2. The fact that farm laborer's wagers are higher than construction laborer's wages throughout the period. This one I'm not so sure about and folks with more knowledge of historical details may correct me at will. But, if one thinks of farm laborers as the workers located in rural areas and construction laborers as workers located in urban areas (of course this has to be true only roughly, on average) then the fact that the Black Death affected urban areas to a greater extent than rural areas is quite at odds with the above data series. If more city folk died than farm folk, then we should see a greater increase in construction worker's wages than farm laborer's wages. But if anything we see a rise in the farm/construction premium which also sort of implies a fall in the urban/rural premium - the opposite of what we would expect if the Black Death affected urban areas more than rural ones.
This particular criticism is weaker than the first one. For example it could be that an ongoing migration from the countryside into the cities (as was the case) equalized the wages between the two areas so that the mortality difference was "split" between the farm and the city and hence we really shouldn't expect any difference in wages between rural and urban workers in pre industrial England (I can write you down a model where this happens but I'm not going to bother right now). Still, the fact that the series implies a different outcome than the Malthusian model with a Black Death shock would imply is a bit troublesome.
In what follows I'm gonna ignore this second criticism (because a full Malthusian model with rural-urban migration is too messy for a blog post) and focus on the first one.
Alright, so what would it take for wages to double in the Malthusian world? In this world, at any point in time wages depend on the size of the population, land and technology level (and we ignore the role of capital since this is pre industrial world. See also Oxonomics on the work by Reed and Frazer, with h/t to Gabriel). Since the amount of available land doesn't change much (well, there's the Dutch and their "fake" land...) we lump in technology and land together. Specifically let the total output of a pre-Industrial economy equal;
}$)
where y(t) is per capita income, A(t) is a "catch-all" factor which includes land, stock of capital and the level of technology, L(t) is population and alpha<1 measures the degree of diminishing returns (if you got a market in land 1-alpha will be land's share in output).
So. How big of a shock was the Black Death? Or, in other words, how big of a shock - in terms of its affect on per capita income - was the wiping out of a third of population of England?
Not much. Because of diminishing returns.
Let y(bbd) be the income before the sudden unset of the Black Death and the y(abd) be the per capita income after the Black Death. We're not gonna be sticklers here and require that the 1/3 drop in population immediately translates into higher wages. But it should translate into the observed doubling at some point within the next 250 years. Can that happen?
Assume that technological growth is low (again, Malthusian Assumption 1) so that there's negligible change in A before and after the shock. Then the ratio of the after-BD and pre-BD incomes is given by
*(\frac {L_{abd}} {L_{bbd}})^{\alpha-1}$)
Since we're assuming technology growth is negligble this just comes down to the ratio of populations pre and after Black Death.
^{1-\alpha}$)
The key parameter here is alpha which measures the rate of diminishing returns to labor. If you've got a labor market then alpha will be the share of total output which goes to labor and 1-alpha the share that is "appropriated" by landowners. Standard estimates for land's share in the pre-Malthusian economy put it t somewhere between 25% and 40%. So let's pick a medium value of 1/3. This means that if population after the black death was 2/3 of that before the black death the ratio of per capita incomes pre and after would be 1.1445.
Or in other words, this "huge" shock - Black Death - would increase percapita incomes by only about 15%. Even if we take land's share to be a very high 50% that still gives us only a 22.5% increase in per capita incomes. Ay. Even with alpha close to zero, a 2/3 shock to population increases per capita incomes by 50%, not by 200%.
Ok. But you wouldn't expect the impact to be immediate and what about those "after shocks"? Perhaps a better model of the shocks would be an initial wiping out of a third or half of England's population, and an overall increase in the mortality rate. This increase would mean that not only would there be an increase in income initially, but also the "steady state" level of per capita income will go up as well. For example if the growth rate of population is given by
Then in steady state y=m/f, so y(abd)/y(bbd)=m(abd)/m(bbd), so all you would need is a doubling of the mortality rate (at initial level of income). If you want to get a bit more realistic a more plausible function for growth rate of population would be
since in this case the mortality rate is bounded between 0 and 1. In this case y=m*(1-f)/f, but with no changes in fertility, y(abd)/y(bbd) still equals m(abd)/m(bbd). Here the mortality rate wouldn't quite double (assuming that pre Black Death income was at its old steady state value and evaluating the mortality rate at that income means that the ratio of pre and after mortality rates would be 2/(1+f)). Close enough.
What does that mean? First let's consider how much would population have to drop in order to get that doubling of per capita incomes:
^{.33}$)
*L_{bbd}$)
which means, that with no technological growth, one way or another (i.e. combining the initial shock to population with a higher mortality rate) you need population to drop by 7/8. In other words, for incomes in 1480 to be twice of those in 1348, the population of England in 1480 needs to be 1/8 of that in 1348. Which of course isn't what happened. In fact, English population in 1480 was slightly higher than in 1348 (about 4 mil compared to 3.5 mil pre Black Death)
But what if there was some technological progress in the intervening years, wouldn't it be possible for incomes to double in those 130 years? Well, if the Malthusian assumption that tech progress (which includes capital accumulation and land expansion here) is slow enough so that over the long period incomes stagnate (between 1348 and 1800) then most of that tech progress would just go into higher populations with only small, if any, impact on per capita incomes. But even ignoring that it's doubtful that tech progress was fast enough to generate these kind of magnitudes.
There's several ways to get an idea of why this won't work. First is to let the labor ratio between 1480 and 1348 be what it was - about 1.15 or just for the sake of argument, about the same (which makes it easier to double the wages), and ask how how much technological progress would be needed to double them wages.
Too skip more equation-editing in blogger let's just assume that 1480 minus 1348 is approximately 140 and then use the good ol' "rule of 70" (actually 72) in which case the time it takes for income to double is 70/g where g is the growth rate. So here this would imply a technological growth rate of about 1/2 percent per year. Compared to the modern world where we see tech growth rates between 1 and 2 % per year this seems paltry. But for the Malthusian world this is huge!
If we assume that England was at steady state in 1348 and again at a steady state in 1811 (i.e. all the adjustment to shocks like the Black Death and its aftermath has worked itself out in that period) then we can estimate the average annual growth rate of technology over the whole period from:
^{\frac {.33} {463}}-1=(\frac {9.5} {3.75})^{\frac {.33} {463}}-1$)
or a measly 0.00066=.066%
But even if technological growth was .5% per year you wouldn't see much of it show up income. Instead it would go into population growth. And in fact this technological growth is the very reason why you have 4 million people in 1480 rather than 3.75, or why you have 9.5 million around 1810 rather than 3.75.
You could keep finanglin' here. Maybe model the Black Death as an initial drop of 1/3 in population followed by an increase in mortality rates until 1450 or so, after which mortality returns to normal. Or maybe later than 1450. But this won't work either. What ends up happening is that you can either match the population levels (but for that you need essentially a stable mortality parameter, higher rate of tech progress won't do it) but not income levels, or you can match the income (with changes in mortality, after shocks to L) but not the population levels (if you increase m or periodically decrease L you will wind up with way too few people in England in 1480, 1601 and 1821.
Of course since this is historical, hence pretty imprecise data, you gotta give it a good bit of leeway. Still a doubling of wages is a lot more than a 15% increase in them so the magnitude is quite a ways off (like I said, I think qualitatively it matches up).
All of which suggests, that if you do believe the numbers in that figure at least somewhat, something else must've been going on in the period 1348 to 1480. Changes in fertility? Maybe, but here you'll run into the same problem as when you monkey around with the mortality rate - too few people if you want high enough income. Acceleration in technological progress during this time? But why 1348 to 1480 as opposed to 100 years later when Enlightenment is beginning to take hold? And even then, the dynamics of the Malthusian economy very strongly suggest that even this higher tech growth would not show up in higher incomes but get eaten up by higher population (you would way over predict population level in 1480). Land expansion? Same as with tech growth and remember that this is before the discovery of the new world? Maybe a change in the share of output going to land and labor respectively - alpha? I wouldn't rule this one out, actually.
(alpha is one parameter that economists don't like to mess with. But it makes a lot more sense to mess with it in the pre-industrial, half-feudal world than in the modern one)
Anyways. I've got a simple excel file which lets you simulate your own toy Malthusian economy based on this which I'll post as soon as I can make it user friendly enough and figure out how to link to excel files.
(Note: There's probably a whole bunch of typos in the above)
In this discussion of Greg Clark's book a similar point comes up (among many others) but without the maths.
Another, um, blogger, links to this paper by Ron Lee (which is also discussed in the above ssha link). This is a JSTOR version but I think I recall seeing an ungated version somewhere which I'll try to find.
This particular graph from Clark's "Farewell to Alms" has been bothering me for awhile:
To see why, let's recap some of the basic ideas behind the pre-industrial Malthusian economy. According to Clark the Malthusian economy was characterized by:
1. Low growth rate of technology.
2. Per capita income at any point in time being a negative function of population size due to diminishing returns to labor which combined with standard "Malthusian pressures" in turn meant that over the "long run",
3. Fertility and mortality rates were the major, if not only, determinants of per capita income.
More specifically, in the context of the above graph we also have that:
1. There was little or no change in real wages between 1200 and 1800 in England.
2. The "hump", or the increase in real wages in England between roughly 1350 and 1600 was due to the Black Death (and the "little Black Deaths" that followed it).
In a qualitative sense (the sign and direction of changes) the Clark story actually matches up pretty well with the data, which is a good portion of the reason why the Malthusian model is a compelling description of the pre-industrial world. But once you start thinking about it, the quantitative implications (the magnitude of these changes) of the Malthusian model are quite a bit at odds with the data.
Specifically, the above graph of the real wages in England has two problematic features (and I'm gonna keep going with the numbered lists here):
1. A doubling of real wages as seen in England between roughly 1350 and, I don't know, 1480, as seen in the graph, is pretty much impossible in the Malthusian world, given some plausible parameter values. To get that order of magnitude would require either a very high rate of technological growth (ruled out by Malthusian Assumption 1) or a HUGE drop in population. Now, of course Black Death, which wiped out perhaps a third of Europe's/England's population at a stroke, may seem to the casual observer like a HUGE shock. But I mean really HUGE. In the Malthusian model (again, assuming some plausible parameter values) something like 7/8 of England's population would have to disappear at a stroke to double incomes. Even with an initial shock which kills 1/3 of population and recurring "after shocks" this pretty much couldn't happen.
The reason for this is that the very logic of Malthusian economy which relies on diminishing returns to labor (Malthusian Assumption 2) basically precludes this kind of an increase (given a reasonable estimate for labor's share in output).
2. The fact that farm laborer's wagers are higher than construction laborer's wages throughout the period. This one I'm not so sure about and folks with more knowledge of historical details may correct me at will. But, if one thinks of farm laborers as the workers located in rural areas and construction laborers as workers located in urban areas (of course this has to be true only roughly, on average) then the fact that the Black Death affected urban areas to a greater extent than rural areas is quite at odds with the above data series. If more city folk died than farm folk, then we should see a greater increase in construction worker's wages than farm laborer's wages. But if anything we see a rise in the farm/construction premium which also sort of implies a fall in the urban/rural premium - the opposite of what we would expect if the Black Death affected urban areas more than rural ones.
This particular criticism is weaker than the first one. For example it could be that an ongoing migration from the countryside into the cities (as was the case) equalized the wages between the two areas so that the mortality difference was "split" between the farm and the city and hence we really shouldn't expect any difference in wages between rural and urban workers in pre industrial England (I can write you down a model where this happens but I'm not going to bother right now). Still, the fact that the series implies a different outcome than the Malthusian model with a Black Death shock would imply is a bit troublesome.
In what follows I'm gonna ignore this second criticism (because a full Malthusian model with rural-urban migration is too messy for a blog post) and focus on the first one.
Alright, so what would it take for wages to double in the Malthusian world? In this world, at any point in time wages depend on the size of the population, land and technology level (and we ignore the role of capital since this is pre industrial world. See also Oxonomics on the work by Reed and Frazer, with h/t to Gabriel). Since the amount of available land doesn't change much (well, there's the Dutch and their "fake" land...) we lump in technology and land together. Specifically let the total output of a pre-Industrial economy equal;
where y(t) is per capita income, A(t) is a "catch-all" factor which includes land, stock of capital and the level of technology, L(t) is population and alpha<1 measures the degree of diminishing returns (if you got a market in land 1-alpha will be land's share in output).
So. How big of a shock was the Black Death? Or, in other words, how big of a shock - in terms of its affect on per capita income - was the wiping out of a third of population of England?
Not much. Because of diminishing returns.
Let y(bbd) be the income before the sudden unset of the Black Death and the y(abd) be the per capita income after the Black Death. We're not gonna be sticklers here and require that the 1/3 drop in population immediately translates into higher wages. But it should translate into the observed doubling at some point within the next 250 years. Can that happen?
Assume that technological growth is low (again, Malthusian Assumption 1) so that there's negligible change in A before and after the shock. Then the ratio of the after-BD and pre-BD incomes is given by
Since we're assuming technology growth is negligble this just comes down to the ratio of populations pre and after Black Death.
The key parameter here is alpha which measures the rate of diminishing returns to labor. If you've got a labor market then alpha will be the share of total output which goes to labor and 1-alpha the share that is "appropriated" by landowners. Standard estimates for land's share in the pre-Malthusian economy put it t somewhere between 25% and 40%. So let's pick a medium value of 1/3. This means that if population after the black death was 2/3 of that before the black death the ratio of per capita incomes pre and after would be 1.1445.
Or in other words, this "huge" shock - Black Death - would increase percapita incomes by only about 15%. Even if we take land's share to be a very high 50% that still gives us only a 22.5% increase in per capita incomes. Ay. Even with alpha close to zero, a 2/3 shock to population increases per capita incomes by 50%, not by 200%.
Ok. But you wouldn't expect the impact to be immediate and what about those "after shocks"? Perhaps a better model of the shocks would be an initial wiping out of a third or half of England's population, and an overall increase in the mortality rate. This increase would mean that not only would there be an increase in income initially, but also the "steady state" level of per capita income will go up as well. For example if the growth rate of population is given by
Then in steady state y=m/f, so y(abd)/y(bbd)=m(abd)/m(bbd), so all you would need is a doubling of the mortality rate (at initial level of income). If you want to get a bit more realistic a more plausible function for growth rate of population would be
since in this case the mortality rate is bounded between 0 and 1. In this case y=m*(1-f)/f, but with no changes in fertility, y(abd)/y(bbd) still equals m(abd)/m(bbd). Here the mortality rate wouldn't quite double (assuming that pre Black Death income was at its old steady state value and evaluating the mortality rate at that income means that the ratio of pre and after mortality rates would be 2/(1+f)). Close enough.
What does that mean? First let's consider how much would population have to drop in order to get that doubling of per capita incomes:
which means, that with no technological growth, one way or another (i.e. combining the initial shock to population with a higher mortality rate) you need population to drop by 7/8. In other words, for incomes in 1480 to be twice of those in 1348, the population of England in 1480 needs to be 1/8 of that in 1348. Which of course isn't what happened. In fact, English population in 1480 was slightly higher than in 1348 (about 4 mil compared to 3.5 mil pre Black Death)
But what if there was some technological progress in the intervening years, wouldn't it be possible for incomes to double in those 130 years? Well, if the Malthusian assumption that tech progress (which includes capital accumulation and land expansion here) is slow enough so that over the long period incomes stagnate (between 1348 and 1800) then most of that tech progress would just go into higher populations with only small, if any, impact on per capita incomes. But even ignoring that it's doubtful that tech progress was fast enough to generate these kind of magnitudes.
There's several ways to get an idea of why this won't work. First is to let the labor ratio between 1480 and 1348 be what it was - about 1.15 or just for the sake of argument, about the same (which makes it easier to double the wages), and ask how how much technological progress would be needed to double them wages.
Too skip more equation-editing in blogger let's just assume that 1480 minus 1348 is approximately 140 and then use the good ol' "rule of 70" (actually 72) in which case the time it takes for income to double is 70/g where g is the growth rate. So here this would imply a technological growth rate of about 1/2 percent per year. Compared to the modern world where we see tech growth rates between 1 and 2 % per year this seems paltry. But for the Malthusian world this is huge!
If we assume that England was at steady state in 1348 and again at a steady state in 1811 (i.e. all the adjustment to shocks like the Black Death and its aftermath has worked itself out in that period) then we can estimate the average annual growth rate of technology over the whole period from:
or a measly 0.00066=.066%
But even if technological growth was .5% per year you wouldn't see much of it show up income. Instead it would go into population growth. And in fact this technological growth is the very reason why you have 4 million people in 1480 rather than 3.75, or why you have 9.5 million around 1810 rather than 3.75.
You could keep finanglin' here. Maybe model the Black Death as an initial drop of 1/3 in population followed by an increase in mortality rates until 1450 or so, after which mortality returns to normal. Or maybe later than 1450. But this won't work either. What ends up happening is that you can either match the population levels (but for that you need essentially a stable mortality parameter, higher rate of tech progress won't do it) but not income levels, or you can match the income (with changes in mortality, after shocks to L) but not the population levels (if you increase m or periodically decrease L you will wind up with way too few people in England in 1480, 1601 and 1821.
Of course since this is historical, hence pretty imprecise data, you gotta give it a good bit of leeway. Still a doubling of wages is a lot more than a 15% increase in them so the magnitude is quite a ways off (like I said, I think qualitatively it matches up).
All of which suggests, that if you do believe the numbers in that figure at least somewhat, something else must've been going on in the period 1348 to 1480. Changes in fertility? Maybe, but here you'll run into the same problem as when you monkey around with the mortality rate - too few people if you want high enough income. Acceleration in technological progress during this time? But why 1348 to 1480 as opposed to 100 years later when Enlightenment is beginning to take hold? And even then, the dynamics of the Malthusian economy very strongly suggest that even this higher tech growth would not show up in higher incomes but get eaten up by higher population (you would way over predict population level in 1480). Land expansion? Same as with tech growth and remember that this is before the discovery of the new world? Maybe a change in the share of output going to land and labor respectively - alpha? I wouldn't rule this one out, actually.
(alpha is one parameter that economists don't like to mess with. But it makes a lot more sense to mess with it in the pre-industrial, half-feudal world than in the modern one)
Anyways. I've got a simple excel file which lets you simulate your own toy Malthusian economy based on this which I'll post as soon as I can make it user friendly enough and figure out how to link to excel files.
(Note: There's probably a whole bunch of typos in the above)
posted by YouNotSneaky! at 6:33 PM
21 Comments:
I can host your Excel file.
Sometimes you write (alpha-1) when you want to write (1-alpha) and vice-versa.
Awesome post! But I need to think about it.
Thanks! I think the alpha's are right - I flip the ratio when I go from one to the other so it's consistent. Maybe I missed it somewhere though.
If beta is the ration of A after to before bb, you don't need much technological change to observe a doubling of income...
Ladd = Lbbd * (beta^3)/8
(This is just from your second equation not assume the ratio of A's are one.)
beta=1.75 or yearly growth less than 0.4% is enough to get us where we need to be.
The first equation... I think you meant
L_t^\alpha
rather than alpha - 1, which is negative.
No no I meant (alpha-1). It's per capita income so it depends negatively on amount of labor.
Total output, Y is A*L^(alpha)
.4% per year is a very very high technological growth rate for the pre-industrial world (like I said, if we take 1348 and 1811 as "steady state" years then the implied growth rate of technology is .066%, which is 1/6 of .4). And that's assuming that ALL of that .4% shows up in higher incomes rather than higher populations.
Maybe there's a problem with the prices data used to deflate the wages data?
Re: problem 2., less people => less demand for housing => lower wages for builders?
But then again, less people => less demand for food => lower wages for farmers?
Maybe it's the relative magnitude of both effects...
Do I read your comments right, that you assume the landowner appropriates a certain percentage from all farmers?
Another way of modelling the wages would be to stipulate that the real wages of peasants would be on par with the output of those working with the lowest quality land (minus the minimum land rent).
As the best available land not currently being labored is (or can be assumed to be) worse than that, a potential competing landowner cannot offer more than he would to those holding the worst currently in use. Thus, if all peasants are 'paid' as much as those doing worst, none of them can actually switch to another farm and be better off.
Cut off part of your labor supply, and the minimum productivity of a laborer rises (because worst land can be left fallow). This creates competition, which pushes the wages to the productivity of the worst land (minus minimum land rent).
The total output does not rise as much as the wages of peasants, because the plague increased the scarcity of laborers more than landowners, which gave the laborers larger share of the total economic pie. Thus, plague was in effect a transfer of wealth from the rich to the poor.
It would seem to me that the above analysis holds well for builders as well.
Mikko,
Yeah that's sort of what I was thinking (though I didn't have a story as detailed and explicit as yours) when I said that I wouldn't rule out changes in labor's share during this period. I think one way or another this has to be a part of the story.
But.
Even with changes in farm laborer's wages in per capita income it's still hard to match a 100% increase. In this case we'll have
% change in farm wages =
% change in farm laborer's share +
% change in per capita income
The thing is, because it's a share, it's bounded between 0 and 1 and it was probably already "high" in the mathematical sense (in the economic sense of course it was low since it meant that a large portion of farmer's output was being appropriated by landlords) - more than .5.
So say we take the min estimate of labor share to be, I dunno .6 as the bbd share and a high one of say .75 as abd. That's a large 25% increase in labor's share.
Additionally with the new 1-alpha of .25 you can stretch the effect on per capita incomes from about a 15% increase to a 20% increase.
So we'd have something close to
% change in farm wages = 45%
which is still much less than 100%.
So while it helps, a good portion of the problem is still there.
I'm very, very late to this discussion, but given that Paul Krugman himself linked to it, perhaps others will join in....
Thanks for a fascinating post. I seem to recall that the mortality rate for the Black Death was greater than you are using here--iirc, for the poor and laboring classes, it was closer to 80%. (And much lower in the wealthier classes, only 10% or so) And that in the 300 years following, it continued to strike adults in their prime working years.
I wonder how that these differences would affect your model?
Oh oh.
Hmmm, I would definitely be interested in seeing some more precise numbers for mortality and population so if anyone knows any good sources, please let me know. 80% sounds crazy high though. There was a lot of people in the 'poor and laboring classes' so I don't see how an 80% drop in that population can easily translate to a 1/3 drop in overall population (maybe 1/2 if we include the after shocks). But yeah, an 80% drop would get you much closer to matching the data.
Great post.
But you use the commonly assumed 1/3 death rate. A recent comprehensive review of the evidence by Ole Benedictow estimates that the death rate was at or above 60%. Google Books has the relevant summary table online:
Ole Benedictow (2004) The Black Death, 1346-1353: The Complete History. See Table 38 on page 383.
http://books.google.com/books?id=pQffKRi5DYIC
While this isn't quite the 'huge' 7/8 shock you describe as necessary, it's definitely BIG.
Also, Benedictow's data does NOT support the view that more peopled did in urban areas, indeed he argues the opposite.
Finally I agree with mikko that it just doesn't make sense to assume labor's share of output per worker remained the same. In much of Europe the black death is said to have been a deciding factor leading to the collapse of involuntary serfdom. It was pretty easy for a landlord to push the laborer's 'wage' down to their reservation utility through a lump sum tribute + labor-service obligation, and the whole idea of serfdom was precisely to keep that reservation utility low by making competition for labor amongst landlords virtually impossible.
It would not surprise me if 'free' laborers would have been able to get at least 25% larger share of output per worker.
I would also give the economy a boost in output per worker coming simply from the fact that resource allocation had to improve with more mobile factors.
So let's add up. Suppose a 10% increase in 'total factor productivity' from improved resource allocation, a 25% increase in laborer's share, and Benedictow's 60% fall in population, plugged into your formula (where your 1/3 is technological paramter 1-alpha). We get:
(1.1)(1.25)*(10/4)^(1/3) = 1.87
Pretty close to 2.0
Very nice. This is the kind of explanation I was looking for.
(Though 60% seems crazy high. Maybe true, but still crazy high)
Maybe the increase in population was allowed by an increase in land amount (deforestation). Then you do not need technology growth in that period.
No, mikko sarela has it right. This is a straight-forward application of Ricardo's theory of rents. Given differences in the natural fertility of agricultural land, the largest plurality of the agricultural labor force might well have been deployed on the poorest quality land, while the division of the surplus between labor and aristocratic landowners would be determined by the productivity/fertility of the poorest quality land. With the elimination of a third or more of the agricultural labor-force, labor would be redistributed to higher quality lands, and it shouldn't be hard to jigger that to result in an increase in per capita output of, say, 50%, and with the withdrawal of the poorest quality land, a shift in the distribution of the surplus from, say, 50-50 to 30-70, such that labor's share becomes 1.5*.7= 1.05 and the landlord's share diminishes slightly to .45 in absolute terms. One could throw in a host of additional factors, such as an increase in labor mobility, a loosening of feudal bonds, an increase in trade and corresponding pareto-improvements, the possibility that through the aforementioned agricultural laborers could accumulate property that would be legally recognized, the discovery of the Grand Banks and the increased supply of salted cod entering the stream of wage goods, enhancing the diet and labor capacity of the work force, which, since the imputed wage is daily rather than hourly, together with the motive of accumulating property, might have led to an increased work effort, etc., such that it shouldn't really be implausible to account for the doubling of the daily agricultural wage. The key point is that the marked decrease in laboring population would have resulted in the focusing of laboring efforts on the largest points of natural fertility/productivity, for which technical development is of little relevance. (The salting of fish was long known, but not the source of supply). Perhaps the problem in your attempt at a math model lies in that alpha-1, which expresses the diminishing returns of labor with an increasing population/labor-force, and yields a negative fractional exponent as a continuous logarithmic curve, when the actual factor is the increasing fertility/productivity of land to constant unit labor, which is not a continuous function.
The Great Deforestation in Europe took place about 200 years before the Black Death (or more, depending on where specifically you're talking about) so the timing doesn't match up.
And yes, the "moving people onto more productive land" is already embodied in the analysis through the production function's diminishing returns. Part of the point is that this by itself cannot account for the increase in observed wages. But I'm sympathetic to the "increase in labor's share do to changes in social arrangements" + "more people died than is commonly believed" explanation.
"when the actual factor is the increasing fertility/productivity of land to constant unit labor, which is not a continuous function"
Can you give a specific example?
I think there should be some more thought given to the variable stock of land. Europe, including England, was characterized by varying cultivation throughout the pre-modern period.
Here's my quick story: the smaller labor force results in greater mobility, since landowners will become more reluctant to honor sanctions against labor that flees its contract. So the landowner will need to pay an increased share of his income to labor. In order to preserve his absolute income, he has to grow his pie by bringing more land into cultivation. Of the income accruing to the additional land, a larger share of it will go to labor than for the inframarginal land, but the landowner will still preserve his absolute income.
Finally, with regard to the second observation, about the "wrong" premium prevailing between agricultural and urban labor, I think the mechanism you're going to want would come from a tradeable/non-tradeable sector model. Agricultural output per capita would benefit from growth of foreign demand relative to domestic.
Oh, one more thing: my story about bringing more land under cultivation would play a role in your 0.5% rate of technological change/capital growth, which you say is suspiciously high. I would agree with you, if it were the case that once cultivated, land could not be left fallow. But it is quite possible that the catch-all variable would turn negative if, in some period after the one I narrated, land were left fallow once again.
Well, the point is that diminishing returns accrue not to labor/population, your L, which occurs in traditional, unimproved form, managerial and technically, and hence is a constant, static unit, but to land, your A, which varies significantly in fertility/output to constant labor, and increased in output sharply to the extent that marginal lands are abandoned and only the most fertile are cultivated: increasing returns to decreases in scale. A crude toy model would look like this: there are 3 kinds of land, A, B, and C, with A twice as productive as B and B twice as productive as C; there are 10 acres of each and 30 laborers, distributed to each segment equally, with the "wage" being determined by the labor output of the least productive land. Hence the total output is 4+2+1=7 units, the average per capita output is .233 per worker, the wage .1 per worker and the landlord's rent is 4 units of output in total. 10 workers die and land C goes out of cultivation. The total output is 6, the per capita output is .3, the wage is .2, and the total rent share is 2. Per capita output has increased 21%, the wage has doubled and the rent share has halved. Not quite the result wanted, but the sort of model in question.
So a variation. A is 4 times as productive as B, and B twice as productive as C, there are 20 acres and laborers on A, 10 acres and laborers on B and 20 acres and laborers on C. Total output is 16+2+2=20 units, per capita output is .4, the wage is .1, and the rental share is 15 units in total. 20 laborer die, 40%, C goes out of cultivation, the total output is 18, the wage is .2, the per capita output is .6 and the rental share is 12. The wage is doubled, per capital output is 50% higher, and the absolute rental share is decreased by 20%. Note that if only 10 laborers dies, C remains in cultivation, the wage remains .1, the rental share 15, and per capita output increase to .475. The upshot with this crude model apparently is that to get the desired results the amount of both top quality and lowest quality land must be relatively high, the population of laborers must be decreased to eliminate the cultivation of the lowest quality land, and the rental share must be quite high to begin with, so as not to markedly diminish in the aftermath.
What is going on here plausibly in this sort of model with respect to "wage" determination and the share of rent? Well, to begin with, the aristocratic lords are pretty dim bulbs, armed thugs, not rational calculators, offering "protection services", not managerial and technical improvements, but they do have an obligation at the root of their traditional "authority" to provide for their serfs. Production is in the hands of the peasant communes and completely traditional, without any attempt to rationalize labor or maximalize output. As population pressures increase, more marginal land is cleared and set into cultivation, and, in effect, the lords tax the output of laborers on fetile lands to provide for the subsistence of laborers on marginal lands, which is what sets the wage as determined by the output of the most marginal lands. (Note that on your chart, the "wage" from 1200 to the BD event decreases by about 25%). With the decrease in laboring population, labor must be attracted to more productive lands, since deaths occur across all labor groups, so that there is some upward pressure on 'wages", but so long as much low quality land remains in cultivation, the landlords will tend to continue to enforce their rental extractions as enforcing "equal" law, as well as their own interest, even as per capita output begins to increase as a continuous function. But when low quality land is entirely and rather suddenly abandonned due to a sharp decrease in laboring population, there is a discontinuous "jump" in "wages" corresponding to the need to restock and shift labor to the better lands, and a corresponding discontinuous fall in the absolute and relative rental share, inspite of the continuing and increasing level of per capita output.
Of course, in the real world, as opposed to the toy model world there would be a patchwork of different kinds and qualities of land, as well as, a patchwork of different feudal hierarchies with their rent transfers that would be involved in adjustment processes. Still a doubling of the "wage' with a marked increase in per capita output, and a relatively modest decrease in the absolute rental share doesn't seem outside the realm of plausibility, provided there is a sharp shift to sharply more productive lands and a high initial rental share. SO the question is whether such a high extraction of agricultural surpluses as rent is historically plausible. I don't know for a fact. Still, the extraction of rents/agricultural surpluses do not just go to the aristocrats and their households and retainers, but indirectly or directly support towns and monasteries, (which would have then been important legal-administrative and productive units). Further, I assume that "wages" and per capita output primarily is being measured through wage goods which is primarily food, and in a little monetized economy where labor is primarily determined through command-and-subvention, it's a bit difficult to determine the "value" of luxury goods, arms, books, and the like in terms of wage goods and relative prices.
Be that as it may, it is noteworthy that the period of rising wages with the BD is coterminous with the Hundred Years War/ War of the Roses, so, while the lordie boys were having their adventures and killing each other off, the serfs were living it up! It's also noteworthy from the chart that "wages" have dropped back by 1600, which is the golden age of Elizabeth, when the state had been consolidated and its authority established and elaborated through legal-administrative means, when the sea trade was robust, when the economy was much more monetized thanks to all that looted Spanish gold and silver, and the beginnings of mercantile capitalism and market commerce were clearly recognizable, though often subsidized through monopoly patents granted by the crown. The great superiority of mercantile capitalism over feudalism, of course, is that, whereas under feudalism surpluses were extracted and stored, under mercantile capitalism stored surpluses could be recirculated, rationalizing and improving their extraction and seeking out new sources of surpluses. So between the two poles of the physiocratic "the fecundity of land is the source of all wealth" and the Malthusian "increases in aggregate wealth are merely absorbed in population increases" lies the extraction of surpluses on behalf of wealthy elites (and the state). I doubt that there was all that much technical improvement in the course of the 17th century economy-wide. Sailing ships were probably pretty much the same at the beginning as at the end of the century, for example, even if there had been improvements in navigational techniques and a great increase in geographical knowledge. What gains there were would largely be attributable to pareto-improvements. But it's likely that improvements in the extraction and control of surpluses were occurring alongside increases in population, and increases in concentrations of wealth were occurring alongside increases in poverty, which is masked by the appeal to average per capita rates on which the Malthusian vision rests. (Even as increases in population were no doubt a factor in driving sea-borne expansions). Current concerns about resource/environmental constraints and the pressures of population expansion shouldn't be an excuse for ignoring the flaws in the physiocratic/Malthusian mirror vision.
Where do you get the figure of 4 million for the population of England in 1480? It is wrong. The population may have been as low as 1 million.
Otherwise great post. I think your point still stands. But the population contracted until 1520. There after England only regained its 14th century population in the 18th century.
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