Debreu's representation theorems

The theorems are usually applied to spaces of finite commodities. However, they are applicable in a much more general setting. These are the general assumptions:

• X is a topological space.
• ${\displaystyle \preceq }$ is a relation on X which is total (all items are comparable) and transitive.
• ${\displaystyle \preceq }$ is continuous. This means that the following equivalent conditions are satisfied:
  1. For every ${\displaystyle x\in X}$, the sets ${\displaystyle \{y|y\preceq x\}}$ and ${\displaystyle \{y|y\succeq x\}}$ are topologically closed in ${\displaystyle X}$.
  2. For every sequence ${\displaystyle (x_{i})}$ such that ${\displaystyle x_{i}\to x_{\infty }}$, if for all i ${\displaystyle x_{i}\preceq y}$ then ${\displaystyle x_{\infty }\preceq y}$, and if for all i ${\displaystyle x_{i}\succeq y}$ then ${\displaystyle x_{\infty }\succeq y}$

Each one of the following conditions guarantees the existence of a real-valued continuous function that represents the preference relation ${\displaystyle \preceq }$. The conditions are increasingly general, so for example, condition 1 implies 2, which implies 3, which implies 4.

1. The set of equivalence classes of the relation ${\displaystyle \sim }$ (defined by: ${\displaystyle x\sim y}$ iff ${\displaystyle x\preceq y}$ and ${\displaystyle x\succeq y}$) are a countable set.

2. There is a countable subset of X, ${\displaystyle Z=\{z_{0},z_{1},...\}}$, such that for every pair of non-equivalent elements ${\displaystyle x\prec y}$, there is an element ${\displaystyle z_{i}\in Z}$ that separates them (${\displaystyle x\preceq z_{i}\preceq y}$).

3. X is separable and connected.

4. X is second countable. This means that there is a countable set S of open sets, such that every open set in X is the union of sets of the class S.

The proof for the fourth result had a gap which Debreu later corrected.[3]

A. Let ${\displaystyle X=\mathbb {R} ^{2}}$ with the standard topology (the Euclidean topology). Define the following preference relation: ${\displaystyle (x,y)\preceq (x',y')}$ iff ${\displaystyle x+y\leq x'+y'}$. It is continuous because for every ${\displaystyle (x,y)}$, the sets ${\displaystyle \{(x',y')|x'+y'\leq x+y\}}$ and ${\displaystyle \{(x',y')|x'+y'\geq x+y\}}$ are closed half-planes. Condition 1 is violated because the set of equivalence classes is uncountable. However, condition 2 is satisfied with Z as the set of pairs with rational coordinates. Condition 3 is also satisfied since X is separable and connected. Hence, there exists a continuous function which represents ${\displaystyle \preceq }$. An example of such function is ${\displaystyle u(x,y)=x+y}$.

B. Let ${\displaystyle X=\mathbb {R} ^{2}}$ with the standard topology as above. The lexicographic preferences relation is not continuous in that topology. For example, ${\displaystyle (5,1)\succ (5,0)}$, but in every ball around (5,1) there are points with ${\displaystyle x<5}$ and these points are inferior to ${\displaystyle (5,0)}$. Indeed, this relation cannot be represented by a continuous real-valued function (in fact, it cannot be represented even by non-continuous functions).

Proofs from.[2]

Notation: for any ${\displaystyle x,y\in X}$, define ${\displaystyle (x,y)=\{z\in X:x\prec z\prec y\}}$, and similarly define other intervals.

Diamond[4] applied Debreu's theorem to the space ${\displaystyle X=\ell ^{\infty }}$, the set of all bounded real-valued sequences with the topology induced by the supremum metric (see L-infinity). X represents the set of all utility streams with infinite horizon.

In addition to the requirement that ${\displaystyle \preceq }$ be total, transitive and continuous, He added a sensitivity requirement:

• If a stream ${\displaystyle x}$ is smaller than a stream ${\displaystyle y}$ in every time period, then ${\displaystyle x\prec y}$.
• If a stream ${\displaystyle x}$ is smaller-than-or-equal-to a stream ${\displaystyle y}$ in every time period, then ${\displaystyle x\preceq y}$.

Under these requirements, every stream ${\displaystyle x}$ is equivalent to a constant-utility stream, and every two constant-utility streams are separable by a constant-utility stream with a rational utility, so condition #2 of Debreu is satisfied, and the preference relation can be represented by a real-valued function.

The existence result is valid even when the topology of X is changed to the topology induced by the discounted metric: ${\displaystyle d(x,y)=\sum _{t=1}^{\infty }{2^{-t}|x_{t}-y_{t}|}}$

These are the general assumptions:

• X, the space of all bundles, is a cartesian product of n commodity spaces: ${\displaystyle X=\times _{i=1}^{n}{X_{i}}}$ (i.e., the space of bundles is a set of n-tuples of commodities).
• ${\displaystyle \preceq }$ is a relation on X which is total (all items are comparable) and transitive.
• ${\displaystyle \preceq }$ is continuous (see above).
• There exists an ordinal utility function, ${\displaystyle v}$, representing ${\displaystyle \preceq }$.

The function ${\displaystyle v}$ is called additive if it can be written as a sum of n ordinal utility functions on the n factors:

${\displaystyle v(x_{1},...,x_{n})=\sum _{i=1}^{n}{k_{i}v_{i}(x_{i})}}$

where the ${\displaystyle k_{i}}$ are constants.

Given a set of indices ${\displaystyle I}$, the set of commodities ${\displaystyle (X_{i})_{i\in I}}$ is called preferentially independent if the preference relation ${\displaystyle \preceq }$ induced on ${\displaystyle (X_{i})_{i\in I}}$, given constant quantities of the other commodities ${\displaystyle (X_{i})_{i\notin I}}$, does not depend on these constant quantities.

If ${\displaystyle v}$ is additive, then obviously all subsets of commodities are preferentially-independent.

If all subsets of commodities are preferentially-independent AND at least three commodities are essential (meaning that their quantities have an influence on the preference relation ${\displaystyle \preceq }$), then ${\displaystyle v}$ is additive.

Moreover, in that case ${\displaystyle v}$ is unique up to an increasing linear transformation.

For an intuitive constructive proof, see Ordinal utility - Additivity with three or more goods.