Gregory Clark on the Malthusian Trap

I've really enjoyed reading Gregory Clark's book A Farewell to Alms. It's been on my must-read-soon list for a long time, but I finally got to it over the last couple of weeks. I'll post a proper review of the book tomorrow, but in the meantime I wanted to focus on Clark's exposition of the Malthusian Trap - an explanation of why income per capita barely changed between ancient times (or even earlier) and the start of the Industrial Revolution. I've been a critic of Malthusian ideas in the past (e.g. see this post), so reading about this model was a good learning experience for me.

The basic model is laid out in the diagrams below (Clark credits this diagrammatic representation to a chapter by Lee and Schofield in this 1981 book). The x-axis on both diagrams is income per person. The top diagram shows the relationships between birth rate (BR) and income per person, and death rate (DR) and income per person. Notice that the birth rate increases with income per person. Higher incomes lead to more births on average. Notice also that the death rate decreases with income per person. Higher incomes lead to fewer deaths. That all seems rather intuitive. When the birth rate is equal to the death rate, the population is in a steady state (neither growing nor shrinking). That happens at an income per person of y*, and a population of N.

The bottom diagram shows the relationship between population and income per person, and this is where things get interesting. If income per person was to increase above y*, then we end up at an income per person of say y0 (with population N0). At that level of income per person, the population starts to increase over time (because in the top diagram, the birth rate is now higher than the death rate). And as population increases, it pulls income per capita down on the bottom diagram, until eventually the economy ends up back at the steady state income per person of y*.

The same thing happens in reverse if income per capita were to fall below y*. In that case, the death rate would exceed the birth rate, and the population would decrease over time. And as the population decreases, it pushes income per capita up until the economy ends up back at the steady state income per person of y*.

You might wonder why income per person and population are negatively related. Clark explains that it relates to diminishing marginal productivity of land:

In the preindustrial era land was the key production factor that was inherently fixed in supply. This limited supply implied that average output per worker would fall as the labor supply increased in any society, as long as the technology of that society remained unchanged. Consequently average material income per person fell with population growth.

In other words, because each additional person employed on the land would be less productive than other workers already working the land (diminishing marginal productivity), average productivity per worker would fall as more workers worked the land, and so average incomes would decrease as well.

And so, because the economy would always end up eventually back at the steady state income per capita of y*, income per capita remained fairly constant for thousands (or even tens of thousands) of years. Clark notes that:

English wages in 1800 on average were about the same as those for ancient Babylon and Assyria, despite the great technological gains of the intervening thousands of years.

However, the model is not just interesting for that fact. You can also use it to show some fairly perverse effects that the Malthusian Trap had on society. Consider what would happen if the general level of health in the population improved (such as through improved sanitation). As shown in the diagram below, that would imply a decrease in the death rate at every level of income - the death rate shifts down from DR0 to DR1. Now at the previous steady state income per person y0* (with population N0), the birth rate is higher than the death rate. The population will start to increase, and as a result income per person falls, eventually reaching the new steady state income per person of y1* and a higher population of N1. The improved sanitation perversely leads to lower income per person - it makes people worse off! As Clark notes:

This Malthusian world thus exhibits a counterintuitive logic. Anything that raised the death rate schedule - war, disorder, disease, poor sanitary practices, or abandoning breast feeding - increased material living standards. Anything that reduced the death rate schedule - advances in medical technology, better personal hygiene, improved public sanitation, public provision for harvest failures, peace and order - reduced material living standards.

On the other hand, consider the impact of an improvement in production technology that raises income per person at any level of population. This shifts the curve in the bottom diagram to the right, as shown in the diagram below. Society would temporarily be at a higher income per person y0 (with population N0), but since at that level of income per person, the birth rate exceeds the death rate, population would increase, with the economy eventually ending up back at the steady state income per person y*, but with a higher population of N1. As Clark notes:

In the preindustrial world, sporadic technological advance produced people, not wealth.

This model is very cool. I wish there was time for me to include it in the first week of my ECONS101 class, as a way of motivating the substantial change in income per person that arose after the Industrial Revolution. Indeed, that is what Clark uses the model for in his book (but more on that tomorrow).

The diminishing marginal returns to classroom experiments

I'm a big fan of using classroom experiments to illustrate concepts to my students. When I taught managerial economics, I specifically organised the workshops and assessment to incorporate many experiments. I use a number of experiments in ECON110, which are worth extra credit for students.

So I was interested to read this recent paper by Tisha Emerson and Linda English (both Baylor University), published in the Papers and Proceedings issue of the American Economic Review (sorry I don't see an ungated version anywhere). In the paper, Emerson and English look at 28 principles of microeconomics classes from Baylor between 2002 and 2013, some of which used experiments (between 6 and 11 experiments) and some of which had no experiments (and act as a control group). Looking at overall performance in the course, they find:

...a statistically significant positive relationship between the number of experiments in which a student participates and their course score. The impact is diminishing, however, as the number of experiments increases.

The impact of experiments is different for different groups:

While older students outperform younger ones, the youthful disadvantage is partially offset by participation in experiments. Similarly, our findings suggest that an achievement gap exists between whites and ethnic minorities, but that experiments serve to help bridge this gap as well. However, experiments do not appear to differentially impact student performance by gender, aptitude, or attendance.

So, the positive effect of classroom experiments is largest for the first experiment, and then diminishes. We can use the coefficients from Table 2 in the paper (shown below) to calculate the optimal number of experiments (see [*] for one example of the calculus if you're feeling adventurous) - that is, the number of classroom experiments that would maximise grades for a student with a given set of characteristics. The optimum differs by group (because of the interactions between the number of experiments and gender, age, nonwhite, SAT total, and absences from class).

For a 19 year old white male (with the mean SAT of 1171 and the mean absences of 2.59), the optimal number of classroom experiments is 5.3, while for a 19 year old nonwhite male (with mean SAT and absences) the optimum is 7.5, and for a 19 year old nonwhite female (with mean SAT and absences) the optimum is 8.0. Interestingly, the optima are much lower for older students. For example, for a 23 year old white male (with mean SAT and absences), the optimum is -0.4. So, most of the gains from classroom experiments in this sample accrue to younger and nonwhite students.

Anyway, overall this tells us that classroom experiments are likely a good thing, and are likely an even better thing for traditionally disadvantaged groups. Indeed, the authors conclude:

...experiments may be providing market experiences that these groups (younger and minority students) may not otherwise have had and thus greater understanding of economic concepts that can’t be gleaned simply from class discussion. As such, the use of classroom experiments may help reach these otherwise disadvantaged groups.

So I guess I should persist with classroom experiments, just not too many!

*****

[*] Using the coefficients for a 19 year old white male (with the mean SAT of 1171 and mean absences of 2.59):

G =  (9.948 - 0.171 - 0.444 * 19 - 0.000186 * 1171 + 0.201 * 2.59) x - 0.154 x^2 + z
    = 1.644 x - 0.154 x^2 + z

where G is the student grade (percentage), x is the number of experiments, and z is all of the other parts of the regression equation that don't include the number of experiments.

Differentiate the function G with respect to x:

dG/dx = 1.644 - 0.308 x

Set this equal to zero, and solve for x:

1.644 - 0.308 x = 0
x = 5.3

So, the percentage grade is maximised for students with these characteristics when they participate in 5.3 experiments. Similar calculations can be made for other combinations of characteristics.

Two books on military and economic history

Earlier in the year I read two books that related to military history, and I've been meaning to write a little bit about them. I've always enjoyed reading about historical events, especially ancient and medieval history. So, I was particularly excited about the first book, Castles, Battles and Bombs: How Economics Explains Military History, by Jurgen Brauer and Hubert van Tuyll. I think I bought this book through a random Amazon recommendation. The second book was Blackett's War: The Men Who Defeated the Nazi U-Boats and Brought Science to the Art of Warfare Warfare, by Stephen Budiansky. I forget who suggested this book to me, but I'm glad they did.

I've found a lot of history books to pretty dry. On the other hand, being able to apply economic theory to history is usually pretty cool. Castles, Battles and Bombs belongs to the dry category, unfortunately. The authors set out to focus on a number of different economic theories, and use one historical case study to illustrate each one: castle building in the High Middle Ages to illustrate opportunity costs; the Condottieri in the Renaissance to illustrate principal-agent problems; the decision to offer battle in the 17th and 18th Centuries to illustrate marginal analysis; the American Civil War to illustrate information asymmetry; strategic bombing in World War II to illustrate diminishing marginal returns; and nuclear armament in France to illustrate substitution. Don't get me wrong - there are lots of really interesting bits, and I got a lot of inspiration for examples to use in next year's ECON110 class. However, the writing probably isn't for a generalist audience - it kind of feels like six journal articles pulled together into a book, with an introduction and concluding chapter to weave it all together.

One part in particular is interesting though, in the chapter on diminishing marginal returns in World War II bombing. Diminishing returns occur when each additional unit of input leads to a lesser amount of additional output. Using data from the USSBS (United States Strategic Bombing Survey), Brauer and van Tuyll show (to take some selected data from their Table 6.3), in September 1944 17,615 tons of bombs dropped on Germany led to a reduction in railroad movements of 3519 ton-kms. In December 1944, 61,392 tons of bombs reduced railroad movements by 6377 ton-kms. So, a 250% increase in bombing led to a 81% increase in damage. Their data are more extensive than that, but that gives you a flavour of how quickly the returns from bombing diminished.

Blackett's War also covers the bombing campaign in World War II, but from an operations research point of view. Operations research is the use of advanced analytical methods (i.e. mathematical and statistical analysis) to decision making. Obviously this is similar to economics, and a lot of economists are involved in operations research today. In the book, Budiansky extensively covers the genesis of operations research, and does so in a way that is both interesting and accessible to ordinary readers. The story follows the physicist Patrick Blackett and a cast of British (mostly) and American scientists, and their contributions to World War II, particularly focusing on how they contributed to the U-Boat war in the North Atlantic.

The book also covers the ineffectiveness of strategic bombing. Budiansky writes:

Blackett talked to Bernal around the same time and did a few pages of penciled arithmetic. In the ten months from August 1940 to June 1941, the Luftwaffe had dropped 50,000 tons of bombs, an average of 5,000 a month. The number of civilian deaths was 40,000, or 4,000 a month; that worked out to 0.8 persons killed per ton of bombs...

He also calculated that the total decline in industrial production from the air attacks on Britain was less than 1 percent: factory output for the country in April 1941 had been affected more by the Easter holiday than by the German bombs that fell during the month.

Blackett argued against strategic bombing, but his arguments fell on deaf ears. However, eventually the operations research prevailed and air attacks on u-boats in the North Atlantic became much more effective as a result.

Overall, Budiansky's narrative style is easy to follow and the choice of anecdotes brings the time period to life. I highly recommend Blackett's War, and I'll certainly be looking to populate my bookshelves with some of Stephen Budiansky's other books, to read in the future.

Why kids should play more Minecraft, but only on PC

Video games are often portrayed as dulling the senses and making gamers dumber. However, a recent paper in the journal Economic Inquiry (gated, earlier ungated version here) by Agne Suziedelyte of Monash University might have just busted that myth.

Suziedelyte argues that:

In order to win a video game, players need to plan their actions, find relevant and discard irrelevant information among all information given to them, and remember their previous actions. Thus, video game playing is most likely to improve such skills as problem solving, abstract reasoning, pattern recognition, and spatial logic, which are part of fluid, or general, intelligence...

In the paper Suziedelyte uses data from the Child Development Supplement to the Panel Study of Income Dynamics to investigate "how game playing affects two skills, the ability to solve practical mathematics problems (mathematics reasoning) and the ability to correctly read English words (reading recognition)". She employed a variety of techniques to overcome endogeneity (time spent playing video games might be correlated with other determinants of cognitive skills), and measurement error in the time spent playing video games.

While similar to the paper I discussed yesterday where the sign on some statistically insignificant coefficients was discussed (when they are not different from zero), in the preferred model specification an additional hour spent playing video games was found to be associated with mathematics reasoning tests scores that were on average higher by 9.3% of a standard deviation. To put this in context, the effect was similar in magnitude to an additional hour spent on "educational activities" (like school or homework), which was associated with scores on average 9.1% of a standard deviation higher. Importantly, video games had no statistically significant effect on reading recognition (a placebo test), which suggests that more video gaming affects problem solving specifically and not cognitive skills more generally (and allays concerns about omitted time-varying variables driving the results).

Before you encourage your kids to spend all day constructing elaborate replicas of Hogwarts using Minecraft, consider these two additional results. First, there were diminishing returns to time spent on video games:

The estimated effect on an additional hour of video game playing on mathematics reasoning ability is largest (21.7% of a standard deviation) when a child does not play any video games... Video game time is found to no longer affect mathematics reasoning ability, when the number of hours played reaches 5.5 hours per day.

So, in other words limit children's video game time to most of their waking, non-school hours. Second:

The effect of computer game playing is estimated to be 6.2% of a standard deviation (significant at the 1% level), whereas the effect of console game playing is smaller (1.1% of a standard deviation) and not significantly different from zero. The difference between the two effects is statistically significant at the 5% level.

So, make sure your children are playing Minecraft on PC, not on XBox.

[HT: Marginal Revolution, back in April]