<p>I think the good way to do that is to introduce the <a href="http://en.wikipedia.org/wiki/Expenditure_function" rel="nofollow">Expenditure function</a> which awesomely solves your problem.</p>

<hr>

<p>The <strong>Expenditure function</strong> gives the money one has to spend to achieve a given level of utility, given a utility function $u$ and the vector of prices $p$.</p>

<p>$e(p,u^*)=min_{x \in \mathbb{R}_+^n,u(x) \geq u^*}p \cdot x$</p>

<p>Then you can express your compensating variation in two equivalent ways:</p>

<p>$CV=e(p_1,u_1)-e(p_1,u_0)=e(p_0,u_0)-e(p_1,u_0)$ where $p_0$ and $p_1$ are the old and new prices and $u_0$ and $u_1$ are the old and new utility. One can note that it can be translated as: if given $CV$ in compensation for the change, then the consumer would be indifferent to that change.</p>

CC BY-SA 3.0
2015-05-20T12:59:34.827Z
2015-05-20T12:59:34.827Z
VicAche
<p>For a single variable price change, </p>

<p>$$CV = - \int_{p_x^o}^{p_x^f} x_H(\rho,p_y,v^o)d\rho$$ </p>

<p>$x_H$ is the Hicksian demand function for good $x$. What happens if both prices change? How does one calculate this? What about for an $n$ dimensional price vector where all the prices are changing? </p>

<p>Please give an example using double integrals. </p>

CC BY-SA 3.0
2015-05-20T12:17:41.153Z
could you use the same idea with a multiple integral
user157623
CC BY-SA 3.0
2015-05-20T12:52:31.130Z
I think you can use multiple integrals or calculate the CV for every price (i.e., $CV((p^0_1,p^0_2),(p^f_1,p^0_2))$ and $CV((p^f_1,p^0_2),(p^f_1,p^f_2))$) and just add them up. I don't know what assumptions you need for that to work, but afair they where not very strong, but it was some time ago that I used CVs.
The Almighty Bob
<microeconomics><consumer-theory>
How does one calculate compensating variation for multiple price change?
Stan Shunpike
# How does one calculate compensating variation for multiple price change?

(Asked by: username_0 on 2015-05-20)

For a single variable price change,

$$CV = - \int\_{p\_x^o}^{p\_x^f} x\_H(\rho,p\_y,v^o)d\rho$$

$x\_H$ is the Hicksian demand function for good $x$. What happens if both prices change? How does one calculate this? What about for an $n$ dimensional price vector where all the prices are changing?

Please give an example using double integrals.

**Tags:** microeconomics, consumer-theory

> **Comments:**
> **username_1**: could you use the same idea with a multiple integral  
> **username_2**: I think you can use multiple integrals or calculate the CV for every price (i.e., $CV((p^0_1,p^0_2),(p^f_1,p^0_2))$ and $CV((p^f_1,p^0_2),(p^f_1,p^f_2))$) and just add them up. I don't know what assumptions you need for that to work, but afair they where not very strong, but it was some time ago that I used CVs.  

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## 1 answers

### Answer by username_3 on 2015-05-20. Score: 2

I think the good way to do that is to introduce the [Expenditure function](http://en.wikipedia.org/wiki/Expenditure_function) which awesomely solves your problem.

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The **Expenditure function** gives the money one has to spend to achieve a given level of utility, given a utility function $u$ and the vector of prices $p$.

$e(p,u^\*)=min\_{x \in \mathbb{R}\_+^n,u(x) \geq u^\*}p \cdot x$

Then you can express your compensating variation in two equivalent ways:

$CV=e(p\_1,u\_1)-e(p\_1,u\_0)=e(p\_0,u\_0)-e(p\_1,u\_0)$ where $p\_0$ and $p\_1$ are the old and new prices and $u\_0$ and $u\_1$ are the old and new utility. One can note that it can be translated as: if given $CV$ in compensation for the change, then the consumer would be indifferent to that change.


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