Microeconomics
Corso E
John Hey
Chapters 23, 24 and 25
• CHOICE UNDER RISK
• Chapter 23: The Budget Constraint.
• Chapter 24: The Expected Utility Model.
• Chapter 25: Exchange in Insurance
Markets.
• (cf. Chapters 20, 21 and 22)
Bet 1
• Will you bet with me?
• We toss a fair coin...
... If it lands heads I give you 100 euros
... If it lands tails you give me 100 euros
• Note... this is a risky choice problem
• ... we know the probabilities...
• ... but we do not know which “state of the world”
will happen.
Bet 2
• Will you bet with me?
• We toss a fair coin...
... If it lands heads I give you 100 euros
... If it lands tails you give me 50 euros
• Note... this is a risky choice problem
• ... we know the probabilities...
• ... but we do not know which “state of the world”
will happen.
Bet 3
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I intend to sell this bet to the highest bidder.
We toss a fair coin...
... if it lands heads I give you 100 euros
... If it lands tails I give you nothing.
We will do an “English Auction” – the student
who is willing to pay the most wins the auction,
pays me the price at which the penultimate
person dropped out of the auction, and I will play
out the bet with him or her.
Contingent Goods
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A bet, for example:
... if it lands heads I give you 100 euros
... If it lands tails I give you nothing.
If you buy this bet, you do not buy
something certain, but something whose
value depends upon the “state of the
world” (heads or tails).
• This is exactly what insurance is.
Bet 1
• We toss a coin...
... if it lands heads (State 1) I give you 100 euros
... if it lands tails (State 2) you give me 100 euros
• Let us denote by m your income.
• If you do not take the bet m is your income...
• ...if you do take the bet then m+100 in State 1
and m-100 in State 2.
• Hence your income is contingent on the state.
Chapter 23
• Description of the situation...
• There are two possibilities – we call them
State 1 and State 2.
• Only one state will happen ... but we do
not know which ex ante.
• We know the probabilities – π1 and π2
• π1 + π2 = 1
• We have to decide ex ante.
Contingent Goods
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Notation:
m1 and m2: incomes in the two states.
c1 and c2: consumption in the two states.
Good 1: income contingent on state 1.
Good 2: income contingent on state 2.
p1 and p2: the prices of the two goods.
For every unit of Good i that you have bought
you receive an income of 1 if state i occurs.
• For every unit of Good i that you have sold you
have to pay 1 if state i occurs.
Insurance
• Consider insurance, for example, against
the theft of your car.
• If no-one steals your car you have to pay
to the insurance company the premium.
• If a thief steals your car the insurance
company pays to you the value of your car
• Insurance provides contingent income
(contingent on the theft of your car).
An example
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Two states of the world:
1 an accident (theft, etc.):
2 no accident (theft, etc.)
Let us suppose each has probability 0.5.
Suppose
m1 = 30 and m2 = 50 e
p1 = 0.5 and p2 = 0.5
Let us go to Maple....
Il Vincolo di Bilancio
• p1c1 + p2c2 = p1m1 + p2m2
Prices in a perfect insurance
market
• If you buy a unit of income contingent on
state 1...
• ...with probability π1 you receive 1,
...with probability π2 you receive 0.
• Your expected income = π1 .1 + π2 .0. = π1
• Hence a fair price is p1 = π1
• Similarly a fair price for Good 2 is p2 = π2
Expected Value
• Consider a variable (income) which takes
the value m1 with probability π1 and the
value m2 with probability π2
• The expected value of the income (or the
expected income) is given by…
... m1π1 + m2π2
Chapter 23
• The budget constraint in a perfect
insurance market is...
... p1c1 + p2c2 = p1m1 + p2m2 ...
...where p1 = π1 and p2 = π2
• Hence…
… π1c1 + π2c2 = π1m1 + π2m2
• Expected consumption is equal to
expected income.
• Has slope = -π1/π2
Chapter 23
• Goodbye!