Tuesday, September 19, 2017

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.

Tuesday, September 5, 2017

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

Friday, August 25, 2017

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.

Sunday, July 23, 2017

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.

Friday, July 14, 2017

East Asia and Economic Convergence

Japan's impressive post-WWII economic growth is a prime example of economic convergence -- Japan's GDP per capita went from a little under 40% of the US in 1960 to over 90% in the early 1990s. This is a classic prediction of the Solow growth model; poorer countries will have quick economic growth as they invest in capital and will slowly catch up to rich countries like the United States.
Then, all of a sudden, a recession in 1996 and the Asian Financial Crisis in 1997 hit, and Japan has been stuck at roughly 73% of US GDP per capita since then (the data from Fred end in 2010, but World Bank has data from 1990 to 2015 that show the same thing). A lot of ink has been spilled in pursuit of an answer to the question of why Japan has settled into a permanently poorer equilibrium, and I'm not sure if I have much to add. I am highly skeptical that demand side factors can depress an economy for more than two decades, especially when better cyclical indicators like unemployment and employment rates tell the opposite story. Japan's demographic transition is also pretty important -- the working age population has been shrinking since the mid '90s meaning that the amount of workers per person (and consequently GDP per person) has had downward pressure for quite a long time.

Regardless, a 20% reduction in GDP per capita relative to the US is pretty huge, and makes me question my expectation that technologically advanced countries with institutions that don't prevent growth from taking place (think North Korea or Zimbabwe) will unconditionally converge the wealthiest countries. Thinking about this led me to the other major wealth East Asian countries: Taiwan, Hong Kong, and South Korea (Hong Kong is technically a special administrative region in China, but some combination of capitalism and former British rule make it both rich and free enough to count as a separate country in this case).
As you can see these three countries look a lot like Japan did at various points in the past. If you compare at which year each country was in about the same position as Japan in 1960 (that is about 40% of US GDP per capita), you can see how far behind Japan they are in terms of convergence. Hong Kong is the furthest along, although it's about 15 years behind in its process of convergence while Taiwan and Korea come in about 16 and 20 years behind Hong Kong, respectively.

Hong Kong and Japan are the two more interesting cases here: both experienced large slumps in the late '90s that lasted well int the 2000s, but then things start to diverge. In the mid 2000s Hong Kong starts to take off while Japan remains plugging along at around three quarters of US GDP per capita. The real question is which is the exception and which is the rule. As a resident of Japan, a small selfish part of me hopes that Taiwan and Korea will eventually get stuck at around the same level as Japan, but it's really more likely that Japan is mired in its own problems and will continue to stagnate while the other countries grow.

This is easier to see when you look at labor productivity -- GDP per hour worked -- instead of GDP per capita, because things that affect hours worked per employee or the overall employment rate can actually misrepresent the state of convergence.
Labor productivity tells a slightly different story than GDP per capita; while Japan still shows stagnation at around 70% of US productivity after 1996, Hong Kong's recent impressive growth in GDP per capita seems to have been caused by a large increase in either employment rates or hours and Koreans have made up for slower productivity growth relative to Taiwan by working long hours.

Japan's collapse in GDP per capita in the late '90s seems to reflect labor market problems unique to Japan as opposed to evidence against convergence. Average hours worked in Japan has been declining for decades as people unable to find full time employment switch to low paying part time jobs ("バイト").
This is probably a symptom of an economy that has been weak for more than 20 years -- the unemployment rate only recently fell back to the level of the late 1990s -- and has little to do with Hong Kong, Korea, or Taiwan. All four regions do face low fertility rates and will likely start being affected by the same demographic transition as Japan over the next few decades, but as long as they avoid a mass transition to part time employment they should look forward to some combination of fewer hours and higher GDP.

The reason for Japan's slowdown in productivity growth still evades me. I find it hard to believe that it's normal for a country to just stop converging with productivity 30% lower than the US, but I guess Hong Kong, Korea, and Taiwan will be a test of this as they either continue to grow or stagnate relative to America over the next few years.

Monday, June 26, 2017

How Healthy is the US Labor Market?

The plunge in labor force participation since the great recession in 2008 has led many to rightly question how well the unemployment rate -- the percent of the labor force that has looked for work in the last 4 weeks -- measures the true level of "slack" in the labor market.

Much (most) of the decline in labor force participation can be explained by the retirement of baby boomers, the oldest of which turned 65 in 2010, and by people between the ages of 16 and 24 choosing to focus on education instead of working.
The decline in labor force participation among young people is something that was occurring before the great recession. Even though it the recession seems to have sped up the decline, I'm pretty certain that most of the people between 16 and 24 who left the labor force in 2008 and 2009 or simply haven't joined since then (I'm in that group) wouldn't rejoin even if there was no slack in the labor market.

While the labor force participation rate for Americans 55 and older didn't actually decline in the great recession, it stopped a decades-long trend upward and has flattened out since then.
The fact that the participation rate for older Americans has settled at a lower level than participation for the general public, and that the share of the civilian noninstitutional population (basically everyone above the age of 16 who isn't deployed in the military or in prison) that is older than 55 years is increasing, has put significant downward pressure on the total labor force participation rate.
That being said, it's hard to be certain whether or not the recession has had lasting cyclical impacts on labor force participation, which warrants using statistics other than the unemployment rate to gauge the strength of the labor market.

One such popular measure is the broadest measure of underemployment put out by the Bureau of Labor Statistics (BLS) -- total unemployed, plus all marginally attached workers plus total employed part time for economic reasons -- or the U-6 unemployment rate.
I don't really like this as a measure of labor market strength though, because it doesn't count people who retired earlier than they wanted to as a result of the recession, people are only counted as "marginally attached" if they have searched for work in the last 12 months, and part time workers aren't really unemployed (there is an alternative measure that excludes involuntary part time workers but it still has the problems I mentioned above).

Some people like to look at the employment rate for the so called "prime age" population who are between the ages of 25 and 54 (I don't know why the cutoff is 54, going up to 65 makes way more sense to me) in order to weed out the effects of aging and lower youth participation on the labor market.
This is a relatively good solution for the years after 1990, and it does show that the labor market is considerably weaker than the unemployment suggests (although not so weak that we need to "prime the pump"), but it has trouble before 1990 because women were still joining the labor force en masse for most of the latter half of the 20th century.

Basically every statistic that you can easily get from BLS data has a problem like this, so it's really hard to get a good measure of how healthy the labor market is, but there is a solution. The Congressional Budget Office (CBO) looks at the demographic composition of the working age population and comes up with what it thinks the labor force participation rate would be at full employment. It calls this measure the "potential labor force", which tries to estimate the movement in labor force participation caused by gender and aging and can then be used to estimate the cyclical component of labor force participation.
It is then possible to find the "adjusted" unemployment rate with the CBO's estimate of the potential labor force. The above chart shows the actual unemployment rate reported by the BLS as well as my calculation of the adjusted unemployment rate using the potential labor force from the CBO's 2007 and 2017 data for "Potential GDP and Underlying Inputs" (all CBO data is available here). The dashed grey line is the natural rate of unemployment -- that is the unemployment rate that is consistent with full employment -- according to the CBO.

Since I was only able to find annual "potential labor force" figures, the estimates only extend to 2016, but both the 2007 and 2017 figures are broadly consistent with the prime age employment rate: the unemployment rate overstates the health of the labor market by between 1 and 2 percent (depending if you use the 2007 or 2017 value for the potential labor force). This is similar to where we were in 2003 or 1994, so while there's no real cause to worry about joblessness right now the recovery isn't completely over yet either. As a side note, tax cuts are even more of a stupid idea now than they were in 2003, but that issue deserves a whole post of its own.

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