intermediate microeconomics w3211 lecture 23: uncertainty and
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Intermediate Microeconomics W3211
Lecture 23: Uncertainty and Information 1: Expected Utility TheoryColumbia University, Spring 2016Mark Dean: [email protected]
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Introduction
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The Story So Far….• We have spent a lot of time modelling the choices made
by • People • Firms
• In doing so, we have always assumed that people knew for sure what the outcome of those choices were• If you buy a commodity bundle, you get those things for sure• If a firm produces y output, they are 100% certain that they will be
able to sell them at price p
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Today• In many cases the outcome of our choices are uncertain
• For example• You are deciding whether or not to buy shares in Apple• You are deciding whether to gamble your student loan on black on
the roulette table• You are deciding whether or not to buy beachfront property in
Miami
• The aim of this lecture is to think about how we model such choices
• Chapter 12 Varian, Chapter 19 Feldman and Serrano
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Choice under UncertaintyWhat should I maximize?
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What Should I Maximize?
When we thought about how consumers behave, we had a very specific model They should make choices to maximize their preference (or utility)
How can we extend this model to choices over things which are uncertain?
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What Should I Maximize?
Here is an example of a very boring fairground game You pay an amount x to play the game If you play, the fairground person flips a (fair) coin If it comes down heads you win $10 If it comes down tails, you win $0
So If you don’t play the game you get $0 for sure If you play the game, with 50% chance you get $10-x and with 50%
chance you get –x
What is the most you would pay in order to play such a game? i.e. how big an x?
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What Should I Maximize?
$5?
This is the expected (or mean) value of the game
Prob($10).10+Prob($0).0
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What Should I Maximize?
Okay, here is a new game The fairground person flips a coin If it comes down tails you get $2 If it comes down heads, the coin gets flipped again If it comes down tails, you get $4 If it comes down heads, the coin gets flipped again If it comes down tails you get $8 If it come down heads, the coin gets flipped again Etc. etc
How much would you pay to play the game?
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What Should I Maximize?
Well, what is the expected value?
Prob($2).2+Prob($4).4+Prob($8).8…
12 . 2
14 . 4
18 . 8
116 . 16
1+1+1+1…. ∞!
Would you pay a million dollars to play this game?
Would you pay a billion?
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What Should I Maximize?
Here is another question
Let’s say a pauper finds a magic lottery ticket, which pays $1,000,000 with 50% chance and $0 otherwise
A wealthy toff offers them $475,000 for the lottery ticket
Should they accept the offer?
If they are maximizing expected value they should not!
Is this sensible?
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Expected Utility
Most of economics assumes that what you maximize is not expected value, but expected utility
So the pauper should figure out the utility they get from $0, the utility they get from $475,000 and the utility they get from $1,000,000
Then compare 0 1,000,000 to 475,000
Compare the expected utility of the gamble to the expected utility of the sure thing
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Expected Utility and Risk Aversion
Question: when would they prefer the sure thing to the gamble, even though the gamble has a higher expected value?
Answer: when their utility function is concave
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Expected Utility and Risk Aversion
Money
Utility
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Expected Utility and Risk Aversion
Money
Utility
1 million
u(1,000,000)
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Expected Utility and Risk Aversion
Money
Utility
1 million
u(1,000,000)
1/2u(1,000,000)+1/2u(0)
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Expected Utility and Risk Aversion
Money
Utility
1 million
u(1,000,000)
1/2u(1,000,000)+1/2u(0)
475,000
• If pauper prefers to get the money for sure, u(475,000) must be above 1/2u(1,000,000)+1/2u(0)
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Expected Utility and Risk Aversion
Money
Utility
1 million
u(1,000,000)
1/2u(1,000,000)+1/2u(0)
475,000
u(475,000)
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Expected Utility and Risk Aversion
Money
Utility
1 million
u(1,000,000)
1/2u(1,000,000)+1/2u(0)
475,000
u(475,000)
• Requires utility function to be concave
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Expected Utility and Risk Aversion
Intuition is as follows
The additional utility from getting $475,000 relative to $0 is huge
The additional utility from getting $1,000,000 relative to $475,000 is much smaller
So the additional utility gained from winning the lottery is relatively small
Not worth the additional risk
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Expected Utility and Risk Aversion
This is the idea of diminishing marginal utility of wealth The utility from an additional dollar is lower when you are rich than
when you are poor
Diminishing marginal utility is exactly the same as saying the utility function is concave Slope of the utility function is decreasing
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Expected Utility and Risk Aversion
Definition: we say someone is risk averse if, for any lottery, they prefer to receive the expected value of that lottery for sure than play the lottery E.g., for a lottery which pays 1/3 $30, 1/3 $15, 1/3 $0, They would prefer $15 for sure
An expected utility maximizer is risk averse if and only if they have a concave utility function i.e. decreasing marginal utility of money
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Summary
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Summary
We have introduced the idea of expected utility as a way of modelling the way people make choices over risky options
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