MicroeconomicsCourse E

John Hey

Chapter 26

• Because we are all enjoying risk so much, I have decided ....

• ... not to cover Chapter 26 (on the labour market) this year...

• ... and so there will be no examination questions on Chapter 26 this year.

• (Not that there were anyhow!)

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)

Contingent Goods (chapter 23)• 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.

Budget Constraint (ch 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

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• m1 = 30 and m2 = 50 and • p1 = 0.5 and p2 = 0.5

Expected Utility Model (ch 24)

• Now we need to put in preferences.• Chapter 24 is difficult.• You do not need to know the detail...• ...only the principles.• The utility of a lottery that yields c1 with

probability π1 and yields c2 with probability π2 is given by…

... U(c1,c2) = π1 u(c1)+ π2 u(c2)• This is the Expected Utility model.

Remember this bet?

• I intend to sell this bet to the highest bidder.• We toss a fair coin...• ... if it lands heads I give you 50 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.

The Expected Utility Model• U(c1,c2) = π1 u(c1)+ π2 u(c2).• … where (c1,c2) indicates a risky bundle which yields c1

with probability π1 and c2 with probability π2.• Suppose your wealth is w and the most that you are

willing to pay for the bet is m. Then you are indifferent between w for sure and a 50-50 bet between 50+w-m and w-m. Hence:

• u(w)=0.5 u(50+w-m) + 0.5 u(w-m)• Note: Expected Profit = 0.5(50+w-m)+0.5(w-m)-w = 25-m• The implications for u(.) depend upon m.

Suppose m = 19 (<25) (risk-averse) and w=30u(30)=0.5 u(61) + 0.5 u(11)

Suppose m = 25 (risk-neutral) and w=30u(30)=0.5 u(55) + 0.5 u(5)

Suppose m = 29 (>25) (risk-loving) and w=30u(30)=0.5 u(51) + 0.5 u(1)

Hence the form of the utility function is important

• If u(.) is concave the individual is risk-averse.

• If u(.) is linear the individual is risk-neutral.

• If u(.) is convex the individual is risk-loving.

• Obviously the function may be concave in some parts and convex in others.

The Expected Utility Model

• U(c1,c2) = π1 u(c1)+ π2 u(c2)• An indifference curve is given by π1 u(c1)+ π2 u(c2) = constant• If the function u(.) is concave (linear,convex)

the indifference curves in the space (c1,c2) are convex (linear, concave).

• The slope of every indifference curve on the certainty line = -π1/π2

Risk neutral

• U(c1,c2) = π1 u(c1)+ π2 u(c2)• u(c)= c : the utility function is linear• An indifference curve is given by π1 c1+ π2 c2 = constant• Hence the indifference curves in the space

(c1,c2) are linear. • The slope of every indifference curve = -π1/π2

Optimal choice π1= π2= 0.5

Risk-averse

• U(c1,c2) = π1 u(c1)+ π2 u(c2)• u(.) is concave• An indifference curve is given by π1 u(c1)+ π2 u(c2) = constant• Hence the indifference curves in the space

(c1,c2) are convex. • The slope of every indifference curve on the

certainty line = -π1/π2

Optimal choice π1= π2= 0.5

Optimal choice π1= 0.6,π2=0.4

Risk-loving

• U(c1,c2) = π1 u(c1)+ π2 u(c2)• u(.) is convex• An indifference curve is given by π1 u(c1)+ π2 u(c2) = constant• Hence the indifference curves in the space

(c1,c2) are concave. • The slope of every indifference curve on the

certainty line = -π1/π2

Optimal choice π1= 0.6,π2=0.4

The Certainty Equivalent

• Consider the lottery (c1,c2) with probabilities = π1

and π2

• The certainty equivalent, EC, is defined by:• u(EC) = π1 u(c1)+ π2 u(c2)• It is a certainty which gives the same utility as the

lottery.• The value of EC depends upon the function u(.).• The individual considers the lottery and the

certainty equivalent as equivalent.

The concavity indicates the risk aversion

• With the Expected Utility Model:• If the function u(.) is concave the

individual is risk-averse.• The more concave is the function the

more risk- averse is the individual – hence the lower the certainty equivalent and the more convex are the indifference curves.

Appello 2 (traccia 1)• In the next two questions you will be asked to consider an individual, taking decisions

under conditions of risk, with Expected Utility preferences and utility function u(x) = x^0.5 (that is, the utility of x is the square root of x). Suppose the individual is faced with two lotteries P and Q as specified below. A lottery is denoted by (a,b;p,1-p) and means that the outcome is a with probability p and b with probability 1-p.

• The lotteries are: P = (25,16;0.25,0.75) Q = (1,36;0.25,0.75)• Question 14: Does the individual prefer P or Q?• P• Q• We cannot tell• The individual is indifferent.• Question 15: What is the individual's certainty equivalent for P?• 27.25• 18.25• 22.5625• 18.0625

Appello 2 (traccia 2)• In the next two questions you will be asked to consider an individual, taking decisions

under conditions of risk, with Expected Utility preferences and utility function u(x) = x^0.5 (that is, the utility of x is the square root of x). Suppose the individual is faced with two lotteries P and Q as specified below. A lottery is denoted by (a,b;p,1-p) and means that the outcome is a with probability p and b with probability 1-p.

• The lotteries are: P = (4,36;0.5,0.5) Q = (1,36;0.75,0.25)• Question 14: Does the individual prefer P or Q?• P• Q• The individual is indifferent• We cannot tell• Question 15: What is the individual's certainty equivalent for P?• 16• 5.0625• 20• 9.75

Summary

• A risk-averse individual in a perfect insurance market always chooses to be completely insured.

Chapter 24

• Goodbye!