This preview shows page 1. Sign up to view the full content.
Unformatted text preview: Econ 121.
Intermediate Microeconomics. Eduardo Faingold
Yale University Lecture 7 Outline of the course
I. Introduction
II. Individual choice
III. Competitive markets
IV. Market failure Outline of the course
I. Introduction
II. Individual choice
Budget constraint (Ch. 2)
Preferences (Ch. 3)
Utility (Ch. 4)
Consumer problem (Ch. 5)
Revealed preference (Ch. 7)
Slutsky equation (Ch. 8)
Endowment income effect (Ch. 9)
Intertemporal choice (Ch. 10)
Choice under uncertainty (Ch. 12) III. Competitive markets
IV. Market failure Demand for Insurance
Suppose consumer initially has $35;000 in wealth.
But his ﬁnal wealth is uncertain and he may loose $10;000, say, because
his car may be stolen. An insurance contract the consumer can buy pays him $1 in case the
loss occurs. To buy insurance the consumer must pay upfront a
premium of dollars per contract he buys. How much insurance will the consumer buy, if any?
How do we even approach this problem? Demand for Insurance
Consumer decides how many contracts to buy
If he buys K insurance contracts his ﬁnal wealth is:
0 wD ( 25;000 C K
35;000 K W car stolen K W car not stolen To simplify matters, assume there is only one consumption good so that
consumer cares directly about wealth. (In reality, with two or more
goods, wealth matters indirectly because of its effect in the ﬁnal decision
of which bundle to consume.) Different choices of K yield different contingent consumption plans.
How do we extend consumer theory to deal with choice under
uncertainty? In terms of standard consumer theory...
There is a way to translate this problem into standard consumer theory.
Two states of nature: state 1 (car stolen) and state 2 (car not stolen).
A bundle is .x1; x2/, that is, consumption (of the same good!) in each
state. That is, we allow the consumer to choose different amounts of a good
contingent on the state. But the choice is made before the true state is realized, like in the
insurance example. In the insurance example, the choice of the amount K of insurance to
buy is, effectively, the choice of a bundle .x1 ; x2/. Choice under uncertainty
So, in principle, our theory could just postulate a utility function u.x1; x2/
and we would be done. However, choice under uncertainty seems to be special in at least two
ways. Perhaps it makes sense to impose more structure on consumer
preferences. First, it sounds intuitive that the probability / likelihood that the car be
stolen should matter. So we would like a theory that takes the effect of
these probability assessments into account explicitly. Second, in terms of actual consumption, either the consumer consumes
x1 or he consumes x2 units of the good! Either the car is stolen or not!
Although this contingent consumption can be formally translated into
standard consumer theory, behaviorally speaking things are different. Expected utility theory will take care of the two points above. In fact, we
will see that they are related. Model basics
Outcomes, states or consequences: C D fc1; c2; : : : ; cng
Lotteries are probability distributions over C , that is, numbers
p1; p2; : : : ; pn (between 0 and 1) which sum to 1. Set of all lotteries over
C denoted L.
pi is the probability of outcome ci
consumer has a preference relation over L
This is similar to consumer theory, except that the choice space is
different. Example
Three (monetary) outcomes: $2;500;000, 500;000 and 0.
The lottery p D .0; 1; 0/ means $500;000 for sure
The lottery p 0 D .1=3; 1=3; 1=3/ means ...
The lottery p 00 D .p1; p2; p3/ means ...
Do you prefer p or p 0 ? Expected Utility
Deﬁnition. A preference relation over lotteries has an expected utility
representation if there exist utility numbers u.ci / such that for all lotteries p ,
p 0 2 L, p p 0 if and only if
0
0
0
p1u.c1/ C p2u.c2/ C : : : C pnu.cn/ p1u.c1/ C p2u.c2/ C : : : C pnu.cn/; that is,
Expected utility under p D Ep u Ep 0 u D Expected utility under p 0: When does a preference relation have an expected utility representation? Expected Utility
Completeness, Reﬂexivity and Transitivity
Independence Axiom. For all lotteries p; p 0 and p 00 and all ˛ 2 Œ0; 1,
p p 0 if and only if ˛p C .1 ˛ /p 00 ˛p 0 C .1 ˛ /p 00: To understand IA need to understand notation ˛p C .1 ˛ /p 00. This is
just a lottery that assigns probability ˛p.ci / C .1 ˛ /p 00.ci / to each
outcome ci .
The axiom makes sense when we assume that the decision maker
views compound lotteries as equivalent to simple lotteries. Expected Utility
Theorem. A preference relation satisﬁes Comp, Reﬂ, Trans and Indep if
and only if it has an expected utility representation.
Proof idea:
Compl., Reﬂ. and Trans. imply that the preference has a utility
representation. IA implies that indifference curves are straight lines, which
means utility is linear in lotteries. Risk Aversion
Consider monetary lotteries only, that is, lotteries over amounts of money.
For a lottery p let Ep denote the lottery that pays the expected value of
lottery p with certainty.
An individual is risk averse if for every monetary lottery p he prefers Ep
than p .
Proposition. An expectedutility maximizer is riskaverse if and only if his
utility function over deterministic monetary outcomes is concave.
Recall that a function f W R ! R is concave if for all x , y 2 R, ˛ 2 Œ0; 1, f .˛x C .1 ˛ /y/ ˛f .x/ C .1 ˛ /f .y/: Demand for Insurance
Consumer decides how many contracts to buy
If he buys K insurance contracts his ﬁnal wealth is:
(
c1 D 35;000 K W w/ prob. 1
wD
c2 D 25;000 C K K W w/ prob.
Budget line:
c2 D 25000 C
Slope of the budget line D
U.c1 ; c2/ D .1 1 .35000 c1/ 1 /u.c1/ C u.c2/. and so
MRS D .1 /u0.c1/
M U1
D
M U2
u0.c2/ Demand for Insurance
Choose K such that MRS D
.1 /u0.c1/
u0 .c2 / D 1 . 1 . Suppose insurance company breaks even, that is, makes zero expected
proﬁt: K K .1 / 0 D 0 ∴ D ; i.e., “fair insurance.” Then, optimal choice of K is such that
u0.c1 / D u0.c2 /;
i.e., “full insurance.” A riskaverse expected utility maximizer who is offered fair insurance
optimally chooses to be fully insured. Investment on a Risky Asset
initial wealth w
two states, “good” or “bad”
risky assets returns rg > 0 or rb < 0, depending on the state
If consumer invests x in risky asset he gets
(
w C xrg W w/ prob.
0
wD
w C xrb W w/ prob. 1 so his expected utility is EU.x/ D u.w C xrg / C .1 /u.w C xrb / Marginal utility of x :
EU 0.x/ D rg u0.w C xrg / C .1 /rb u0.w C xrb / Investment on a Risky Asset
Consider second derivative:
2
EU 00.x/ D rg u00.w C xrg / C .1 2
/rb u00.w C xrb /; which is negative if consumer is risk averse. Look at marginal utility at x D 0:
EU 0.0/ D u0.w/ . rg C .1 /rb /
„
ƒ‚
…
expected return A risk averse consumer does not invest in a risky asset with negative
expected return; A risk averse consumer always invests a positive amount in a risky asset
with positive expected return. Taxing a Risky Asset
after tax returns: .1 t /rg in the good state and .1 t /rb in the bad state The FOC determines his optimal investment level:
EU 0.x/ D 0;
that is, .1 t /rg u0.w C x.1 t /rg / C .1 /.1 t /rb u0.w C x.1 t /rb / D 0: Does imposing a tax encourage or discourage investment in the risky
asset? Taxing a Risky Asset
The answer, surprisingly, is it encourages investment in the risky asset!
Let x be the optimal investment level when t D 0, and x the optimal
O
investment under t > 0.
Then we must have:
xD
O x
1 t And the proof comes from the FOC:
.1 t /rg u0.w C x.1
O t /rg / C .1 /.1 t /rb u0.w C x.1
O t /rb / D 0; see blackboard. The tax reduces the expected return but it also reduces the risk,
because in the bad state the tax is actually a subsidy. By increasing investment, consumer can get the same consumption
pattern as before, offsetting the tax. ...
View
Full
Document
This note was uploaded on 03/01/2012 for the course ECON 121 taught by Professor Samuelson during the Spring '09 term at Yale.
 Spring '09
 SAMUELSON
 Microeconomics

Click to edit the document details