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My professor says the expenditure function and the indirect utility function are inverses of each other. But how is this possible? Consider each.

$$v:(p_x,p_y,m) \rightarrow \Bbb{R}$$

$$e:(p_x,p_y,\bar{U}) \rightarrow \Bbb{R}$$

These are both mappings from $\Bbb{R^3}\rightarrow \Bbb{R}$. How can one be the inverse of the other?

Only if one was a scalar field and the other a vector field would this work.

My Question:

What is the relationship between the expenditure minimization function and the indirect utility function?

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Fix prices at some arbitrary positive values $p_x^0,p_y^0$ and view the functions $v$ and $e$ as $v:\mathbb{R}\rightarrow \mathbb{R}$ and $e:\mathbb{R}\rightarrow \mathbb{R}$. Then they are inverses.

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  • $\begingroup$ Why do we have to fix the prices? $\endgroup$ – Stan Shunpike Apr 19 '15 at 5:57
  • $\begingroup$ To get the dimensions right. Can't invert a function from $\mathbb{R}^3$ to $\mathbb{R}$. $\endgroup$ – Sander Heinsalu Apr 19 '15 at 5:58
  • $\begingroup$ But how does fixing it solve that problem? Oh, I see, its like mapping $(1,2,3)$ to $(1,2,6)$ in that suddenly you are only affecting one dimension. Okay I get it. Accepted :D $\endgroup$ – Stan Shunpike Apr 19 '15 at 5:59
  • $\begingroup$ I think this has to do with the fact that the EMP is solved it he dual space of where you are solving the UMP, therefore a solution to the EMP will be the evaluation of a linear operator on the minimizing x. The linear operator is determined as the inner product with P but in the dual space each of these operators has a fixed P. Might be confusing but I think that thats the reason why P is fixed. $\endgroup$ – BVJ Apr 20 '15 at 12:43

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