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)