Binary Preferences

1. Binary Preferences Zhaochen He

2. Would You Rather? OR Have a nice teacher who is bad at teaching Have a mean teacher that is great at teaching.

3. Would You Rather? OR Time travel 200 years into the past Time travel 200 years into the future

4. The Big Picture • We will be spending the next few lectures discussing the most fundamental model in microeconomic theory: the theory of consumer choice. • Consumer choice theory is a mathematical description of how people might make purchasing decisions, but can be generalized to much broader situations.

5. Meet Mark’s Dilemma ?

6. The Big Picture • The mathematics of consumer choice theory can make a prediction about choice a person will make, but it needs two pieces of “given” information. • A description of the person’s preferences, usually in the form of a utility function • A description of the person’s financial situation (the money he has available, and how expensive his various options are); usually called a budget constraint.

7. The Big Picture A description of a person’s preferences usually comes in the form of a utility function. By the end of this lecture, we’ll begin to talk about utility functions. But utility functions themselves are based off of a even more fundamental way to represent preferences. It all begins with binary preferences.

8. A binary preference is a preference between two distinct options. • This is in some sense the simplest form of preference we could consider. • When faced with a binary preference A vs B, an agent could prefer A to B, B to A, or be indifferent between the two. • From now on, we’ll write these possibilities as: • A p B • B p A • A i B

9. Of course, we often have more than two options when we make a choice.

10. However, we could reduce your preferences over multiple items to a series of binary comparisons. vs vs 1 2 3 vs

11. A good way to represent this set of binary preferences is with a table. vs i i i

12. vs i i i This collection of all binary preferences over a group of items is called a preference relation over those items.

13. vs i i i 1. Reflexivity – Any good is indifferent with itself

14. vs i i i 2. Symmetry - The table is symmetric across the diagonal of indifference

15. vs i i i 3. Transitivity: If A p B, and B p C, then A p C

16. vs i i i 3. Transitivity: If A p B, and B p C, then A p C

17. vs i i i 3. Transitivity: If A p B, and B p C, then A p C

18. vs i i i 3. Transitivity: If A p B, and B p C, then A p C

19. vs i i i 3. Transitivity: If A p B, and B p C, then A p C

20. vs i i i

21. Alice Bill

22. A C B D C D A B With transitive preferences, we can reduce all of the above to a simple list, or ranking.

24. E B C D A

25. Utility Functions • A utility function simply assigns a numerical value to each option. The SIZE of these numerical value fullyrepresent the consumer’s binary preferences over all choices. • For example, if he prefers A to B, then the utility of A will be higher than the utility of B.

26. Utility Functions • IMPORTANT: The magnitudes given by a utility function are not unique – that is, many different utility functions could describe the same set of binary preferences. • Another way of saying this: A utility of 10 isn’t necessarily “twice as good” as a utility of 5. • Utility functions are ordinal, not cardinal.

27. Towards Mark’s Dilemma • So far, we’ve looked at multiple goods, but with a quantity of one. • We could also look at only one good, but allow any quantity. • Or, we could look at multiple goods, and allow any quantity.

28. One good, any quantity 1 2 3 4 5 5 Etc…

29. One good, any quantity 4 3, 5 2, 6 1, 7 0, 8 9 10 11 Etc.