Binary Preferences Zhaochen He

Would You Rather? OR Have a nice teacher who is bad at teaching Have a mean teacher that is great at teaching. Consider the question, but don’t answer yet.

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

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.

Meet Mark’s Dilemma ?

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.

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.

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

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

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

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

vs i i i What are some things you notice about the pattern of preferences displayed on this table? This collection of all binary preferences over a group of items is called a preference relation over those items.

i i i vs 1. Reflexivity – Any good is indifferent with itself What are some things you notice about the pattern of preferences displayed on this table? 1. Reflexivity – Any good is indifferent with itself

vs i i i What are some things you notice about the pattern of preferences displayed on this table? 2. Symmetry - The table is symmetric across the diagonal of indifference

i i i vs 3. Transitivity: If A p B, and B p C, then A p C What are some things you notice about the pattern of preferences displayed on this table? 3. Transitivity: If A p B, and B p C, then A p C

i i i vs 3. Transitivity: If A p B, and B p C, then A p C What are some things you notice about the pattern of preferences displayed on this table? 3. Transitivity: If A p B, and B p C, then A p C

i i i vs 3. Transitivity: If A p B, and B p C, then A p C What are some things you notice about the pattern of preferences displayed on this table? 3. Transitivity: If A p B, and B p C, then A p C

i i i vs 3. Transitivity: If A p B, and B p C, then A p C What are some things you notice about the pattern of preferences displayed on this table? 3. Transitivity: If A p B, and B p C, then A p C

i i i vs 3. Transitivity: If A p B, and B p C, then A p C What are some things you notice about the pattern of preferences displayed on this table? 3. Transitivity: If A p B, and B p C, then A p C

vs i i i What are some things you notice about the pattern of preferences displayed on this table?

A B C D i I A B C D i I Alice Bill What are some things you notice about the pattern of preferences displayed on this table? Alice Bill

A B C D A B C D i I C D A B With transitive preferences, What are some things you notice about the pattern of preferences displayed on this table? C D A B With transitive preferences, we can reduce all of the above to a simple list, or ranking.

A B C D E i Activity: ask students to create the ordered ranking. Answer: E B C D A

Option Utility E ? B C D A E B C D A Answer: E B C D A

Utility Functions A utility function simply assigns a numerical value to each option. The SIZE of these numerical value fully represent 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.

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.

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.

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

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