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