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 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 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 – π 1 and π 2 π 1 + π 2 = 1 We have to decide ex ante.

Contingent Goods Notation: m 1 and m 2 : incomes in the two states. c 1 and c 2 : consumption in the two states. Good 1: income contingent on state 1. Good 2: income contingent on state 2. p 1 and p 2 : 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 Two states of the world: 1 an accident (theft, etc.): 2 no accident (theft, etc.) Let us suppose each has probability 0.5. Suppose m 1 = 30 and m 2 = 50 e p 1 = 0.5 and p 2 = 0.5 Let us go to Maple....

Il Vincolo di Bilancio p 1 c 1 + p 2 c 2 = p 1 m 1 + p 2 m 2

Prices in a perfect insurance market If you buy a unit of income contingent on state with probability π 1 you receive 1,...with probability π 2 you receive 0. Your expected income = π π 2.0. = π 1 Hence a fair price is p 1 = π 1 Similarly a fair price for Good 2 is p 2 = π 2

Expected Value Consider a variable (income) which takes the value m 1 with probability π 1 and the value m 2 with probability π 2 The expected value of the income (or the expected income) is given by…... m 1 π 1 + m 2 π 2

Chapter 23 The budget constraint in a perfect insurance market is p 1 c 1 + p 2 c 2 = p 1 m 1 + p 2 m where p 1 = π 1 and p 2 = π 2 Hence… … π 1 c 1 + π 2 c 2 = π 1 m 1 + π 2 m 2 Expected consumption is equal to expected income. Has slope = -π 1 /π 2

Chapter 23 Goodbye!