## 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).