Unit 1 Prosperity, inequality, and planetary limits

1.6 Explaining the flat part of the hockey stick: Production functions and the diminishing average product of labour

In Italy, for example, the proportion of the labour force in agriculture stayed almost constant at around 60% between 1300 and 1800. In 2019, it was less than 4%.

Before 1800, most people in most countries made their living in agriculture. In 1798 an English clergyman, Thomas Robert Malthus, published An Essay on the Principle of Population, which presented a pessimistic view of the potential for technological progress to raise living standards.1

His argument was based on a simple model of the economy, in which agricultural output and living standards depend on the population employed in agriculture, and in turn, those living standards affect the growth of the population. Malthus argued that a sustained increase in income per capita would be impossible. We now know that he was wrong, but his analysis still helps us to understand why living standards had remained low for hundreds of years before his time.

A model is a simplified way of understanding the economy. We explain more about models in Unit 2.

In this section, we explain his model of the relationship between employment and output, which introduces concepts that are used widely in economics—in particular the production function and the diminishing average product of labour.

A model of production

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.

Imagine an agricultural economy that produces just one good: grain. Suppose that grain production is very simple—involving only farm labour to work the land. Ignore the need for spades, combine harvesters, grain elevators, silos, and other types of buildings and equipment; labour and land are the factors of production in this model, and therefore the inputs into the production process.

But the quantity of land is fixed. So if employment is increased, there will be more workers on the same quantity of land, and the average output of grain per worker will fall. In other words, the average product of labour diminishes.

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.

To be more precise, suppose there are 800 farmers, and the land is divided equally between them. All land is of the same quality. Each farmer works the same total number of hours during a year. Together, the 800 farmers produce a total of 504,000 kg of grain. The average product of a farmer’s labour is:

\[\begin{align*} \text{average product of labour} &= \frac{\text{total output}}{\text{total number of farmers}} \\ &= \frac{\text{504,000 kg}}{800 \text{ farmers}} \\ &= 630\text{ kg per farmer} \end{align*}\]

How does grain production in this economy change if the number of farmers changes? Figure 1.8a lists some values of the labour input and corresponding grain output. In the third column, we have calculated the average product of labour.

Labour input (number of farmers) Grain output (kg) Average product of labour (kg/farmer)
200 171,000 855
400 300,000 750
600 409,000 682
800 504,000 630
1,000 587,000 587
1,200 659,000 549
1,400 723,000 516
1,600 778,000 486
1,800 825,000 458
2,000 864,000 432
2,200 895,000 407
2,400 919,000 383
2,600 935,000 360
2,800 944,000 337
3,000 946,000 315

Figure 1.8a The relationship between the input of labour and the output of grain.

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.

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.

We call the relationship between the inputs to a production process and the amount of output a production function. In this example there are two inputs: labour and land. But since the amount of land is fixed, we can represent the production function as in Figure 1.8b–by plotting a graph that shows the number of farmers on the horizontal axis, and the corresponding output of grain on the vertical axis (assuming that the relationship holds for the numbers in between those shown in the table).

You can think of the production function as an ‘if–then’ statement: if there are X farmers, then they will harvest Y kg of grain. Expressing it mathematically, we say that ‘Y is a function of X’ :

\[Y = f(X)\]

X is the amount of labour devoted to farming, Y is the output in grain that results from this input, and the function f(X) describes the relationship between X and Y, represented by the curve in Figure 1.8b. Work through the steps in the figure to see how to interpret the graph.

The supply curve for books.: In this diagram, the horizontal axis shows the number of farmers, and ranges between 0 and 2,800. The vertical axis shows thousands of kilograms of grain produced, and ranges between 0 and 1,000. Coordinates are (number of farmers, thousands of kilograms of grain produced). An upward-sloping, concave curve connects the origin with point (2,800, 950). This is the farmers’ production function. This shows how the number of farmers working the land translates into grain produced. Points A (800, 504) and B (1,600, 778) lie on the production function. At point A, the average product of labour is 504,000 divided by 800, which is 630 kilograms of grain per farmer. At point B, the average product of labour is 778,000 divided by 1,600, which is 486 kilograms of grain per farmer.
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The supply curve for books.

Figure 1.8b The farmers’ production function.

The farmers’ production function: In this diagram, the horizontal axis shows the number of farmers, and ranges between 0 and 2,800. The vertical axis shows thousands of kilograms of grain produced, and ranges between 0 and 1,000. Coordinates are (number of farmers, thousands of kilograms of grain produced). An upward-sloping, concave curve connects the origin with point (2,800, 950). This is the farmers’ production function. This shows how the number of farmers working the land translates into grain produced.
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The farmers’ production function

The production function shows how the number of farmers working the land translates into grain produced at the end of the growing season.

Output when there are 800 farmers: In this diagram, the horizontal axis shows the number of farmers, and ranges between 0 and 2,800. The vertical axis shows thousands of kilograms of grain produced, and ranges between 0 and 1,000. Coordinates are (number of farmers, thousands of kilograms of grain produced). An upward-sloping, concave curve connects the origin with point (2,800, 950). This is the farmers’ production function. This shows how the number of farmers working the land translates into grain produced. Point A (800, 504) lies on the production function.
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Output when there are 800 farmers

Point A on the production function shows the output of grain produced by 800 farmers.

Output when there are 1,600 farmers: In this diagram, the horizontal axis shows the number of farmers, and ranges between 0 and 2,800. The vertical axis shows thousands of kilograms of grain produced, and ranges between 0 and 1,000. Coordinates are (number of farmers, thousands of kilograms of grain produced). An upward-sloping, concave curve connects the origin with point (2,800, 950). This is the farmers’ production function. This shows how the number of farmers working the land translates into grain produced. Points A (800, 504) and B (1,600, 778) lie on the production function.
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Output when there are 1,600 farmers

Point B on the production function shows the amount of grain produced by 1,600 farmers.

The average product: In this diagram, the horizontal axis shows the number of farmers, and ranges between 0 and 2,800. The vertical axis shows thousands of kilograms of grain produced, and ranges between 0 and 1,000. Coordinates are (number of farmers, thousands of kilograms of grain produced). An upward-sloping, concave curve connects the origin with point (2,800, 950). This is the farmers’ production function. This shows how the number of farmers working the land translates into grain produced. Points A (800, 504) and B (1,600, 778) lie on the production function. At point A, the average product of labour is 504,000 divided by 800, which is 630 kilograms of grain per farmer. At point B, the average product of labour is 778,000 divided by 1,600, which is 486 kilograms of grain per farmer.
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The average product

At A, the average product of labour is 504,000/800 = 630 kg of grain per farmer. At B, the average product of labour is 778,000/1,600 = 486 kg of grain per farmer.

The calculations in the third column of Figure 1.8a demonstrate that the more farmers there are, the lower is the average product of labour. Figure 1.8c shows that the average product of labour at each point on the graph corresponds to the slope of the line from the point to the origin. For example, the slope at point B is the vertical distance between the origin and B (778,000) divided by the horizontal distance (1,600), and the average product when there are 1,600 farmers is 778,000/1,600 = 486.

In this diagram, the horizontal axis shows the number of farmers, and ranges between 0 and 2,800. The vertical axis shows thousands of kilograms of grain produced, and ranges between 0 and 1,000. Coordinates are (number of farmers, thousands of kilograms of grain produced). An upward-sloping, concave curve connects the origin with point (2,800, 950). This is the farmers’ production function. This shows how the number of farmers working the land translates into grain produced. Points A (800, 504) and B (1,600, 778) lie on the production function. At point A the average product of labour is 504,000 divided by 800, which is 630 kilograms of grain per farmer. At point B, the average product of labour is 778,000 divided by 1,600, which is 486 kilograms of grain per farmer. A straight line connects the origin with point A. Another straight line connects the origin with point B, and its slope is 778,000 divided by 1,600, which is 486.
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Figure 1.8c The diminishing average product of labour.

The line from A to the origin is steeper (the slope is higher). So without doing any calculations, we can tell from the graph that the average product is higher at A. Also, if you draw the lines to the origin from other points they will get flatter as you move along the graph—that is, as the number of farmers increases. This happens because of the concave shape of the production function: it continues to rise but gets flatter as more farmers work the land.

Our grain production function is hypothetical, but it has two features that are plausible assumptions about farming:

  • Labour combined with land is productive. The more farmers there are, the more grain is produced (up to 3,000 farmers, at least).
  • As more farmers work on a fixed quantity of land, the average product of labour falls. This diminishing average product of labour is one of the foundations of Malthus’s model (and many other models).

The diminishing average product of labour worried Malthus.

To understand why, imagine that a generation later, each of the 800 farmers has had many children. So instead of a single farmer working each farm, there are now two farmers. The total labour input into farming was 800, but is now 1,600. The average grain produced per farmer has fallen from 630 kg to 486 kg.

You might argue that in reality, as the population grows, more land can be used for farming. But Malthus pointed out that earlier generations of farmers would have picked the best land, so any new land would not be as productive. This also reduces the average product of labour.

A diminishing average product of labour in agriculture can be caused by:

  • more labour devoted to a fixed quantity of land
  • more (inferior) land brought into cultivation

Because the average product diminishes as more labour is devoted to farming, the amount of grain each person can consume (their income or living standard) inevitably falls.

Exercise 1.4 The farmer’s production function

Think of farming biologically.

  1. Find out how many calories per hour a farmer burns on average, and how many calories are contained in 1 kg of grain.
  2. Does farming produce a surplus of calories—more calories in the output than used up in the work input—at points A and B on the production function in Figure 1.8b? Clearly state any assumptions you make in your calculations.

Question 1.5 Choose the correct answer(s)

Figure 1.8b depicts the production function of grain for farmers under average growing conditions with the currently available technology. Based on this information, we can ascertain that:

  • In a year with exceptionally good weather conditions, the production function curve will be higher and parallel to the curve above.
  • A discovery of new high-yielding crop seeds would tilt the production function curve higher, pivoted anticlockwise at the origin.
  • In a year of bad drought, the production curve can slope downwards for large numbers of farmers.
  • If there is an upper limit on the total amount of grain that can be produced in the country, then the curve will end up horizontal for large numbers of farmers.
  • Zero farmers means zero output. Therefore, all curves must start at the origin, and cannot shift upwards or downwards in a parallel manner.
  • Such a discovery would increase the kilograms of grain produced for any given number of farmers (except zero); this can be represented graphically as an anti-clockwise pivot in the production function curve.
  • A downward-sloping curve implies decreasing output as the number of farmers increases. This would only be the case if the additional labourers have negative effects on the productivity of the existing labourers, which we normally rule out.
  • An upper limit implies that additional farmers would not yield any additional kilograms of grain, which would be represented graphically by a flat production function past the upper limit.
  1. Thomas R. Malthus. 1798. An Essay on the Principle of Population. Library of Economics and Liberty. London: J. Johnson, in St. Paul’s Church-yard.