# Complementarity in CES Production Function

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).

• This is not about a single-output "CES production function" and its inputs. It is about complementarity in "production decisions" (increases in the production of one capital good are accompanied by increases in the production of the other capital good). The language of Fisher is admittedly confusing at some points. I believe I will be able to post a proper answer also here. Oct 11 '17 at 17:43
• Thanks. While the language is now clear, what is still unclear is how such a production function leads to the two moving together wrt a productivity shock. At first I thought this must act through utility. The agent would like a mix of both consumption and housing capital, but this contradicts the fact that complementarity disappears with K=1 (utility unchanged). Oct 11 '17 at 18:12

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.

• Ah, I thought $(I_b^K+I_h^K)^{\frac{1}{k}}$ was the output not the input. Thanks for clarifying. Oct 12 '17 at 1:18