1
$\begingroup$

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.

$\endgroup$
1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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 '15 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. $\endgroup$ – Alecos Papadopoulos Apr 17 '15 at 20:08

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.