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2016-09-05T01:58:36.803Z
2016-09-05T01:58:36.803Z
Royun
<p>Letting $f_i(\mathbf{x})=\partial f(\mathbf{x})/\partial x_i$, ($\mathbf{x}$ is a vector, a commodity bundle, and $x_i$ is a scalar, commodity $i$ in the bundle) show that,</p>

<p>$\sigma_{ij}(\mathbf{x})\equiv -\frac{x_if_i(\mathbf{x})+x_jf_j(\mathbf{x})}{f^2_j(\mathbf{x})f_{ii}(\mathbf{x})+2f_i(\mathbf{x})f_j(\mathbf{x})f_{ij}(\mathbf{x})+f_i^2(\mathbf{x})f_{jj}(\mathbf{x})}\frac{f_i(\mathbf{x})f_j(\mathbf{x})}{x_ix_j}$</p>

<p>Then using the above formula, show that $\sigma_{ij}(\mathbf{x})\geq0$ whenever $f$ is increasing and concave.</p>

<hr>

<p>PS: I know $\sigma_{ij}$ can be written as</p>

<p>$\sigma_{ij}=-\frac{d (x_i/x_j)}{x_i/x_j} \frac{f_i(\mathbf{x})/f_j(\mathbf{x})}{d (f_i(\mathbf{x})/f_j(\mathbf{x}))}$</p>

<p>How can we derive the result from this known fact? Thank you.</p>

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<microeconomics><mathematical-economics><elasticity><production-function><producer-theory>
Elasticity of substitution in Jehle and Reny Advanced Micro (3rd ed) exercise 3.8
Royun
# Elasticity of substitution in Jehle and Reny Advanced Micro (3rd ed) exercise 3.8

(Asked by: username_0 on 2016-01-23)

Letting $f\_i(\mathbf{x})=\partial f(\mathbf{x})/\partial x\_i$, ($\mathbf{x}$ is a vector, a commodity bundle, and $x\_i$ is a scalar, commodity $i$ in the bundle) show that,

$\sigma\_{ij}(\mathbf{x})\equiv -\frac{x\_if\_i(\mathbf{x})+x\_jf\_j(\mathbf{x})}{f^2\_j(\mathbf{x})f\_{ii}(\mathbf{x})+2f\_i(\mathbf{x})f\_j(\mathbf{x})f\_{ij}(\mathbf{x})+f\_i^2(\mathbf{x})f\_{jj}(\mathbf{x})}\frac{f\_i(\mathbf{x})f\_j(\mathbf{x})}{x\_ix\_j}$

Then using the above formula, show that $\sigma\_{ij}(\mathbf{x})\geq0$ whenever $f$ is increasing and concave.

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PS: I know $\sigma\_{ij}$ can be written as

$\sigma\_{ij}=-\frac{d (x\_i/x\_j)}{x\_i/x\_j} \frac{f\_i(\mathbf{x})/f\_j(\mathbf{x})}{d (f\_i(\mathbf{x})/f\_j(\mathbf{x}))}$

How can we derive the result from this known fact? Thank you.

[![enter image description here](https://i.sstatic.net/zM19k.png)](https://i.sstatic.net/zM19k.png)

**Tags:** microeconomics, mathematical-economics, elasticity, production-function, producer-theory

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## 1 answers

### Answer by username_0 on 2016-09-05. Score: 0 | Accepted answer

[![enter image description here](https://i.sstatic.net/4ldMG.png)](https://i.sstatic.net/4ldMG.png)

[![enter image description here](https://i.sstatic.net/UqV2S.png)](https://i.sstatic.net/UqV2S.png)

I emailed the question to one of the authors. He said the solution manual will come soon!


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