Kimball (1995) defines production of the final good ($Y$) with intermediate goods $y_l$ in his equation (1) as

$$ 1 = \int_0^1 G\left(\frac{y_l}{Y}\right) dl $$

with $G(1) = 1$, $G'(x) > 0$ and $G''(x) < 0$.

I have never seen such a construct before. What does the $1$ on the left-hand-side mean, is that a normalization or some cost construct? An introduction into this type of production function would be greatly appreciated.


1 Answer 1


Some input:
Kimball assumes a constant-returns-to-scale (CRS) production of the final good $Y$ from the intermediate goods (and no other inputs are involved in this function). Turn to discrete space for a moment, and this means that we would have something like

$$Y = F(y_1,...,y_l,...,y_m)$$

Since this is a CRS function we have

$$1 = F\left(\frac {y_1}{Y},...,\frac {y_l}{Y},...\frac {y_m}{Y}\right)$$

But also, from Euler's theorem for homogeneous functions we have

$$Y = \sum_{i=1}^m \frac {\partial F}{\partial y_i}\cdot y_i \implies 1 = \sum_{i=1}^m \frac {\partial F}{\partial y_i}\cdot\frac{ y_i}{Y}$$

Combining and manipulating the index into $[0,1]$-continuity we get something like

$$1 = F\left(\frac {y_1}{Y},...,\frac {y_l}{Y},...\frac {y_m}{Y}\right) =\sum_{i=1}^m \frac {\partial F}{\partial y_i}\cdot\frac{ y_i}{Y}\rightarrow \int_0^1\left[\frac {\partial F}{\partial y_l}\cdot\frac{ y_l}{Y}\right]{\rm d}l$$

In a sense, $G(y_l/Y)$ is the elasticity of final output with respect the the $l$-th intermediate good. Given the assumptions on $G()$, it rules out a Cobb-Douglas CRS production function, where the elasticities not only sum to unity but they are constant, and it looks, say, to a C.E.S. production function with constant returns to scale, where the elasticities are variable but always sum up to unity.

  • $\begingroup$ So the issue with CD is that it is not symmetric. If we had $K^\alpha L^\alpha$, we could summarize it as $G(K/Y) = G(L/Y) = \alpha$ $\endgroup$
    – FooBar
    Apr 17, 2015 at 15:59
  • $\begingroup$ @FooBar The main issue I see with CD here is that it would make the $G()$ function a constant, not depending on $y_l/Y$, and so $G' = 0$, turning essentially $G$ into a useless identity, rather than an effective constraint that co-determines the subsequent firm optimization problem. $\endgroup$ Apr 17, 2015 at 20:08

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