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Binary Preferences Zhaochen He

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**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.

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

Time travel 200 years into the future

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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.

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Meet Mark’s Dilemma ?

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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.

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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.

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**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

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

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**However, we could reduce your preferences over multiple items to a series of binary comparisons.**

vs vs 1 2 3 vs

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

vs i i i

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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.

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**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

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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

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**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

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**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

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**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

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**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

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**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

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

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**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

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**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.

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

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

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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.

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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.

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**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.

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

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

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