1 Uncertainty and Insurance

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1 Uncertainty and Insurance Reading: Some fundamental basics are in Varians intermediate micro textbook (Chapter 2). A good (advanced, but still rather accessible) treatment is in Kreps A Course in Microeconomic Theory.. Expected Utility: Setup Let;. X be a set of prizes. While not necessary, it is convenient to assume that X is nite. 2. P the set of probability distributions (lotteries, gambles) De nition Let p and q be two probability distributions in P and a number in [0; ] : Then p + ( ) q is called a compound lottery. Example Let X = fx; y; z; wg and take p = 4 ; 4 ; 2 ; 0 and q = 0; 2 3 ; 0; 3 ; and let = 3 : Then p + ( ) q = = 2 ; ; 6 ; ; 9 36 ; 6 36 ; 8 : 36 One way to think about this is that one creates a new lottery by rst ipping a coin that selects lottery p with probability = 3 and lottery q with probability = 2 3 simply runs lottery p or q depending on the outcome of the initial coin ip. p + ( and then ) q is then the probability distribution over prizes. A maintained assumption in expected utility theory is that only the probability distribution over nal prizes matter. This is often refuted in experiments.

2 .2 Axioms We assume that the decision maker has a transitive and complete preference ordering over all probability distributions P (de ned over X). We write for the weak preference ordering and recall that; De nition 2 is transitive if p r for every p; q; r 2 P such that p q and q r De nition 3 is complete if p q or q p holds for every p; q 2 P All standard rational theoreies of choice insists of transitivity and completeness, so these properties have nothing to do with the probabilistic structure in the setup. However, the focus on lotteries gives us some additional structure that makes some additional assumptions reasonable.we will refer to these additional assumptions as axioms (which means a proposition that is not susceptible of proof or disproof; its truth is assumed to be self-evident). The philosophy here is that if the axioms are obviously true, then the consequences of these (which may be derived by logic) follow. For this reason, any decision theory that is based on axioms is viewed as having a foundation, whereas theorists often view non-axiomatic theories of choice with suspicion, as it is much less evident what the underlying assumptions on choces are with such a theory. Axiom Suppose that p and q are two probability distributions such that p q [i.e., p q and not q p meaning p is strictly better than q]. Also suppose is a number in (0; ) [means that 0 < a < : whenever an interval is written as (a; b) that means that the boundaries a and b are not part of the set]. Also suppose that r is some probability distribution. Then, p + ( ) r q + ( ) r: Called the substitution axiom. The idea is that if you like p better than q and are o ered a lottery where with some probability you get p and with the complementary you get r; then you should like that better than if you replaced the p-lottery with the q lottery, but kept the probability of the r-lottery the same. 2

3 Axiom 2 Suppose that p; q; and r are lotteries such that p q r: Then there exists ; 2 (0; ) such that p + ( ) r q p + ( ) r probability This is a continuity axiom (which is called the Archimedean axiom for reasons that are somewhat obscure). It says that no matter how bad r is, as long as p is strictly better than q there is some (presumably small enough when r is really bad) probability on r that makes you willing to go with the compound lottery that involves the better lottery p with the complementary probability. This is the interpretation of the rst inequality, the second is symmetric and says that no matter how much better p is than q; a su ciently large probability of the worst outcome (which could be only slightly worse than q) would make q better than the compound lottery. For example, let p be 000 dollars for sure, q be 000 dollars with probability and 0 2 dollars with probability 2 and r be certain death. Take as granted that the decision maker prefers the 000 dollars for sure to the 50% chance to win 000 dollars, and that certain death is the worst outcome. According to the axiom there then exists some probability > 0 such that 000 dollars with probability and death with probability is better than a fty/ fty coin ip over 000 dollars or none. Unreasonable? Suppose I framed it as either i) ip a fair coin over 000 dollars or 0 dollars, or; ii) drive downtown to pick up a 000 dollar check...driving downtown increases the probability of death ever so slightly, so if you are willing to take the second option you are behaving in accordance with the axiom..3 The Theorem Theorem A is transitive and complete preference ordering [de ned over the set of probability distributions P:::so we are also making the assumption that that compound lotteries can be reduced to lotteries over prizes ] satis es axioms and 2 id and only if there exists 3

4 a real valued function u de ned over the set of prizes X such that. p q if and only if X u (x) p (x) > X u (x) q (x) : x2x x2x Moreover if both u and v represents the same preference order then there are constants a; b with a > 0 such that v (x) = au (x) + b holds for every x 2 X (uniqueness of the function up to a ne transformations). What is remarkable about this representation is that it means that the utility function is linear in probabilities, which is analytically very convenient and also provides a testable hypothesis (which of course is routinely rejected on humans)..4 The Proof Denote by x the lottery that gives outcome x for sure. The preference relation is de ned over these degenrate lotteries as well, and we assume (only for simplicity) that X is nite. This implies that we may order the elements is X so that the rst element is weakly preferred to all elements that follow, the second is weakly preferred to all elements that follow etc (a formal proof uses completeness and transitivity and proceeds by contradiction). That is, we can write X as a list X = fx ; x 2 ; :::; x i ; :::; x n g : where x x 2 :::: x i x i+ :::: x n Notice that if x x n and x n x (which is usually written as x n x ); then there is indi erence between all prizes and the representation is trivial. Hence, suppose that x x n in what follows. Writing i (instead of xi ) for the lottery that gives x i for sure we can now show that; Lemma Let 2 [0; ] and 2 [0; ] : Then + ( ) n + ( ) n if and only if > : 4

5 Proof.. Substitution axiom) + ( ) = + ( ) n () [where you should think of p as ; q as n and r as ] 2. Let p = ; q = + ( ) n : By the () we have that + ( ) n ; so we may apply the substitution axiom again to conclude that " + ( ") [ + ( ) n ] + ( ) n (2) = " [ + ( ) n ] + ( ") [ + ( ) n ] for any " 2 (0; ) [this time we take p to be ; and q = r to be + ( ) n ]. Rearrange the left hand side of (2) and write the relation as [" + ( ") ] {z } + ( ") ( ) {z } n + ( ) n ; which holds for every " > 0: Hence + ( ) n + ( ) n for every > (since for every < we can nd some " 2 [0; ] such that = " + ( ") ): 3. OTHER DIRECTION: EXERCISE! Lemma 2 For every p in P there exists a number 2 [0; ] such that p + ( ) n Proof. If p or p n the result is trivial (pick = or = 0), so assume p n : De ne = inf f 2 [0; ] j + ( ) n pg 5

6 NOTE: Let A be a subset of the real line. Then inf A means the largest number (on the real line) that is smaller than everything in A: Unlike the minimum, the in nium always exists. Want to show that p + ( ) n : Suppose not. Then + ( ) n p: Moreover (continuity axiom applied on ; p and n ), we can always nd some 2 (0; ) such that p + ( ) n : That is + ( ) n p + ( ) n By continuity axiom (now applied on + ( ) n ; p; and + ( ) n ) we can nd " such that ( ") [ + ( ) n ] + " [ + ( ) n ] p [( ") + "] + [ ( ") + "] p But since > (this we proved in the previous lemma) we have that ( ") + " < ; which violates the de nition of (since the 0 = ( ") + " is in the interval f 2 [0; ] j + ( ) n pg by de nition of ; for every in the set and we therefore conclude that 0 = ( ") + " < ; which is an impossibility) Lemma 3 If p q and r is any other lottery and any number between 0 and, then p + ( ) r q + ( ) r This looks just like the substitution axiom, but with indi erence instead of strict preference. Oddly enough it is not that easy to show it (sometimes this is taken as an axiom and it isn t any less primitive than the actual axiom) 6

7 Proof. Suppose that the Lemma is false. By choice of names for p and q we may without loss of generality assume that there exists some 2 (0; ) and r such that p + ( ) r q + ( ) r By an application of the previous lemma there is some 2 [0; ] and some! 2 [0; ] such that p + ( ) r + ( ) n : q + ( ) r! + (!) n Hence + ( ) n p + ( ) r q + ( ) r! + (!) n n where + ( ) n if and only if = and where + ( ) n n if and only if = 0: If n we obviously have a contradiction (transitivity fails), so assume that n : There are two cases to consider;. p q 2. p q n Either or 2 must be true (both may obviously be true, but that is ne) I assume that p q: Then, for every 2 (0; ) we have that s = + ( ) q q p: Using the substitution axiom this imples that s + ( ) r p + ( ) r q + ( ) r; where the second strictly preferred is by assumption (the one we are trying to disprove). By the continuity axiom there exists such that [s + ( ) r] + ( ) [q + ( ) r] p + ( ) r: 7

8 But [s + ( ) r] + ( ) [q + ( ) r] = s + ( ) r + ( ) [q + ( ) r] = s + ( ) q + ( ) r = [ + ( ) q] + ( ) q + ( ) r 2 3 = ( ) + ( 7 ) 5 q + ( ) r {z } 2 = =[ ] 3 = 4 + ( {z ) q5 + ( } ) r = t + ( ) r =t That is, since + ( ) = we can view the bracketed expression as a randomization over and q: To sum up we have that t + ( ) r p + ( ) r (3) But, p; which implies that t = + ( ) q q p; which by the substitution axiom implies that t + ( ) r p + ( ) r: (4) Obviously, (3) and (4) is a contradiction. The rest is easy. To complete the proof, de ne u (x i ) as the number i u (x i ) + ( u (x i )) n The lemma we just proved shows that such a number exists and it is unique by the rst lemma. Then show that Lemma 4 For u de ned above, any lottery p is indi erent to the one that gives x with probability P n i= u (x i) p i and x n with probability P n i= u (x i) p i : 8

9 SIMPLE. DO IT. Hence, lottery;. p is exactly as good as a lottery that gives x with probability P n i= u (x i) p i and x n with the complementary probability 2. q is exactly as good as a lottery that gives x with probability P n i= u (x i) q i and x n with the complementary probability 3. The higher the probability on x the better)representation in theorem. 2 Utility over Money Bets Suppose that X is the real line or an interval on the real line. Then, if we make the assumption that the consumer gets happier the more money he or she gets we have that Proposition Suppose that x y for every x > y (where x and y denotes the degenerate lotteries that give x and y for sure. Then, the utility function u is strictly increasing. R x However, the more interesting issue is attitudes towards risk. Let Ep = P x2x xp (x) (or xp (x) dx with p being a probability density function). A natural de nition is then; De nition 4 The decision maker is;. risk averse if Ep p for every lottery p: 2. risk neutral if Ep p for every lottery p: 3. riskloving if Ep p for every lottery p Also recall; De nition 5 A real valued function f is called;. concave if f (x + ( ) y) f (x) + ( ) f (y) for all x; y and 2 [0; ] 9

10 2. convex if f (x + ( ) y) f (x) + ( ) f (y) for all x; y and 2 [0; ] We then have that; Proposition 2 The decision maker is. risk averse if and only if u () is concave. 2. risk loving if and only if u () is convex. 3. risk neutral if both concave and convex=linear (a ne). 6 u(x 2 ) u( 2 x + 2 x 2) 2 u(x ) + 2 u(x 2) u(x ) x 2 c + 2 x 2 x 2 - Figure : Utility of Expected Value vs Expected Utility of Gamble for a Risk Averse Agent Economists usually assume agents are risk averse (otherwise there wouldn t be any rationale for a market for insurance) or risk neutral. Firms are often modelled as risk neutral (idea-lot s of independent risks) rm can pool risks and thereby eliminate them). Many models also assume that consumers are risk neutral, but this is not because we think it is 0

11 a particularily descriptive assumption, rather, that the models focus on other things and therefore often want to get rid of insurance aspects of problems. 3 Insurance Example Loss P PPPPPPPPPPPPPPq No Loss Figure 2: Possible Outcomes For simplicity we will think of a situation with two possible outcomes. With probability ; the consumer su ers a loss : (bad state) With probability ; the consumer su ers no loss : (good state) m b for the income in the bad state (probability ) m g for the income in the good state (probability ) ) \Loss = m g m b consumer can purchase insurance, so the consumption need not be m b or m g let x b be consumption in the bad state let x g be consumption in the good state Expected utility function is U (x b ; x g ) = u (x b ) + ( ) u(x g );

12 Notice, that if you want to tie this up with intermediate micro theory, we may draw indifference curves in the usual way (since there are two goods ). The slope of the indi erence curves would be (assuming di erentability) MRS (x g ; x b ) = CASE ; Risk neutrality; u (x) = ax + b ;x b ;x g) = ( ) u0 (x g ) u 0 (x b b u(c) 6 c b 6 ) Z ZZ Z Z Z Z Z Z Z Z Z - Z Z ZZ Z Z Z Z - c c g Figure 3: Indi erence Curves for u(c) = ac + b In this case, the slope of the indi erence curve is ; so we have the indi erence curves as in Figure 3. The slope is constant and equal to the negative of the ratio of the respective probabilities. Preferences of this sort are called risk neutral preferences. To see why note that x b + ( ) x b is the expected value of consumption and that (ax b + b) + ( ) (ax g + b) = a (x b + ( ) x b ) + b; so that the consumer is indi erent between all bundles (or contingent plans) that has the same expected consumption. 2

13 CASE 2: Risk Aversion; u concave u(c) 6 c b 6 2 p 2 u u u pu ) @ u - 2 c c g Figure 4: Indi erence Curves for u(c) = p c Begin with a concrete example, say that u (x) = p xin this case, the slope of the indifference curve is To actually depict them, let = 2 ( ) p x g p x b and consider the curve though (; ) : p + 2p = ; gives k = for the curve going through (; ) p 4 + 2p 0 = ) (4; 0) and (0; 4) on curve 2 2 Thus, the indi erence curves look like in Figure 4. Hence, the example generates nice convex preferences in the sense used in basic micro theory. This is true for all concave functions u (x) that have a slope which is decreasing in x (that is concave functions u) 3. A ordable Consumption Plans in the Presence of Insurance Now suppose that consumer can buy insurance: Let z be quantity insurance (units paid to consumer in the event of a loss) Let p be the price per unit of insurance 3

14 ) x b = m b + z pz is the consumption in bad state x g = m g pz is the consumption in good state Now, we can eliminate z from this to write budget constraint as ( p) x g + px b = ( p) m g + pm b c b 6 slope p p m b u m g Z- c g Figure 5: The Budget Set 3.2 The Choice Problem Graphically, given convex preferences, the solution will be characterized in the usual way as a nice tangency between (the highest possible) indi erence curve and the budget set. Note that, at the diagonal line (the certainty line ) the slope of the indi erence curve is 4

15 c b 6 T TT c b m b T TT slope T TT Zu ZZ T Z T ZZ u TT p p slope : c g T T T T m g Z- c g Figure 6: Graphical Solution with p > to be compared with the slope of the budget line so if p > ; then p < p p p ; ; meaning that at the diagonal the indi erence curves must be steeper than the budget line meaning that the tangency must occur somewhere below the diagonal as in Figure 6. Characterizing the solution using calculus it is convenient to keep z as the choice variable (although you may use c g or c b if you want). FOC is 0 max u (m b + z pz) + ( ) u (m g pz) z u b + z pza B C ( p) + ( ) u {z g pza ( p) = 0 {z } c b c g 5 0 p p m = ( )u0 c g u 0 (c b ) ;

16 which we sort of knew already from the picture since this just says that the slope of the indi erence curve must equal the slope of the budget line. However note that:. If p = ; then the consumer pays pz = z the insurance company gives the consumer z with probability the insurance company gives the consumer 0 with probability )Expected pro t for insurance company pz z = 0 This is called a fair premium since the insurance company breaks even on average. Note that the solution in this case is c g = c b ; since u0 (c) is a decreasing function. The conclusion is clear: if a risk averse consumer can buy insurance at a fair price the consumer will fully insure. 2. p > )partial insurance (or no insurance). 3. p < )overinsurance. 3.3 A Simple Portfolio Problem The same ideas as the ones for modelling an insurance problem can be applied to the choice between a risky asset and a safe bond:. Suppose that the safe bond earns interest r s (there is no chance of the issuer defaulting) 2. Suppose that the risky asset earns a return r g in the good state (probability ) and r b in the bad state (probability ), where r g > r b to make sense of the labeling of the states. We will abstract away from the issue of how much will be invested in total and simply assume that the investor has wealth w that he or she will invest. 6

17 If the agent can choose how much to eat in the current period and how much to invest the current analysis will still apply for whatever wealth w the agent decides not to eat. Hence it makes lots of sense to ignore the intertemporal issues. The most straightforward way to set the problem up is to let 0 x w be the investment in the risky asset (so that w x is invested in the safe bond). Consumption is then c g = (w x) ( + r s ) + x( + r g ) = w ( + r s ) + x(r g r s ) c b = (w x) ( + r s ) + x( + r b ) = w ( + r s ) + x(r b r s ): For the purposes of drawing the budget set in c g c b space we may observe that the budget constraint reads r b c g + r g c b = w ( + r s ) + w ( + r s ) r g r s : r s r b r s r b You should try to plot this, but before doing that it is good to pass a while to think:. If nothing is invested in the risky asset, then c b = c g = w ( + r s ) 2. If everything is invested in the risky asset, then c b = w ( + r b ) and c g = w( + r g ): 3. ) r s < r g needed in order to not have a corner solution where everything is put in the safe bond. 4. ) r b < r s needed in order to not have a corner solution where everything is put in the risky asset. 5. That is the interesting investment problem is when r b < r s < r s ; which also implies r b that rg r s r b is a positive number. 6. It is an open question whether one should assume that it is possible for c g > w( + r g ) or c b < w ( + r b ) : If you allow this, you are actually assuming that the investor can issue a safe bond at rate r s : It is a good exercise to draw the relevant budget set both under the assumption that borrowing is possible and when it is not. 7

18 Like in the insurance problem, it is more convenient to keep the variable x when setting up the optimization problem. Assuming that the investor can not issue a bond the problem is max u (w ( + r s ) + x(r b r s )) + ( ) u (w ( + r s ) + x(r g r s )) x The rst order condition is u 0 (w ( + r s ) + x(r b r s )) (r b r s ) + ( ) u 0 (w ( + r s ) + x(r g r s )) (r g r s ) = 0 We will not really solve this problem although the solution will look like the usual tangency condition. Qualitatively, an interesting question is simply to ask when will the investor invest at all in the risky asset. To answer this question we only need to check when to expected utility function is increasing at x = 0; that this when is 0 < u 0 (w ( + r s )) (r b r s ) + ( ) u 0 (w ( + r s )) (r g r s ) = u 0 (w ( + r s )) (r b + ( ) r g r s ): We conclude that: If r b + ( ) r g > r s then some part of the wealth should be invested in the risky asset. In words, this means that if the expected return for the risky asset exceeds the expected return on the safe bond, then it cannot be optimal not to take any risk at all. The reasoning is that making the calculation for the rst dollar it is almost as if the agent is risk neutral, even if she actually is risk averse. The idea for this is that the variability in income that one creates with the rst dollar is so small that it can be ignored, making the agent locally risk neutral. 8