In the revealed preferences arguments that we used in solving Question 1 in Consumer Theory Problem Set, even after we assume local nonsatiation, the question is still tricky because we are never sure if we can observe the entire choice set. We had to argue through many results relying on the fact that the optimal bundle on the budget line is strictly preferred to any interior bundle within the budget set.

What if we also assume strict convexity of preferences? Then there would be a unique optimal bundle? Under strictly convex preferences, there is only one unique optimal bundle, and we observe only one chosen bundle. So they must be the same thing. Therefore the strict convexity assumption is equivalent to the assumption that we observe all elements (only one) of the choice function C(B). And the comparison would be easier because we are sure the unique chosen bundle is strictly preferred to anything else on or within the budget line. Under a new set of prices, then, the new optimal choice must not be any of the feasible bundles under the old set of prices. So assuming that preferences are strictly convex would save us a lot of trouble thinking whether the bundle chosen by the consumer is one of the many bundles that are equally preferred or is the unique most preferred bundle.

I think this is also related to a Choice Theory Problem Set question 4 where we are asked to show that preference based models are not falsifiable, i.e. any observation could be rationalized by the preference that everything is equally preferred. If we can somehow assume away this situation so that we observe the entire choice set, life would be so much easier.

Would you say my thought is correct?

## Revealed preference and strictly convex preferences

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