Nested CES Production Function

Question

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.

Answer 1

Consider the CES production function 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