Unit 9 Uneven development on a global scale

9.4 How much does capital accumulation contribute to the growth rate of living standards?

The examples in the previous section suggest that the growth of capital per worker played an important part in the rise in output per worker and living standards over the nineteenth and twentieth centuries, but that technological progress was important too, to mitigate the problem of the diminishing productivity of capital goods. To try to disentangle these effects using data, and allow for the contribution to GDP of population (and hence labour force) growth, we return to the production function.

Allowing for more factors of production

For an example of a production function with more than two factors of production, read Section 2.4 of The Economy 2.0: Microeconomics.

factor of production

Any input into a production process is called a factor of production. Factors of production may include labour, machinery and equipment (usually referred to as capital), land, energy, and raw materials.

To interpret the data, we need to recognize that labour and capital goods are not the only inputs to production. One way to write a more general production function would be to continue to think of \(F\) as the production technology, but include a longer list of factors of production—the broad categories of resources that are combined to produce output. For example:

\[Y = z F(K, N, E, R, L, …)\]

where \(E\) is energy, \(R\) is raw materials, \(L\) is land, and so on. For a full description we could go further, and take into account that the aggregate labour input, \(N\), depends not only on the size of the labour force, but also on health, education, worker effort, and hours of work.

The \(z\)-term represents anything that affects the amount of output other than through the listed factors of production—that is, total factor productivity (TFP). TFP includes the state of technology, but we can interpret it more widely. For example, the quality of management in firms in the economy, the degree of competition between firms, and the availability of public goods (for example, knowledge) and infrastructure (for example, transport and communication systems) will all affect the amount of output that can be produced with given amounts of the inputs.

Ideally, if we had data on all of the inputs and could measure \(z\), we could use this type of production function to work out how the growth rate of output depends on the growth of all of these components. Typically, however, we only have data on aggregate capital goods and labour, measured as the number of workers. We can still obtain some insight into their contributions to output growth using an ‘approximate’ production function:

\[Y = \tilde{z}\tilde{F}(K, N)\]

This is just like the theoretical production function we started with, but we now interpret it differently. The function, \(\tilde{F}\) (pronounced ‘F-tilda’), captures the way in which capital goods and labour are combined to produce output. The TFP term, \(\tilde{z}\), captures the contributions of everything else (the other inputs, the quality of management, the state for technology, adjustments for education, and so on.).

Decomposing output growth​​

Just as in the previous section, we assume that the function \(\tilde{F}\) has constant returns to scale, and deduce that when \(Y = \tilde{z}\tilde{F}(K, N)\):

• If \(\tilde{z}\) increases from one year to the next, while capital, \(K\), and the labour force, \(N\), remain the same, output increases in the same proportion as \(\tilde{z}\). In other words, the growth rate of output, \(g^Y\), is equal to the rate of improvement of TFP, \(g^z\).
• If \(K\) grows while \(\tilde{z}\), \(N\), and other inputs remain constant, we expect output to grow, but at a slower rate because of falling capital productivity: \(g^Y < g^K\).

Similarly if the labour force, \(N\), grows without any increase in \(\tilde{z}\) or \(K\), we would expect falling labour productivity and hence \(g^Y < g^N\).

In general, when \(\tilde{z}\), \(K\), and \(N\) are all changing at different rates, we can analyse how output grows over time by adding all the effects together:

\[\begin{align*} \text{output growth rate} &= \text{rate of TFP growth} \\ & ~+ \text{weighted average of growth rates of } K \text{ and } N \\ g^Y &= g^z + \beta g^K + (1-\beta) g^N \end{align*}\]

Here, \(𝛽\) (the Greek letter, beta) represents the effect on output of increasing the capital stock while everything else remains constant. It depends on the properties of the production function, but we expect it to be positive, and less than 1 (capturing the falling productivity effect). The extension to this section explains in more detail how to derive this equation.

We can use this relationship to study the data on growth in different countries. But there is still a problem: we have no practical way of measuring TFP growth, \(g^z\). However, if we assume a plausible value for \(𝛽\), we can determine how much of the output growth is accounted for by the growth rates for capital and labour. The remaining part—the residual—gives us an indirect estimate of TFP growth. Figure 9.7a shows an example, assuming that \(𝛽 = 0.4\).

The table shows, in the second-last column, how much output growth can be accounted for by a combination of capital and labour growth, and in the last column the residual the part of growth that is unaccounted for. In China, for example, output grew by an average of 5.9% per year for a period of 60 years; of this, a substantial part (4.6%) can be attributed to capital and labour. In other words, just under three-quarters of the growth that has taken place over this 60-year period can be interpreted as resulting from increases in the labour force and the capital stock. The remaining 1.3% output growth must be attributed to other sources: technological progress, together with growth in other inputs such as energy, or contributions from education or infrastructure.

South Korea is broadly similar, although the average growth rate is higher, and TFP accounts for a larger proportion of it. The US (which had much higher GDP per worker in 1960) grew more slowly, and (relatively) the residual is higher still: capital and labour account for less than half of the growth in output.

Figure 9.7b shows an alternative way of analysing the same data: how much of the growth of output per worker is accounted for by the growth of capital per worker. In Extension 9.4 at the end of this section, we show that the growth accounting equation from earlier can be rearranged to say that the growth rate of output per head is equal to TFP growth \(g^z\) plus \(𝛽\) times the growth rate of capital per head. Figure 9.7b shows these growth rates and the residual when \(𝛽\) is 0.4.

These estimates of the residual are almost, but not exactly, the same as in the previous table. (They differ because we are working in averages, and increases in capital do not always move exactly in step with increases in the labour force.) The per worker table captures the growth paths illustrated in Figure 9.8.

Figure 9.8 presents data for China and South Korea in the same format as Figure 9.6b. It zooms in on the period from the 1950s for two countries that experienced rapid ‘hockey stick’ growth in these years. Between 1953 (the end of the Korean War) and 1970, capital goods per person in South Korea fell, but output per capita rose, giving evidence of the importance of the introduction of new technologies, increased education, and the factors contributing to total factor productivity growth.

The data series for China runs to 2019, by which time its capital labour ratio was as high as it had been in South Korea a quarter century earlier in 1992–1993. But China’s level of productivity was only three-quarters as high. Growth in China is much more capital-intensive than in South Korea. From the growth accounting analysis over the 1960–2019 period, the contribution of ‘the residual’ to growth in South Korea is greater than in China. One interpretation is that China has been less successful in accessing the best available technology and organizational knowledge at a given level of capital intensity than South Korea.

A puzzle: Why does investment not flow to poor countries?

The analysis in this section poses a big puzzle that we explore in the coming sections: why do some countries fail to catch up to the technology leaders? From the model production function figures, when capital per worker is low, capital productivity is high. The average product of capital is even higher when new technology is used (the production function rotates upwards as in Figure 9.5).

In a world in which the owners of firms in rich countries can choose where they invest their profits and in addition, they understand the new technology and how to organize production processes, it would seem logical to invest their profits in poor countries where the productivity of capital is higher. The production function suggests that for a given investment of capital, the increase in output per worker and therefore in profits would be higher.

But what is observed around the world is that instead of flowing mainly to lower income countries, investment by firms in machinery and equipment in foreign countries (called foreign direct investment) is directed mainly to countries that have a similar (high) level of per capita GDP and wages.

Figure 9.9 shows that more than 50% of US foreign direct investment in the first decade of this century flowed to countries with higher wages in manufacturing than in the US. Before returning to the puzzle about why investment does not flow from rich to poor countries to produce catch up by (all) poor countries, we need to explain more about saving and investment in a closed economy, where goods, services, and capital do not flow across borders.