# Is the implicit function theorem needed to relate the EMP to UMP?

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?

## 1 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.

• Why do we have to fix the prices? – Stan Shunpike Apr 19 '15 at 5:57
• To get the dimensions right. Can't invert a function from $\mathbb{R}^3$ to $\mathbb{R}$. – Sander Heinsalu Apr 19 '15 at 5:58
• 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 Apr 19 '15 at 5:59
• 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 Apr 20 '15 at 12:43