# Robots as Factor-Eliminating Technical Change

NOTE: The Growth Economics Blog has moved sites. Click here to find this post at the new site.

A really common thread running through the comments I’ve gotten on the blog involve the replacement of labor. This is tied into the question of the impact of robots/IT on labor market outcomes, and the stagnation of wages for lots of laborers. An intuition that a lot of people have is that robots are going to “replace” people, and this will mean that wages fall and more and more of output gets paid to the owners of the robots. Just today, I saw this figure (h/t to Brad DeLong) from the Center on Budget and Policy Priorities which shows wages for the 10th and 20th percentile workers in the U.S. being stagnant over the last 40 years.

The possible counter-arguments to this are that even with robots, we’ll just find new uses for human labor, and/or that robots will relieve us of the burden of working. We’ll enjoy high living standards without having to work at it, so why worry?

I’ll admit that my usual reaction is the “but we will just find new kinds of jobs for people” type. Even though capital goods like tractors and combines replaced a lot of human labor in agriculture, we now employ people in other industries, for example. But this assumes that labor is somehow still relevant somewhere in the economy, and maybe that isn’t true. So what does “factor-eliminating” technological change look like? As luck would have it, there’s a paper by Pietro Peretto and John Seater called …. “Factor-eliminating Technical Change“. Peretto and Seater focus on the dynamic implications of the model for endogenous growth, and whether factor-eliminating change can produce sustained growth in output per worker. They find that it can under certain circumstances. But the model they set is also a really useful tool for thinking about what the arrival of robots (or further IT innovations in general) may imply for wages and income distribution.

I’m going to ignore the dynamics that Peretto and Seater work through, and focus only on the firm-level decision they describe.

****If you want to skip technical stuff – jump down to the bottom of the post for the punchline****

Firms have a whole menu of available production functions to choose from. The firm-level functions all have the same structure, ${Y = A X^{\alpha}Z^{1-\alpha}}$, and vary only in their value of ${\alpha \in (0,\overline{\alpha})}$. ${X}$ and ${Z}$ are different factors of production (I’ll be more specific about how to interpret these later on). ${A}$ is a measure of total factor productivity.

The idea of having different production functions to choose from isn’t necessarily new, but the novelty comes when Peretto/Seater allow the firm to use more than one of those production functions at once. A firm that has some amount of ${X}$ and ${Z}$ available will choose to do what? It depends on the amount of ${X}$ versus the amount of ${Z}$ they have. If ${X}$ is really big compared to ${Z}$, then it makes sense to only use the maximum ${\overline{\alpha}}$ technology, so ${Y = A X^{\overline{\alpha}}Z^{1-\overline{\alpha}}}$. This makes some sense. If you have lots of some factor ${X}$, then it only makes sense to use a technology that uses this factor really intensely – ${\overline{\alpha}}$.

On the other hand, if you have a lot of ${Z}$ compared to ${X}$, then what do you do? You do the opposite – kind of. With a lot of ${Z}$, you want to use a technology that uses this factor intensely, meaning the technology with ${\alpha=0}$. But, if you use only that technology, then your ${X}$ sits idle, useless. So you’ll run a ${X}$-intense plant as well, and that requires a little of the ${Z}$ factor to operate. So you’ll use two kinds of plants at once – a ${Z}$ intense one and a ${X}$ intense one. You can see their paper for derivations, but in the end the production function when you have lots of ${Z}$ is

$\displaystyle Y = A \left(Z + \beta X\right) \ \ \ \ \ (1)$

where ${\beta}$ is a slurry of terms involving ${\overline{\alpha}}$. What Peretto and Seater show is that over time, if firms can invest in higher levels of ${\overline{\alpha}}$, then by necessity it will be the case that we have “lots” of ${Z}$ compared to little ${X}$, and we use this production function.

What’s so special about this production function? It’s linear in ${Z}$ and ${X}$, so their marginal products do not decline as you use more of them. More importantly, their marginal products do not rise as you acquire more of the other input. That is, the marginal product of ${Z}$ is exactly ${A}$, no matter how much ${X}$ we have.

What does this possibly have to do with robots, stagnant wages, and the labor market? Let ${Z}$ represent labor inputs, and ${X}$ represent capital inputs. This linear production function means that as we acquire more capital (${X}$), this has no effect on the marginal product of labor (${Z}$). If we have something resembling a competitive market for labor, then this implies that wages will be constant even as we acquire more capital.

That’s a big departure from the typical concept we have of production functions and wages. The typical model is more like Peretto and Seater’s case where ${X}$ is really big, and ${Y = A X^{\overline{\alpha}}Z^{1-\overline{\alpha}}}$, a typical Cobb-Douglas. What’s true here is that as we get more ${X}$, the marginal product of ${Z}$ goes up. In other words, if we acquire more capital, then wages should rise as workers get more productive.

The Peretto/Seater setting says that, at some point, technology will progress to the point that wages stop rising with the capital stock. Wages can still go up with general total factor productivity, ${A}$, sure, but just acquiring new capital will no longer raise wages.

While wages are stagnant, this doesn’t mean that output per worker is stagnant. Labor productivity (${Y/Z}$) in this setting is

$\displaystyle \frac{Y}{Z} = A \left(1 + \beta \frac{X}{Z}\right). \ \ \ \ \ (2)$

If capital per worker (${X/Z}$) is rising, then so is output per worker. But wages will remain constant. This implies that labor’s share of output is falling, as

$\displaystyle \frac{wZ}{Y} = \frac{AZ}{A \left(Z + \beta X\right)} = \frac{Z}{\left(Z + \beta X\right)} = \frac{1}{1 + \beta X/Z}. \ \ \ \ \ (3)$

With the ability to use multiple types of technologies, as capital is acquired labor’s share of output falls.

Okay, this Peretto/Seater model gives us an explanation for stagnant wages and a declining labor share in output. Why did I present this using ${X}$ for capital and ${Z}$ for labor, not their traditional ${K}$ and ${L}$? This is mainly because the definition of what counts as “labor”, and what counts as “capital”, are not fixed. “Capital” might include human as well as physical capital, and so “labor” might mean just unskilled labor. And we definitely see that unskilled labor’s wage is stagnant, while college-educated wages have tended to rise.

***** Jump back in here if you skipped the technical stuff *****

The real point here is that whether technological change is good for labor or not depends on whether labor and capital (i.e. robots) are complements or substitutes. If they are complements (as in traditional conceptions of production functions), then adding robots will raise wages, and won’t necessarily lower labor’s share of output. If they are substistutes then adding robots will not raise wages, and will almost certainly lower labor’s share of output. The factor-eliminating model from Peretto and Seater says that firms will always invest in more capital-intense production functions and that this will inevitably make labor and capital substitutes. We happen to live in the period of time in which this shift to being substitutes is taking place. Or one could argue that it already has taken place, as we see those stagnant wages for unskilled workers, at least, from 1980 onwards.

What we should do about this is a different question. There is no equivalent mechanism or incentive here that would drive firms to make labor and capital complements again. From the firms perspective, having labor and capital as complements limits their flexibility, because they then depend on the other. They’d rather have the marginal product of robots and people independent of one other. So once we reach the robot stage of production, we’re going to stay there, absent a policy that actively prohibits certain types of production. The only way to raise labor’s share of output once we get the robots is through straight redistribution from robot owners to workers.

Note that this doesn’t mean that labor’s real wage is falling. They still have jobs, and their wages can still rise if there is total factor productivity change. But that won’t change the share of output that labor earns. I guess a big question is whether the increases in real wages from total factor productivity growth are sufficient to keep workers from grumbling about the smaller share of output that they earn.

I for one welcome….you know the rest.

# Oberfield and Raval on Capital/Labor Elasticity of Substitution

NOTE: The Growth Economics Blog has moved sites. Click here to find this post at the new site.

I was in Boston for the NBER summer institute on Friday, sitting in on what it typically called either “Growth day” or “Jones/Klenow” after the organizers. Regardless, here’s the program. It’s a chance to see what is some of the cutting/bleeding edge research in economic growth.

The first paper I saw was by Ezra Oberfield of Princeton and Devesh Raval of the Federal Trade Commission (I missed the Grossman/Helpman paper because I like to sleep, and didn’t get to Boston until 10:15am – sue me). They were doing two things. (1) providing an estimate of the aggregate elasticity of substitution (EOS) between capital and labor and (2) using that to try and account for the decline in labor’s share of income over the last 30-40 years.

On (1), they made the point that the aggregate EOS is not a technological constant, but rather is an artifact of the micro-level EOS. Specifically,

$\displaystyle \sigma^{agg} = (1-X)\sigma^{micro} + X \epsilon \ \ \ \ \ (1)$

where ${\sigma^{micro}}$ is the EOS at the plant level. The weighting term ${X}$ reflects the variation in capital shares across firms. ${\epsilon}$ is the elasticity of demand for plant output. The demand elasticity is in to account for the fact that some of the response to a change in factor prices is to move demand away from the plants that tend to use the more expensive factor.

Regardless, Ezra and Devesh provide evidence that ${X}$ is really close to zero, so essentially this demand adjustment is negligible, and the aggregate EOS is roughly equivalent to the micro EOS. They estimate this from plant-level data, and find something like 0.52, meaning that capital and labor are not easily substituted for each other. Over time, the aggregate EOS is roughly stable at around 0.70, based on their values for ${X}$ and ${\epsilon}$.

On (2), given their aggregate EOS, the implication is that the decline of labor’s factor share is biased technical change. Increased automation, IT investment, and offshoring, among other things, have driven down labor’s share of output down over time.

Changes in factor prices alone (wages and rental rates) would have raised labor’s share of output over this period, they find. The force of biased technical change was so strong it overcame that tendency.

It’s worth noting how important finding the EOS1, then firms can switch easily from labor to capital. Relatively cheap capital is substituted for labor, and labor’s share drops. If EOS>1, then the decline in labor share is driven in part by more expensive labor, and hence the implied degree of biased technical change is smaller.

NOTE: The Growth Economics Blog has moved sites. Click here to find this post at the new site.

This group of papers is one of the first that I cover in class, because it’s useful to frame much of the growth/development research. The concept is that real GDP per capita is produced using a function something like ${y = A F(k,h)}$. Real GDP thus depends on total factor productivity (${A}$), capital (${k}$), and human capital/labor (${h}$). So variation in real GDP per capita depends on variation in ${A}$, ${k}$, and ${h}$ across countries. All your favorite theories about institutions, geography, culture, innovation, etc.. must operate through one of these three proximate factors. To focus ourselves on what is important, we’d like to know which of the three proximate factors are actually responsible for the variation in real GDP per capita we see.

One way to do this is to first assume a Cobb-Douglas production function for ${F()}$ and take logs

$\displaystyle \ln{y}_i = \ln A_i + \alpha \ln{k}_i + \beta \ln{h}_i. \ \ \ \ \ (1)$

Conceptually, one could then run a regression of ${y_i}$ (the ${i}$ index specifies the country) on ${k_i}$ and ${h_i}$. We don’t have information on ${A_i}$ directly, so we could treat that as the error term. We could get even fancier and replace ${k_i}$ and ${h_i}$ with some terms based on savings rates or human capital accumulation rates, consistent with theory. Regardless, we’d then look at the R-squared or partial R-squared’s to tell us how important each factor was. This is, in a nutshell, what Mankiw, Romer, and Weil (1992) are up to.

One problem with this is that TFP (${A}$) is not uncorrelated with ${k}$ and ${h}$, so the regression estimates of ${\alpha}$ and ${\beta}$ are going to be biased, and hence so are our R-squares. I wrote a whole post about this here.

So rather than run the regression, we could pull values for ${\alpha}$ and ${\beta}$ from some other source and just calculate the R-squares without actually running the regression. This is essentially what the development accounting literature is doing, with Hall and Jones (1999) and Klenow and Rodriguez-Clare (1997) being the classic examples. The upshot of these papers is that variation in ${A}$ accounts for at least 50% of the differences in ${y}$ across countries, and maybe more. ${k}$ accounts for maybe 30-40%, and ${h}$ only 10-20%. So TFP is the most important proximate factor.

The other papers are then riffs on this basic idea. Gollin (2002) is about whether ${\alpha}$ or ${\beta}$ themselves vary across countries (they do) and whether they are correlated with real GDP per capita (they are not). Caselli (2005) shows that differences in how exactly you account for ${k}$ and ${h}$ are not necessarily important for overall result that TFP matters most. You can also do this kind of accounting for a single country over time, to see the sources of growth. The Young (1995) and Hsieh (2002) papers are a back and forth over how to do this for several East Asian countries, differing in technique and data sources. Hsieh and Klenow (2007) is included in this section of the class because it helps establish that domestic savings rates do not vary much across countries, and so we cannot expect capital variation to matter a lot either.

The reading list here is light on human capital. I talk about Hendricks (2002) work on trying to measure ${h}$ more accurately using immigrant data from the U.S., and Weil’s (2007) paper on including health as part of human capital. The reason for the light coverage is that German Cubas, one of our junior faculty, is going to be teaching a graduate course this year that focuses a lot of human capital. So I only touch on it in my course.

As usual, PDF and Bibtex files with the reading lists are on the “Papers” page.

# Wealth and Capital are Different Things

NOTE: The Growth Economics Blog has moved sites. Click here to find this post at the new site.

Piketty’s book is like a giant attention-sucking vortex. I can’t seem to escape it. This time I’m thinking about the criticism of Piketty’s analysis that has to do with rates of return on capital. Piketty says that if ${r > g}$, where ${r}$ is the return to capital, and ${g}$ is the growth rate of aggregate GDP, then wealth will become more and more concentrated.

Critiques of Piketty have questioned the assumptions underlying this conclusion. The most recent one I’ve seen is in Larry Summers’ review piece. Let’s let him sum up the issues:

This rather fatalistic and certainly dismal view of capitalism can be challenged on two levels. It presumes, first, that the return to capital diminishes slowly, if at all, as wealth is accumulated and, second, that the returns to wealth are all reinvested. Whatever may have been the case historically, neither of these premises is likely correct as a guide to thinking about the American economy today.

With respect to the first assumption regarding the rate of return, here is what Summers says:

Economists universally believe in the law of diminishing returns. As capital accumulates, the incremental return on an additional unit of capital declines.

But Summers has fallen into what I think is a really common trap for economists. He presumes that his second statement (“As capital accumulates, the incremental return on an additional unit of capital declines”) contradicts Piketty’s assumption (“that the return to capital diminishes slowly, if at all, as wealth is accumulated”). These two statements are not mutually exlusive.

The issue is that Summers is confounding wealth and capital. This is not helped by Piketty, who uses “capital” in his title and in the book the way that normal people use it, as a synonym for “wealth”. But from the perspective of an economist, these two concepts are not the same thing. The capital that Summers refers to in his critique (often denoted ${K}$) is a subset of the measure of national wealth (${W}$, as I’ll call it) that Piketty documents.

Without going too deep into this, Piketty’s measure of wealth consists of three parts: real estate, corporate capital, and financial assets. Only real estate and corporate capital are what economist have in mind when they say capital (${K}$). Wealth, however, consists of all three parts, so that Piketty’s wealth is ${W = K + F}$, where ${F}$ is the value of financial assets. Asserting that the return to capital falls as the capital stock increases – as Summers does – does not imply that the return to wealth falls as the stock of wealth increases. Even if we assume that financial markets work so efficiently that the return to capital and the return to financial assets are identical, this does not mean that the return to wealth necessarily falls as wealth accumulates.

To see this, consider a really slimmed down version of the “bubble asset” model from Blanchard and Fischer (1989, p. 228). We have that the return on capital is ${r = f'(K)}$, where ${f'(K)}$ is the marginal product of capital. The ${f'(K)}$ is the derivative of the production function, and represents the marginal increase in output we’d get from adding one more unit of capital. Under our typical assumptions about diminishing returns, as ${K}$ goes up ${r}$ goes down. This is what Summers is using as his critique.

An efficient financial market would ensure that financial assets (F) would also have a return of ${r}$. If they did not, then people would buy/sell financial assets until the return was equal. (Yes, I’m ignoring risk entirely, but that doesn’t change the main point here). So the return on all wealth is equal to ${r}$, and note that this is pinned down by the value of ${K}$ alone.

Now, we have assumed that ${r}$ falls as ${K}$ increases. Does this imply that ${r}$ falls as wealth (${W}$) increases? No. The relationship between ${r}$ and ${W}$ depends entirely on the composition of the change in ${W}$. If ${W}$ rises because ${K}$ rises (say ${F}$ stays constant), then the rate of return on wealth falls because the marginal product of capital has declined. This is what Summers and others have in mind.

However, it’s perfectly plausible that ${W}$ rises even though ${K}$ falls, because the value of financial assets (${F}$) are increasing even more quickly. In this case, the marginal product of capital has increased, and the rate of return on wealth has increased. In this case, the rate of return rises with wealth.

Is it reasonable for an economy to experience falling capital but a rising value of financial assets? Sure. The point of Blanchard and Fisher’s model of bubbles is that even though all individuals are acting rationally at all times, the economy can take off onto a weird path where the stock of capital (${K}$) gets run down while the value of financial assets (${F}$) rises. Eventually this is unsustainable, as we’d run out of capital, but there is no reason that a situation like this cannot persist for a while.

Will the return to wealth necessarily rise as wealth accumulates? No. There are other equally reasonable paths that the economy could take where wealth accumulation is driven mainly by capital accumulation and the rate of return falls as wealth accumulates, consistent with the Summers critique. The point I want to make is that there is no particular reason to believe in a fixed relationship between wealth and the return on capital. They can move completely independently of each other.

So Piketty can easily be right that we are currently in a world where both the wealth/income ratio is increasing and the rate of return on wealth is rising (or remaining roughly constant), and that this could persist for some indefinite period. On the other hand, it was not inevitable that this was going to happen, and it could just as easily end tomorrow as in 100 years.

I think the story that is milling around beneath the surface of Piketty’s book is that recent wealth accumulation has been primarily of financial assets, not capital. Hence the return has stayed high and the concentration of wealth has continued. If the returns on that wealth are continually reinvested in financial assets as opposed to capital, then Piketty’s death spiral of wealth concentration would likely be the outcome. To avoid that death spiral, you’d want to get the returns on wealth reinvested into real capital so that the return on capital (and hence wealth) gets pushed down.

# Piketty and Growth Economics

NOTE: The Growth Economics Blog has moved sites. Click here to find this post at the new site.

Reviews of Thomas Piketty’s “Capital in the 21st Century” are second only to cat videos on the internet, it seems. Not having any cats, I am unable to make a video, so you’re stuck with a review of Piketty’s book.

I was particulary struck by the implications of this work for economic growth theory. The first section of the book studies capital/output ratios, one of the core elements of any model of growth that includes capital. Piketty provides a long time series of this ratio, showing that in Europe it tended to hover around 7 during the 1800’s and early 1900’s, then dropped dramatically following World War I, stayed at around 3 until the 1970’s, and now is rising towards 6. In the U.S., it has been less variable, going from around 4.5 in the 1800’s to about 3 in the 1960’s, and now is back up to about 4.5.

The projection that Piketty makes is that the capital/output ratio will tend to be about 6-7 across the world as we go into the future. The main reason is that he expects population growth to decline, and the capital/output ratio is inversely related to population growth. In a standard Solow model with a fixed savings rate ${s}$, the capital/output ratio is ${K/Y = s/(n+\delta+g)}$, where ${n}$ is population growth, ${\delta}$ is depreciation, and ${g}$ is the growth of output per worker. You can see that as ${n}$ goes down, the ${K/Y}$ ratio rises.

By itself, this doesn’t imply much for growth theory, in that the expected ${K/Y}$ ratio in the future is entirely consistent with Piketty’s claim regarding population growth. He might be wrong about population growth, but if ${n}$ does in fact fall, then any growth model would have predicted ${K/Y}$ will rise.

The interesting implication of Piketty’s work is on the returns to capital. In particular, the share of national income that goes to capital. His figures 6.1-6.3 document that this share has changed over time. From a share of about 35% in the 1800’s in both Britain and France, the share dropped to about 20-25% in both countries by the mid-20th century. Most recently, the capital share is starting to rise across many countries, going up about 10 percentage points between 1970 and 2010.

One of the bedrock assumptions made in most growth models is a Cobb-Douglas production function, which implies (under conditions of perfect competition) that capital’s share in output is fixed by a technological parameter, typically called ${\alpha}$ and typically assumed to be ${\alpha = 0.3}$. Over time, the share of output going to capital is constant at this value of ${\alpha}$. Growth economists lean on this assumption because of work done by Nicholas Kaldor, who established as a “stylized fact” that capital’s share in output is constant at about 0.3–0.35. As Piketty points out, though, Kaldor established this fact using a very small time series of data from a particularly unusual time period (roughly the mid-20th century).

The fact that capital’s share of output has changed distinctly over long time frames means that this baseline assumption is called into question. What does it mean? I have two immediate thoughts.

• Perfect competition is not a good assumption. This is probably trivially true; there is no such thing as a perfectly competitive economy. But what Piketty’s data would then indicate is that the degree of imperfection has possibly changed over time, with economic profits (not accounting ones) rising in the late 20th century. We have lots of models of economic growth that allow for imperfect competition (basically, any model that involves deliberate research and development), but we do not talk much about changes in the degree of that competition over time.
• The production function is not Cobb-Douglas. Piketty talks about this in his book. The implication of rising capital shares that coincide with rising capital/output ratios is that the elasticity of substitution between capital and labor is greater than one. For Piketty, this contributes to increasing inequality because capital tends to be owned by only a small fraction of people. For growth economists, this raises interesting possibilities for what drives growth. With a sufficiently large elasticity of substitution between capital and labor, then growth can be driven by capital accumulation alone. To see this, imagine perfect substitutability between capital and labor in production, or ${Y = K + AL}$, where ${A}$ is labor-specific productivity. Output per worker is ${y = K/L + A}$. As the capital/labor ratio rises, so does output per worker. This continues without end, because there are no longer decreasing returns to capital per worker. Even if technology is stagnant (${A}$ does not change), then output per worker can go up. We tend to dismiss the role of capital per worker in driving growth, but perhaps that is because we are wedded to the Cobb-Douglas production function.

The remainder of Piketty’s book is very interesting, and his own views on the implications of rising inequality have been subject to an intense debate. But from the perspective of growth economics, it is the initial section of the book that carries some really interesting implications.

# 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.

# The Solow Model

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This is another idea for modifying how to teach the Solow model. One thing I’d like to do is go immediately to including productivity – it follows cleanly from the simplest growth model. Second, I think it might be nice to work with the K/Y ratio immediately. In this way, I think you can actually skip using the whole “k-tilde” thing. And, *gasp*, do away with the traditional Solow diagram.

The simplest growth model doesn’t allow for transitional growth, and this due to the fact that it does not allow for capital, a factor of production that can only be slowly accumulated over time. The Solow Model is a standard model of economic growth that includes capital, and will be better able to account for the transitional growth that we see in several countries.

Production in the Solow Model takes place according to the following function

$\displaystyle Y = K^{\alpha}(AL)^{1-\alpha}. \ \ \ \ \ (1)$

${K}$ is the stock of physical capital used in production, and ${A}$ and ${L}$ are defined just as they were in our simple growth model. So the production function here is just a modification of the simple model to include capital. The coefficient ${\alpha}$ is a weight telling us how important capital or ${AL}$ are in determining output.

To analyze this model, we’re going to rewrite the production function. Divide both sides of the function by ${Y^{\alpha}}$, giving us

$\displaystyle Y^{1-\alpha} = \left(\frac{K}{Y}\right)^{\alpha} (AL)^{1-\alpha} \ \ \ \ \ (2)$

and then take both sides to the ${1/(1-\alpha)}$ power, which gives us the following expression

$\displaystyle Y = \left(\frac{K}{Y}\right)^{\alpha/(1-\alpha)} AL. \ \ \ \ \ (3)$

In per capita terms, this is

$\displaystyle y = \left(\frac{K}{Y}\right)^{\alpha/(1-\alpha)} A. \ \ \ \ \ (4)$

Output per worker thus depends not just on ${A}$, but also on the capital-output ratio, ${K/Y}$.

So to understand the role of capital in economic growth, we need to understand the capital-output ratio and how it changes over time. We’ll start by looking at the balanced growth path, and then turn to situations where the economy is not on the balanced growth path (BGP).

One fact about the BGP is that the return to capital, ${r}$, is constant. The return to capital is ${r = \alpha Y/K}$, which depends (negatively) on the capital-output ratio (the return to capital is just the marginal product of capital). If ${r}$ is constant on the BGP, then it must be that ${K/Y}$ is constant on the BGP as well. What does this mean? It means that ${K/Y}$ can have a level effect on output per worker, but has no growth effect. To see this more clearly, take logs of output per worker,

$\displaystyle \ln y(t) = \frac{\alpha}{1-\alpha} \ln\left(\frac{K}{Y}\right) + \ln A(t) \ \ \ \ \ (5)$

and then plug in what we know about how ${A(t)}$ moves over time,

$\displaystyle \ln y(t) = \frac{\alpha}{1-\alpha} \ln\left(\frac{K}{Y}\right) + \ln A(0) + gt. \ \ \ \ \ (6)$

The capital-output ratio affects the intercept of this line — a level effect — alongside ${A(0)}$. The slope of this line — the growth rate — is still ${g}$.

The capital/output ratio is constant along the BGP, and has no effect on the growth rate on the BGP. But what if the economy is not on the BGP? Then it will be the case that ${K/Y}$ affects the growth rate of output per worker, because the ${K/Y}$ ratio will not be constant. More precisely, the growth rate of capital/output is

$\displaystyle \frac{\dot{K/Y}}{K/Y} = \frac{\dot{K}}{K} - \frac{\dot{Y}}{Y}. \ \ \ \ \ (7)$

So the ${K/Y}$ ratio will change if capital grows more quickly or more slowly than output. First, capital accumulates as follows

$\displaystyle \dot{K} = s Y - \delta K \ \ \ \ \ (8)$

where ${\dot{K}}$ is the change in the capital stock. ${s}$ is the savings rate, the fraction of output that the economy sets aside to invest in new capital goods, so that ${sY}$ is the total amount of new investment. ${\delta}$ is the depreciation rate, the fraction of the existing capital stock that breaks or becomes obsolete at any given moment.

To find the growth rate of capital, divide through the above equation by ${K}$ to get

$\displaystyle \frac{\dot{K}}{K} = s\frac{Y}{K} - \delta. \ \ \ \ \ (9)$

You can see that the growth rate of capital depends on the capital/output ratio itself.

The growth rate of output is

$\displaystyle \frac{\dot{Y}}{Y} = \alpha \frac{\dot{K}}{K} + (1-\alpha)\frac{\dot{A}}{A} + (1-\alpha)\frac{\dot{L}}{L}. \ \ \ \ \ (10)$

Now, with (7), and using what we know about growth in capital and output, we have

$\displaystyle \frac{\dot{K/Y}}{K/Y} = (1-\alpha)\left(s\frac{Y}{K} - \delta - g - n \right) \ \ \ \ \ (11)$

where we’ve plugged in that ${\dot{A}/A = g}$, and ${\dot{L}/L = n}$.

Re-arranging a bit, the capital output ratio is growing if

$\displaystyle \frac{K}{Y} < \frac{s}{\delta + n + g}, \ \ \ \ \ (12)$

and growing if the capital/output ratio is larger than the value on the right-hand side. In other words, if the capital stock is relatively small, then it will have a tendency to grow faster than output, raising the ${K/Y}$ ratio. Eventually ${K/Y = s/(\delta+n+g)}$, the steady state value, and the ${K/Y}$ ratio stops changing.

What is happening to growth in output per worker? If ${K/Y < s/(\delta+n+g)}$ then the ${K/Y}$ ratio is growing, and so output per worker is growing faster than ${g}$. So the temporarily fast growth in output per worker in Germany or Japan would be because they found themselves with a ${K/Y}$ ratio below their steady state value. How would this occur? It’s easier to see how this works if we re-write the ${K/Y}$ ratio slightly

$\displaystyle \frac{K}{Y} = \frac{K}{K^{\alpha}(AL)^{1-\alpha}} = \left(\frac{K}{AL}\right)^{1-\alpha}. \ \ \ \ \ (13)$

From this we can see that the ${K/Y}$ ratio would be particularly low if the capital stock, ${K}$, were to be reduced. This is what happened in Germany, to a large extent, after World War II. The capital stock was destroyed, so ${K/AL}$ fell sharply. This made ${K/Y}$ fall below the steady state value, which meant that there was growth in the ${K/Y}$ ratio, and so growth in output per worker greater than ${g}$.

A slightly different situation describes South Korea. There, we can think of there being a level effect on ${A}$, an advance in productivity. This also makes ${K/AL}$ fall sharply, and again causes growth in ${K/Y}$ and growth in output per worker faster than ${g}$. But in both this case and in Germany’s, as the ${K/Y}$ ratio grows it approaches the steady state value and growth in output per worker slows down to ${g}$ again.