Microeconomics Course E

1. MicroeconomicsCourse E John Hey

2. 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!)

3. 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)

4. Contingent Goods (chapter 23) • Notation: • m1and m2:incomes in the two states. • c1and c2:consumption in the two states. • Good 1: income contingent on state 1. • Good 2: income contingent on state 2. • p1and 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.

5. 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

6. 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 andm2 = 50 and • p1= 0.5 and p2 = 0.5

8. 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.

9. 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.

10. The Expected Utility Model • U(c1,c2) = π1 u(c1)+ π2 u(c2). • … where(c1,c2) indicates a risky bundle which yields c1with probabilityπ1 andc2with probabilityπ2. • 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.

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

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

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

14. 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.

15. 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

16. 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

17. Optimal choice π1= π2=0.5

18. Risk-averse • U(c1,c2) = π1 u(c1)+ π2 u(c2) • u(.) isconcave • 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

19. Optimal choice π1= π2=0.5

20. Optimal choice π1=0.6,π2=0.4

21. Risk-loving • U(c1,c2) = π1 u(c1)+ π2 u(c2) • u(.) isconvex • 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

22. Optimal choice π1=0.6,π2=0.4

23. 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.

24. 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.

25. 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

26. 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

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

28. Chapter 24 • Goodbye!