Question for the Theorists

all the hoopla

Where would you like to see your favorite piece of Economic Theory in, say, 50 years time? If you're into the General Equilibrium thing what kind of progress do you think needs to be made? If you're a Game Theory fool, what else can be said? Where is DSGE going to go that doesn't involve just another tweak of parameters/assumptions that we all only half heartedly believe in? I'm not asking for specific theorems (or applications)... well, maybe for conjectures. More along the lines of a road map. Not quite like Hilbert's program because I'm sure we all can agree that would be asking too much at this point. But a goal. What would an ideal economic methodology look like? At least in a specialized kind of way, within your favorite approach?

Ok, ok, it is possible that there may be some who think that their preferred method, whatever it is, already describes actually existing economies, Sraffa Shrugged and all, and nothing more needs to be said. I'm asking the other peoples.

And mind you, this is meant to be a question for the theorists. So answers like "we will be able to model agent learning better" are not satisfactory since they deal with a specific problem within a methodological approach (mostly DSGE in this case) rather than the big questions that have been forgotten in these past years. If it's about learning, for example, than it's gotta be something that criss crosses the methodological approaches.

We can bring in the applied folks in on this (of whom I am one, just sort of keeping an eye on the theory as I think all applied folks should) - what kind of theoretical results do you think would have a big impact on that Kudzu sub field of economics called "Applied Micro"? In the sense that it would stop a lot of people from doing it. As long as we're on this topic, what is your Bayesian estimate of the optimal ratio of theory to empirics? In our hearts of hearts, that little place where our irrationality resides, we know that "there's no theory without data"! Of course in our soul of souls we also know that "empirical work unbacked by theory is just data mining"! But if you leave those special places then you do realize that - since we're all economists here - it's probably an interior solution. So what's the break down? Gimme a number folks.


(As an aside, I'm sort of pessimistically rooting for the whole Agent-Based-Modeling approach. Rooting for it cuz I think it does have potential - integrate it with standard GE! And maybe it can get at some of the questions that Alex keeps asking about dynamics and so on. Pessimistically because even at this early stage (and my knowledge of the field is slightly greater than epsilon) I think it seems pretty obvious that it's bound to run into all the standard problems of other methodological approaches; indeterminacy, arbitrariness (there's a million ways to be heterogeneous. Why pick a particular one?) and ultimately, well, hopefully, generality.)

Americans are right - soccer is a ridiculous game

"the ball bounces around randomly for about an hour an a half, then suddenly it hits some guy's head and accidentally goes into the net, everybody goes crazy, and somebody's lost really bad and somebody's achieved a great victory" - a paraphrase of an American columnist, possibly PJ O'Rourke, describing their attempt at trying to appreciate soccer, done from memory.

Austria:Poland - 1:1


The setup: Euro 2008. Poland has lost to Germany. Austria has lost to Croatia. The looser of the Poland/Austria game is eliminated from the competition. Austria dominates for first 30 minutes. The Poles get a lucky, though very elegant goal ... which might have been an offside (similar situation occurs in the Poland-Germany game with Poland the worse off). Energized by the goal the Poles control the rest of the game although they do not manage to score again. In the 92nd minute, at the very very very end of injury time, the ball is kicked into the Polish penalty box and an Austrian player very clearly either slips or takes a dive. The referee awards a spurious penalty kick to the Austrians which they take advantage off to tie it up. The Poles, or at least this one, feel totally cheated and for all practical purposes both teams are eliminated.

In American football, basketball or baseball, a bad decision by a referee will give one team a slight advantage (in critical situations perhaps a crucial slight advantage) in terms of winning the game. In soccer a bad decision by a referee not only can very easily determine the outcome of a match but in many cases the entire standing in a group/division and even advancement to the next round (i.e. play offs). There are huge discontinuities in terms of the marginal effects of a referee's decisions.

To be exciting games we like to watch and play need a mixture of uncertainty and reward to skill and effort, unless one's a gambling sort who thinks that roulette is an intellectually sophisticated game. In this respect however soccer is worse than pinball. True, skill plays a role - you can rely on Argentina to beat up on The United Arab Emirates every time. But if the teams are even remotely evenly matched in terms of the randomness/skill content it's not even a pinball game. It's a pinball game with broken flippers, a roulette wheel where a referee occasionally decides to magically and arbitrarily move the silver ball closer or farther from the number you picked.

The great power wielded by the referee and the impossibility of double checking his decision (by design, since the technology exists to do this and American Football, for example, has managed to implement it successfully) also creates great potential for corruption. And any honest assessment of team sports is pretty much bound to conclude that FIFA and its various national counterparts are rife with it. From how locations are chosen, to how groups are selected, to how matches are adjudicated.

In addition to corruption there's other problems that this huge amount of uncertainty combined with power of no appeal on part of the refs creates (in addition to the fact that soccer refs are the least respected among their type/proffesion across the sports). First there's an obvious incentive for players to learn how to "game the ref" rather than focus on skill - the dives, the fake injuries, the prima donna acting that characterizes many a "star" team. Second, since with this much uncertainty it really is hard to objectively rank teams in terms of their skill, the all to human referees come to rely on certain rules-of-thumb, strength of tradition, and "I think thats right because it seems to have been right in the past". Germany, Brazil, Argentina, Italy win because... well, they're supposed to win, particularly if in terms of fundamental objective skill the game's close.

Of course this begs the question - why is soccer popular in the rest of the world, but not so much in US? Here everyone's got their favorite answers which either boil down to assertions that Americans are too crude and brutish to appreciate the grace and beauty of the game, or, on the other side to mistaken assessments that "soccer is a game for teenage girls" (as anyone who's ever played it knows, soccer is as physically demanding as American football, probably more than basketball, and certainly more than baseball). But the simple truth is that it's all about tribalism - in its national or regional forms. Without the tribalism the game of soccer looses most of its appeal. It can best be seen as either a very beautiful excercise in synchronization -in which case it should be judged in the same way as ice skating competitions and water ballet. Or as a simple spin of a roulette wheel. Or perhaps both - a spin of a very beautiful, ornamented and elegant roulette wheel.

And US is a very non-tribal place. While some folks in the South may still smart about those damn Yankees, for the most part a Wyomingian, or a Rhode Islander tend to have fairly weak connections to their place of residence. This is compounded by the fact that often times, for real tribalism to be strong, a Nebraskan-of-Guatemalan-origin would have to feel very strongly about Nebraska, a Delawerian-of-Vietnamese-descent very strongly about Delaware and a some white bread Oregon hippy about Oregon. To be sure, these kinds of feelings do exist but compared to ROW they are very muted, for fairly obvious reasons.

The outcome of the Austria-Poland game wasn't even that unfair. Both teams played well and they were ... evenly matched. Both goals had a good bit of arbitrariness about them (though of course I think the Poles were "border line offsides" when they scored their goal whereas the Austrian penalty kick was clearly undeserved). But the game was a really good illustration of why soccer is a very ridiculous sport.




Some notes:
1. American College football competition actually shares some of the defects of soccer competitions - mostly due to the institutional design where the (top 25) winners are chosen by voting by folks who obviously have vested interest in the outcome. And given the large number of college football team you get intrinsic intransitivity (haha - alliterative joke alert) - A beats B, B beats C, C beats A, so who's 1st, 2nd and 3rd?. As a result, much like refs in soccer, the NCAA voters choose teams who are "supposed to win" (or in this case, be ranked high) even when facts on the ground - given that there's room for interpretation - speak otherwise. So USC, Florida, Ohio State ... a few others ... are pretty much always guaranteed spots in Top 10/Bowls even if they get stumped once or twice by gutsy underdogs (which actually happens quite often in college football, which makes it more exciting than soccer (which is too random) and NFL football (which is too predetermined)).

2. I don't mean to imply above that there's a HUGE amount of corruption in soccer competitions, or in particular in regard to this particular game. I don't think the ref was "bought" or even incompetent or anything like that. I do think there is enough corruption in it, created by the nature of the incentives of the game as she is practiced today, and enough uncertainty which makes this corruption possible to pretty much make it a coin flip.

3. I got a post on the difference between games that we enjoy playing and watching and the games that game theorists analyze coming up, but I think I need to cool down a bit. Go outside and smoke three more cigarettes in a row or something. So later.

4. Sorry, Michael, though I don't know if this is your thing.

5. Also note that what American Football, Baseball and Basketball all have in common is that there's a lot more "points" scored in each game than in a soccer game. From point of view of Economics of Sports this makes a lot sense. If each "point scored" is a random draw from some distribution (which could favor one team over another) then with the absolute outcome of a game depending on a large number of points scored that outcome is more likely (by sort of wavy-handy law of large numbers) to reflect the "true" underlying skill of the two competing teams. Of course if the number of points required/time allowed is too large, then not only do we as spectators loose our attention span, but the outcome becomes too predetermined. You want some variance in there to make it exciting. But soccer's got too much of it.

Remittances

For a while there it seemed like "Remittances" were going to be the next big panacea in development economics. You know, you've all heard the stories about how Haiti's remittances are more than 100% of its GDP or how remittances trump the flow of private foreign aid in Uzbekistan, or how in Lilliputan nobody works anymore, they all live off remittances now.

And there's a lot of truth in that in the sense that it's definitely a phenomenon that is of a magnitude that begs for careful study. But honestly I've actually been a bit disappointed with a good bit of the relevant literature.

On one hand you've got the remittance cheerleaders who extol the great virtues of these cross border transfers. They're actually right to do so in fact but they tend to miss the point - I'll explain what I mean in a second, and in fact that is what this post is all about. On the other hand, it's almost too easy to say "hey, poor people are getting sent money now so they're better off". So people look for "counter intuitive" effects of remittance flows because "counter intuitive" effects are all the rage these days. As a result you get a somewhat sizable literature on the moral hazard effects of remittances. For example, this IMF working paper takes this position. The New Economist blog discussed similar results awhile back here (and btw that is a really good blog). The basic idea here is that when workers in destination countries send money back home to grandma in Azerbaijan, grandma doesn't work as much anymore... ha ha you foolish grandson, grandma got one over on you!

So let's dispose of the latter, "Remittances are bad" argument first since that's more straight forward. The basic problem here is that it doesn't pass the common sense test. Yeah, of course when folks send money to their grandma, she'll work less. That maybe the whole point. In fact, based on personal experience of a family which has sent much money back to the old world, this ain't "moral hazard". Quite simply, it's remittances doing what they're supposed to do. And while it's definitely possible to set up a model where it is moral hazard which leads to a decline in labor supply of remittance recipients, it is even more straightforward to set up a model where a fall in labor supply is the purpose that the remittance sender is trying to achieve. Since, empirically, all we observe is that (sometimes) labor supply falls when remittances go up there really isn't a way to distinguish between these two models and it certainly is no evidence for the "moral hazard" story.

Also, please note that the paper linked to above develops a model where an increase in the LEVEL of remittances is supposed to generate a decrease in the per capita growth rate (under some assumptions). Yet, the researchers find no evidence of that so they skip very quickly to a regression of the effect of the CHANGE in remittance flows on the level of growth. Which is a different thing all together. (pg 18 for model specification. I'm not sure what the offered explanation: "This specification better captures the dynamics of private transfers" actually means in this context. Table 4 shows the lack of relationship between level of remittances and growth. Table 5 shows the fudged model where there is a negative relationship between change in remittances and change in output (could it be that when output grows faster, wage differentials get smaller, so there's less of a need to send remittances?).

Ok. But what about the argument that remittances are all biscuits and gravy with tiny bits of sausage in it? Welllll, no. Sort of. I mean, yes, but, let's think about things more carefully here. (This is basically the gist of this WB report, which doesn't mean that agree with everything in it)

Bottom line is that most of the so called "gains from remittances" are straight up gains from IMMIGRATION. Or in other words, they are gains from the fact that some person from a poor household in a poor county has managed to make their way to a rich country and now has a richer income. Strictly speaking the gain from remittances is just the gain from INTER-HOUSEHOLD reallocation of income between the migrant and those who stay behind, not the overall increase in household income due to migration.

Let's illustrate this with some standard econ graphs. Take a household which is comprised of two individuals, say, brothers. And suppose that initially both brothers live in a poor country where they both earn the poor country wage of W(P). Now, we're going to take "household utility" as out metric here, and assuming that each brother's utility has diminishing returns to income and that each brother cares about the other we basically get that the two brothers should divide their total income (2*W(P)) equally between themselves (this is sufficient, not necessary. All we need is that there is some altruism between household members here). This is illustrated in a standard budget constraint/indifference curve below, where the utility of brother 1 is on the x-axis and the utility of brother 2 is on the y-axis;



Now suppose one of the brothers has the good luck to be able emigrate to a richer country where he earns the wage w(Rich). If there is no possibility of remittances here what is the highest utility level that the two brother household can attain here? Welp, it's represented by the indifference curve which crosses the new endowment point;



But we got that new blue budget constraint there - which illustrates the possible income/utility levels which are attainable IF there is a costless way to transfer resources from the lucky immigrant brother to the unlucky stay at home brother. Hence, if remittances are allowed the household will be able to reach a higher utility level as illustrated below;



Given all that we can actually decompose the benefit "from remittances" into that which is due to immigration of a particular household member and that which is due to, well, actual remittance flows:



All that basically means that the observed benefits from remittances that people rave so much about are mostly just straight up benefits from LABOR MIGRATION. Which are huge, but somehow that just isn't being said.

Now. I actually think the pure gains from remittances (moving from u2 to u3 in the graph above) are actually quite substantial as well. But it is important to keep two things in mind here:
1. These gains would not be possible without the migration of some household members in the first place. Hence, the ability to MIGRATE is a necessary condition for the REMITTANCE benefits to take place. The graphical analysis above misses this fact but it's there.
2. The size of the two effects are complementary. The bigger the gain from migration, the greater the disparity between those who migrate and those who stay behind. If there is diminishing returns to income, this means the gains from reallocation from the lucky migrants to those who stay behind are greater as well.

But all of this seems extremely simplified. I'm basically assuming that all these wages that people make are just consumption. But a strand of this literature emphasizes the fact that remittances can serve as basis for capital for households in poor countries. Something like; the lucky immigrant sends money to your grandma back in Lesotho and she invests that and starts her own business and boom, growth in Lesotho happens.

Again. There's some truth to that. But there's also problems with that kind of logic. First, we can take the above analysis and extent it to some kind of a dynamic setting where investment today pays of in growth in the future. But we're not missing much (I actually think there's all kinds of good reasons for economists to stay with static models - very often there's a simple way to reduce a dynamic model to a static one). So let's say that the sending of remittances is not just a pure transfer of resources but it also increases the income of the brother left behind. Maybe it provides funds for him to invest in a personal business etc. This is basically a shift/kink of the budged constraint above, and only a change in our labeling of our axis from "one period income" to "lifetime income":



So yeah, there's some additional benefits there to be had if the household members back home are more credit constrained (in very broad definition of that phrase) then the ... migrants in destination countries ... ? But wait a minute! Does that really make sense? After all, if anything, the return to investment, just like the return to labor tends to be higher in destination countries than in source countries. If the return in the destination country is higher then why shouldn't migrant invest in a business in his new found home, make more money, and then send it back? A hot dog stand in New York will make an immigrant more money than a similar business venture back in Nairobi. So why not invest in "capital" in the destination country for the household as a whole rather than using up resources in a relatively inefficient (compared to what you can get in US or EU) start up venture run back home? And then send the proceeds back so that all can enjoy it?

Again. There may be some truth to the view that remittances can facilitate investment back home. Usually a successful investment requires a good amount of local/tacit knowledge. Hence a newly arrived immigrant might not be able to take advantage of the investment opportunities in the host countries. You don't know how to speak English so you can't take advantage of the high rate of return to investment in US. But, if you send enough money back to Moldova, your relatives back home (who know the local language and economy well) will be able to turn that money into a productive business.

Except that the difference in rates of return is too huge to really make that story make sense. Again. Yeah, sometimes. For some people. Occasionally. But on average the brother who has been lucky enough to migrate is also likely to be the one lucky enough to have more and better investment opportunities.

And all that means is that most likely the majority of "remittances" flows that we observe are in fact meant to facilitate consumption. In other words, they're there for grandmas and brothers in poorer countries to be able to enjoy a standard of living which otherwise would be unattainable. And if you increase somebody's income they might work less (since there's no change in the wages they face this would be a pure income effect). But that's a good thing. In a similar way, the recipients may not spend their received remittances on investment (rationally, since the return to these investments in typical home countries is low) but instead buy themselves a cell phone or just some extra food. Is there anything wrong with that?

Finally, two caveats. First, you might actually see a spike in investment due to remittances if these are used to finance construction of new homes. I remember visiting a village in my own home country in the early 90's and it seemed like everyone there was building themselves a new house. The typical explanation offered by the relatives was "they got family abroad". But of course this is more of a consumption of a durable good rather than "investment" as that word is commonly used.

Second, there might actually be additional benefits from remittances (to consumption) which are not captured above. The graphs above all implicitly assume that both brothers face the same price levels in host and home countries. But of course the Balassa-Samuelson theorem (which is well supported empirically) tells us that price levels are lower in poor countries. Hence a dollar sent back to Bolivia, when converted to local currency, can go farther than if that dollar were to be spent in US. The graph below illustrates this:



(There's some caveats and details here - basically it matters how you label the axis (utility or income) and what is the exact shape of each individual's utility. I was too lazy to redraw the whole thing and basically it works out as illustrated above).

Oh and by the way, all the above generalizes easily to the case where the household welfare function assigns unequal weights to the two brothers, or where the household is comprised of more than two individuals, only few of whom are migrants, and other possible extensions.

Economists and their non-Economics hobbies

Keeping up with the theme of the last post.

So, Danny Quah kicks innocent wooden boards for the sake of global equality, and Kenneth Rogoff (former Chief Economist at the World Bank) is an International Grandmaster in Chess (and former under 21 US Chess Champion) (I recall seeing one or two of his games in the British Times a few years back as one of the "Chess puzzles"). William Baumol, as it turns out is a pretty good artist. Here's his cover of Thijs ten Raa's "The Economics of Input-Output Analysis" (thanks to Daniel for being enough of a pain to make me get that book):



Here's Baumol's art page. (www.econ.nyu.edu/user/baumolw) It loads weird, at least for me. Basically, you wait for it to load and then it disappears. But if you click on a particular (absent) image while it's loading you get it and then just navigate from there. I think the stuff is really good and it certainly looks fitting on an economics book cover - the one on the ten Raa book makes me think of people standing on the shoulders of other people in order to put something up, which makes me think of intermediate inputs into production and also fits in very well with the idea behind (some of) input/output analysis that ultimately labor is the only factor of production (i.e., as far as I can tell, none of them folks are standing on any machines, "neoclassical" capital). On the other hand some of the pictures on that website look like they loose something in the translation into a computer format or maybe Baumol got a little carried away with photoshop at some point.

The picture in the upper right of this blog, right under "About Me", is a portrait of Amadeo Modigliani. The non-economic Modigliani. Then came grad school, then came job, so maybe after tenure I'll have time to bust out the oil tubes again. Also, if you paint in oils you pretty much get it on everything, it doesn't come off, and you smell like turpentine all the time, which'd probably get you weird looks at conferences. Also at one point I had a 2000+ rating on Yahoo Chess, but that's Yahoo Chess.

Randy Wright, of the Search Models With Monetary Stuff In'Em fame, is a hesher and also has the Contractions.

Now everyone else needs to fess up. My suspicion is that Mark Thoma could track a butterfly in a snowstorm while living off of roots, berries and Oregon mushrooms.

Kung Fu Development Economics

I just wanted to link to this post by Danny Quah, which in addition to being really interesting and informative (and helpful when you get into internet fights with people who assert, with no backing data, that the growth in China and India "has only benefited the rich there and multinational corporations") also has Kung Fu Development Economics in it!

(Alright, alright, I don't know exactly which martial art Danny practices (actually I think it's Taekwondo) but Kung Fu Development Economics just sounds, you know, TOUGH.)

In fact, just in general I wanted to recommend Danny Quah's blog which is excellent and which as soon as I get around to updating my bloglist I'll add there along with a couple of others (and some are getting the boot!)

Update: Yes, it's Taekwondo.

Malthusian Simulation

The model in the simulation is as follows:

Per capita income at time t is



where A is land/technology, L is amount of labor and alpha<1. A grows at the rate g (possibly 0) so



Growth of population is births minus deaths or



where f is the crude birth rate (CBR) and m/m+y is the crude death rate. So here we're assuming that the fertility rate is independent of income (which isn't too much at odds with empirical evidence) and the mortality rate is a declining function of income. If per capita income is zero, everybody dies and if per capita income income is very very large then very very few people die. Each period.

That's pretty much the model and if we got numbers for the parameters we can simulate it. Some parameters, like the fertility rate we can get more or less from the data. Right at the start we can pick a value for alpha, or labor's share in output. Others we can pin down by using observed data and assuming that at particular points in time the economy is at its long run steady state. To do that we need to solve for it:

The change in per capita income from one period to next is given by






If technological progress is slow enough relative to rate at which the population adjustment takes place (whole another post - but the condition will be evident in a second) then in steady state per capita income will be constant even with on going technological progress. So dy=0, or



or winding that puppy backwards



where L(0) is the initial level of population (assuming we start out in steady state) and L(t) is our final level of population (assuming we end in steady state). So from that we can calculate/calibrate the annual growth rate of technological progress as



Then we calculate the steady state level of per capita income from the population growth equation:



solving for y (and dropping the time sub script since this is steady state per capita income) we have



where



from which you can also see the necessary condition for the steady state to exist (if g is too high then you will get ever increasing income) and with a bit of thought can also figure out the approximate rate of convergence to the steady state.

From this, if we know the steady state income at some point in time (basically you want to pick to points in time where income is the same and which are far enough apart so that any kind of transitory shocks, like, um, the Black Death, have time to sort themselves out) we can calibrate m, the mortality rate parameter. Alternatively if we observe a mortality level in what we think is a steady state year than we can calibrate the income. In practice the former turns out to be easier. In particular if we only care about the magnitude of changes rather the magnitude of levels (as in the previous post) we can just "normalize" the steady state level of income to 1 and see how much of a rise we can get in % terms with various shocks.

Then all that's left is to calibrate the initial level of technology which is easy if we have steady state income:



And that's pretty much it.

The nature of this calibration excercise and the model is that if we shock the model (by, say, killing off a third of population of England in 1350) and then leave it alone then it will be pretty good in matching the population data.



But this is pretty much by construction so it doesn't really constitute a "test" of the model. The problem, as noted in the post below, is that it is pretty much impossible to match the 100% increase in per capita incomes (or wages, but that might be even worse) that occurred between 1350 and 1480. Here's what it looks like with just an initial shock of wiping out a third of the population:



The fact that in the model the shock has an immediate impact on incomes while this doesn't happen until 130 years later historically is not that important. You know, sticky wages, market frictions and all that, means that we shouldn't expect an immediate adjustment. The important aspect is that at its peak income rises only 14.5% above its steady state level, rather than the 100% seen in the data.

Like I said below, you can try to mess with it in order to get those higher incomes;
1. Increase mortality rate or decrease the fertility rate. But then you'll screw up that nice match up of population data in the above graph, winding up with way too few people in 1570 and on.
2. Increase the technological growth rate. But then you'll screw up that nice match up of population data in the above graph, winding up with way too many people in 1570 and on.

Of course you can try a combination of these to try to match it up. Or play with the other parameters. That's why here is the Excel file used for the simple simulations (Thanks to Gabriel).

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



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)