Unit 2 Technology and incentives

2.5 Modelling a dynamic economy: Technology and costs

The Industrial Revolution offered new choices to firms—in particular, the possibility of adopting technologies that raised labour productivity by means of machines powered by non-renewable energy (that is, by coal). In this section, we consider how a firm can evaluate the cost of different technologies.

Suppose we ask an engineer to report on the technologies that are available to produce 100 metres of cloth, where the inputs are labour (number of workers, each working for a standard eight-hour day) and energy (tons of coal). The answer is represented in the diagram and table in Figure 2.5. The five points represent five different technologies.

These are fixed-proportions technologies with constant returns to scale, like the two olive oil examples in Figure 2.4. Instead of drawing each technology as a ray from the origin, we can compare them by marking only the point on each one that shows the input requirements for 100 metres of cloth. This tells us everything we need to know about each technology: for example, technology E uses ten workers and one ton of coal to produce 100 metres of cloth, so it would need 20 workers and two tons of coal to produce 200 metres.

Different technologies for producing 100 metres of cloth.
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Technology Number of workers Coal required (tons)
A 1 6
B 4 2
C 3 7
D 5 5
E 10 1

Figure 2.5 Different technologies for producing 100 metres of cloth.

Technology Number of workers Coal required (tons)
A 1 6
B 4 2
C 3 7
D 5 5
E 10 1

Five different technologies

Five different technologies for producing 100 metres of cloth.

The table describes five technologies that we compare in the rest of this section. They use different quantities of labour and coal as inputs for producing 100 metres of cloth.

Technology A: energy-intensive: In this diagram, the horizontal axis shows the number of workers, ranging from 1 to 10, and the vertical axis shows tons of coal, ranging from 1 to 10. Coordinates are (number of workers, tons of coal).  Technology A, the point (1, 6), uses 1 worker and 6 tons of coal. An upward-sloping, straight line passes through point A and shows that the coal-labour ratio is 6:1.
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Technology Number of workers Coal required (tons)
A 1 6
B 4 2
C 3 7
D 5 5
E 10 1

Technology A: energy-intensive

Technology A is the most energy-intensive, using one worker and six tons of coal.

Technology B: In this diagram, the horizontal axis shows the number of workers, ranging from 1 to 10, and the vertical axis shows tons of coal, ranging from 1 to 10. Coordinates are (number of workers, tons of coal). Technology A, the point (1, 6), uses 1 worker and 6 tons of coal. Technology B, the point (4, 2), uses 4 workers and 2 tons of coal. Technology C, the point (3, 7), uses 3 workers and 7 tons of coal. Technology D, the point (5, 5), uses 5 workers and 5 tons of coal.
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Technology Number of workers Coal required (tons)
A 1 6
B 4 2
C 3 7
D 5 5
E 10 1

Technology B

Technology B uses four workers and two tons of coal: it is more labour-intensive than technology A, because the ratio of workers to coal required to produce the given output is higher for B than A.

Three more technologies: In this diagram, the horizontal axis shows the number of workers, ranging from 1 to 10, and the vertical axis shows tons of coal, ranging from 1 to 10. Coordinates are (number of workers, tons of coal). Technology A, the point (1, 6), uses 1 worker and 6 tons of coal. An upward-sloping, straight line passes through point A and shows that the coal-labour ratio is 6:1. Technology B, the point (4, 2), uses 4 workers and 2 tons of coal. An upward-sloping, straight line passes through point B and shows that the coal-labour ratio is 1:2. Technology C, the point (3, 7), uses 3 workers and 7 tons of coal. Technology D, the point (5, 5), uses 5 workers and 5 tons of coal. Technology E, the point (10, 1), uses 10 workers and 1 ton of coal. An upward-sloping, straight line passes through point E and shows that the coal-labour ratio is 1:10.
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Technology Number of workers Coal required (tons)
A 1 6
B 4 2
C 3 7
D 5 5
E 10 1

Three more technologies

We have added technologies C, D, and E to the diagram. Technology E is the most labour-intensive of the five technologies: it uses ten workers and one ton of coal.

Technology E is relatively labour-intensive and technology A is relatively energy-intensive. If an economy using technology E shifted to technology B, for example, we would say that they had adopted a labour-saving technology, because the labour requirement for 100 metres of cloth had fallen. This is what happened during the Industrial Revolution.

Which technology will the firm choose? We can rule out technologies that are obviously inferior. Comparing A and C shows that technology C is inferior: it uses more workers and more coal to produce the same amount of cloth. We say that technology C is dominated by technology A: assuming all inputs must be paid for, no firm will use C when A is available. The steps in Figure 2.6 show you how to determine which technologies are dominated, and which technologies dominate.

In this diagram, the horizontal axis shows the number of workers, ranging from 1 to 10, and the vertical axis shows tons of coal, ranging from 1 to 10. Coordinates are (number of workers, tons of coal). Technology A, the point (1, 6), uses 1 worker and 6 tons of coal. Technology B, the point (4, 2), uses 4 workers and 2 tons of coal. Technology C, the point (3, 7), uses 3 workers and 7 tons of coal. Technology D, the point (5, 5), uses 5 workers and 5 tons of coal. Technology E, the point (10, 1), uses 10 workers and 1 ton of coal.
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Figure 2.6 Technology A dominates C; technology B dominates D.

A dominates C: In this diagram, the horizontal axis shows the number of workers, ranging from 1 to 10, and the vertical axis shows tons of coal, ranging from 1 to 10. Coordinates are (number of workers, tons of coal). Technology A, the point (1, 6), uses 1 worker and 6 tons of coal. Technology B, the point (4, 2), uses 4 workers and 2 tons of coal. Technology C, the point (3, 7), uses 3 workers and 7 tons of coal. Technology D, the point (5, 5), uses 5 workers and 5 tons of coal. Technology E, the point (10, 1), uses 10 workers and 1 tons of coal.
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A dominates C

First, consider point A. Technology C is in the shaded rectangle above and to the right of A. So technology A dominates technology C: C uses more workers and more energy to produce the same amount of cloth. This means that if A is available, you would never use C.

B dominates D: In this diagram, the horizontal axis shows the number of workers, ranging from 1 to 10, and the vertical axis shows tons of coal, ranging from 1 to 10. Coordinates are (number of workers, tons of coal). Technology A, the point (1, 6), uses 1 worker and 6 tons of coal. Technology B, the point (4, 2), uses 4 workers and 2 tons of coal. Technology C, the point (3, 7), uses 3 workers and 7 tons of coal. Technology D, the point (5, 5), uses 5 workers and 5 tons of coal. Technology E, the point (10, 1), uses 10 workers and 1 tons of coal.
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B dominates D

The shaded rectangle at point B shows that technology B dominates D: B can produce the same amount of cloth with less labour and energy.

E does not dominate: In this diagram, the horizontal axis shows the number of workers, ranging from 1 to 10, and the vertical axis shows tons of coal, ranging from 1 to 10. Coordinates are (number of workers, tons of coal). Technology A, the point (1, 6), uses 1 worker and 6 tons of coal. Technology B, the point (4, 2), uses 4 workers and 2 tons of coal. Technology C, the point (3, 7), uses 3 workers and 7 tons of coal. Technology D, the point (5, 5), uses 5 workers and 5 tons of coal. Technology E, the point (10, 1), uses 10 workers and 1 tons of coal.
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E does not dominate

Technology E does not dominate any of the available technologies. None of them are in the area above and to the right of E. Technology E is also not dominated by any other available technology.

Using only the engineering information about inputs, we have narrowed down the choices: technologies C and D would never be chosen. But how does the firm choose between A, B, and E? We assume that the firm’s goal is to make as much profit as possible, which means producing the most cloth at the smallest possible cost.

So the firm also needs economic information about relative prices—the cost of hiring a worker relative to that of purchasing a ton of coal. Intuitively, the labour-intensive technology E would be chosen if labour was cheap relative to the cost of coal; the energy-intensive technology A would be preferable in a situation where coal is relatively cheap. An economic model helps us be more precise than this.

How does a firm evaluate the cost of production using different technologies?

The firm can calculate the cost of any combination of inputs that it might use by multiplying the number of workers by the wage and the tons of coal by the price of coal. We use the symbol w for the wage, N for the number of workers, p for the price of coal, and R for the tons of coal:

\[\begin{align*} \text{cost} &= (\text{wage} \times \text{workers}) + (\text{price of a ton of coal} \times \text{number of tons}) \\ &= (w\times N) + (p\times R) \\ \end{align*}\]
isocost line
A line that represents all combinations of inputs that cost a given total amount.

Suppose the wage is £10 and the price of coal is £20 per ton. Then, for example, this formula tells us that the cost of employing two workers and three tons of coal is £80. This corresponds to combination P1 in Figure 2.7. If the firm were to employ more workers—say, six—but reduce the input of coal to one ton (point P2), that would also cost £80. Isocost lines join all the combinations of workers and coal that cost the same amount: iso is the Greek for ‘same’. A simple way to draw any line is to find the endpoints. For example, the £80 line joins point J (no workers, four tons of coal) and point H (eight workers, no coal). The steps in Figure 2.7 explain how to construct isocost lines to compare the costs of all combinations of inputs.

In this diagram, the horizontal axis shows the number of workers, ranging from 1 to 15, and the vertical axis shows tons of coal, ranging from 1 to 10. Coordinates are (number of workers, tons of coal). There are four downward-sloping parallel lines, representing the isocosts for £40, £80, £120, and £150. The isocost for £40 connects the points (0, 2) and (4, 0). The isocost for £80 has four labelled points: the vertical intercept J at (0, 4), P1, at (2, 3), P2 at (6, 1), and the horizontal intercept H at (8, 0). The isocost for £120 connects the points (0, 6) and (12, 0). The isocost for £150 has two labelled points: Q1, with coordinates (3, 6), and Q2, with coordinates (5, 5). All the technologies above the JH line cost more than £80.
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Figure 2.7 Isocost lines for different technologies when the wage is £10 and the price of coal is £20.

The total cost at P1: In this diagram, the horizontal axis shows the number of workers, ranging from 1 to 15, and the vertical axis shows tons of coal, ranging from 1 to 10. Coordinates are (number of workers, tons of coal). Point P1, with coordinates (2, 3), costs £80.
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The total cost at P1

The total cost of employing two workers with three tons of coal is (2 × 10) + (3 × 20) = £80.

P2 also costs £80: In this diagram, the horizontal axis shows the number of workers, ranging from 1 to 15, and the vertical axis shows tons of coal, ranging from 1 to 10. Coordinates are (number of workers, tons of coal). Point P1, with coordinates (2, 3), and point P2, with coordinates (6, 1), both cost £80.
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P2 also costs £80

If the number of workers is increased to six, thereby costing £60, and the input of coal is reduced to one ton, then the total cost will still be £80.

The isocost line for £80: In this diagram, the horizontal axis shows the number of workers, ranging from 1 to 15, and the vertical axis shows tons of coal, ranging from 1 to 10. Coordinates are (number of workers, tons of coal). Point P1, with coordinates (2, 3), and point P2, with coordinates (6, 1), both cost £80. A downward-sloping straight line connecting both points is the isocost for £80.
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The isocost line for £80

The straight line through P1 and P2 joins together all the points where the total cost is £80. We call this an isocost line. When drawing the line, we simplify by assuming that fractions of the inputs can be purchased.

A higher isocost line: In this diagram, the horizontal axis shows the number of workers, ranging from 1 to 15, and the vertical axis shows tons of coal, ranging from 1 to 10. Coordinates are (number of workers, tons of coal). There are two downward-sloping parallel lines, representing the isocosts for £80 and £150. The isocost for £80 has two labelled points: P1, with coordinates (2, 3), and P2, with coordinates (6, 1). The isocost for £150 has two labelled points: Q1, with coordinates (3, 6), and Q2, with coordinates (5, 5).
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A higher isocost line

At point Q1 (three workers, six tons of coal) the total cost is £150. To find the £150 isocost line, plot another point that costs £150: if two more workers are employed, the input of coal should be reduced by one ton to keep the cost at £150. This is point Q2.

Isocost lines are parallel: In this diagram, the horizontal axis shows the number of workers, ranging from 1 to 15, and the vertical axis shows tons of coal, ranging from 1 to 10. Coordinates are (number of workers, tons of coal). There are four downward-sloping parallel lines, representing the isocosts for £40, £80, £120, and £150. The isocost for £40 connects the points (0, 2) and (4, 0). The isocost for £80 has four labelled points: the vertical intercept J at (0, 4), P1, at (2, 3), P2 at (6, 1), and the horizontal intercept H at (8, 0). The isocost for £120 connects the points (0, 6) and (12, 0). The isocost for £150 has two labelled points: Q1, with coordinates (3, 6), and Q2, with coordinates (5, 5).
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Isocost lines are parallel

We can draw isocost lines through any point in the diagram. We find that the isocost lines are parallel. For any isocost line, like the £80 one joining points JH, all points above the line cost more, and all points below cost less.

Points above an isocost line cost more: In this diagram, the horizontal axis shows the number of workers, ranging from 1 to 15, and the vertical axis shows tons of coal, ranging from 1 to 10. Coordinates are (number of workers, tons of coal). There are four downward-sloping parallel lines, representing the isocosts for £40, £80, £120, and £150. The isocost for £40 connects the points (0, 2) and (4, 0). The isocost for £80 has four labelled points: the vertical intercept J at (0, 4), P1, at (2, 3), P2 at (6, 1), and the horizontal intercept H at (8, 0). The isocost for £120 connects the points (0, 6) and (12, 0). The isocost for £150 has two labelled points: Q1, with coordinates (3, 6), and Q2, with coordinates (5, 5). All the technologies above the JH line cost more than £80.
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Points above an isocost line cost more

For any isocost line, like the £80 one joining points, all points above the line cost more, and all points below cost less.

All the isocost lines are parallel. This happens because at any point, if you increase the number of workers by one, your costs rise by £10 (the wage). But the price of coal is £20, so if you decrease the coal input by 0.5 tons at the same time, costs will stay the same. The slope of the isocost line is –0.5 (the change in energy divided by the change in labour).

The slope just depends on the relative prices of labour and energy:

\[\text{slope of isocost lines } = -\frac{w}{p}\]

We can use isocost lines to compare the technologies that are not dominated. The table in Figure 2.8 shows the cost of producing 100 metres of cloth using each technology when w = £10 and p = £20. Clearly technology B allows the firm to produce cloth at lower cost.

In the diagram, we have drawn the isocost line through the point representing technology B. This shows immediately that, at these input prices the other two technologies are more costly.

In this diagram, the horizontal axis shows the number of workers, ranging from 1 to 10, and the vertical axis shows tons of coal, ranging from 1 to 7. Coordinates are (number of workers, tons of coal). Line HJ represents the isocost for £80 when the wage is £10 and coal costs £20 per ton, and connects the points J at (0, 4), B at (4, 2), and H at (8, 0). Points A (1, 6) and E (10, 1) represent technologies that cost above £80.
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Technology Number of workers Coal required (tons) Total cost (£)
A 1 6 130
B 4 2 80
E 10 1 120
Wage £10, coal price £20 per ton

Figure 2.8 The costs of using different technologies to produce 100 metres of cloth.

When w = 10 and p = 20, B is the least-cost technology. The other available technologies will not be chosen at these input prices. Importantly, it is the relative price, w/p, that matters for the choice. If both prices doubled, the diagram would look almost the same: the isocost line through B would have the same slope, although the cost would be £160.

Exercise 2.5 Isocost lines

Suppose the wage is £10 but the price of coal (per ton) is only £5.

  1. What is the relative price of labour?
  2. On a diagram with the number of workers on the horizontal axis and tons of coal on the vertical axis, draw the isocost line that shows all combinations of inputs that cost £60. (Hint: Find and connect both endpoints of this line.)
  3. On the same diagram, draw the isocost lines corresponding to costs of £30 and £90. Compare your diagram to Figure 2.7. How does the set of isocost lines for these input prices compare to the ones for w = 10 and p = 20?

Extension 2.5 The equation of an isocost line

Using simple algebra, we show how to rearrange the equation of an isocost line to see the slope and the intercept with the vertical axis.

We can represent the isocost lines for any wage \(w\) and coal price \(p\) as equations. To do this, we write \(c\) for the cost of production. The equation for calculating \(c\) is:

\[c = wN + pR\]

This is one way to write the equation of the isocost line for a particular value of \(c\).

But to draw an isocost line, it can help to express it in the form:

\[y = a + bx\]

where \(a\), which is a constant, is the vertical axis intercept and \(b\) is the slope of the line. In our model, \(R\) is on the vertical axis and \(N\) is on the horizontal axis. So we rearrange the equation for the cost to put \(R\) on the left-hand side:

\[pR = c-wN\]

and divide by \(p\):

\[R = \frac{c}{p} - \frac{w}{p}N\]

This tells us that the slope of the isocost line is the relative price of labour, −(\(w/p\)). The slope is negative, so the line slopes downward. It crosses the vertical axis at the point (\(c/p\)).

When \(w\) = 10 and \(p\) = 20, the isocost line for \(c\) = 80 has a vertical axis intercept of 80/20 = 4 and a negative slope equal to −(\(w/p\)) = −0.5.

Exercise E2.2 Writing isocost lines as equations

Figure 2.5 (reproduced below) shows five possible technologies that can produce 100 metres of cloth.

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Suppose the wage is £4 and the price of coal is £6.

  1. Calculate the cost of producing 100 metres of cloth, and rank the technologies from cheapest to most expensive.

For each technology:

  1. Use your answer from Question 1 of this exercise to write down the equation of the isocost line that passes through that technology.
  2. In a diagram like Figure 2.7, draw the isocost line that passes through that technology.

Now suppose that the wage increases to £8.

  1. Repeat Question 1 for each technology. How do the cost rankings of the technologies change?
  2. Draw a new diagram with the isocost lines passing through each technology to visually confirm your answer to Question 4.