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Two questions:

  1. Given $v(\vec{p},m)$ and $e(\vec{p},\bar{U})$, is there only a single point at which these are inverses of each other?
  2. Does an inverse always exist for a given price vector $\vec{p}$, income $m$ and $\bar{U}$, $v$ and $e$?

These are points my mates and I are arguing about as we prepare for our midterm.

Also, I prefer rigorous answers so if you can, feel free to be as thorough as you want.

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Regarding question 1:

From the assumption that the consumer has a single income value $m$, there can only be one value for $m$. So by the notion that $e$ and $v$ are inverses, there can only be one value $e$ for which $m$ corresponds to and thus only one $u$. So there is only one point.

Regarding Question 2:

If we have (1) a utility function that is continuous and locally non-satiated and (2) if $m > 0$ and (3) if both UMP and EMP exist, then they are equivalent. And $e$ and $v$ are inverses.

Claim 1: Solving UMP solves EMP.

Proof:

Suppose bundle $c^*$ solves the UMP but not EMP. Let $c'$ solve EMP. Then (1) amount spent on $c^*$ greater then that on $c'$ and so $u(c') \geq u(c^*) $ because obviously spending more means worse off or same utility. But (2) by local nonsatiation a consumption there exists a bundle $c''$ close enough to $c'$ such that amount spent on $c''$ is less than amount spent on $c^*$ and $u(c'') > u(c^*)$. (3) Contradiction because we assumed $c^*$ solves UMP.

Claim 2: Solving EMP solves UMP.

Proof:

Suppose bundle $c^*$ solves EMP but not UMP. Let $c'$ solve UMP. Then (1) $u(c') > u(c^*)$ even though we spend the same amount on each. That is, $c^*$ doesn't solve UMP since less utility for same $m$. Because amount spent positive, we can find $t$ where $0<t<1$ such that amount spent on $tc'$ is lower than that spent on $c^*$ yet $u(tc') > u(c^*)$ because $c^*$ does not solve the UMP. (3) Contradiction because we said $c^*$ solved EMP.

Summary:

  1. If solutions exist either to UMP or EMP, then solutions exist for the other and they are inverse functions.
  2. If an inverse exists, it must be a point.
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  • $\begingroup$ This may need a little clean up to become more rigorous. I may have the time to edit it in a couple of hours from now. By the way, It would be good if you would provide some feedback on my excavation activities, here, economics.stackexchange.com/questions/3020/… $\endgroup$ – Alecos Papadopoulos Apr 30 '15 at 10:06
  • $\begingroup$ Ah yes. Sorry, been busy with school. I shall respond with specifics but it was excellent. $\endgroup$ – Stan Shunpike May 1 '15 at 1:23
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As regards

QUESTION 1
The concept of "inverse" is usually used for a function. Here we are dealing not just with two functions, but with two optimization problems: "Duality" is, in a sense, the analogous concept of "function inverse", for optimization problems.

If two optimization problems are dual (in a more or less rigorous sense), then the solution of the one is, in a sense, the "inverse" of the solution of the other.

But what is a "solution" in an optimization problem? When the analysis is abstract, the solution does not take on a specific numerical value, but it is a value function, i.e. a function that will give us the solution for any values of the inputs on which the solution depends.

For the Utility Maximization Problem (UMP), the value function is the Indirect Utility function, $v(\mathbf p,m)$. Abusing notation (but we gain something from this abuse) we can write

$$v(\mathbf p,m) = \bar U \implies m = v^{-1}(\mathbf p,\bar U)$$

For the Expenditure Minimization Problem (EMP), its value function is the Expenditure function, $e(\mathbf p,\bar{U})$, and

$$e(\mathbf p,\bar{U}) = m \implies \bar U = e^{-1}(\mathbf p,m)$$

Looking at the two we see that

$$v(\mathbf p,m) = e^{-1}(\mathbf p,m),\;\;\; v^{-1}(\mathbf p,\bar U) =e(\mathbf p,\bar{U})$$

So, I wouldn't think that it is correct to say that "they are inverses for a single point". Stretching the concept of "inverse", for any given set $(\mathbf p,m)$, the "inverse" Expenditure function is the the Indirect Utility function, and for any given set $(\mathbf p,\bar U)$ the "inverse" Indirect Utility function is the Expenditure function.

Note: When the concept of "inverse" is treated in the proper manner (i.e. for univariate functions), then we examine the property point-by-point to conclude that it holds.

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  • $\begingroup$ +1 The definition of value function you have given matches what I have read in the literature. But I don't understand how this is different from the defintion of a function. Can you elaborate on this? One thing that does strike me is the following: consider $$v = U(x^*(p_x,p_y,m),y^*(p_x,p_y,m))$$ This is a map between $$\Bbb{R^3} \mapsto \Bbb{R}^2 \mapsto \Bbb{R}^1$$ so it is really a composition of functions in some ways except that the optimal demand function is a vector-valued function $\endgroup$ – Stan Shunpike May 10 '15 at 21:07
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    $\begingroup$ The "value function" of an optimizing problem is the objective function of the problem, but where the argument(s) of itare no longer the decision variables of the problem, but the optimal determination rules of these decision variables, expressed in terms of the exogenous parameters/variables. $\endgroup$ – Alecos Papadopoulos May 10 '15 at 22:05

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