# Are the Solow residual and scale effects related?

On the one hand, "the cross country variation in output by inputs [is] termed the Solow residual" (Acs et al. 2014). (This also applies across time to single countries.) In my own words, economic output is only partially explained by inputs of capital, labour and knowledge, and the Solow residual is the residual of that regression. (The Solow residual concept was updated to include knowledge based on the later endogenous growth theory.)

On the other hand, the scale effect is the idea that "models in which growth is driven by the accumulation of non-rival knowledge predict that larger economies (measured by a larger labour force) grow faster" (Peretto & Smulders 2002). A major research question is why scale effects cannot be found empirically.

Both streams of research come under endogenous growth theory and revolve around roughly the same concept of (technological) knowledge. Thus, my main question is whether the Solow residual and the absence of scale effects are actually two names/approaches for the same basic problem. A secondary question is why a quick search for the combination of those two does not yield any significant results. Is it because they are actually unrelated or only remotely related, because they come from disconnected sub-disciplines, because scale effects are trendy as opposed to the Solow residual, or for some other reason?

EDIT: I've just realised that the Solow residual and the absence of scale effects are in fact opposed, so I guess my main question boils down to whether they can be reconciled while my secondary question stands.

• Whether or not it can be shown that Solow residual is an increasing function of population level seems like an interesting paper. I can not remember any model off the top of my head. All the best if you choose to write one.
– erik
Mar 15 at 12:41

Thus, my main question is whether the Solow residual and the absence of scale effects are actually two names/approaches for the same basic problem.

No I do not think that is accurate. Solow residual also always exists regardless of scale effects or not. Solow residual is literally a residual from linear regression:

$$y= \beta_0 + \beta_1 k + \beta_2 l + e \tag{1}$$

Where $$y,k$$ and $$l$$ are log of output, capital and labor and $$e$$ is the error term that will become Solow residual once we estimate (1).

If there are scale effects or if growth is endogenous the error term that we are trying to estimate will still exist (Solow residual is just total factor productivity), however it won’t anymore be independent of other variables meaning the model above would be misspecified. But that’s not same as ceasing to exist.

However, interpretation of Solow residual would change. Currently Solow residual is typically being interpreted as the portion of growth that cannot be attributed to changes in factors of production (so in essence the total factor productivity). If there are scale effects then Solow residual no longer captures effect on growth that is not attributable to other factors as now it will capture indirect effect of labor as well.

A secondary question is why a quick search for the combination of those two does not yield any significant results. Is it because they are actually unrelated or only remotely related, because they come from disconnected sub-disciplines, because scale effects are trendy as opposed to the Solow residual, or for some other reason?

Yes they can. If scale effects are present then simply the original regression becomes:

$$y= \beta_0 + \beta_1 k + \beta_2 l + e(l) \tag{2}$$

Now the error term of (2) that we want to use to estimate Solow residual is function of labor, the problem is that now you cannot get unbiased estimates anymore using naive regression model. However, in principle you could run some more fancy structural model that could fix it.

This being said, people will typically prefer to run different types of estimations for endogenous growth models (the example above is quite ad hoc just showing it’s possible, not necessarily the appropriate way to run model with scale effect), so that’s the reason why probably literature on scale effects does not talk about Solow residual.

• The part about consistent estimates is not true without further assumptions on the structure. The variables on the left- and right are jointly determined, so there is a large potential for an omitted variable bias in any model of that kind. But as we are only interested in the unexplained Solow residual, not the true error term or for that matter the parameter estimates, we don't care. Mar 15 at 15:06
• @jpfeifer we do care because even though you are right that there are likely other omitted variables - that’s the point we can’t observe the technology - we do not want the residual to capture additional effect of L. In that case the Solow residual no longer has the interpretation of being the growth caused by all other factors than capital and labor
– 1muflon1
Mar 15 at 15:21
• You defined the Solow residual (correctly) as the residual of an OLS regression where you account for observed factor inputs. But your talk about consistency confuses residuals and error terms. The Solow residual will not be a function of labor. By construction, it will be orthogonal to the regressor l. It is the error term that is correlated with the regressor. So the estimated Solow residual will be a biased and inconsistent estimator for the error term (technology). Mar 15 at 15:37
• @jpfeifer ok I tried to make it clearer in the text, thanks for pointing it out
– 1muflon1
Mar 15 at 15:53
• @syre scale effects literally say that the total factor productivity (Error term that we estimate as Solow residual) is function of labor so that’s the justification. It’s not unspecified factor it’s total factor productivity and it’s not function of unspecified factors but of labor because scale effects say it should be
– 1muflon1
Mar 17 at 6:22