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.

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.

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.

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.

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.