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Binary Preferences Zhaochen He

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

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

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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 12 3

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

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i i i 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

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i i i vs 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

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i i i vs 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

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i i i vs 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

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i i i vs

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ABCD AiACD BAiCB CCCiC DDBCI ABCD AiACD BAiCD CCCiC DDDCI AliceBill

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ABCD AiACD BAiCD CCCiC DDDCI 1.C 2.D 3.A 4.B With transitive preferences, we can reduce all of the above to a simple list, or ranking. A B C D

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ABCDE AiBCDE BBiBBE CCBiCE DDBCiE EEEEEi

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1.E 2.B 3.C 4.D 5.A Option Utility 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 i i i i i Etc…

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One good, any quantity , 5 3.2, 6 4.1, 7 5.0, Etc.

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