<p>Consider the <a href="https://en.wikipedia.org/wiki/Constant_elasticity_of_substitution" rel="nofollow noreferrer">CES production function</a> over four goods defined as follows:
$$ x = (\alpha j^{\gamma} + k^{\gamma})^{1/\gamma}$$
$$ y = (\delta l^{\beta} + m^{\beta})^{1/\beta }$$
$$ z = (\zeta x^{\xi} + y^{\xi})^{1/\xi }$$
If $\gamma$ is near one then $j$ and $k$ are near perfect substitutes. If $\beta$ is near one then $l$ and $m$ are near perfect substitutes. If $\xi$ is a large negative value then the composite goods $x$ and $y$ are near perfect complements</p>

CC BY-SA 4.0
2018-08-10T00:46:31.293Z
2018-08-10T00:46:31.293Z
BKay
<p>If I have four input factors (a, b, c, b) and I want to construct a nested CES production function such that (a, b) are substitutes, (c, d) are substitutes and [(a, b), (c, d)] are complements, I.e. a, b together are complements of c, d together. How would such a production function look like? </p>

<p>I saw some people include a Cobb Douglas there to simplify the function. So, under what situations do people use Cobb Douglas in a nested CES? Cause the factor price of a here in my model will decrease asymptotically to zero, if use the classic CES, b will be completely substituted, which I don’t want.</p>

<microeconomics><production-function><ces-function>
Nested CES Production Function
user10158324
# Nested CES Production Function

(Asked by: username_0 on 2018-07-31)

If I have four input factors (a, b, c, b) and I want to construct a nested CES production function such that (a, b) are substitutes, (c, d) are substitutes and [(a, b), (c, d)] are complements, I.e. a, b together are complements of c, d together. How would such a production function look like?

I saw some people include a Cobb Douglas there to simplify the function. So, under what situations do people use Cobb Douglas in a nested CES? Cause the factor price of a here in my model will decrease asymptotically to zero, if use the classic CES, b will be completely substituted, which I don’t want.

**Tags:** microeconomics, production-function, ces-function

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## 1 answers

### Answer by username_1 on 2018-08-10. Score: 1 | Accepted answer

Consider the [CES production function](https://en.wikipedia.org/wiki/Constant_elasticity_of_substitution) over four goods defined as follows:
$$ x = (\alpha j^{\gamma} + k^{\gamma})^{1/\gamma}$$
$$ y = (\delta l^{\beta} + m^{\beta})^{1/\beta }$$
$$ z = (\zeta x^{\xi} + y^{\xi})^{1/\xi }$$
If $\gamma$ is near one then $j$ and $k$ are near perfect substitutes. If $\beta$ is near one then $l$ and $m$ are near perfect substitutes. If $\xi$ is a large negative value then the composite goods $x$ and $y$ are near perfect complements


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