# 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? – Herr K. Jul 31 '19 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? – Frank Swanton Jul 31 '19 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.