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My question concerns exercise 2.4 from Hal Varian's Microeconomic Analysis.

Please note that I am not requesting a solution to the exercise, and only asking how to understand what I am being asked to do here.

2.4. Let $f(x_1, x_2)$ be a production function with two factors and let $w_1$ and $w_2$ be their respective prices. Show that the elasticity of the factor share $(w_2 x_2 / w_1 x_1)$ with respect to $(x_1 / x_2)$ is given by $1/\sigma - 1$.

Here $\sigma$ is defined earlier as the elasticity of substitution $$\frac{\text{TRS}}{x_2 / x_1}\frac{d(x_2 / x_1)}{d \text{TRS}}$$

where TRS is the technical rate of substitution $\frac{-\partial f / \partial x_1}{\partial f / \partial x_2}$.

The text has not yet defined the factor share or its elasticity yet, but in general, I interpret the phrase "the elasticity of $A$ with respect to $B$" as referring to the quantity $\frac{B}{A}\frac{dA}{dB}$. It is not hard to show using the chain rule that this quantity is equal to $d (\ln A) / d (\ln B)$.

Using the first definition, I take $A = w_2 x_2 / w_1 x_1$ and $B = x_1 / x_2$. Then $A = \frac{w_2}{w_1}\frac{1}{B}$, and $$\frac{B}{A}\frac{dA}{dB} = \frac{B}{A}\frac{w_2}{w_1} \frac{-1}{B^2}= -1$$

Using the second definition, I get the same result. We know $$\ln A = \ln (w_2 x_2 / w_1 x_1 ) = \ln (w_2 / w_1) - \ln (x_1 / x_2) = \text{const.} - \ln B $$ Differentiating both sides with respect to the symbol $\ln B$ yields $d (\ln A) / d (\ln B) = -1$.

Is my math wrong, or have I misunderstood the question?

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Your math is not wrong.

To get to the correct solution, you should not treat $(w_2/w_1)$ as a constant. If $x_1/x_2$ changes then the prices will have to change to keep the factor shares optimal for a given level of output (I know that the question is not clear in this respect).

Use the first order conditions $\dfrac{w_2}{w_1} = \dfrac{\partial f(x_1,x_2)/\partial x_2}{\partial f(x_1, x_2)/\partial x_1}= -\dfrac{1}{TRS}$ to subsitute out.

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  • $\begingroup$ Thank you. Is there any justification for treating $w_2/ w_1$ as a function of $x$? The question comes from a chapter on profit maximization, which assumes a competitive market. If the firm is a price taker, it seems strange to talk about moving (market) prices around in order to accommodate firm-based optimality. $\endgroup$
    – Max
    Nov 8 '21 at 4:57
  • $\begingroup$ Maybe it is better to see it from a macro perspective (in this case, $w_1$ and $w_2$ do change). If $x_1$ is labour and $x_2$ is capital then $w_2 x_2/w_1 x_1$ gives the renumeration of capital versus labour. $x_1/x_2$ then gives the 'supply or usage' of labour versus capital. So a positive elasticity means that if the share of labour increases, the share of total renumeration that goes towards labor (vis a vis) capital in fact decreases. $\endgroup$
    – tdm
    Nov 8 '21 at 16:20

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