<p>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.</p>

CC BY-SA 3.0
2015-04-19T05:57:21.880Z
Why do we have to fix the prices?
Stan Shunpike
CC BY-SA 3.0
2015-04-19T05:58:06.137Z
To get the dimensions right. Can't invert a function from $\mathbb{R}^3$ to $\mathbb{R}$.
Sander Heinsalu
CC BY-SA 3.0
2015-04-19T05:59:33.470Z
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
Stan Shunpike
CC BY-SA 3.0
2015-04-20T12:43:16.153Z
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.
BVJ
CC BY-SA 3.0
2015-04-19T05:56:10.817Z
2015-04-19T05:56:10.817Z
Sander Heinsalu
<p>My professor says the expenditure function and the indirect utility function are inverses of each other. But how is this possible? Consider each.</p>

<p>$$v:(p_x,p_y,m) \rightarrow \Bbb{R}$$ </p>

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

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

<p>Only if one was a scalar field and the other a vector field would this work.</p>

<p><strong>My Question:</strong></p>

<p>What is the relationship between the expenditure minimization function and the indirect utility function? </p>

<consumer-theory>
Is the implicit function theorem needed to relate the EMP to UMP?
Stan Shunpike
# Is the implicit function theorem needed to relate the EMP to UMP?

(Asked by: username_0 on 2015-04-19)

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?

**Tags:** consumer-theory

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## 1 answers

### Answer by username_1 on 2015-04-19. Score: 2 | Accepted answer

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.

> **Comments:**
> **username_0**: Why do we have to fix the prices?  
> **username_1**: To get the dimensions right. Can't invert a function from $\mathbb{R}^3$ to $\mathbb{R}$.  
> **username_0**: 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  
> **username_2**: 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.  

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