# Question about skill-biased technical change, Acemoglu

I am writing about skill-biased technical change, and in particular the 2001 Acemoglu (http://economics.mit.edu/files/283) paper on the topic, in which there is a part I am very stuck on.

In the paper he suggests there is a market size effect and a price effect which directs technological change. The price effect directs technological change to areas which are relatively more expensive, and the market size effect to areas which are relatively bigger. The elasticity of substitution between two factors plays an important part, so that a low elasticity between two inputs will result in the price effect dominating, since in this situation scarce factors are relatively more expensive and innovation will intend to economise on this. With a high elasticity, the market size effect dominates, since scarce factors do not have a different relative price, so innovation will make widely used factors more efficient.

He applies this to skill biased technical change, in which he says one of the reasons the skill premium has increased so much over the past 60 years is because of the influx of skilled workers (i.e. the market size effect is dominating), so that with more skilled workers, innovation is directed at making them more efficient and they enjoy a higher wage. On page 18 under section 4.2 he states that this will happen with no reference to the degree of substituability between skilled and unskilled workers.

This is where I am stuck. I am trying to explain the theory, in words like I have above, so that it would read something like this: with a high degree of substitutability, technical change favours the more abundant factor (the market size effect). Skilled and unskilled labour have a high degree of substitutability, so an increase in the supply of skilled workers will direct innovation towards complementing that factor. But surely skilled and unskilled workers have a low degree of substitution? In which case surely the price effect would dominate?

I would greatly appreciate any help on this, I am sure I am making a big error somewhere, since it makes sense for the market effect to dominate here due to the empirical evidence of the increasing wage premium over the years... I also hope what I am asking makes sense. I wouldn't have written such a long question, but I feel it wouldn't have made sense without the context of the theory.

Acemoglu's argument as to when a high relative supply of skilled labor will lead to skill-biased technological change is laid out in page 13 (below eq. $(18)$), it includes the all-important condition eq. $(19)$ and ends in the middle of page 14. Here the author explains what must hold in order to observe this phenomenon.

Then at the the bottom of page 18 the author writes

"Moreover, recall that when condition ($19$) is satisfied, equation $(18)$ traces an upward-sloping long-run relative demand curve for skills. Therefore, the induced skill bias of new technologies can be sufficiently pronounced that the skill premium may increase in response to a large increase in the supply of skills."

So here, the author simply invokes the necessary theoretical condition obtained previously. So what is this condition? It reads

$$\sigma > 2- \delta \tag{19}$$

where $\delta$ is an index raging in $[0,1]$ (and serving some purpose) and $\sigma$ is the elasticity of substitution between skilled and unskilled labor.

So we see that in order to obtain skill-biased technological change together with high relative supply of skilled labor in the model (as appears to be the historical experience), namely, in order for condition $(19)$ to hold, we must have, as a necessary but not sufficient condition $\sigma >1$ according to this model.

Now, at the bottom of page 9, we see

$$\sigma \equiv \varepsilon - (\varepsilon-1)(1-\beta), \;\;\; \beta \in (0,1)$$

and where $\varepsilon$ is the elasticity of substitution of the two goods that are combined in private consumption (though not directly in the utility function as the author writes somewhere -see eq. $(6)$).

So for our necessary condition $\sigma > 1$ we must have

$$\sigma > 1 \implies \varepsilon - (\varepsilon-1)(1-\beta) > 1$$

$$\implies (\varepsilon-1) > (\varepsilon-1)(1-\beta)$$

which will hold iff $\varepsilon >1$.

So the real-world rationalization is shifted

from "is there high substitutability between skilled and unskilled labor"

to "is there high substitutability in final product/consumption between intermediate products made using skilled labor and intermediate products made using unskilled labor"?

...because if there is such high substitutability at this "second" stage, then indirectly, skilled and unskilled labor acquire a high degree of subsitutability, even though looking directly at a single production function, it would appear reasonable that they should not be highly substitutable. In other words, here "substitutability" emerges as an even more economic, rather than mainly an engineering/technical concept.

I will leave to the OP the task of contemplating further.