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I'm reading MWG's Microeconomics, and I'm a bit confused about the utility representation theorem. It states that a rational and continuous preference relation can be represented by a continuous utility function, but in the proof it gives, monotonicity is used as a condition.

Further, in the next chapter considering Walrasian correspondence, it says if the relation is locally nonsatiated, then the correspondence $x(p,w)$ satisfies Walras'law.

I am very confused here. So utility representation theorem only require rationality and continuity? But the proof even used strong monotonicity which is stronger than monotonicity, let alone locally nonsatiated.

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The result holds without any nonsatiation assumption. However, proofs of the result in full generality tend to be very long and often use mathematical methods outside of the scope of MWG.

For simplicity, MWG only give a proof (essentially due to Wold) under the assumption that the domain of the preferences is $\mathbb{R}^l_+$ and that preferences are monotone. This is mentioned in the first sentence of the proof. And no, the proof does not use strong monotonicity; monotonicity suffices.

Without local non-satiation, Walras' law will, in general, not hold.

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