# Does the growth rate of a neoclassical production function converge as all input factors grow with constant, but different growth rates?

Suppose you have a neoclassical production function with N-inputs

$F(x_t^1,...,x_t^N)$

All input factors grow in continuous time with constant, but not identical growth rates $g^j$. Assume $g^1 \leq g^2 \leq ... \leq g^N$. The growth rate of $F$ is then

$\hat{F}=\sum_{j=1}^N \varepsilon_{F,x^j} g^j$

with $\varepsilon_{F,x^j}$ being the elasticity of $F$ with respect to $x^j$. Since $F$ is linear homogenous I know that $\sum_{j=1}^N \varepsilon_{F,x^j}=1$ holds. $\frac{\partial F}{\partial x^j}>0$ and $\frac{\partial^2 F}{\partial {x^j}^2}<0$ imply $\varepsilon_{F,x^j}>0$. Hence

$g^1 \leq \hat{F} \leq g^N$ $\forall t$

My question is: Is $\hat{F}$ going to converge as $t \to \infty$? I find it hard to imagine that the elasticities fluctuate (periodically) around some value as all input factor ratios go to zero or infinity (or stay constant the whole time in case of equal growth rates).

I tried to show that the elasticities all converge. I suspect that the result might hinge on the propety that $\frac{\partial^2 F}{\partial x_j \partial x_k}>0$ for $k \neq j$ which always holds in the two goods case, but I'm not sure about that.

Thank you all in advance for your help! If my English seems a little awkward, it's because I'm German. But I hope you understand the problem anyway. :)

I am not sure that elasticities converge for arbitrary growth rates.

Here is a treatment of the issue. Subscripts in $F$ denote partial derivatives. I will omit the time subscript. By definition, the elasticity of the $i$-th input is the ratio of marginal product ($F_i$) over Average Product $(F/x_i)$. So for an elasticity to stabilize intertemporally, it must be the case that the growth rate of the marginal product is equal to (or equalizes a the limit with) the growth rate of the average product.

For input $x_{1}$ we have

$$\frac {d}{dt} F_1 = F_{11}\dot x_{1} +F_{12}\dot x_{2} +...+F_{1n}\dot x_{n}$$

All inputs are strictly positive so we can multiply and divide, and also divide by $F_1$ to obtain the growth rate (denoted by a tilde)

$$\tilde {F_1} = \frac {dF_1/dt}{F_1} = \frac{F_{11} x_1}{F_1}\frac{\dot x_{1}}{x_1} + ... +\frac{F_{1n} x_n}{F_1}\frac{\dot x_{n}}{x_n}$$

The first ratio in each term is the elasticity of the $F_1$ with respect to each input, say $\eta_{1k}$. So,

$$\tilde {F_1} = \sum_{k=1}^n\eta_{1k}g_k$$

and analogously for the other inputs.

Turning to the average product of the first input, we have

$$\frac {d}{dt} (F/x_1 ) = \frac {F_1x_1 - F}{x_1^2}\dot x_1 + \frac {F_2}{x_1}\dot x_2+...+\frac {F_n}{x_1}\dot x_n$$

Manipulating,

$$\frac {d}{dt} (F/x_1 ) = \left[F_1-\frac {F}{x_1} \right]g_1 + \frac{F}{x_1}\frac{F_2x_2}{F} \frac {\dot x_2}{x_2}+...+\frac{F}{x_1}\frac{F_nx_n}{F} \frac {\dot x_n}{x_n}$$

$$= \frac {F}{x_1}\cdot \Big( (e_1-1)g_1 + e_2g_2 + ...+ e_ng_n\Big)$$

$$\implies \frac {d(F/x_1)/dt}{(F/x)} = \tilde F - g_1 = \sum_{k=1}^ne_kg_k - g_1$$

So for all elasticities to stabilize (and so for $\tilde F$ to stabilize) we want, at least eventually, that

$$\forall i=1,...,n\;\;\; \sum_{k=1}^n\eta_{ik}g_k = \sum_{k=1}^ne_kg_k - g_i$$

Denoting $\mathbf H$ the $n\times n$ the matrix of the elasticities of the marginal products, $\mathbf I_n$ the identity matrix, $\mathbf e$ an $n \times 1$ vector containing the output elasticities $e_i$, $\mathbf i$ a column vector of ones, and $\mathbf g$ the $n \times 1$ column vector of growth rates of inputs, we can write the conditions for all elasticities to be stable compactly , as

$$\big(\mathbf H + \mathbf I_n - (\mathbf i \oplus \mathbf e')\big)\mathbf g =0$$

I note that the sum-of-matrices term does not depend at all on the growth rates vector. So the above orthogonality condition cannot hold for arbitrary $\mathbf g$ (except if I am neglecting some property stemming from the homogeneity of $F$).