Convergence Conditional On What?

About two months ago, I wrote a post about the weird lack of economic convergence in the G7 -- Canada, France, Germany, Italy, Japan, the United Kingdom, and the United States. Neoclassical growth theory suggests that countries with similar economic institutions, levels of investment, and population growth (e.g., the G7) should all converge to similar levels of labor productivity. The fact that this doesn't happen, or has stopped happening for different countries at different times (Canada in the 1980s, Japan and Italy in the 1990s, the UK since 2010), flies in the face of the 'absolute convergence' hypothesis.

Unlike the evidence against absolute convergence from other countries, however, members of the G7 haven't been diverging from each other in terms of GDP per capita over the last few decades, per se. Instead, there seems to be something affecting the level of each countries productivity over time. In terms of GDP per hour worked, the UK is consistently around 75% as productive as the US, Japan is about 65% as productive, and Germany and France roughly match the US. The Solow model does have predict some variability in productivity levels based on the savings rate, the population growth rate, and depreciation. However, all of these differences should be controlled for if you look at total factor of productivity, which is what I tried to do in my previous post. Long story short, the Solow model fails almost entirely in explaining productivity differences across the G7.

Economists from the '90s are probably screaming right now that I should look at human capital and test for convergence conditional on that. My issue with looking at human capital is that 1) nobody even knows what it is and 2) the way it's used in theories is really strange. For instance, in the paper I linked above, Mankiw et al famously model human capital as just another kind of capital in a Cobb-Douglas production function and then assert that it accumulates in the same way are normal capital, i.e. through saving/investment. They then assert that the savings rate for human capital is equal to the share of the working age population that is in school (even though in their model human capital investment is a portion of income).

Still, the idea of conditional convergence is intriguing, so why not regress growth from 1981 to 2015 on the initial level of productivity and some crude measure of human capital (percentage of population with a post-secondary degree)? For the heck of it, let's throw in R&D spending as a percentage of GDP.

Interestingly enough, the fit is actually astonishingly good, especially considering that I only used data from seven countries. I was also surprised to see that the (statistically insignificant) coefficient on tertiary degree attainment is negative. There's probably a better measure of 'human capital', but I still think there's something strange about using anything as a proxy for an inherently unmeasurable theoretical construct.

If you exclude the education variable, then the results are as follows:

The coefficient of R&D spending does change, but the main result here is consistent: strong support for the hypothesis that productivity in the G7 converges conditional on R&D spending. The equation that I tested was

growth from 1981 to 2015 = a * GDP per hour in 2015 + b * average R&D spending from 1981 to 2015 + c

All values are relative to the US, so you can predict the relative level of GDP per hour in the long run by plugging in a value R&D spending and then solving for GDP per hour. Assuming a level of R&D spending equal to the US, the regression predicts complete convergence: 43.1548/0.4155 ≈ 104% of US productivity.

Alternatively, you can compare actual productivity in 2015 to productivity predicted by the model and a counterfactual with R&D spending equal to the US:

Country	Productivity	Prediction	Counterfactual
Canada	76.915034	81.727441	94.292790
France	94.021332	89.476058	95.644441
Germany	93.728856	93.321678	94.395491
Italy	75.604129	75.233253	96.770322
Japan	65.538401	66.124831	63.949169
United Kingdom	75.644250	73.969879	83.190180
United States	100.000000	101.604800	101.604800

This approach predicts actual productivity levels surprisingly well (removing R&D spending from the regression reduces the r-squared value to 0.522). It also seems to suggest that Japan's low productivity levels should be of no surprise; it just started from such a low level that we should expect it to take a while longer to converge completely. Similarly, Canada, Italy, and the UK look to be suffering simply from low R&D spending.

Even though R&D does appear to explain productivity differences in the G7, I am hesitant to say I've solved my puzzle yet. Looking specifically at Japan, it's period of high growth occurred when R&D spending was low, and subsequently higher R&D spending has coincided with lower growth. Another problem is the tiny sample size. Seven countries is far too small to make any serious conclusions (if I calculated correctly, the standard deviation on the estimate for the amount of convergence in the long run is 35.2).  So I end this post barely more sure of what causes productivity differences between developed countries than I was two months ago, but maybe a slightly better idea of where to go from here.

Are there scientific facts?

The response to this tweet has been so large that it even touched my economics-packed Twitter feed (via Noah Smith). Noah has a relatively reasonable response to the idea that "scientific facts are social constructs," although I am still annoyed about his misguided use of the word "anti-rationalism" to describe anti-intellectual or anti-science arguments.

Noah's ignorance of philosophy is fitting, because I (unlike Jason Smith, who called the ensuing argument an "utter philosophical mess") think that most people have failed to understand that this argument is really all about philosophy; specifically, philosophy of science. I think both Jason (and somewhat Noah) focused too much on the word "social" and too little on what "social" and "constructs" mean together -- namely that scientific facts (which are really theories or their predictions) are socially constructed and therefore do not refer to things in the real world.

This is scientific anti-realism, or the idea that scientific theories don't or can't interact with reality. It's probably better to look at anti-realism through the lens of scientific realism. A scientific realist would say that scientific theories are either true or will eventually converge to truth given enough time and resources. Anti-realists are skeptical of the truth of scientific facts for various reasons -- for instance, the anthropology professor would probably say that society influences the scientific process such that the results it achieves are not true. Thomas Kuhn (or at least many readers of The Structure of Scientific Revolutions) would argue that "scientific facts" are inherently influenced by the prevailing paradigm.

The folly of scientists in this case has been to unwittingly invoke their realism in attempts to mock anti-realists. Neil deGrasse Tyson's tweet was particularly ironic:

Dr. Tyson argued against the mere assertion of scientific anti-realism by asserting his own philosophical position (without making any arguments for it, I might add).

Jason concedes that scientific facts in the past have been social constructs:

One of my [favorite examples] is the aether. That was a "scientific fact" that was a "social construct": humans thought "waves" traveled in "a medium", and therefore needed a medium for light waves to travel in. This turned out to be unnecessary, and it is possible that someone reading a power point slide that said "scientific facts are social constructs" might have gotten from the aether to special relativity a bit faster.

Another example is the geocentric model of the solar system, which is relatively empirically accurate yet no longer accepted as an explanation for planetary movements. The anti-realist argument is simply that there is no reason to believe we are completely right this time, let alone that we will eventually be right about everything.

The problem with Jason's blog post is that he did precious little to defend his scientific realism before asking if we can "get away from the philosophical argy bargy." As the only economics blogger I can remember referring to Popperian falsificationism, I am more than a little bit disappointed (and frankly annoyed) at his dismissal of the philosophical argument here.

Personally, I don't even agree with "scientific facts are social constructs." I just hate the arrogance of people in their ignorance. There are better arguments for why science is worthwhile, and reasonably objective than just assuming that to be the case. My favorite is pragmatism (which should appeal to the effective-theory-espousing Jason Smith), which basically sees the empirical successes of science as reasons for us to act as if scientific realism is true.

I think Pierce's formulation of truth, while being as far from succinct as it is possible to be, is a good description of a pragmatic view of science:

Truth is that concordance of an abstract statement with the ideal limit towards which endless investigation would tend to bring scientific belief, which concordance the abstract statement may possess by virtue of the confession of its inaccuracy and one-sidedness, and this confession is an essential ingredient of truth.

Basically, truth (in the pragmatic sense) is what science would come up with given infinite resources and infinite time.

Honestly, though, any defense of scientific realism (even Noah Smith's argument that anti-realism is counterproductive because it fosters anti-rationalism -- more accurately anti-scientific beliefs) is better than "but scientific facts exist because we have smart phones."

Economic Growth is All About Increasing Returns to Scale

Jason Smith has written a response to my previous post in which he brings up a few interesting criticisms of growth economics. Namely, he questions the attachment to constant returns to scale in the Solow model, which made me realize (or at least clarified my thinking about the fact that) growth theory is really all about increasing returns to scale.


The original aim of neoclassical growth theory was to provide a rudimentary explanation for why some countries are poorer than other, or really why some countries produce less output per capita than others. Income differences can be explained by 1) differences in the skills of workers in each country and 2) differences in the amount of capital per worker (or per hour worked).

This is because the factors of production (ignoring land) are generally considered to be capital (e.g. tools, machines, or computers) and labor, but the conventional theory of production presents a problem: economists like to assume constant returns to scale so that doubling the factors of production will double output. As Smith admits in his post, this intuitively makes sense, at least when dealing with real quantities:

Constant returns to scale is frequently justified by ‘replication arguments’: if you double the factory machines (capital) and the people working them (labor), you double output. Already there's a bit of a 19th century mindset going in here: constant returns to scale might be true to a decent approximation for drilling holes in pieces of wood with drill presses.

The problem with this formulation is that economic growth with constant returns to scale is impossible because you can never increase output by more than the amount of increase in inputs. More specifically, if you adopt a production function with constant returns to scale, e.g. Solow’s $Y = K^\alpha L^{1-\alpha}$, then
$$\frac{dY}{Y} = \frac{\partial K}{\partial Y}\frac{dK}{Y} + \frac{\partial L}{\partial Y}\frac{dL}{Y}$$
 Which is
$$\frac{dY}{Y} = \alpha \frac{dK}{K} + (1-\alpha)\frac{dL}{L}$$
Since $0 < \alpha < 1$ by assumption, growth in output ($\frac{dY}{Y}$) will always be less than the growth in capital or labor. This means that the only two ways to have exponential growth with constant returns to scale are 1) have labor grow forever (resulting in infinitesimal output per capita) and 2) have capital grow forever (resulting in an infinite capital to income ratio).

Obviously the first option is inconsistent with exponential growth in GDP per capita, so we can reject it immediately as an explanation for economic growth while the second option implies infinite capital accumulation, which won’t happen because (since capital depreciates over time) that would imply an increasing share of income going to savings over time.

The solution to this problem is to add some mechanism for increasing returns to scale. The Solow model leaves this process implicit — much to Smith’s chagrin — by calling it technological progress and assuming constant growth but the rest of growth theory is just attempts to augment production to allow for increasing returns to scale.

The simplest way of doing this, which is similar to what Smith does near the end of his post, is to assume that Solow’s Total Factor of Productivity is just some function of capital and labor. This is the logic behind the AK model, which takes the neoclassical production function $Y = BK^\alpha L^{1-\alpha}$ and assumes $B = AK^{1-\alpha}L^{\alpha-1}$. Plugging $B$ in results in
$$Y = AK$$
Other models are more sophisticated; they try to add things like human capital or research and development. But the underlying principle remains the same: growth theory is basically about finding ways to justify increasing returns to scale. Smith’s approach (ignoring his focus on nominal values) is just a much more explicit way of adding increasing returns to models. In this sense, Smith is right that the original assumption of constant returns to scale “leads to the invention of "total factor productivity" to account for the fact that the straitjacket we applied to the production function (for the purpose of explaining growth, by the way) makes it unable to explain growth.” The real difference is that economists want to model the underlying process that allows for increasing returns while Jason is content with allowing increasing returns to scale from the get go.


Update: I know the AK model is really just constant returns to scale for capital, but the real point is that, for sustained economic growth, there cannot be decreasing returns to scale for a non-labor factor of production. Otherwise, output per worker can't increase along a balanced growth path (which is when the other factor(s) of production don't grow faster or slower than output in the long run).

The G7 Productivity Puzzle

With the exception of the US (and maybe Canada and Germany), all of the countries in the G7 have pretty similar levels of GDP per capita. In constant price PPP terms, Japan, France, and the UK are all around 38,000 USD, while Italy is a bit lower at 34,000 USD (the Great Recession really hurt Italy, which has also been in a long term decline for a couple decades), and Germany and Canada are both about 44,000 USD.

This clustering in GDP per capita strikes me as a little bit strange, since employment-population ratios and average hours per employee vary drastically across countries. People in France work a lot fewer hours than their neighbors across the English channel for the same amount of output, while people in Japan work infamously long hours and seemingly get nothing out of it.

I guess I would expect labor productivity (GDP per hour worked) to be differ between countries a little bit, but it seems strange to me that unconditional convergence hasn't even held between western democracies. America seems to have some magical ability to be more productive than every country but France and Germany, and the UK is inexplicably much poorer than its European neighbors. Italy has understandably been a mess since the days of Silvio Berlusconi, but Japan's perennially low productivity does not match its reputation as a paragon of efficiency.

I also tried adjusting for the size of the capital stock in each country, with little success. First, I assumed output in each country of the G7 is produced with a typical neoclassical/Cobb-Douglas production function, i.e.
$$Y_t = A_t K_t^\alpha H_t^{1-\alpha}$$
where $Y_t$ is GDP, $A_t$ is total factor of productivity (TFP), $K_t$ is capital, and $H_t$ is total hours worked. Working with the somewhat unrealistic assumption that $\alpha$ (capital's share of output) is constant at $0.34$ for all of the G7, I calculated TFP by dividing $Y_t$ by $K_t^\alpha H_t^{1-\alpha}$[1]:

I didn't include Germany or Italy in the chart because they are mostly similar to France (Italy started falling to the level of the UK and Japan in the mid-nineties, though) and they crowded out the more interesting information. Looking at TFP instead of hourly output presents about as many questions as it answers. First, it becomes clear that Canada, France, and Germany are equally as productive, but that Canada and Germany do a much better job of ensuring full employment than France does (this isn't just a difference in hours either, Germans work fewer hours per year than French people). But why did Canada slow down relative to the US in the eighties? And what the heck is going on with Japan here?

My partiality to the UK makes me happy that at least Britain is better off than Japan (incidentally my second favorite country in the G7), but I'm a little bit skeptical that even two world wars and decolonization made the UK lose 40% of its productive capacity relative to the United States, especially when the end of French empire didn't have the same consequences. Also, everyone talks about Japan's lost decade starting with the recession in the late 1990s, but the decline seems to have actually begun in the seventies, and the cause completely evades me.

Whenever I read about growth/development economics, it's usually taken for granted that America is at the technological frontier and that explains its unusually high productivity, but Canada's relative smallness looks like the only thing that could prevent us from giving it that title, at least from 1950 to 1980.

Unfortunately most of the growth/development economics research I encounter doesn't really care about 30% gaps between rich countries, but instead (and probably rightfully so) focuses on the 90% gaps between rich countries and poor ones -- heck, I'm looking into those gaps in Vietnam, Cambodia, Laos, and China for my school's version of a senior thesis. Maybe there is no real good explanation for why Japan or Britain haven't converged to America's level of GDP per capita in the 70 years since World War II, or why Canada has been losing ground for the last 30 years. It just really, really bothers me that there isn't.


[1] All the data I used comes from the Penn World Tables. Capital is the capital stock at constant national prices, GDP is GDP at constant national prices (I would use PPPs, but capital is only available at constant national prices and current PPPs, and the data seem to match World Bank's idea of constant PPPs in that they are at PPP for the base year), and Hours is the product of average hours worked per person engaged and total persons engaged.

Econ 101 should at least do math right

This is a small break from my normal type of post, but I've become a TA for my school's AP economics (AP = Advanced Placement, for those unfamiliar with the American and Canadian education systems) course, which has left me with a couple of takeaways:

1. Calculus should be a prerequisite for economics
2. AP Econ/Econ 101 resorts to a lot of inconsistent nonsense in order to explain things to people who don't understand calculus

To remedy my annoyance at introductory economics (which I have confirmed from friends taking the same course at other schools universally explains the demand curve differently from the way it should/the way that is mathematically consistent with the rest of Econ 101), I decided to write down a derivation of the Econ 101 demand and supply curves in consistent way.

Under normal circumstances, I would probably rather criticize the theory for being unrealistic, but being clear about the math going on behind the scenes is all I choose to care about for the moment.

I should preface the math with an explanation of the way Econ 101 usually deals with the demand curve:

There are a lot of people who come to a market that sells one item. Each person is willing to buy the item at any price lower than some arbitrary price, so if the owner of the market comes out and declares a high price, relatively few people will buy the item. Similarly, if the owner declares a low price, many people will buy it.

This explanation results in a weird demand curve with 'steps' at different prices whose width is determined by the number of people with their maximum price at that level. This is entirely different from the smooth curves instructors like to draw to illustrate demand, and inconsistent with the math used when teaching firm behavior (marginal revenue doesn't make sense when the demand curve is a bunch of steps).

Anyway, this is how Econ 101 students (with at least an understanding of derivatives) should be taught supply and demand:

Demand Curve Derivation

Consumers derive a certain amount of utility when they buy units of the good. This utility can be expressed as the function U(Q) where U stands for utility and Q is the quantity of the good that consumers purchase.

Consumers pay the same price for each unit of the good that they buy, so their total cost is PQ where P is the price of the good.

People want to maximize the net benefit they derive from buying units of the good. Mathematically this means maximizing U(Q) - PQ.

We know from calculus that setting the derivative to zero will give us the maximum, so the net benefit maximizing quantity satisfies

U'(Q) - P = 0 or U'(Q) = P

This is the demand curve. The reason it is downward sloping is because of diminishing marginal utility -- the notion that each additional unit of the good is less valuable than the last. This means that U'(Q) is a negative function of Q, necessitating a downward sloping demand curve.

Supply Curve Derivation:

Firms want to maximize profits, which are defined at their total revenue (PQ) minus total costs (C(Q)). They do this given what they know about the demand for their product, so they replace the P in PQ with U'(Q) from the demand curve. Thus, firms maximize

U'(Q)Q - C(Q)

meaning that

d/dQ U'(Q)Q - C'(Q) = 0
which is the same as
U''(Q)Q + U'(Q) - C'(Q) = 0

The supply curve needs to be written as a function of P, so we can just substitute P in for U'(Q) above, yielding

P = C'(Q) - U''(Q)Q

This is the supply curve.

Let's derive the demand and supply curves given example utility and cost functions:

U(Q) = aln(Q)

C(Q) = 1/3(Q-b)^3 - cQ^2 + dQ

In this case, the demand curve should be
P = U'(Q) = a/Q
and the supply curve should be
P = C'(Q) - U''(Q)Q = (Q-b)^2 - 2cQ + d + a/Q

This example gives fancy curves similar to those you might draw as examples, but a simpler example does a better job of showing the types of the linear curves you might see in econ 101/AP Micro

U(Q) = Qa - 0.5bQ^2

C(Q) = Q^2 + cQ

This gives the demand curve
P = a - bQ
and the supply curve
P = 2Q + c - bQ = (2-b)Q + c

Automation and Job Loss

The prospect of automation, or more generally huge productivity improvements in different sectors of the economy, has a lot of people worried that millions of people will lose their jobs over the course of the next century. What will all the taxi drivers do, the reasoning goes, when driver-less cars are perfected? Alternatively, what happens to all the manufacturing workers when automation makes their jobs obsolete?

This line of thinking has a serious problem: it assumes that aggregate demand for goods and services remains constant in the face of productivity improvements. Normally this won't be the case, because people generally want more, or at least better, stuff. If productivity improvements mean that society can now produce a 4k TV with half the amount of labor as it could two years ago, people will probably start buying more 4k TV's. Of course, some goods are inferior goods (people buy less of them as their incomes increase), but in aggregate Say's Law -- that supply creates its own demand -- seems to ring true, at least in the long run.

Maybe, at some point in the future, economy-wide productivity will be so high that people consciously choose to work fewer hours, or some parents will choose to stay at home instead of work full-time, but this would be nothing to worry about. In this case, lower employment is just the consequence of people acting in their best interests. With much higher wages, people can afford to spend more time doing leisurely activities, which they very well might prefer to more income.

Fear about automation is not entirely unfounded, though. In many industries, such as manufacturing, demand really does reach a ceiling -- each person only wants to buy so many refrigerators, for instance. This is part of the reason that manufacturing employment has fallen from over 17 million in 2000 to about 12 million last year. At some point, demand for certain goods and services stops growing with income.

People who formerly had well-paying manufacturing jobs might be forced to take a low-paying service sector job, meaning that they will end up with a real pay cut while most consumers reap the benefits of cheaper manufactured goods. At this point, though, the problem is no longer about people losing jobs; it's about distribution of income. Policies that increase incomes for people who work in the service sector -- whether they take the form of direct transfers, minimum wage increases, or something else -- would go a long way toward solving the problem posed by technological enhancements or productivity growth.

Needless to say, at least for the foreseeable future, automation need not necessarily be that big of a concern. We shouldn't worry about millions of people losing their jobs; they will probably find work elsewhere. Instead, we need to make sure that no one is left behind as we steadily proceed toward a world without scarcity.

The Price of Health Care

Even if you are only a little bit familiar with different health care systems in the world, you probably know that America spends more on health than any other country in the OECD in terms of both per capita and percentage of GDP. With such high spending, you would expect outcomes -- such as life expectancy or amenable mortality (basically preventable deaths) -- to be much better than other countries that spend less. Strangely, as data from a recent paper on the German health care system shows, this is not the case.

In spite of massive spending increases and a relatively high baseline in 2000, the US remains significantly behind other developed countries in terms of preventable deaths. On top of this, the improvement in amenable mortality for each dollar of new spending is a lot lower than the other countries.

This is where purchasing power parities (PPPs) come in. High prices for various health care related goods and services such as prescription medication or MRI scans could explain much of America's elevated health care costs, rather than high quantity/quality of care. If this were the case, that would explain why American health care spending continues to rise rapidly without significant improvement in outcomes.

Finding PPP data for different countries would shed light on this because it would give us a good comparison of the quantity of health care that each country consumes as opposed to the amount of money it spends. If the quantity of health care per capita in the US was similar to or less than other countries, then that would explain the lackluster outcomes it experiences.

Until recently there was no data that I could find for health care specific PPPs outside of Europe, but apparently in May the OECD and Eurostat published a report that updated the previous estimates with data from the US and a few other non-European countries. Figure 4 in the report shows that higher prices explain some, but by no means most of all, of the discrepancy between outcomes and spending in the US health care system.

Alternative explanations as to why quality of health care lags spending so much in the US are necessary. Wasteful spending brought on by the gratuitous use of expensive tests and procedures and drugs probably makes a big difference here. Also, if there was a single payer insurance market, the government would have a significant amount of leverage in lowering prices, but it's unclear how much can be gained from fixing incentives and switching to single payer.

Health care spending in America is also highly concentrated among high spenders, suggesting that programs that increase spending on people who currently don't have insurance (and therefore don't spend much right now) won't necessary do much to solve the problem. Reducing total spending might require curtailing superfluous spending on things like cosmetic surgery and rationing expensive procedures that many people depend on.

Ultimately, the US has a lot to gain from health care reform that increases coverage and -- hopefully -- reduces costs, but we should all be wary of thinking we can get a free lunch on health care.