Econ 604 Advanced Microeconomics
DavisSpring 2006, March 30
Lecture 9
Reading.Chapter 8 (pp. 198-210
Problems:8.3, 8.5, 8.7
Next timeChapter 11, pagers 268 -278
REVIEW
VII. Appendix: Separable Utility and the Grouping of Goods. We consider the theoretically-predicted effects of some stronger assumptions about the nature of substitution between products. In particular we considered Separable utility. (6.9).
Notice that the assumption of separability often facilitates very considerably the analysis of certain problems. Further, separability is less restrictive than it may first appear. Recall, that our condition for quasi-concavity requires
(U11 U22 - 2U12 U1 U2 + U22 U12)<0
Separability implies only that U12 = 0. Sufficient variability movement in own first and second derivatives will allow, for example, convex indifference curves
VIII. Chapter 7 Market Demand and Elasticity
A. Market Demand Curves
1. The two consumer case
2. The n consumer case.
In general market demand equals individual demand aggregated across price (7.3)
Recall, there are problems of aggregation across income (7.1)
B. Elasticity
1. Motivation and a general definition
2. Price Elasticity of Demand
Insight: For linear demand, elasticity is the ratio of P* to Po – P* (Problem 7.5)
3. Income Elasticity of Demand
4. Cross Price Elasticity of Demand
C. Relationships Between Elasticities
1. Sum of Income Elasticities for all Goods
Recall: PXX + PYY=I,and that market demands for X and Y are function of Prices and Income
X=dX(PX, PY, I) and Y=dY(PX, PY, I).
Differentiating the budget constraint w.r.t. I,
PX(X/I) + PY(Y/I)=1
Multiplying through I/X yields
sxeX,I+ sYeY,I=1
where
PXX/I = sxand PYY/I = sy
Thus, the sum of income elasticities equals 1. For all goods where consumption increases more than proportionally with income, offsetting goods must exist where consumption increases less than proportionally. This is sometimes referred to as Engel’s Law
2. Slutsky Equation in Elasticities
Given
X=X |-X X
PXPX |U = constant I
Multiply through by PX/X
X PX=X PX |-X[XI ] PX
PXXPX X |U = constant I X I
Solving yields
eXP= eSXP-eXI sx
3. Homogeneity
Euler’s theorem states that, for a function that is homogenous of degree m
f1X1+ f2X2 +… fnXn = mf(X1, X2, …. , Xn)
Demand functions are homogenous of degree zero. Thus
(X/PX) PX + (X/PY)PY+(X/I)I=0
eX,Px + e X,Py + e X,,I = 0
(Problem 7.9)
D. Types of Demand Curves
1. Linear Demand
2. Constant Demand Elasticity
PREVIEW
IX. Chapter 8 Expected Utility and Risk Aversion
A. Probability and Expected Value
B. Fair Games and the Expected Utility Hypothesis
C. The Von Neumann-Morgenstern Theorem
D. Risk Aversion
E. Measuring Risk Aversion
Lecture______
IX. Chapter 8 Expected Utility and Risk Aversion. To this point, we have concentrated on optimal decisions for consumers who confronted no uncertainty of any kind when making their decisions. Of course, in normal circumstances, consumer decisions are characterized by a variety of different kinds of uncertainty, including outcome uncertainty, or uncertainty as to given outcomes, imperfect information (uncertainty regarding the quality and availability of choices), and strategic uncertainty (that is uncertainty regarding the actions of rivals). Much of modern microeconomics focuses on incorporating these three kinds of uncertainty into decision-making models. In this course, we will focus only on one of these, outcome uncertainty. We will explore why consumers generally dislike uncertainty, and then how measures of uncertainty can be incorporated into our analysis.
A. Probability and Expected Value. Two terms are critically importantly when discussing outcome uncertainty: Probability and expected value.
Probability is the relative frequency with which an event will occur. For example, the probability of a head in a toss of a fair coin is .5. The probability of a 2 from a roll of a 6-sided die is 1/6.
In general for any “lottery,” with possible outcomes 1 to n, we may write
i = 1. That is, the sum of probabilities for all possible outcomes must equal one.
Expected Value: Is a measure of the average value of a lottery. The expected value of a lottery is simply the probability weighted value of each outcome, that is,
E(X)=1X1+1X1+ … +nXn
Example. Suppose Jones and Smith flip a fair coin. If the coin comes up heads, Jones will pay Smith $1. If the coin is a tail, Jones earns $1 from Smith. Jones expected value is
E(X)=(1/2)(-$1)+(1/2)($1)=0
That is, on average Jones would expect to breakeven in repeated play of such a game. If Jones wins $10 in the case of a tails, then expected value becomes
E(X)=(1/2)(-$1)+(1/2)($10)=$4.50
The former of these is referred to as an actuarially fair game. The notion of risk aversion is the observation that individuals often refuse to play actuarially fair games.
B. Fair Games and the Expected Utility Hypothesis. A convincing illustration of people’s reluctance to play ‘fair’ games is the St. Petersburg Paradox. Suppose a game is flipped until a head appears. When a head appears, the game ends, with a payoff of 2n. (where n is the number of tosses until the first head). Thus, for example, if the game consisted of 3 tails before a heard, (4 flips), the player would earn 24 = $16.
How much would you pay to play this game?
Consider its expected value. The probability of a head on the first toss is ½. On a second toss, the probability of a head is (½)2. (That is, a tail on the first toss, and head on the second.) Reasoning similarly, the probability of a head on the nth toss is (½)n (e.g, n-1 consecutive tails, followed by a head). Thus, expected value becomes
E(X)=1X1+1X1+ … +nXn
=(½)(2)1+ (¼)(2)2 + ….+ (1/2n)(2)n
=1+ 1+….+ 1
=
Of course, few people would take me up on an offer to play the game for, say, $1 billion (even though the game is worth considerably more.)
Bernoulli (the first person to analyze this game) argued that individuals cared not for the expected value of a game, but for the expected well being (or utility) that the game generated. If the utility of a dollar amount increases less rapidly than the dollar amount itself, the game will have a finite value.
Bernoulli used a utility function U(Xi) = ln(Xi). (The natural log is a reasonable, but arbitrary choice, since it simply damps out linear equations). In this case the expected utility of the game is
E(U(X))=1ln(X1)+2 ln(X2)+ … +n ln(Xn)
=(½)1ln(2)+ (¼)2ln(2) + ….+ (1/2n)nln(2)
=(i/2i) ln(2)
=2 ln(2)
=1.39
(Given this utility function, this implies the gamble is worth a certain wealth of about X = $4, since ln(4) =1.39.
C. The Von Neumann-Morgenstern Theorem. Jon von Neumann and Oscar Morgenstern pioneered the mathematical modeling of uncertainty. They provide an axiomatic (formal) basis for constructing a utility index (something a bit less ambitious than a utility function), and then develop a theorem for assigning values. The idea is as follows.
Consider a lottery consisting of prizes X1, X2 … Xn. Without loss of generality, rank these prizes in order of ascending preference, from least preferred to most preferred. Now, assign an arbitrary value to the worst and to the best outcomes (Since we can’t see utilities, our choice is arbitrary. 0 and 1 work fine.) Thus,
U(X1) =0and U(Xn) =1.
The point of the von Neumann-Morgenstern Theorem is to assign utility index values to the other outcomes. Consider an outcome Xi. Their approach was to elicit from the player the probability i that the player would trade outcome Xi for a gamble between the
U(Xi) =(1-i)U(X1) + iU(Xn)
Our choice of scale implies that this can be reduced to
U(Xi) =(1-i)0 + i1
=i
Thus the normalization of initial and terminal values at 0 and 1 is convenient, as it allows us to construct a utility index where the utility of a prize equals the minimum probability of winning that a player would accept for a gamble between the most-desired and least-desired prizes.
This expression of a utility index in terms of probabilities is extremely useful, because it allows us to assign preferences over any gambles involving combinations of element pairs in ( X1… Xn). To see this, consider an individual’s preferences between two gambles.
Gamble 1 yields X2 with probability q and X3 with probability (1-q). Gamble 2 yields X5 with probability t and X6 with probability (1-t).
Using the expected von Neumann-Morganstern utility index, we can write
Expected utility (1)=qU(X2) +(1-q)U(X3)
Expected utility (2)=tU(X5) +(1-t)U(X6)
The first gamble will be preferred if
qU(X2) +(1- q)U(X3) > tU(X5) +(1- t)U(X6)
Recall, now that each i may be expressed in terms of U(X1) =0 and U(Xn)=1
Specifically,
U(Xi) =iU(Xn) =i(1) = i
So the two sides of the above inequality are both simply probability-weighed utilities of winning most valued prize Xn Thus, we may substitute index values i for U(Xi) to yields
Expected utility (1)=q2 +(1-q)3
Expected utility (2)=t5 +(1-t)6
and rewrite the expected utility equation as
q2 +(1-q)3t5 +(1-t)6
An individual will prefer gamble 1 to gamble 2 iff gamble 1 offers a higher expected index value. This is the von Neumann- Morganstern expected utility theorem
Expected Utility Maximization: If individuals obey the von Neumann- Morganstern axioms of behavior in uncertain situations, they will act as if they choose the option that maximizes the expected value of their von Neumann- Morganstern utility index.
D. Risk Aversion. Now let’s use the von Neumann- Morganstern utility index to make more concrete the notion of risk aversion. In general risk is a measure of the variability of an outcome. Casually speaking, the higher the variability of payoffs, the more risky the outcome. For example, we expect that most people would be much more reluctant, to trade evening the following coin-toss gambles
Gamble 1: $1,000 earnings for an H and $1,000 losses for a T
Gamble 2: $1 earnings for an H, $1 losses for a T.
Even though, in either case, the expected value if the gamble is 0, the variability of outcomes is very considerably higher for gamble 1, and as a consequence, we might expect most people to pay less for the gamble.
The figure on the next page illustrates the relationship between the variability of outcomes and utility In the figure an individual chooses between a certain return W*, yielding a U(W*), and two gambles.
Gamble 1:W*+h with probability .5 and W*-h with probability .5, or
Gamble 2:W*+2h with probability .5 and W*-2h with probability.5.
As the figure illustrates, a concave utility function (that is a diminishing marginal utility of income) suffices to generate an inverse relationship between the utility of a gamble and the variability of possible outcomes. The utility of a 50/50 chance of having earnings of W*+h or W*-h yields an expected utility Uh(W*) below the certain outcome U(W*). Similarly for U2h(W*), this despite the fact that all outcomes have and identical expected value (W*).
In other wordsU(W*) > Uh(W*) >U2h(W*).
Thus, the assumption of risk aversion is just an alternative way to say that individuals have a diminishing marginal utility of income: The pain of losing dollars from a negative outcome exceeds the joy of gaining dollars from a positive outcome.
Risk aversion and insurance: Observe that a risk averse individual is indifferent between an income of Woand gamble 1 (involving a potential gain or loss of h from W*). Thus, a person would be willing to pay slightly less than W* - Wo to avoid having to play the gamble. This explains why many people purchase insurance.
Definition: Risk Aversion: An individual who always refuses fair bets is said to be risk averse. If individuals exhibit a diminishing marginal utility of wealth, they will be risk averse. As a consequence, they will be willing to pay something to avoid taking fair bets
Example 8.2. Willingness to Pay for Insurance In this example we explore the connection between risk aversion an insurance. Consider a person with a current wealth of $100,000 who faces a 25% chance of losing $20,000 (say via automobile theft). Suppose that the utility index is U(W) = ln(W).
Without insurance, expected utility for this person is
E(U)=.75 U(100,000) +.25 U(80,000)
=.75 ln(100,000)+.25 ln(80,000)
=11.457
How much might an individual pay to avoid this risk of loss?
Suppose the premium for an insurance policy is $4000. Then the individual would net $96,000, yielding utility
ln(96,000)=11.4721
What might a fair insurance premium be? To answer this, find the certain income that yields a utility of 11.457
11.457=ln(94,570)
Thus, the fair insurance premium (assuming no processing costs is 100,000 – 94,570 = $5,230.
E. Measuring Risk Aversion. In studying economic choices in risky situations, it is sometimes convenient to have a quantitative measure of risk aversion. The most standard measure r(W) is
r(W)=- U”(W)/U’(W)
Given a U”<0 (as is consistent with diminishing marginal utility of wealth), r(W)>0. This measure is invariant to linear transformations of the utility function, and thus, it doesn’t depend on the values chosen for the minimum and maximum wealth states. (However, r(W) is sensitive to the choice of the utility function itself.)
Risk Aversion and Insurance Premiums. One useful feature of r(W) is that it is proportional to the amount an individual will pay for insurance against a fair bet.
Consider a fair bet, with winnings h. Since the bet is fair E(h) = 0. Now define p as an insurance premium that would make an individual indifferent between the fair bet h, and certain income. That is, pick p so that
E[U(W+h)]=U(W-p)
Where W equals current wealth. To solve for p, use a Taylor’s series expansion
(A Taylor’s series expansion provides a way of approximating any differentiable function around a point. If f(x) has derivates of all orders,
f(x+h)=f(x) + hf’(x) + (h2/2)f’’(x) + higher order terms.)
Here, since the right side involves a fixed amount, a linear expansion will do.
U(W-p)=U(W)-pU’(W)+ higher order terms
On the left side, a quadratic expansion is necessary
E[U(W+h)]=E[U(W)+ hU’(W) + (h2/2)U”(W) + higher order terms]
=U(W)+ E(h)U’(W)+E(h2/2)U’’(W) + higher order terms
Ignoring higher order terms, set the two sides equal (and remember that E(h) = 0)
U(W)-pU’(W)=U(W)+E(h2/2)U’’(W)
thusp=kU”(W)/U’(W)
where k = E(h2/2).
Thus, an individual’s willingness to payan insurance premium is proportional to r(W). In many naturally occurring applications, analysts attempt to measure r(W) from insurance premia. In this way, we can learn a considerable amount about risk attitudes from insurance data.
Risk Aversion and Wealth Another important question regards the effects of wealth on risk aversion. Intuitively, one might suspect that increasing wealth would tend to make individuals less risk averse, since a given loss affects wellbeing less seriously as wealth increases. This intuition, however, is not necessarily correct. With diminishing marginal utility, it is also the case that income increases also increase utility less than at lower wealth levels. Thus, the effect of wealth on risk aversion depends on the curvature of the utility function.
To illustrate, consider three utility functions.
1. Quadratic utility. Suppose that utility is quadratic in wealth.
U(W)=a+ bW+cW2
b>0 and c<0.
Then
r(W) = -U”(W)/U’(W)=-2c/[b+cW]
Here, r(W) increases with W.
2. Logarithmic Utility. On the other hand, suppose thatutility is logarithmic in Wealth
U(W)=ln(W)
Then
r(W)=(1/W2)/(1/W)
=1/W
which moves inversely with wealth.
3. Exponential Utility,
U(W)= - e-AW=- exp(-AW)
(A>0). Here
r(W)=-U”(W)/U’(W)=-A2exp(-AW)
-Aexp(-AW)
=A
Exponential utility illustrates the property of constant absolute risk aversion. This can be a useful property, at least for purposes of illustration.
Example: Constant Absolute Risk Aversion. Consider an individual with initial wealth W0 facing a 50/50 gamble of losing $1,000. Suppose utility is given by
U(W)= - e-AW
How much (F) would the individual pay to avoid the risk of the gamble?
-exp[-A(Wo - F)])=-.5exp[-A(Wo + 1000)] - .5exp[-A(Wo-1000]
Factoring out –exp(-AWo) this reduces to
exp(AF)=.5exp(-1000A) +.5exp(1000A)
This is a relationship between A and F. For example, if A = .0001, F = 49.9 – a person would pay roughly $50 to avoid the risk. If A = .0003, this more risk averse person would pay F = 147.8 to avoid the gamble. Because the results are not unreasonable, they are sometimes used in empirical investigations.
Relative Risk Aversion It seems unlikely that willingness to avoid a gamble is independent of wealth. A more appealing assumption is that willingness to pay is inversely proportional to wealth, and that the expression
rr(W)=Wr(W)=-W[U”(W)/U’(W)]
might be approximately constant. The rr(W) term is labeled relative risk aversion. The power utility function
U(W)=WR/R(R<1, 0)
and
U(W)=ln W (for R = 0)
Exhibits diminishing absolute risk aversion:
r(W)=- U”(W)/U’(W)=(R-1)WR-2/WR-1=- (R-1)/W
but constant relative risk aversion
rr(W)-Wr(W)=-(R-1)=1-R
Empirical evidence is generally consistent with R {-3, -1}.
Example 8.4: Constant Relative Risk Aversion. An individual with constant relative risk aversion will be concerned about proportional gains or loss of wealth. Thus, we can ask what fraction of initial wealth (F) a person would be willing to give up to avoid a fair gamble, of say 10% of initial wealth. First set R = 0 and use logarithmic wealth
ln[(1-F)Wo]=.5ln(1.1 Wo) + .5 ln(.9Wo)
Factoring out ln Wo yields
ln(1-F)=.5[ln(1.1)+ ln(.9)]=ln(.99).5.
Thus(1-F)=.99.5=.995
So F = .005.
Hence a person will sacrifice up to half of 1% of wealth to avoid a 10% gamble. With R=-2, a similar gamble yields F = .015. Thus, this more risk averse person would be willing to give up 1.5% of wealth
1