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In: Economics

Lexicographic Preferences: Suppose that a consumer has lexicographic preferences over bundles of non-negative amounts of each...

Lexicographic Preferences: Suppose that a consumer has lexicographic preferences over bundles of non-negative amounts of each of two commodities. The consumer’s consumption set is R2+. The consumer weakly prefers bundle a = (a1,a2) over bundle b = (b1,b2) if either (i) a1 > b1, or (ii) both a1 = b1 and a2 ? b2. In any other circumstance, the consumer does not weakly prefer bundle a to bundle b. (Note that these preferences are not continuous. Furthermore, they cannot be represented by a utility function.)

1. Under what circumstances will bundle c be strictly preferred to bundle d?

2. Under what circumstances will bundle c be indifferent to bundle d?

3. Are these preferences weakly complete? Explain why.

4. Are these preferences reflexive? Explain why.

5. Are these preferences strongly complete? Explain why.

6. Are these preferences transitive? Explain why.

7. Are these preferences rational? Explain why.

8. Show that these preferences are strongly monotone. (Note that this means that they are also monotone and locally non-satiated.)

9. Show that these preferences are strictly convex. (Note that this means that they are also convex.)

10. More challenging and non-examinable: Show that these prefer- ences are not continuous.

Expert Solution

Answer (1): Bundle C will be strictly preferred to bundle D if c1>d1 and c2>d2.

Answer (2): Bundle C will be indifferent to bundle D if c1>d1 and c2<d2.

Answer (3): These preferences are weakly incomplete, as these do satisfy some other situations such as what will happen when c1=d1 or c2=d2, or c1<d1 and c2>d2.

Answer (4): The preference are reflexive, as they reflects the nature of c1 and d1. These preferences do not interfere.

Answer (5): No, these preferences are not strongly complete, as they do inform as to what will happen when c1=d1 or c2=d2, or c1<d1 and c2>d2.

Answer (6) : These preferences are transitive, as they safely keep distance from each other and do not get intermixed.

Answer (7): The preferences are rational, as they explain logical and true nature of c and d or a and b

Answer (8): Since, these preferences give us the information of one type, i.e. who is greater than who, or who is lesser than who, therefore these information are all of one type. Hence they are monotonous in nature.

Answer (9): Convex nature implies that the preferences are bounded in a certain group or collection of ideas. Since all these preferences are of one type and monotonous, they can be grouped under one collection, i.e. who is greater than who or who is lesser than whom. Hence these are also convex in nature.

Answer (10): Continuous preferences are always of one nature or one type such as a>b, b>c, c>d etc. Here, these preferences are not continuous because they also show pattern where b>c or c>d. Therefore these preferences are not continuous.

Rahul Sunny answered 2 years ago

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