Economics Stack Exchange is a question and answer site for those who study, teach, research and apply economics and econometrics. It only takes a minute to sign up.
Sign up to join this community
Two questions:
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.
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:
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.
