Math Problem Statement

Prove or provide counter-example: any complete and transitive binary relation > over X can be represented by a utility function, if: e e In X X (1), jis is if. finite; countable. [2+5] X = R* and & is continuous over X in addition to complete- ness and transitivity, is it possible to find a continuous utility function representing >?

Solution

Let's break down the problem into two parts:

1. Case 1: XX is Finite or Countable

Given:

  • >> is a complete and transitive binary relation over XX.
  • XX is either finite or countable.

Claim: Any complete and transitive binary relation >> over XX can be represented by a utility function u:XRu: X \to \mathbb{R}.

Proof:

  • Finite Case: If XX is finite, any complete and transitive binary relation can be represented by a utility function, as discussed previously. You can enumerate the elements of XX in order of the relation >> and assign increasing real numbers as utilities.

  • Countable Case: If XX is countable, similar reasoning applies. The completeness and transitivity of >> ensure that XX can be well-ordered according to >>. This allows us to assign utility values that preserve the order.

In both cases, a utility function can represent >>.

2. Case 2: X=RX = \mathbb{R}^* and >> is Continuous, Complete, and Transitive

Given:

  • X=RX = \mathbb{R}^* (the set of all non-negative real numbers, excluding 0, often denoted R+\mathbb{R}^+ or R\mathbb{R}^*).
  • >> is a complete, transitive, and continuous binary relation over XX.

Claim: Is it possible to find a continuous utility function u:XRu: X \to \mathbb{R} representing >>?

Explanation:

  • Continuity: The continuity of >> means that if a sequence xnx_n converges to xx, then the ordering of the elements in the sequence should reflect in the utility values smoothly.

  • Debreu's Theorem: According to the Debreu Representation Theorem, if >> is a continuous, complete, and transitive binary relation on a topological space (like R\mathbb{R}^*), there exists a continuous utility function u:XRu: X \to \mathbb{R} that represents >>.

Thus, in this case, it is possible to find a continuous utility function that represents the relation >>.

Conclusion:

  • For finite or countable XX, a utility function can always represent a complete and transitive binary relation.
  • For X=RX = \mathbb{R}^*, if the relation >> is also continuous, then there exists a continuous utility function representing >>.

Follow-up Questions:

  1. How does the result change if the space XX is uncountable but not continuous?
  2. What happens if XX is R2\mathbb{R}^2 or higher-dimensional, with a continuous relation >>?
  3. Can you construct a specific example of a continuous utility function for X=RX = \mathbb{R}^*?
  4. How does the assumption of continuity influence the possibility of constructing a utility function?
  5. What if >> is not continuous—can a utility function still be constructed?

Tip:

When dealing with relations on real spaces, continuity is a powerful assumption that allows the application of theorems like Debreu's to ensure the existence of utility functions.

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Math Problem Analysis

Mathematical Concepts

Binary Relations
Utility Functions
Complete and Transitive Relations
Continuous Functions
Debreu Representation Theorem

Formulas

-

Theorems

Debreu Representation Theorem

Suitable Grade Level

Advanced Undergraduate