By Atle Selberg
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C. P is transitive. d. P is negatively transitive. 9. Assume the same conditions as in Exercise 8, above, except this time assume: (∀x, y ∈ X) : f (x) > f (y) ⇒ xP y. 3. Preference Relations and Utility Functions 19 Answer the same questions as in Exercise 8. 10. Let G be the relation deﬁned on R by: xGy ⇐⇒ x ≥ f (y), where f : R → R. Can you provide suﬃcient conditions for G to be: a. reﬂexive? b. total? c. transitive? d. asymmetric? e. antisymmetric? ) 11. Show that if G is a weak order on a ﬁnite set, X, then the following function represents G on X: u(x) = #X − #P x, where P is the asymmetric part of G.
Because this model explains the standard theory (that is, its assumptions imply those of the standard theory, with the appropriate speciﬁcations of the primitives) it is, in eﬀect, a special case of the standard theory; and, it should be emphasized, there are other special cases of the standard economic theory of demand in which the consumer’s daily preferences would be well-deﬁned. On the other hand, the model to be developed here [which we will call the sequential consumption plan (SCP) model] seems suﬃciently plausible and interesting as to merit the time which we will spend on its development.
We will consider this question in the next section. 8 Sequential Consumption Plans It was noted in Section 5 that a consumer might satisfy all the assumptions of the standard economic theory of demand, and yet not have well-deﬁned preferences over daily consumption. Thus, a given consumer may satisfy all the assumptions of the standard economic theory of demand, and yet not satisfy the Soft Drink Model; at least not in the sense of having invariant preferences from one day to the next. In order to establish this fact, and to explore the reasons for it, we will develop a model in this section which explains the standard economic theory of demand.