Microeconomics 2
John Hey
Chapters 23, 24 and 25
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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 riskaverse 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 riskloving 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 riskloving 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
75
If State 2 occurs
50
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!