Unit 9 Uneven development on a global scale

9.3 Capital goods and technology

production function

A production function is a graphical or mathematical description of the relationship between the quantities of the inputs to a production process and the amount of output produced. When used to represent output in the whole economy, it is described as an aggregate production function.

labour productivity, productivity of labour

A measure of the effectiveness of the labour input in a production process. Typically it is total output divided by the number of units of labour (e.g. hours, or workers) used to produce it, or in other words the average product of labour.

To think about how and why a country’s income and living standards may grow over time, we can start with the aggregate production function, which summarizes how the inputs to production are converted into aggregate output (and income). In the macroeconomic model of the economy developed in Units 1 to 3, we use the simplifying assumptions that labour is the only input, and output per worker, \(\lambda\), is fixed. The production function is \(Y=\lambda N\), where \(Y\) is output and \(N\) is total employment. In that scenario, output can only increase if either more workers are employed, or labour productivity, \(\lambda\), increases—for example, due to an improvement in the quality of education and training. Section 2.4 shows how this affects output, unemployment, and inequality.

For a fuller picture of how the economy changes, we need to include other factors of production. How much output is produced also depends on inputs such as:

• capital equipment, such as tools (including digital ones), machines, and buildings
• energy: coal, oil, gas, or renewable energy such as wind and solar
• land
• other environmental resources, such as water, forests, or minerals
• public infrastructure, such as roads, railways, and digital networks.

And it depends on the technologies used to convert inputs into outputs.

The production function: Labour, capital goods, and technology

capital goods, capital

Capital goods (sometimes shortened to ‘capital’) are the durable and costly non-labour inputs used in production (e.g. machinery, equipment, buildings). They do not include some essential inputs (e.g. air, water, knowledge) that are used in production at zero cost to the user.

Initially we will imagine that there are just two inputs: capital (\(K\)) and labour (\(N\)). We could then write the production function as \(Y = F(K, N)\), and think of the function \(F\) as the production technology: using this technology, different amounts of \(K\) and \(N\) are combined to produce output. However, we also want to allow for technological change. So instead, we will write the production function as:

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

total factor productivity, TFP

In a production process, if the amount of output that can be produced increases, without any changes in the factors of production (inputs) we say that total factor productivity (TFP) has increased. For example, if the inputs are capital (\(K\)) and labour (\(N\)), we can write the production function as \(Y = zF(K,N)\), where \(z\) represents TFP.

where \(z\) represents the current state of technology. Technological progress causes \(z\) to increase, meaning that more output can now be produced using the same amounts of capital and labour as before. We say that total factor productivity has risen. For example, if \(z\) increases by 1% but \(K\) and \(N\) remain constant, the economy produces 1% more output (it grows by 1%).

With this production function, there are three reasons the economy may grow: technological progress (an increase in \(z\)), capital accumulation (\(K\) increases), and employment growth (\(N\) increases). The labour input could rise as a result of an increase in labour force participation, for example, or a rise in the number of migrant workers.

For the aggregate economy it is reasonable to assume that, at a given state of technology, if all inputs to production increase in the same proportion, output will do the same. For example, if the labour force doubles, and so do the number of factories where the workers are employed and the number of machines that they use, then the economy will produce twice as much. In other words, the function \(F\) has constant returns to scale.

An increase in \(z\) raises output in the same proportion, but what about a rise in capital equipment, \(K\)? What happens, for example, if \(K\) increases but \(z\) and \(N\) stay constant? With a constant returns technology \(F(K, N)\), we can answer this question by focusing on output per worker, \(Y/N\)—in other words, labour productivity:

When \(K\) rises but \(N\) doesn’t change, capital per worker—the number of machines operated by each worker, say—increases. We would expect output to rise, but by less than the increase in \(K/N\): for example if workers now operate double the number of machines, they spend less time with each one, so the amount of output does not double. And as \(K/N\) rises further, extra equipment has less and less effect: workers are too busy operating the equipment they already have to make good use of the new ones.

average product

The average product of an input is total output divided by the total amount of the input. For example, the average product of a worker (also known as labour productivity) is total output divided by the number of workers employed to produce it.

Figure 9.4 shows the relationship between output per worker (\(Y/N\)) on the vertical axis and capital equipment per worker (\(K/N\)) on the horizontal axis—that is, the production function per worker. At point A, each worker operates equipment costing $16,000, and produces $12,800 of output. The average product of capital (APK) is \(12,800/16,000 = 0.8\). At B, capital per worker is more than twice as high ($36,000), and output has risen—but only to $19,200, so the productivity of capital is much lower: \(\text{APK} = 0.53\).

The APK at B is shown on the diagram: it is the slope of the line from the origin to B. It is clear without doing any calculations that it is lower than the APK at A—the line from the origin gets flatter as you move along the production function. The curved shape of the production function captures the ‘Charlie Chaplin’ property: as capital intensity, \(K/N\), increases, the average product of capital diminishes.

As a result of the diminishing APK, an economy will not be able to sustain growth in output per worker simply by adding more of the same type of capital. Eventually, the productivity of capital becomes so low that it is not worth investing any further, given the current state of technology.

Technological progress

innovation rent

Profits in excess of the opportunity cost of capital that an innovator gets by introducing a new technology, organizational form, or marketing strategy.

creative destruction

Joseph Schumpeter’s name for the process by which old technologies and the firms that do not adapt are swept away by the new, because they cannot compete in the market. In his view, the failure of unprofitable firms is creative because it releases labour and capital goods for use in new combinations.

But technological change that increases the productivity of capital allows sustained economic growth. Firms have incentives to develop and adopt new technologies: they can earn innovation rents by adopting them ahead of their rivals. Firms that fail to innovate (or copy other innovators) are unable to sell their product for a price above the cost of production, and eventually fail. Unit 2 in the microeconomics volume explains how this process of creative destruction⁠ has led to sustained increases in living standards. Technological progress and the accumulation of capital goods⁠ are complementary: each provides the conditions necessary for the other to proceed.

• New technologies require new machines: The accumulation of capital goods is a necessary condition for the advance of technology, as in the case of the spinning jenny, for example.
• Technological advance is required to sustain the process of capital goods accumulation: It means that the introduction of increasingly capital-intensive methods of production continues to be profitable.

Figure 9.5 illustrates the effect of a technological improvement—modelled as an increase in \(z\). This rotates the production function upwards. Each worker produces more output for a given level of capital equipment—the productivity of capital rises. (Specifically, in this example, output rises by 50% at each level of capital.)

For example, starting at point A with $16,000 capital per worker and APK of 0.75, adopting the new technology increases the productivity of capital, encouraging net investment. Businesses can raise capital per worker as high as $36,000 without loss of productivity—at point C, output per head is much higher than at A, and the APK is still 0.75. The economy has grown through a combination of technological progress and capital investment.

New technology can also refer to new ways of organizing work. The managerial revolution in the early twentieth century called Taylorism⁠ is a good example: labour and capital equipment were reorganized in a streamlined way, and new systems of supervision were introduced to make workers work harder. More recently, the information technology (ICT) revolution allows one engineer to be connected with thousands of other engineers and machines all over the world. The ICT revolution therefore rotates the production function upward, increasing the average product of capital at every level of capital per worker.

catch-up growth

When an economy with relatively low GDP experiences a period of rapid growth that brings incomes closer to those in high-income countries, this is described as catch-up growth. See also: convergence.

convergence

If a country where GDP is initially low experiences catch-up growth until its growth path is similar to that in high-income countries, this is described as convergence. See also: catch-up growth.

To examine how technological progress and capital accumulation shaped the world, we compare countries that have been technology leaders with follower countries. Figure 9.6a shows the rise of GDP per capita over 150 years in four countries, using a ratio scale so that the slope of each line measures the growth rate. We know from Unit 1 of the microeconomics volume that economies moved up the hockey stick at very different times. Britain was the technological leader, from the Industrial Revolution until the eve of the First World War, when the US took over leadership. Until the end of the Second World War the standard of living in Japan and Taiwan remained much lower, while growing at a similar rate to the leaders; both countries then experienced a period of rapid catch-up growth until their paths converged with the leaders at the end of the twentieth century. Over the past quarter of a century, Taiwan has virtually closed the gap in GDP per capita with the US, while Japan’s growth path has diverged from that of the US.

Figure 9.6b shows the relationship between output per worker and capital per worker, in the same countries for comparison with the model in Figure 9.5.

Focusing first on Britain, the figure shows the path traced out over time as both capital intensity and productivity rose, beginning in 1760 (the bottom corner of the chart) and ending in 1990 with output per worker more than seven times as high. In the US, productivity overtook the UK by 1910 and has remained higher since. In 1990, the US had higher productivity and capital intensity than the UK.

Figure 9.6b shows that the economies that are rich today have had their labour productivity (output per worker) rise over time as they became more capital-intensive. For example, if we consider the US, capital per worker (measured in 1985 US dollars) rose from $4,325 in 1880 to $14,407 in 1953, and $34,705 in 1990. Alongside this increase in capital intensity, US labour productivity rose from $7,400 in 1880 to $21,610 in 1953, to $36,771 in 1990.

Now consider Japan and Taiwan. By 1990, capital per worker in Japan was not only higher than in the US, but also almost twice as high as in Britain. Japan had reached this level in less than half the time it took Britain to do so. Taiwan in 1990 was also more capital-intensive than Britain. The lead in mass production and science-based industries that the US had established was eroded as other economies invested in education and research, and adopted American management techniques.¹

In contrast to the model in Figure 9.5, which shows the shift in the production function and an increase in capital intensity separately, the data in Figure 9.6b combines both effects. In the model, the effect of the increase in \(K/N\) on its own is to reduce the average product of capital (APK), which is partially offset by the upwards rotation of the production function. This is why the historical data graphs in Figure 9.6b are less curved (less concave) than the ones we drew in Figure 9.5.

Interpreting Figure 9.6b using the model of the production function in Figure 9.4 shows that all the economies adopted more capital-intensive methods of production as they became richer. However, while Japan and Taiwan both experienced substantial technological progress, the fact that output per worker at each level of capital intensity remained below that of both the US and Britain means that they remained on a lower production function.

To summarize:

• The increase in the amount of capital goods per worker contributes to increases in the output per worker, which permits increases in living standards.
• Technological progress rotates the production function upwards, stimulated by the prospect of innovation rents.
• This partially offsets the diminishing average returns to capital that would otherwise limit the accumulation of capital.