I have this production function:


I am trying to find the elasticity of substitution, and I found this:

$$\sigma = -\frac{d \ln (\frac{x_2}{x_1})}{d \ln(\frac{x_1}{1+x_2})}$$

Then I have these conditions:

$x_1 >0$ and $x_2=0$

When putting this numbers in, I have ln (0) which is not possible, what is the interpretation of this elasticity of substitution?

Will the elasticity be zero or infinity?

  • 1
    $\begingroup$ You did not finish the derivation. Please do so, this site is not a calculator. $\endgroup$ – Giskard Oct 19 '17 at 9:02
  • $\begingroup$ Also this is not about interpretation, at best it is about limit calculus. $\endgroup$ – Giskard Oct 19 '17 at 9:03

The ratio of marginal products $\frac{\partial P}{\partial x_1}/\frac{\partial P}{\partial x_2}$ (the marginal rate of tech. substitution) is equal to $1/x_1+r$ where $r\equiv x_2/x_1$.

The derivative of the MRS wrt to $r$ ($\frac{d MRS}{d r}$) is equal to $1-\frac{g}{MRS-r}$, where $g\equiv \frac{dln(x_1(r))}{dr}$.

Note how $\frac{dln(MRS)}{dr}$ which is equivalent to $\frac{d MRS/d r}{MRS}$ is equal to $\frac{MRS-r-g}{MRS(MRS-r)}$. Also note how $\frac{dln(r)}{dr}=1/r$.

Rearranging the last two relations in order to acquire expressions for $dln(MRS)$ and $dln(r)$ and substituting into the definition of the elasticity of substitution yields $-\frac{MRS(MRS-r)}{(MRS-g-r)r}$.

Finally, note how $r=0$ when $x_2=0$. Also, $MRS=1/x_1$ when $x_2=0$. The behavior of σ depends of the value of $g\equiv \frac{dln(x_1(r))}{dr}$ at $x_2=0$. If the value of $g$ is defined and is finite and constant then $σ$ approaches $- \infty$.

Intuitively, the behavior of $g$ near $x_2=0$ depends on the shape of the isocurves. $g$ denotes the (proportionate) rate of change in $x_1$ as a result of an infinitesimal change in $r$. The following graph shows that for values of $x_2$ near zero (notice the values on the vertical axis), the isocurves are really steep.

enter image description here

Very informally, in this setting, (really) small changes in $r$-which is equal to the ratio of $x_2/x_1$ (which is equivalent to the tangent of angle $θ$)- have almost no effect on $x_1$. For $Δθ$ approaching zero $Δx_1$ is also small and probably close to zero too. Equivalently and very loosely speaking, in this region $g$ should be close to zero too.