Complementarity in CES Production Function

Question

I'm reading Fisher (1997, Journal of Monetary Economics). From the intermediate goods produced ($Y_t$), the final goods firm allocates into consumption ($C_t$), business capital investment ($I_{b,t}$), and household investment ($I_{h,t}$) s.t....

$C_t + (I_{b,t}^K + I_{h,t}^K)^{\frac{1}{K}} \le Y_t$

Fisher claims when $K>1$ there is complementarity in production, but my intuition is failing me. If $K>1$, $I_t = (I_{b,t}^K + I_{h,t}^K)^{\frac{1}{K}}$ is maximized when I invest all savings into either business or investment capital, while $I_t$ is minimized when I split my savings equally between the two. To me this sounds like anti-complementarity.

Can someone point one where my intuition is going awry? If it helps, on the utility side of things is log-utility w.r.t to $C_t$, $K_{h,t}$, and $L_t$ (leisure).

Answer 1

Total investment in terms of how much capital is augmented, is always $I = I_{b} + I_{h}$.

$(I_{b}^K + I_{h}^K)^{\frac{1}{K}}$ is equal to the amount of the intermediate good $Y$ that we need to allocate for investment. And given the formulation and when $K>1$ we see that we economize on the amount of the intermediate good $Y$,

$$Y_I = (I_{b}^K + I_{h}^K)^{\frac{1}{K}} < I_{b} + I_{h} = I$$

Further, if we split investments equally, we achieve the maximum possible economy (i.e. the minimum $Y$ required for given total $I$ to be achieved).

So inherently, the formulation "pushes us" to keep the two investment levels close, because the further apart they are (for given total $I$), the more $Y$ we will require, reducing consumption. This "closeness of levels" is what the author refers to as "complementarity" (perhaps confusingly, if one has the micro/demand-production theory meaning of the term in mind).

Fisher, J. D. (1997). Relative prices, complementarities and comovement among components of aggregate expenditures. Journal of Monetary Economics, 39(3), 449-474.