“The Human Capital Stock: A Generalized Approach,” B. Jones (2012)

Back to the main article in today’s post, Ben Jones’ Human Capital Stock paper. Measuring human capital is difficult. We think of human capital as an input in a production function. A general production function is Y=f(K,H,A) where A is a technology scalar, K is a physical capital aggregator, and H (a function of H(1),H(2), etc., marking different types of human capital) is a human capital aggregator. Every factor is paid its marginal product if firms are cost minimizers. Let H(i)=h(i)L(i) be the quantity of some class of labor (like college educated workers) weighted by the flow of services h(i) provided by that class. We can measure L, but not h. The marginal product of L(i), the wage received by laborers of type i, is df/dH*dH/dH(i)*h(i). That is, wage depends both on the amount of human capital in workers of type i, as well as contribution of H(i) to the human capital aggregator.

Consider the ratio of wages w(i)/w(j)=[dH/dH(i)*h(i)]/[dH/dH(j)*h(j)]. Again, we need to the know how each type of human capital affects the aggregator to be able to go from wage differences to human capital differences. If the production function is constant returns to scale, then the human capital aggregator can be rewritten as h(1)*H(L(1),[w(2)*dH/dH(1)]/[w(1)*dH/dH(2)]…). If wages w and labor allocations L were observed, we could infer the amount of human capital if we knew h(1) and we knew the ratios of marginal contributions of each type of human capital to the aggregator. Traditional human capital accounting assumes that h(1), the human capital of unskilled workers, is identical across countries, and that the aggregator equals the sum of h(i)L(i). Implicitly, this says each skill-adjusted unit of labor is perfectly substitutable in the production function: a worker with wage twice the unskilled wage, by the above assumptions, has twice the human capital of the unskilled worker. If you replaced her with two unskilled workers, the total productive capacity of the economy would be unchanged.

You may not like those assumptions. Jones notes that, since rich countries have many fewer unskilled workers, and since marginal product is a partial equilibrium concept, the marginal productivity of unskilled workers is likely higher in rich countries than in poor ones. Also, unskilled worker productivity has complementarities with the amount of skilled labor; a janitor keeping a high-tech hospital clean has higher marginal product than an unskilled laborer in the third world (if you know Kremer’s O-Ring paper, this will be no surprise). These two effects mean that traditional assumptions in human capital accounting will bias downward the relative amount of human capital in the wealthy world. It turns out that, under a quite general function form for the production function, we only need to add the elasticity of unskilled-skilled labor substitution to our existing wage and labor allocation data to estimate the amount of human capital with the generalized human capital function; critically, we don’t need to know anything about how different types of skilled labor combine.

How does this matter empirically? There seems to be a puzzle in growth accounting. Highly educated countries almost always correlate with high incomes. Yet traditional growth accounting finds only 30% or so of across-country income difference can be explained by differences in human capital. However, empirical estimates of the elasticity of substitution of unskilled and skilled labor are generally something like 1.4 – there are complementarities. Jones calculates for a number of country pairs what elasticity would be necessary to explain 100% of the difference in incomes with human capital alone. The difference between Israel (the 85th percentile of the income distribution) and Kenya (the 15th percentile) is totally explained if the elasticity of substitution between skilled and unskilled labor is 1.54. Similar numbers prevail for other countries.

So if human capital is in fact quite important, why explains the differences in labor allocation? Why are there so many more skilled workers in the US than in Congo? Two things are important to note. First, in general equilibrium, workers choose how much education to receive. That is, if anyone in the US is not going to college, the difference in wages between skilled and unskilled labor cannot be too large. For the differences in wages to not grow too large, there must be a supply response: the amount of unskilled laborers shrinks, causing each unskilled worker’s marginal product to rise. Israel has a ratio of skilled to unskilled labor 2300% higher than Kenya, but the skilled worker wage premium is only 20% higher in Israel than in Kenya. If the elasticity of substitution is 1.6, service flows from skilled workers in Israel are almost 100 times higher than in Kenya, despite an almost identical skilled-unskilled wage premium. That is, we will see high societal returns to human capital in the share of skilled workers rather than in the wage premium.

Second, why don’t poor countries have such high share of human capital? Adam Smith long ago wrote that the division of labor is limited by the size of the market. At high levels of human capital, specialization has huge returns. Jones gives the example of a thoracic surgeon: willingness to pay for such a surgeon to perform heart surgery is far higher than willingness to pay a dermatologist or an economics professor, despite similar levels of education. Specialization, therefore, increases the societal return to human capital, and such specialization may be limited by small markets, coordination costs, low levels of existing advanced knowledge, or limited local access to such knowledge. A back of the envelope calculation suggests that a 4.3-fold difference in the amount of specialization can explain the differences in labor allocation between Israel and Kenya, and that this difference is even lower if rich countries have better ability to transmit education than poor countries.

This is all to say that, in some ways, the focus on TFP growth may be misleading. Growth in technology, for developing countries, is very similar to growth in human capital, at least intuitively. If the Solow residual is, in fact, relatively unimportant once human capital is measured correctly, then the problem of growth in poor countries is much simpler: do we deepen our physical capital, or improve our human capital? This paper suggests that human capital improvements are most important, and that useful improvements in human capital may be partially driven by coordinating increased specialization of workers. Interesting.

2011 working paper, which appears to be the newest version; IDEAS page.