CES v. Leontief Aggregator in Production

Question

Consider a production process with two distinct capital types such that there is a capital aggregator. You could view $k_v$ as a more versatile capital (e.g. can be converted into many different production processes) as opposed to $k_u$ which is a unique factor of production exclusive to a particular production process. Hence $J=\{v,u\}$.

A CES aggregator is: ($\gamma$ is substitution coefficient, $\sigma_j$ is factor share such that $\sum_\limits{j\in J}\sigma_j=1$) $$k\equiv(\sum_{j\in J}\sigma_j k_j^\gamma)^{\frac{1}{\gamma}}.$$

Similarly,

a Leontief aggregator is:
$$k\equiv\min\{\frac{k_v}{\sigma_v},\frac{k_u}{\sigma_u}\}.$$

My Question:

When and why (i.e. economic intuition) is it appropriate to use one aggregator over the other? Are there other aggregators that are also commonly used?

Answer 1

I guess it depends on the application in question. Leontief function presumes that there is no substitution between the arguments, i.e. no amount of increase in one argument can compensate the decrease in another to keep output at some original level. In contrast, the general CES does allow some degree (as captured by the parameter $\gamma$) of substitution between the arguments. Note also that Leontief is obtained by taking $\gamma\to\infty$, thereby rendering the elasticity of substitution between the arguments $0$.

If you were doing a simulation exercise, you should usually "let the data speak" first, namely, estimating the value of $\gamma$ (and $\sigma_j$) from some existing data. However, if you have some a priori reason to believe that $\gamma$ is of a particular value, then you would assume that value and carry on with the analysis.

Answer 2

This is an old thread that I came across while searching on CES aggregators.

My understanding of CES aggregators is that they are mainly used to aggregate outputs. I don't know whether they have been used to aggregate, or create an index of, production inputs. Technically, I don't see why you can't do this - the aggregator will give you a number; but I doubt that this in itself would be considered as a solution to the well-known problems of capital aggregation. (I would be interested in what you found out about this outside of this thread.)

My first stop on capital aggregation has been Charles Hulten's NBER handbook article. https://www.nber.org/system/files/chapters/c5974/c5974.pdf

But recent work, most notably by David Baqaee and the late Emmanuel Farhi, has gone in the other direction: how to use discrete capital stocks in a disaggregated production function. http://www.nber.org/papers/w24684

Dietrich Vollrath explores the Baqaee-Farhi approach in this blog post. https://growthecon.com/blog/Elasticity/

These examples may be far from your particular interest - but they show that your question is not a dead-and-buried topic!