# Can an overall improvement in technology make the owners of one factor worse off?

In a closed two-factor economy, suppose there is a biased technological improvement that increases the productivity of both factors, but one more than the other. Can this reduce the aggregate returns (and not just the share of returns) to one factor? If so, how is that consistent with the usual rules of factor pricing? Doesn't (or can't) the marginal productivity of both factors rise with their respective total productivity?

Finally, if biased technology improvement can result either in increased payments to both factors (in real terms) or in increase to one and decrease to the other, how might one characterize quantitatively the conditions for (partially) immiserizing technological growth?

1x2 model

Consider a mode where production of a single good is given by a constant returns to scale CES production function:

$$Y=A(\alpha L^\rho +(1-\alpha)K^\rho)^{\frac{1}{\rho}}$$

where the elasticity of substitution between the two factors is

$$\sigma = \frac{1}{1-\rho}$$

It can be shown that the marginal product of labour (equal to the real wage in a competitive economy) is

$$MP_{L} = A\alpha \left( \frac{Y}{L}\right)^{1-\rho}$$

To understand the consequences of biased technical change on real wages, we need to look at the three components separately.

For simplicity, let's assume a fixed supply of labour, $\bar{L}$. Now, a technical change that improves productivity of both factors, but more that of capital is such that $A$ increases (the neutral component), and $\alpha$ falls (the bias). On top of this, any increase in productivity (regardless of its bias) will increase $Y$. Yet, labour is fixed. Then, you have that the three determinants of the real wage change. In particular, two are increasing and one is falling. The change in the real wage - and the nature of growth (enriching, immiserizing, neutral) depend on how these play-out. In particular, the greater the bias, the smaller the neutral technical change, and the higher the elasticity of substitution (higher $\rho$), the more likely you will have immiserizing growth. If $\rho = 1$, the third term disappears altogether.

2x2 model

You can get similar results by having a 2x2 model (2 goods and 2 inputs). For instance, say that capital productivity increases more than labour productivity. Furthermore, say that the elasticity of consumption among the two goods ($Y_{a}$ and $Y_{b}$) is very high. For example, individuals are willing to consume a lot of the former and little of the latter. If the production of $Y_{a}$ is very intensive on capital, and the technology has a high degree of substitution between capital and labour (so firms could produce by expanding capital and not much expanding labour), then you will end up with a reduction in aggregate demand for labour. This will lead to a fall in the real wage, and an absolute loss of income for workers (i.e. immiserizing growth).

Thus, in the end you can get any type of behaviour depending on:

• technology used in either good, and in particular, the degree of substitution between factors
• preferences of consumers, and in particular, the degree of substitution between goods

Think of robots increasing the productivity of capital by much more than that of workers. If consumers are happy to rebalance their consumption bundles toward more robot-produced goods and away from labour-intensive goods, then the wage rate will fall, affecting workers.