# CES v. Leontief Aggregator in Production

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?

• Are you aware that Leontief is a special case of the CES function? Jul 31, 2019 at 16:24
• @HerrK. Yes! A special case representing a fixed proportion case. But why would someone use the Leontief and in some cases a general CES? Jul 31, 2019 at 16:49

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.

• Hi Herr, thanks for the response. I was trying to get to a real-life scenario where a production process would require two distinct types of capital to be used in fixed proportions. Can you think of an example? I am trying to think why it would be in such case... Aug 1, 2019 at 12:51
• @FrankSwanton: I'm not sure what you mean by types of capital, but a common example of a Leontief production technology would be a car, which requires 4 tires and 1 steering wheel, in fixed proportion. Aug 1, 2019 at 21:23
• Hi Herr, what I meant is that I am more interested in the justification of capital "aggregation". When you have more than 1 capital types and instead of putting them into the production function as individual arguments, often there is a justification for aggregation of such capital. Do you know any good reference to how theoretically macroeconomists justify this type of aggregation? An example could be an expensive and cheap capital or durable and non-durable or foreign exchange rate risky and non-risky, etc. Oct 20, 2019 at 15:34
• @FrankSwanton: I'm afraid I'm not very familiar with the literature you're referring to. Sorry. Oct 21, 2019 at 1:40

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!

• Excellent response, Tom. Thanks a lot! Jun 2 at 11:18