# Is Capital Important?

There is kind of a disconnect in teaching economic growth. We spend a lot of time telling students about the Solow model and capital accumulation, but at the same time the general consensus among growth economists is that total factor productivity is more important to understanding levels of output per worker.

Why do we think that capital isn’t terribly important to levels of output per worker? Basically, because the correlation between capital per worker and output per worker is low – or rather, we assume that it is low. Here’s a way of thinking about this in terms of simple regressions. If I was interested in how important capital per worker was in explaining output per worker, I could run this regression for a sample of countries (${i}$)

$\displaystyle \ln{y}_i = \beta_0 + \beta_1 \ln{k}_i + \epsilon_i \ \ \ \ \ (1)$

where I’ve put output per worker (${y_i}$) and capital per worker (${k_i}$) in logs. Logs keep countries with very small or very big values of capital per worker from being so influential, and in logs this regression will have an obvious interpretation for the coefficient ${\beta_1}$.

If I run this regression, I’ll get some estimated coefficient ${\hat{\beta}_1}$, which is the elasticity of output per worker with respect to capital per worker. Moreover, I could look at the R-squared of this regression. This R-squared will tell me what fraction of the variance of log output per worker (${Var(\ln{y}_i)}$) is explained by variation in log capital per worker (${Var(\ln{k}_i)}$). The R-squared is really what I want; it’s the answer the question “How important is capital in explaining differences in output per worker?”. The coefficient by itself doesn’t tell us that answer.

Now, there are some big problems with this regression. Most importantly, it is almost certainly the case that ${\ln{k}_i}$ is correlated with ${\epsilon_i}$, the residual. The residual captures things like technology levels, institutions, human capital, etc.. etc.. and capital per worker tends to be large when these things are “big”, meaning that they have a big positive effect on output per worker.

So that means we cannot trust our estimate ${\hat{\beta}_1}$, and cannot trust our value of R-squared. It’s worth writing out what the “true” R-squared is if we in fact had the right estimate of ${\beta_1}$. I’ll pre-apologize for the fact that this involves a lot of steps, but I’m writing them all out so it is easier to follow.

$\displaystyle \begin{array}{rcl} R^2 &=& \frac{{\beta}_1^2 Var(\ln{k}_i)}{Var(\ln{y}_i)} \\ \nonumber &=& \beta_1 \frac{Cov(\ln{k}_i,\ln{y}_i)}{Var(\ln{k}_i)}\frac{Var(\ln{k}_i)}{Var(\ln{y}_i)} \\ \nonumber &=& \beta_1 \frac{Cov(\ln{k}_i,\ln{y}_i)}{Var(\ln{y}_i)} \\ \nonumber &=& \frac{Cov({\beta}_1\ln{k}_i,\ln{y}_i)}{Var(\ln{y}_i)} \\ \nonumber &=& \frac{Cov({\beta}_1\ln{k}_i,{\beta}_1 \ln{k}_i + \epsilon_i}{Var(\ln{y}_i)} \\ \nonumber &=& \frac{ Var({\beta}_1\ln{k}_i) + Cov({\beta}_1 \ln{k}_i,\epsilon_i)}{Var(\ln{y}_i)}. \nonumber \end{array}$

The last line is identical to what Pete Klenow and Andres Rodriguez-Clare (1997, and KRC hereafter) use to evaluate the importance of capital in explaining cross-country output per worker differences. In other words, KRC are just looking for an R-squared. But as they point out, they cannot simply run the regression I proposed above and get the R-squared from that, because almost certainly ${\hat{\beta}_1 \neq \beta_1}$.

Rather than run the regression, KRC suggest that we use some alternative means of estimating ${\beta_1}$. They propose using the share of total output that gets paid to capital. Why? Because under perfect competition and constant returns to scale, that share should be precisely equal to ${\beta_1}$. In data from the U.S., capital’s share of output is usually something between 0.3–0.4, and KRC use ${\hat{\beta}_1 = 0.3}$. The rest of their data (${\ln{k}_i}$ and ${\ln{y}_i}$) is exactly the same data that one would use to run the regression. The only thing they are doing differently is plugging in their outside estimate of ${\hat{\beta}_1}$. What KRC find is that their R-squared is about 0.30, or that only 30% of the variation in log output per worker across countries is accounted for by variation in capital per worker across countries. This is a big reason why growth economists don’t think capital is of primary importance in explaining cross-country differences in output per worker.

It’s interesting to consider, though, what could rescue capital as an important explanatory variable. KRC use the idea that capital’s share in output is equal to ${\beta_1}$ under perfect competition and constant returns to scale. But what if there is not perfect competition and/or constant returns to scale? There is a neat little relationship that holds if we assume that firms are cost-minimizers. That is

$\displaystyle s_K = \frac{\beta_1}{\mu} \ \ \ \ \ (2)$

where ${s_K}$ is capital’s share in output (which KRC say is about 0.3) and ${\mu \geq 1}$ is the markup over marginal cost for firms. ${\mu = 1}$ only under perfect competition, and if there is imperfect competition or increasing returns to scale then markups are greater than one, meaning that the price charged by firms is greater than their marginal cost. From this we see that capital’s share may understate the value of ${\beta_1}$ if ${\mu>1}$. In particular, if there are increasing returns to scale at the firm level (i.e. fixed costs) but perfect competition (i.e. free entry/exit) then ${s_K}$ still measures the payments to capital accurately, but ${\mu}$ will be greater than one as firms with increasing returns need to charge more than marginal cost in order to cover the fixed costs.

Practically, if ${\beta_1 = 0.55}$, meaning that ${\mu = 1.83}$, or a markup of 83%, then the R-squared for capital goes to one. That is, with ${\beta_1 = 0.55}$, capital perfectly explains the varation in output per worker. Even with ${\beta_1 = 0.45}$, the R-squared is 0.67, meaning capital explains 2/3 of the variation in output per worker. So a relatively slight adjustment in the value of ${\beta_1}$ changes the conclusion regarding capital’s importance for output levels.

The issues with this line of thinking are (1) if there are increasing returns to scale at the firm level, why don’t we see increasing returns to scale at the aggregate/country level? (2) even if capital explains most of the variation in output per worker, there isn’t any data showing that savings rates actually vary across countries meaningfully. The differences in capital are probably the result of different technologies/institutions, and so those are the more fundamental source of variation.