4
$\begingroup$

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.

Improve this question
$\endgroup$
2
  • $\begingroup$ could you use the same idea with a multiple integral $\endgroup$
    – user157623
    Commented May 20, 2015 at 12:17
  • $\begingroup$ 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. $\endgroup$
    – The Almighty Bob
    Commented May 20, 2015 at 12:52

1 Answer 1

Reset to default
2
$\begingroup$

I think the good way to do that is to introduce the Expenditure function which awesomely solves your problem.

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.

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.