Microeconomics 2 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 Markets for Risk (for contingent goods). (cf. Chapters 20, 21 and 22: and compare chapter 25 with chapter 8.) Remember the Health Warning: this is one of John H’s research areas... Remember also the message from Chapter 8: exchange is almost always mutually beneficial.

Expected Utility Model (ch 24) This is a model of preferences. Suppose a lottery yields C which takes values c1 with probability π1 and c2 with probability π2 (where π1 + π2 = 1). Expected Utility theory says this lottery is valued by its Expected Utility: ... Eu(C) = π1 u(c1)+ π2 u(c2) where u(.) is the individual’s utility function. In intuitive terms the value of a lottery to an individual is the utility that the individual expects to get from it.

EU and a risk-averter A risk-averter’s utility function over consumption u(.) is concave; the more risk-averse the more concave. A risk-averter’s indifference curves in (c1,c2) space are convex; the more risk-averse the more convex. A risk-averter’s Certainty Equivalent for some gamble is less than the Expected Value of the gamble; the more risk-averse the smaller; the riskier the gamble the smaller. A risk-averter’s Risk Premium for some gamble is positive; the more risk-averse the bigger; the riskier the gamble the bigger. A risk-averter is always willing to pay to get rid of risk; the more risk-averse the more he/she will pay; and the riskier the risk the more he/she will pay. With fair insurance, a risk-averter will always choose to be fully insured.

EU and a risk-neutral A risk-neutral’s utility function over consumption u(.) is linear. A risk-neutral’s indifference curves in (c1,c2) space are linear. A risk-neutral’s Certainty Equivalent for some gamble is always equal to the Expected Value of the gamble. A risk-neutral’s Risk Premium for some gamble is always zero. A risk-neutral would not pay to get rid of risk; nor would he/she pay any money to accept some risk.

EU and a risk-lover A risk-lover’s utility function over consumption u(.) is convex; the more risk-loving the more convex. A risk-lover’s indifference curves in (c1,c2) space are concave; the more risk-loving the more concave. A risk-lover’s Certainty Equivalent for some gamble is always greater than the Expected Value of the gamble; the more risk-loving the smaller; the riskier the gamble the larger. A risk-lover’s Risk Premium for some gamble is negative; the more risk-loving the more negative; the riskier the gamble the more negative. A risk-lover is always willing to pay to get risk; the more risk-loving the more he/she will pay; and the riskier the risk the more he/she will pay.

Chapter 25 We consider the exchange of risk between two individuals. We use the same framework as in Chapters 8 (the Edgeworth Box) and 22. We get the same results: ...exchange is almost always beneficial. ...competitive exchange is efficient... ...but does not necessarily imply actuarially fair prices (this is new).

Exchange of risk The Maple file contains lots of examples asking if exchange is possible and, if so, what form it takes. First example: 2 individuals A and B; both risk-averse, B more than A; both start with same risk (75,50); both states equally likely. But first I want to revise Price-Offer curves, as I get the feeling that there may be some confusion. A Price-Offer curve of an individual is the locus of points to which the individual would want to move for different prices. Let us go to Maple, first Lecture 8, then Lecture 25...

Competitive equilibrium in this first case Starting positions Finishing Positions Trades Individual A Individual B If State 1 occurs 75 If State 2 occurs 50 Individual A Individual B If State 1 occurs 83 67 If State 2 occurs 45 55 Individual A Individual B If State 1 occurs 8 -8 If State 2 occurs -5 5

Chapter 25 There are lots of other examples in the Maple file. Smell them and get intuition. They are very entertaining. You are not expected to know the detail... ...just the principle.

Chapter 25 Goodbye!