<p>Some input:<br>
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</p>

<p>$$Y = F(y_1,...,y_l,...,y_m)$$</p>

<p>Since this is a CRS function we have</p>

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

<p>But also, from Euler's theorem for homogeneous functions we have</p>

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

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

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

<p>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.</p>

CC BY-SA 3.0
2015-04-17T15:59:50.267Z
So the issue with CD is that it is not symmetric. If we had $K^\alpha L^\alpha$, we could summarize it as $G(K/Y) = G(L/Y) = \alpha$
FooBar
CC BY-SA 3.0
2015-04-17T20:08:45.503Z
@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.
Alecos Papadopoulos
CC BY-SA 3.0
2015-04-17T15:30:06.213Z
2015-04-17T15:30:06.213Z
Alecos Papadopoulos
<p><a href="http://www.nber.org/papers/w5046" rel="nofollow">Kimball (1995)</a> defines production of the final good ($Y$) with intermediate goods $y_l$ in his equation (1) as</p>

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

<p>with $G(1) = 1$, $G'(x) &gt; 0$ and $G''(x) &lt; 0$.</p>

<p>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.</p>

<microeconomics><industrial-organisation><production-function>
Kimball (1995) Specification of Final Good Production
FooBar
# Kimball (1995) Specification of Final Good Production

(Asked by: username_0 on 2015-04-17)

[Kimball (1995)](http://www.nber.org/papers/w5046) 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.

**Tags:** microeconomics, industrial-organisation, production-function


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## 1 answers

### Answer by username_1 on 2015-04-17. Score: 3 | Accepted answer

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.

> **Comments:**
> **username_0**: So the issue with CD is that it is not symmetric. If we had $K^\alpha L^\alpha$, we could summarize it as $G(K/Y) = G(L/Y) = \alpha$  
> **username_1**: @username_0 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.  

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