Is elasticity of substitution defined for non-homogeneous production functions?

Question

The elasticity of substitution between two inputs $x_1$ and $x_2$ is typically given as $$\frac{d \ln \left( \frac{x_2}{x_1} \right)}{d \ln(\mathrm{MRTS}_{21})}.$$

As these notes show, if the production function is homogeneous and continuously differentiable, then the marginal rate of technical substitution (MRTS) depends solely on the ratio of the two inputs $x_2/x_1$, and therefore equals $g(x_2/x_1)$ for some function $g$ of a single variable. Moreover, if the production function is strictly quasi-concave, then this function $g$ is strictly decreasing, and so can be inverted to another function $h$ of a single variable. The elasticity of substitution is simply the elasticity of this new function $h$.

But all this assumes that the production function is homogeneous. If it isn't homogeneous, then I don't understand what the formula defining the elasticity of substitution is even supposed to mean, since the MRTS and the input ratio $x_2/x_1$ are not single-valued functions of each other.

Is the elasticity of substitution defined for a non-homogeneous production function? If so, how?

Answer 1

Let a function $y=f(x_1,x_2)$, and where $f_1$ and $f_2$ denote first partial derivatives and $f_{11}$ etc second partials.

The elasticity of substitution was defined by Hicks in order to answer the question

"How does the input factor ratio changes as the ratio of factor prices changes?"

To be an elasticity, it had to be dimensionless, a relative measure. So Hicks' original concept starts conceptually with the expression

$$\sigma = -\frac{\frac{d(x_1/x_2)} {(x_1/x_2)}}{\frac{d (w_1/w_2)} {(w_1/w_2)}}, \tag{1}$$

where $f$ is now a production function, $w_1, w_2$ are the factor prices, and the operator "$d$" is best viewed as a differential, linking also to its discrete analogue "$\Delta$".

Making it even more an economic concept, assume now that the firm is a cost-minimizer.

Then, at optimized behavior we have $$\frac{w_1}{w_2} = \frac{f_1}{f_2}, \tag{2}$$

...and this explains why we should care about such a metric, and why it is a measure of "substitutability" (a measure not the measure): as relative factor prices change, how much do the factor ratio changes? This takes into account both the economic, behavioral aspect of cost minimization, and the available production technology.

So we can write

$$\sigma = -\frac{\frac{d(x_1/x_2)} {(x_1/x_2)}}{\frac{d (f_1/f_2)} {(f_1/f_2)}}. \tag{3}$$

Note that, if one doesn't know the economic motivation, then eq. $(3)$ (which is exactly the same as the one used by the OP), can be seen as a purely mathematical metric of the bivariate function: the relative change of the ratio of the two variables over the relative change of the ratio of their first derivatives.

The economic mid-way, is to say, "the relative change of input ratio over the relative change of the MRS". But this retains economic meaning only if we keep reminding ourselves that inputs are chosen so that their ratio of marginal products, the MRS, equals the factor price ratio at optimized behavior.

After certain mathematical manipulations (see Silberberg The Structure of Economics 2nd ed., ch. 9.4), one gets

$$\sigma = \frac{-f_1f_2(f_1x_1 + f_2x_2)}{x_1x_2(f_2^2f_{11}-2f_1f_2f_{12}+f_1^2f_{22})}.$$

All the above do not require the function $f$ to be homogeneous, nor $\sigma$ to be a constant.

The above expression of course simplifies greatly if the production function is homogeneous of degree one.

Answer 2

The reference that Alecos Papadopoulos gives in his answers (Silberberg The Structure of Economics 2nd ed., ch. 9.4) clears up my confusion.

What was not clear (to me) in the formula given in the question is that the differentials are taken along the isoquant passing through the base point of the derivative. That is, $x_1$ and $x_2$ are not varied independently; instead, the output $f$ is held fixed as we vary the quantities in the derivative. Under this constraint (plus convexity assumptions), the MRTS and the input ratio are indeed one-to-one functions of each other. This constraint effectively reduces the problem to a 1D problem, and we can apply the usual 1D chain rule.

More explicitly, we can consider one input (WLOG, $x_1$) to be an independent variable and fix the other input $x_2$ to be a function $x_2(x_1)$ given by the isoquant that passes through the base point. Then the derivative $$\frac{d\left( \frac{x_1}{x_2} \right)}{d\left( \frac{f_1}{f_2} \right)}$$ in the expanded formula for the elasticity of substitution is interpreted to mean $$\frac{\frac{d}{dx_1} \left( \frac{x_1}{x_2 \left(x_1\right)} \right)}{\frac{d}{dx_1} \left( \frac{f_1(x_1,x_2(x_1))}{f_2(x_1,x_2(x_1))} \right)}.$$