# Lecture 4: Directed Technical Change

In this lecture we will discuss the notion of directed technical change. Up until now, we have been treating the technology frontier as a unidimensional object, along which we progress at an endogenously determined rate. Now we will also treat the direction of technical change as endogenous. The exact nature of the possible directions we can take and the forces that will determine the outcome will vary based on application, but there are lessons we can learn from the general theory.

The main source of material here will be , whose notation we will for the most part follow. For a more explicit discussion of directed technological change and the skilled wage premium, consult .

## Production Environment

We'll be using the generalized CES production function once again in this setting out of necessity. There is one final good $Y$ and two intermediate goods $Y_L$ and $Y_Z$, which are combined according to

Each intermediate is produced according to the so called **lab equipment** model. There are $N_L$ and $N_Z$ machine types that are used as inputs. Each is produced by a monopolist at constant marginal cost $\psi$ and rented to intermediate producers at respective prices $\chi_L^j$ and $\chi_Z^j$. The production functions are given by

Let the price of the respective intermediates be $p_L$ and $p_Z$. Assuming competitive production of the final good, we can derive demand functions of the form

Now that we've established that outcomes will be symmetric at the machine level, we can dispense with differentiation. Plugging into , we find
**price effect** whereby goods that demand a higher price pass on higher returns to their input machines, which results in increased incentives to innovation; and (2) the **market size effect** whereby innovations, being non-rival, are more profitable in larger markets. The final term represents an attempt to discern the net impact of these two effects.

Of course, the above distinction could be said to be somewhat arbitrary given that intermediate prices are an endogenous object that are themselves functions of factor and machine prevalence. Nonetheless, it is a useful distinction to make in the real world. Additionally, one could instead decompose the effect into contributions from market size and factor prices. Combining and , we can find the analogous ratio

Turning to the factor price ratio, the short term effect of an increase in the relative abundance of $Z$ will be to decrease this quantity. However, the long-run effect will depend on the endogenous innovation response through it's effect on $N_Z/N_L$. So long as $\sigma \gt 1$, there will be some "rebound," but the net long-run change is of interest too. For that we need to be more specific about the innovation technology.

Also of interest may be the share of income going to each of the factors. The ratio of these quantities will satisfy

## Innovation Structure

The paper raises some interesting points regarding state dependence in the path of innovation. That is, there is the possibility that doing innovation in a particular direction today changes the cost (or ease) of innovation in that or another direction tomorrow. However, for now we will focus on the base case of no path dependence. In particular, given research inputs $R_L$ and $R_Z$, let the respective rates of machine invention be

### State Dependence

Allowing for path dependence calls for a slightly generalized functional form for the cost of innovation, which is now specified by

Furthermore, when considering possible outcomes for the economy, state-dependence introduces some degree of instability through positive feedback. It can be show that if $\delta \gt 1/\sigma$, this will result in an extreme outcome in which one type of good completely takes over. Otherwise, there will be an interior solution.