Econ 604 Advanced Microeconomics

Econ 604 Advanced Microeconomics

Davis Spring 2004, March 30

Lecture 10  

Reading. Chapter 8 (pp. 198-210

Problems: 8.3, 8.5, 8.7

Next time. Chapter 8, 211-224, Chapter 11, pagers 268 -278

REVIEW 

VIII. Chapter 7 Market Demand and Elasticitity

      A. Market Demand Curves

            1. The two consumer case

            2. The n consumer case

      B. Elasticity

            1. Motivation and a general definition

            2. Price Elasticity of Demand

            3. Income Elasticity of Demand

      4. Cross Price Elasticity of Demand

      C. Relationships Between Elasticities

            1. Sum of income Elasticities for all Goods

            2. Slutsky Equation in Elasticities

            3. Homogeneity

      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

      F. The State-Preference Approach to Choice Under Uncertainty 

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 uncertainty as to choice 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+ … +_nX _n 

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 break even 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. People often refuse to play actuarially fair games. A convincing example is the St. Petersburg Paradox.  

Suppose a game is flip until a head appears.  When a head appears, the game ends, with a payoff of 2ⁿ. 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 an head)

      Thus, expected value becomes

      E(X) = ₁X₁+₁X₁+ … +_nX _n 

            = (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(X_i) = ln(X_i).  (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(X_n)

                  = (½)1ln(2)`+ (¼)2ln(2) + ….+  (1/2ⁿ)nln(2) 

                  = (1/2ⁱ)

                  = 2 ln(2)

      = 1.39 


      (Given this utility function, this implies X = $4. 


      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 … X_n.  Without loss of generality, rank these prizes in order of ascending preferences, 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) = 0 and  U(X_n) = 1.

The point of the von Neumann-Morgenstern Theorem is to assign utility index values to the other outcomes.  Consider an outcome X_i. Their approach was to elicit from the player the probability _i that the player would trade outcome X_i for a gamble between the  


      U(X_i)  = (1-_i)U(X₁) + _iU(X_n) 


      Our choice of scale implies  that this can be reduced to  


      U(X_i)  = (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- and least- desired prizes.

      This expression of a utility index in terms of probabilities is extremely useful, as it allows us to assign preferences over any gambles involving combinations of in X₁ to X_n.  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)n the case of a win. 

      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) 


      Substituting index values _i for U(X_i) yields  


      Expected utility (1) = q₂ +(1- q) ₃

       

      Expected utility (2) = t₅ +(1- t) ₆ 


      Now we wish to show that the individual will prefer gamble 1 to gamble 2 iff 


            q₂ +(1- q) ₃ > t₅ +(1- t) ₆ 


      Recall, however, that each _i may be expressed in terms of U(X₁) =0 and U(X_n)=1

      Specifically, 


      U(X_i)  = _iU(X_n). Thus, the two sides of the above inequality are both simply probability weighed expressions expected utilities of winning most valued prize X_n .  Thus, an individual will prefer gamble 1 to gamble 2 iff gamble 1 offers a higher expected utility.  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 a H and $1,000 losses for a T

Gamble 2: $1 earnings for a H, $1 losses for a T. 


Even though, in either case, the expected value if the gamble is 0. In the figure at the top of the next page, for example, 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. 

Now

      This can be easily represented graphically by assuming a concave utility function (that is a diminishing marginal utility of income). As seen in the figure to the left, the utility of a 50/50 chance of having earnings of W*+h or W*-h

yields an expected utility U^h(W*) below the certain outcome U(W*). Similarly for U^2h(W*), this despite the fact that all outcomes have and identical expected value (W*).

      In other words

U(W*) > U^h(W*) > U^h(W*).  Intuitively, 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 W^o and gamble 1 (involving a potential earnings or loss of h from W*).  Thus, a person would be willing to pay slightly less than W* - W^o 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  Consider 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 a premium of $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 utlity 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 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) 


thus p = kU”(W)/U’(W) 


where k = E(h²/2).

      Thus, an individual’s willingness to pay an 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 an 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 well being 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 risk aversion 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 Utility is logarithmic in Wealth 


      U(W) = ln(W)

      Then

      r(W) = (1/W²)/(1/W)  

            = 1/W

      which does move 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 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. 
 


       

      of in addition to reductions in wealth as income increase 


to Perhaps the most useful m  
 


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) = W^R/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)W^R-2/W^R-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)W_o] = .5ln(1.1 W_o) + .5 ln(.9W_o) 


Factoring out ln W_o 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 


      F. The State-Preference Approach to Choice Under Uncertainty

      Next time.