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Consequently, one is lead to the absurd result that the willingness to gamble ... such as F and G denote distribution, and f and g the associated density .... So how can we dare to use a discontinuous function in the proof of one of the ..... x F(t) ?G(t)]dt. 0 for all z 2 x; c), where c denotes the crossing point. We now show that this.
Stochastic Dominance: Theory and Applications Elmar Wolfstetter1 March 1996 Humboldt{Universitat zu Berlin Institut f. Wirtschaftstheorie I Wirtschaftswissenschaftliche Fakultat Spandauerstr. 1 10178 Berlin Germany e{mail [email protected]{berlin.de

Research support by the Deutsche Forschungsgemeinschaft (DFG), Sonderforschungsbereich 373, \Quanti kation und Simulation O konomischer Prozesse", Humboldt{Universitat Berlin, is gratefully acknowledged. Comments by Murray Brown, Andras Dorr, Uwe Dulleck, Roman Inderst, Peter Kuhbier, Michael Landsberger, and Aloys Prinz are gratefully acknowledged. 1

Abstract This is a chapter of a book manuscript entitled Topics in Microeconomics . The chapter starts with basic stochastic dominance theorems. These results are useful in a large range of economic applications. Several applications are developed: the theory of labor supply under uncertainty, the theory of the rm under price uncertainty, auction theory, the theory of optimal portfolio selection, oligopoly theory, and the analysis of consistent rankings of income distributions.

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\Only risk is sure."

William Shakespeare (The Merchant of Venice)

1 Introduction Decision problems under uncertainty concern the choice between random payo s. For a rational agent with a known utility function, one random variable is preferred if it maximizes expected utility. This is easy enough in theory. However, in practice it is often dicult to nd an agent's utility function. Therefore it would be most useful to know whether a random variable is the dominant choice because it is preferred by all agents whose utility functions share certain general characteristics. In this chapter we introduce two such rankings, known as rst and second order stochastic dominance. These notions apply to pairs of random variables. They indicate when one random variable ranks higher than the other by specifying a condition which the di erence between their distribution functions must satisfy. Essentially, rst order stochastic dominance is a \stochastically larger", and second order stochastic dominance a \stochastically less volatile" or \less risky" relationship. While the \larger" random variable is preferred by all agents who prefer higher realizations, the \less volatile" random variable is preferred by all agents who also dislike risk. In this sense, stochastic dominance theory provides unanimity rules, provided that utility functions have certain common properties. Stochastic dominance theory has a bearing on the old issue whether one can judge a random variable as more risky than another, regardless of who is the judge, provided that utility functions belong to a class with certain common properties. For many years, economists thought that mean and variance are a satisfactory measure of comparative risk. But this was not quite right. Indeed, mean and variance can only serve this purpose if agents' utility functions are quadratic or if all probability distributions are normal distributions. Surely, normal distributions are far too restrictive.1 And quadratic utility For example, studies have shown that stock price changes are inconsistent with the assumption of normal probability distributions. See Mandelbrot, B. [1963]. \The variation of certain speculative prices", Journal of Business, 36: 394-419, and Breen, W. and J. Savage [1968]. \Portfolio distributions and tests of security selection models", Journal of 1

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functions are unsatisfactory. Not only do they imply that utility reaches a maximum; they also entail that the absolute degree of risk aversion is increasing in wealth, approaching in nity as utility approaches its maximum. Consequently, one is lead to the absurd result that the willingness to gamble for a bet of xed size should decrease as wealth is increased. Stochastic dominance is interesting in itself. But we are mostly interested in what it contributes to the solution of complex decision problems and their comparative statics of risk. Keeping this in mind, we will not only give a detailed exposition of the theory, but also elaborate on a number of applications, from various elds in economics. Among other examples, we address the following issues:

 Will the competitive rm increase its supply if the random product    

price becomes \larger", and what if it is subject to less risk? Consider the labor supply and saving decision if the return from saving is risky. Will labor supply increase if the return from saving becomes more volatile? Consider the entry decision in an oligopoly market. Is there a symmetric equilibrium in mixed entry strategies, and how does the equilibrium entry probability change if more rms contemplate entry? Can one unanimously rank distributions of wealth in a society if all agents agree on some general ethical principles, represented by a social welfare function with certain properties? Does it pay to diversify a nancial portfolio, and should one always invest more in an asset if its return becomes \larger" or if it is subject to less risk?

2 Assumptions and de nitions We use capital letters, such as X and Y , to denote random variables, and the corresponding lower case letters to denote realizations. Capital letters such as F and G denote distribution, and f and g the associated density functions. For convenience, all random variables have the same (extended) Finance , 23: 805-819.

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support I := [x; x]; ?1 < x < x < +1. All expected values are assumed to be nite. The decision maker has a preference ordering over all possible outcomes, represented by a von Neumann{Morgenstern utility function. Two properties of the utility function are emphasized: monotonicity (more is better than less), and concavity (risk aversion). The set of utility functions that are monotone increasing is denoted by U1, and the set of functions that are monotone increasing and concave by U2. Di erentiability requirements will be invoked as needed. All utility functions are bounded. Obviously, U 2 U1 if and only if the argument of U is regarded as a good, and U 2 U2 if and only if the decision maker is also risk averse.

3 First order stochastic dominance We say that the random variable X rst order stochastically dominates (FSD) the random variable Y , written X % FSD Y , if PrfX > zg  PrfY > zg for all z; (3.1) or equivalently if F (z)  G (z) for all z; (3.2) (where F (z) := 1 ? F (z) and G (z) := 1 ? G(z)). If the inequality is strict for some z, the dominance relationship is \strict". In the literature, (3.2) is often written in the form F (z)  G(z); for all z: (3.3) However, since it is slightly easier to remember that the dominant random variable is stochastically larger , we nd formulation (3.2) slightly more appealing. An illustration of rst order stochastic dominance is in Figure 1.

3.0.1 Main results The following result is useful in several proofs. Also, it indicates that Y cannot FSD dominate X if its expected value is lower; this o ers a quick test for FSD.

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Lemma 1 X % FSD Y =) E [X ]  E [Y ]. Proof 1) Suppose X and Y are nonnegative random variables. Then, by a known property of expected values2 the assumed FSD implies

E [X ] =

x

Z

0

F (t)dt 

x

Z

0

G (t)dt = E [Y ]:

(3.4)

2) If x < 0 one has:3

E [X ] =

x

Z

0

Z

x

dx ?

dx ? = E [Y ]:



0

Z

x

x Z x  x

F (x)dx G(x)dx

Proposition 1 (FSD{theorem) X is unanimously preferred to Y by all agents with monotone increasing utility functions if and only if X % FSD Y .

X % FSD Y () E [U (X )]  E [U (Y )] for all U 2 U1: (The preference ranking is reversed if utility functions are decreasing.)

Proof 1) Suppose X % FSD Y . Then,4 PrfU (X ) > zg = PrfX > U ?1(z)g  PrfY > U ?1 (z)g = PrfU (Y ) > zg: Therefore, U (X ) % FSD U (Y ), and Lemma 1 implies E [U (X )]  E [U (Y )]. Consult rule R6 in Appendix D, on page ??. Consult rule R5 in Appendix D, on page ??. De ne U ?1(z ) := inf fxjU (x) = z g. This may seem at bit pedantic. But it allows you to cover the case when U is monotone but not strict monotone increasing. Notice, however, that continuity is required to make this work. 2 3 4

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2) Suppose E [U (X )]  E [U (Y )] for all U 2 U1. For each z consider the particular monotone increasing function:5 (

x>z Uz (x) := 10 ifotherwise ; Evidently, E [Uz (X )] = PrfX > zg, E [Uz (Y )] = PrfY > zg, and therefore X % FSD Y .  G F; 1

6

.................................. ... ....... ..... ................................... ................ . ............. . . .......... .. . . . ............ . . ................... ........ ..... . . . ......... . . . . ......... ....... .. . . . . ............ . . . . ........ ...... .. . . . . . ........ . . . . ............ ....... .. . . . . . . ............. . . . . ....... ... ...... . . . . . . . . ...... . . . . ........... .. ...... . . . . . . . . ......... . . . ............ ...... ...... . . . . . . . . . . ....... . . . . ....... ...... . . . . . . . . . . . ....... . . ........... .. ....... . . . . . . . . . . . ....... . . . ......... .. .. . . . . . . . . . . . . ............. . . ....... .. ...... . . . . . . . . . . . . . . ....... . . ....... .. .. . . . . . . . . . . . . . . ............ . ....... . ....... . . . . . . . . . . . . . . . . ...... . ........ .. .. . . . . . . . . . . . . . . . . ............ ....... ...... .... . . . . . . . . . . . . . . . . . ............. ........ ..

G

F

x

- x; y

Figure 1: First order stochastic dominance, FSD (dotted area = E[X])

3.0.2 FSD and stochastically \larger" relationship* Proposition 1 links the FSD stochastic order relationship and a preference order, shared by all agents who prefer \more to less". This already indicates that FSD can be viewed as a \stochastically larger" relationship. The following result makes this interpretation even more compelling. It shows that if X % FSD Y , then it is possible to represent X and Y by probabilistically equivalent gambles, X  and Y , in such a way that X  is unambiguously larger than Y , for all events. Generally, we assume continuous utility functions (see the remark in the preceding footnote). So how can we dare to use a discontinuous function in the proof of one of the key results? Don't worry about this. One can always approximate this function by a continuous function, as closely as one desires. 5

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Proposition 2 If X % FSD Y one can nd random variables X  and Y  that

have the same probability distributions as X and Y , and satisfy the \larger" relationship

PrfX   Y g = 1:

(3.5)

Proof Notice that the random variable Y  := G?1 (F (X )) has distribution

G, just like random variable Y , because PrfY   xg = = = = = =

PrfG?1 (F (X ))  xg PrfF (X )  G(x)g PrfX  F ?1(G(x))g F (F ?1(G(x))) G(x) : PrfY  xg:

Now set X  = X , and observe that PrfX   Y g = PrfX  G?1 (F (X ))g = PrfG(X )  F (X )g = 1, as asserted.

3.0.3 Some examples There are many examples of rst order stochastic dominance. Some of the more frequently used examples are listed below: 

1. The binomial distribution, with the density function f (x; N; p) := p)N ?x, increases stochastically both as N and as p are increased. 2. The normal distribution increases stochastically with its mean. 3. X % FSD Y if the likelihood ratio f (x)=g(x) is not decreasing.

N x

px(1?



3.0.4 A reminder If you only skimmed through the main result you may mistakenly equate FSD with the monotonicity of utility functions. Therefore, we remind you that the FSD{theorem is an unanimity result. It gives a necessary and sucient condition under which all agents with monotone increasing utility functions agree on a preference ranking of two random variables. At the risk

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of unnecessary repetition, we stress that for random variables that do not satisfy FSD, one can always nd monotone increasing utility functions that give higher rank to either the one or the other, as in the following discrete example. But this cannot occur if random variables can be ranked according to FSD.

x1 x2 x3 PrfX = xig 0.2 0.2 0.6 PrfY = xig 0.1 0.4 0.5 Table 1: Two probability distributions that violate FSD

Example 1 Consider the probability distribution functions over the three outcomes, x1; x2 ; x3, stated in Table 1, and two monotone increasing utility functions: a) U (x1) = 0:5, U (x2) = 1, U (x3) = 2, and b) U (x1 ) = 0, U (x2) = 0:5, U (x3) = 2. Obviously, a) =) E [U (Y )] > E [U (X )], and b) =) E [U (Y )] < E [U (X )].

4 Second order stochastic dominance We say that random variable X second order stochastically dominates (SSD) random variable Y , written X % SSD Y , if Z

k

x

PrfX > xgdx 

which is equivalent to Z

k

x

x Z

F (x)dx 

k

Z

k

x

PrfY > ygdy for all k;

(4.1)

G (y)dy for all k:

(4.2)

If the inequality in (4.1) is strict for some k, the dominance relationship is \strict". In the literature, condition (4.2) is often written in the form Z

k

x

F (x)dx 

Z

k

x

G(y)dy; for all k:

An illustration of SSD dominance is in Figure 2.

(4.3)

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 G F; 1

6

.......................................... .... . . . .................. .... .... ..... . . . .................. .... . . . . . . ......... .... . . . . . . ....... ..... ...... . . . . . .............. ...... . . . . . . ........ .... . ..... . . . . . . ............ ....... . . . . . . . ....... ...... . . . . . . . ..... .. ...... ....... . . . . . . . ........ ...... . . . . . . . .... ....... . . . . . . . ..... ....... ...... . . . . . . ........ ....... . . . . . . ..... ........ . ......... . . . . ....... .......... . . . . ..... ........... . . . ... ............. .................... . ....... ................................ .... ................... .... . ..... . ...... . .............................. ....... . . . ............ ... ....... ........ . . . ............ ...... . . . ..... ........ ............ . . ....... ............. . . ... ..................... ...

G

F

+

?

x

- x; y

Figure 2: Second order stochastic dominance (SSD)

4.0.5 Main results Of course, FSD =) SSD | but not vice versa. Also, computing expected values o ers a simple test for SSD:

Lemma 2 X % SSD Y =) E [X ]  E [Y ]. Proof Follows immediately by the fact that E [X ] = 0x F (t)dt, if x  0, and E [X ] = 0x dx ? xx F (x)dx otherwise. Proposition 3 (SSD{theorem) X is unanimously preferred to Y by all R

R

R

agents with monotone increasing and strictly concave utility functions if and only if X % SSD Y X % SSD Y () E [U (X )]  E [U (Y )] for all U 2 U2: (The preference ranking is reversed if utility functions are decreasing.)

Proof 1) Suppose E [U (X )]  E [U (Y )], for all U 2 U2. Consider the following function k Uk (x) := xk ifif xx > k: (

Obviously, U 2 U2. Suppose x  0. Then,

E [Uk (X )] =

Z

k

0

F (x)dx; for each k;

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and therefore, the assumption implies 0k F (x)dx  0k G (x)dx for all k, and hence X % SSD Y . Similar reasoning applies if x < 0. 2) We show that X % SSD Y =) U (X ) % SSD U (Y ) for all utility functions U 2 U2. Since monotonicity implies di erentiability almost everywhere, and since risk aversion entails that U 0 is monotone decreasing, we can invoke di erentiability with U 0 > 0; U 00  0 almost everywhere. Using a transformation of random variables, from X to U ?1(X ), and integration by parts, one obtains R

Z

k x

PrfU (X ) > xgdx =

Z

k

x Z k

R

PrfX > U ?1(x)gdx

 (U ?1(x)dx F x z = F (~z)U 0(~z)dz~ z z 0 d v F (~v)dv~dz~ U (~z) dv = z z z z 0 = U (z) F (~v)dv~ ? U 00(~z) =

Z

Z

Z

Z

Z

z

z

Z

z

v

F (~v)dv~dz:

Since U 0 > 0; U 00  0, the assumed X % SSD Y implies immediately that U (X ) % SSD U (Y ). Combined with Lemma 2, we conclude E [U (X )]  E [U (Y )], for all U 2 U2.6

4.0.6 SSD and stochastically \more risky" relationship* Proposition 3 links the SSD stochastic order relationship and a preference order, shared by all agents who prefer more to less and are risk averse. This already indicates that SSD can be viewed as a \stochastically more risky" relationship. The following result makes this interpretation even more compelling. It shows that if X % SSD Y , then it is possible to represent X and Y by probabilistically equivalent gambles, X  and Y , in such a way that Y  is more risky than X , for all events. 6 Sketch of alternative proof (using integration by parts, twice): Let Z (x) := F  (x) ? Rx Rx  0   G(x) = G(x) ? F (x): Then, E [U (X )] ? E [U (Y )] = x U (x)dZ (x) = ? x U (x)Z (x)dx = R R R ?U (x) xx Z (y)dy + xx U 00 (x) xx Z (y)dyd  0.

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Proposition 4 If X % SSD Y one can nd random variables X  and Y  that

have the same probability distributions as X and Y , and satisfy the \more risky" relationship

PrfE [Y jX ]  X g = 1:

(4.4)

4.0.7 Examples The most frequently used example of SSD are two single crossing distributions; the mean preserving spread is a special case. Two distribution functions are called single{crossing, if they intersect exactly once, as in Figure 2. An important implication of single crossing is that one random variable has more probability mass on both tails of the distribution. Another implication of the single is that the relationship between expected values and SSD can be strengthened, as follows:

Proposition 5 (Single crossing and SSD) Suppose X and Y are nonnegative random variables that are single crossing, with F (x)  G (x) for low and F (x)  G (x) for high x. Then, X % SSD Y () E [X ]  E [Y ]. Proof Lemma 2 already shows that X % SSD Y =) E [X ]  E [Y ]. Therefore, we need only prove that E [X ]  E [Y ] =) X % SSD Y . By the assumed single crossing property it follows immediately that xz [F (t) ? G (t)]dt  0 for all z 2 [x; c), where c denotes the crossing point. We now show that this relationship extends also to all z 2 [c; x]. Therefore, assume z 2 (c; x]; by R

the assumption one has

0  E [X ] ? E [Y ] x = 0 [F (t) ? G (t)]dt z = [F (t) ? G (t)]dt + Z

Z



0 Z

0

z

[F (t) ? G (t)]dt:

x

Z

z

[F (t) ? G (t)]dt

Hence, 0z [F (t) ? G (t)]dt  0 holds for all z. Another example concerns normal distributions. R

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Proposition 6 (Normal distributions) Suppose X and Y are normally

distributed, and Y has a higher variance or a lower expected value than X . Then, X % SSD Y .

Proof If the variance is the same, but the expected value of X is greater than

that of Y , then we know already that X % FSD Y , which of course implies SSD. If the variance of X is lower while expected values are the same, the two distributions are single crossing.7 Therefore, X % SSD Y , by Proposition 5. If X has both a higher expected value and a lower variance than Y , the assertion follows by the transitivity of stochastic dominance relationships (see Proposition 7 below). This last fact proves that the mean{variance analysis in nance is justi ed if the returns of nancial securities are normally distributed.

5 An invariance property* Stochastic dominance is preserved when the original random variables are multiplied by a constant or when another independent random variable is added.8

Proposition 7 (Transitivity) Consider three independent nonnegative random variables, X; Y and W , and the linear combination aX + bW and aY + bW , with a > 0, b  0. Then: 1. X % FSD Y =) (aX + bW ) % FSD (aY + bW ) 2. X % SSD Y =) (aX + bW ) % SSD (aY + bW ). Proof See Hadar and Russell.9

The crossing point is z = E [X ] = E [Y ], which is where the density function has its peak. 8 This result is needed in the starred application on optimal portfolio selection. You may skip it if you do not cover this application. 9 Hadar, J. and W. R. Russell [1971]. \Stochastic dominance and diversi cation", Journal of Economic Theory, 3:288-305. 7

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6 Ranking transformations of random variables Suppose the random variable X is transformed, using the deterministic function t(x). This transformation maps each possible outcome of the original random variable into a new value. Let t be monotone increasing (to assure that the transformation does not reverse the preference ranking), and di erentiable. We begin with two results that provide necessary and sucient conditions for FSD and SSD dominance of t(X ) over X . In the light of what you have already learned, these results are plausible. Therefore the proofs are omitted.

Proposition 8 Consider the transformation t(X ), and de ne k(X ) := t(X )?

X . Then,

t(X ) % FSD X () k(x)  0; 8x t(X ) % SSD X ()

Z

x

0

k(t)dF (t)  0; 8x:

(6.1) (6.2)

Strict dominance requires strict inequalities for some x.

Proof To prove the FSD relationship note that Prft(X ) > xg = PrfX > t?1(x)g > PrfX > xg ? 1 () t (X ) < x () X < t(x): The proof of the SSD relationship is left as an exercise. Frequently used transformations are ane transformations, t(x) := + x. But these are often more specialized than necessary to derive unambiguous comparative statics of risk.

Example 2 (Stretching) Consider the ane transformation of X t(X ) := + X;

(6.3)

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with  0,  1, at least one strict. Then t(X ) FSD X . Obviously t(X ) can be viewed as a \stretching" of X . The direct proof follows from a simple assessment, as follows:

Prft(X ) > zg = PrfX > z ? g  PrfX > zg:

But, it is already a consequence of Proposition 8.

Example 3 (Single{crossing) Consider a single crossing transformation t(x), with t(x)  x for x  x, and t(x)  x otherwise. Also assume the mean{preserving property E [t(X )]  E [X ]. Then the integral condition in (6.2) is satis ed; therefore, t(X ) % SSD X .

7 Comparative statics of risk Economic models of uncertainty usually examine optimal decisions in the face of random outcomes. In comparative statics we explore how the optimal action responds to parameter changes. Comparative statics of risk concerns changes in the nature of the underlying uncertainty. In this section we lay the foundations for the comparative statics of risk, for a class of decision problems. The results will then be used in the analysis of various typical applications.

Framework Suppose the decision{maker has the objective function V (a; X ) := U ((a; X )); where a is his action variable, and (a; X ) the random outcome, codetermined by his action a and the nonnegative10 random variable X . The function  (which could be the pro t function of a rm) is strictly concave in a, and a = 0 is satis ed for some nite a, for each x. The decision maker maximizes expected utility max E [V (a; X )]: a0 In the following nonnegativity is invoked only in order to avoid cumbersome case distinctions. 10

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Existence of an interior solution a that solves

E [Va(a; X )] = 0 and E [Vaa(a; X )] < 0;

(7.1)

is assumed.

Key issue We want to nd out how a changes when the random variable

X is altered by a FSD or SSD transformation. For this purpose consider the convex linear combination of X and t(X ) Z () : = t(X ) + (1 ? )X;  2 [0; 1] = X + k(X ); where

k(X ) := t(X ) ? X: We will analyze the solution function a() a() := arg max a E [V (a; Z ()]; in particular its derivative at the point  = 0. The sign of a0 (0) determines the e ect of the transformation t(X ) on the optimal action. Since a() solves the rst{order condition, E [Va] = 0, one has by the implicit function theorem (7.2) a0 (0) = ? EE[V[Vaxk]] ; aa where all functions are evaluated at the point (a; X ). By the second order condition E [Vaa] < 0. Thus, sign a0 (0) = sign E [Vaxk];

(7.3)

and the comparative static problem is reduced to determine the sign of E [Vaxk].

Proposition 9 (FSD transformations I) The optimal action increases (decreases), a0 (0)  0 ( 0) for all FSD transformations if Vax (a; X )  0 ( 0) everywhere.

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Proof Follows immediately by (7.3) and the fact that k(X )  0 (FSD). Proposition 10 (SSD transformations I) The optimal action increases 0  (decreases), a (0)  0 ( 0), for all SSD transformations if Vaxx(a; x)  0 ( 0) everywhere, in addition to the condition stated in Proposition 9. Proof Using integration by parts one obtains d x k(t)dF (t) dx Vax E [Vaxk] = dx 0 0 x x  = Vax(a ; x) k(x)dF (x) ? Vaxx(a; x) Z

x

(a; x)



Z

Z

Z

0

Z

0

0

x

k(t)dF (t)dx

Since 0x kdF  0; 8x, by SSD, the sign of E [Vaxk] is unambiguous when Vax(a; x) < 0 and Vaxx  0 for all x, and the assertion follows immediately. R

For an interpretation of these results we introduce what has been called the generalized risk aversion11 00 R := ?UU0 ax : Then, Vax and Vaxx can be rewritten as follows

Vax = U 0(ax ? R)

(7.4)

Vaxx = U 0(axx ? Rx) + U 00x(ax ? R):

(7.5)

Therefore, the conditions in Propositions 9 and 10 can be replaced by requirements concerning the  function, and the generalized risk aversion R. For example, if (a; X ) := aX , one has ax = 1 and therefore a0 (0) > 0 for all SSD transformations if R(a; x) < 1; Rx(a; x) > 0. The conditions summarized in Propositions 9, 10 are sucient to sign a0 (0), but not also necessary. This suggests that there is room for other sucient conditions. We close with two further results that are useful in

Notice, if ax = 1, the measure reduces to the absolute risk aversion, and if ax = ax to the partial relative risk aversion proposed by Menezes, C. F. and D. L. Hanson [1970]. \On the theory of risk aversion", International Economic Review, 11: 481-487. 11

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several applications. Key requirements are that k0(X )  0 and that the absolute risk aversion, A := ?U 00 =U 0 , must be monotone decreasing. Whereas the emphasis of the previous conditions is on the degree of risk aversion, the emphasis of the following conditions is on the functional form of the transformation rule.

Proposition 11 (FSD transformations II) The optimal action increases for all FSD transformations if all of the following conditions hold everywhere 1. U 0 > 0; U 00  0; A0  0 2. x > 0; xx  0; ax  0 3. k 0 (x)  0.

Proof Using the absolute risk aversion A, Vax can be written in the form Vax = U 0(ax ? Axa): (7.6) Therefore,

E [Vaxk] =

Z

0

x 0 U axkdF

?

Z

0

x 0 U A

xa kdF:

(7.7)

The rst integral is obviously nonnegative, for all FSD transformations. In order to determine the sign of the second integral, notice two facts. 1) Axk is positive and decreasing in x; 2) U 0a is negative for low x, say for all x < x1, and positive for all x > x1, in such a way that the rst{order condition U 0adF = 0 holds. Combining these facts, one has R

x1

Z

0

Z

x

x1

(Akx)U 0adF  (Akx)jx=x1 (Akx)U 0adF  (Akx)jx=x1

Z

0

x1 0 U

a dF

x 0 U adF: x1

Z

(7.8) (7.9)

Therefore, by the rst{order condition concerning a x

Z

0

AkxU 0adF

 (Akx)jx=x

Z

1

0

x 0 U adF

= 0;

(7.10)

and hence E [Vaxk]  0. Finally, a similar result applies to all SSD transformations, if one makes the stronger requirement that the \marginal product" a must not only increase but must also be concave in x.

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Proposition 12 (SSD transformations II) The optimal action increases for all SSD transformations if axx  0, in addition to the conditions stated in Proposition 11. Proof Look at the rst integral term on the RHS of (7.7). Using integration by parts this can be written as x 0 U axkdF x

Z

= =

x 0 d Z x k(y)dy dF (x) U  ax dx x x Z x U 0ax k(y)dy x x=x Z Z x  x 00 0 ? x (U ax + U axx) x k(y)dydF (x): !

Z

(7.11)

(7.12)

Using an argument similar to the one developed in the previous proof, the assertion follows easily.

Summary table Since we make extensive use of these results in 0 the following applications, we summarize these sucient conditions for a (0)  0

in Table 2.

Propositions FSD{transformations SSD{transformations 9 and 10 Vax  0 : : : and Vaxx  0 0 0 11 and 12 A ; k ; xx  0, x; ax  0 : : : and axx  0 Note: A := ? UU , R := Aa x, Vax = U 0 (ax ? R), 00 0

Vaxx = U 0(axx ? Rx) + U 00 x (ax ? R)

Table 2: Sucient conditions for a0 (0)  0

8 Applications In the introduction we claimed that stochastic dominance theory is useful in many areas of economics. We now put this claim to a proof, with a sample of applications | ranging from a simple portfolio selection problem to the measurement of income inequality in society.

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9 Portfolio selection I A risk averse investor considers to divide given wealth w between two assets: one riskless asset, with the sure return r, and one risky asset, with the random return r + X . Short sales are not feasible. The decision problem is to choose that investment in the risky asset a 2 [0; w], that maximizes expected utility max a E [U ( (a; X ))]; where  (a; X ) := wr + aX:

(9.1)

To assure an interior solution, a 2 (0; w), E [X ] > 0 is assumed.12 Evidently, one has 00 (9.2) a = X; ax = 1; axx = 0; x = a; xx = 0; R = ? UU 0 aX: Therefore,

Vax = U 0(1 ? R); Vaxx = ?U 0Rx + U 00a(1 ? R):

(9.3)

Proposition 13 Investment in the risky asset will increase if 1. its return is subject to a FSD transformation, and R < 1 everywhere; 2. its return is subject to a SSD transformation, and R < 1, Rx > 0 everywhere.

Proof See Table 2 and eqs. (9.2), (9.3). The two requirements in Proposition 13 are fairly restrictive. Indeed, R < 1 excludes some commonly used utility functions, and the alternative requirement Rx > 0 contradicts the usual assumption of diminishing risk aversion. However, by restricting the permitted transformations of the return of the risky asset, one can show that the demand for the risky asset tends to increase also under more reasonable assumptions concerning the degree of risk aversion.

The rst{order condition for an interior solution is E [U 0X ] = 0. Since E [U 0 X ] = E [U 0]E [X ]+ Cov (U 0 ; X ) it follows that an interior solution exists if and only if E [X ] > 0. Notice, Cov (U 0; X ) < 0 () a > 0, since U 0 is strict monotone decreasing. 12

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Proposition 14 Investment in the risky asset will increase if its return is subject to a SSD transformation, with t0(x)  1 (equivalently k 0 (x)  0), and the absolute risk aversion is diminishing (A0  0). Proof See Table 2 and eqs. (9.2). As an illustration, suppose the \premium" on the risky asset is increased by a xed amount , at all realizations of X . This leads to the transformation t(X ) = X + . Hence, k(X ) = ; k0(X ) = 0, and the demand for the risky asset will increase, by Proposition 14, Part 1. As another illustration suppose that the earnings on the risky asset are subject to an income tax. Suppose the tax is reduced in such way, that the tax reduction is high at low and low at high realizations x. Then k(X ) = t(X )?X > 0 and k0(X ) = t0(X )?1 < 0. Again, the demand for the risky asset will increase, by Proposition 14, Part 2. Interestingly, if t0(x) > 1 (equivalently, if k0 > 0), the higher return of the risky asset may lead the investor to hold less of it. An interpretation of this paradoxical \Gi en good e ect" makes use of the distinction between an income and substitution e ect, adapted from traditional demand theory. The fact that t(x)  x (k(x)  0) implies that the substitution e ect is positive, which indicates that a should increase. However, if t0 > 1 (k0 > 0), the transformed random variable is riskier, and the income e ect may be negative and larger than the substitution e ect.13

10 The competitive rm under price uncertainty Consider a risk averse competitive rm with the convex cost function, C (a), de ned on output a, with xed costs lima!0 C (a) > 0. Production is subject If the demand for the risky asset may diminish when its return increases, you may rightly suspect that the demand for the riskless asset may also increase if its (safe) return diminishes. This possibility was noticed by Fishburn and Porter. Necessary and sucient conditions for this to occur are in Fishburn, P. C. and L. B. Porter [1976]. \Optimal portfolios with one safe and one risky asset", Management Science, 22:1064{1073, and in Pratt, J. [1989]. \Utility functions, interest rates, and the demand for bonds", in: Fomby, T. B. and T. K. Seo (eds.), Studies in the Economics of Uncertainty: In Honor of Josef Hadar, Springer Verlag, 225{231. 13

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to a time lag; therefore, production decisions have to be made before the market price is known. The rm is a price{taker, in the sense that the product price is a nonnegative random variable X . To assure an interior solution, E [X ] > mina C (a)=a is assumed. The rm maximizes the expected utility of pro ts  by choosing nonnegative output a, max E [U ((a; X ))] where (a; X ) := Xa ? C (a): (10.1) a0 Evidently,14 a = X ? C 0(a); ax = 1; xx = axx = 0; x = a:

(10.2)

Proposition 15 The competitive rm willx increase output if the product price is subject to a SSD transformation ( 0 kdF  0), with k 0(x) = t0(x) ? 1  0, and the absolute risk aversion is decreasing in x. Proof See Table 2, and eq. (10.2). R

As an illustration, suppose output is subsidized at a xed rate  > 0. Then t(x) = x + , and k(x) = t(x) ? x =  > 0; k0(x) = 0. Alternatively, let this subsidy be degressive, in the sense that the subsidy rate is positive but monotone decreasing in the price x. Then, k(x) > 0 and k0(x) < 0. In both cases, output will increase, provided absolute risk aversion is not increasing.15 Finally, suppose the government introduces a price support program, where it buys or sells from its inventory in such a way that the price is con ned to a narrower range, say [x1; x2]  [x; x]. Such price support or bu er stock programs are observed in many markets for natural resources and agricultural products. Stated formally, they lead to the transformation (see Figure 3) X + (x1 ? X ) if X  x1 t(X ) = X if X 2 (x1; x2) X ? (X ? x2) if X  x2: 8 > < > :

x.

14

Note, R = ? UU (X ? C 0 )a; which is why R is positive for high x, and negative for low 00 0

Notice, however, that the e ect of an ad valorem subsidy is ambiguous, because in this case t(x) = x;  > 0, and therefore k0 (x) =  > 0. 15

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k(x)

6

x2 x1

?x

... .....

. .......

.. ......

... .....

.. ......

... .....

... .....

... .....

.. .....

.. ...... .. ...... ....... ...... ...... .. ....... ...... ...... ... ....... ...... ...... .. ...... ...... ...... ...... .. ...... ...... ...... .......... ....... ...... .. ...... ...... ...... ...... .. ...... ...... ..... . . . ...... . . . .. ....... ...... . ...... ....... ..... .. ....... ...... .......................................................................................... ...... ...... ...... ....... ...... ...... ...... ....... ...... .

x1

x2

x

-x

Figure 3: E ect of price support program Therefore,

x1 ? X if X  x1 k(X ) = 0 if X 2 (x1; x2) x2 ? X if X  x2: If x1 is chosen suciently high or x2 suciently low, one can assure that t(X ) % SSD X . The exact condition is x21 > (x ? x2)2 (see Proposition 8 and the illustration), which is of course always satis ed if the price support intervenes only to assure a minimum price. In this case, Proposition 15 indicates that the price support program leads to higher output (provided the absolute risk aversion is not increasing), which in turn may lead to a build{ up of public inventories, epitomized by the infamous \butter mountain" of the European Community. 8 > < > :

11 Labor supply A risk averse household chooses labor supply, a, facing a xed real wage rate, w > 0, and random earnings on nonhuman capital, X  0. The endowment with time is one unit; time spent without working is leisure. All income is spent on a xed basket of consumption goods; therefore, the number of units

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consumed is

(a; X ) := wa + X: The household maximizes expected utility max E [V (a; X )]: a2[0;1]

22 (11.1)

The objective function is based on the twice continuously di erentiable and concave utility function U , with the two arguments a and (a; X ) V (a; X ) := U (a; (a; X )): Of course, Ua < 0; U > 0; U  0. To assure an interior solution, a 2 (0; 1), the usual Inada conditions are assumed. The important di erence between this and all previous applications is that the utility function U now has two arguments, (a; x) and a. This must be kept in mind when one applies the general theory of the comparative statics of risk. We want to nd out under what conditions labor supply tends to either increase or diminish if random property income is subject to FSD or SSD transformations. Unlike in previous applications we now also need assumptions concerning the behavior of the cross derivative of the utility function. To obtain clear{cut results, we admit only utility functions that display either (Ua + wU ) < 0 or > 0 everywhere. In the well{known deterministic labor supply model these inequalities correspond to leisure being either a normal or inferior good, respectively. We will stick to this interpretation of the sign of (Ua + wU ). As you know from section 7, the signs of Vax and Vaxx play an essential role in explaining the labor supply response. The nice feature of the present model is that there is a one{to{one correspondence between the sign of these functions and the nature of the good leisure. Therefore the supply response is unambiguously determined by the income elasticity of the demand for leisure. Proposition 16 Labor supply diminishes (increases) for all FSD transformations of property income if leisure is a normal (inferior) good everywhere. Proof Follows immediately by Proposition 9, and the fact that Vax = Ua + wU S 0 if leisure is normal, neutral, inferior: (11.2)

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This result can be pushed a bit further if one adds assumptions concerning the behavior of the cross derivative of the utility function. For example, suppose the consumption of leisure does not a ect the household's willingness to take gambles. Speci cally, suppose the absolute risk aversion A is independent of the consumption of leisure d A (a; (a; x)) = 0 everywhere: (11.3) da Then, the property of leisure as a normal resp. inferior good determines not only the sign of Vax, but also that of Vaxx:

Lemma 3 Suppose (11.3) holds. Then, sign Vaxx = ?sign Vax. Proof Since x = 1, one has dxd (ln U ) = U =U = ?A. Therefore, by the

assumed invariance of A with respect to a, and the fact that the order of di erentiation does not matter, one has d d (ln U ) 0 = da dx  d d (ln U ) = dx da  d Ua + wU = dx U V d ax = dx U  V axx + Vax A = U : Therefore, Vaxx = ?VaxA, and the assertion follows immediately. Using this result one can extend the labor supply response described in Proposition 16 to all SSD transformations of property income. 







Proposition 17 The labor supply diminishes (increases) for all SSD trans-

formations of property income if leisure is a normal (inferior) good.

Proof Follows immediately by Proposition 10 and Lemma 3.

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12 Entry in Cournot oligopoly Consider the mixed strategy equilibrium of the entry game in a Cournot oligopoly, explained in the chapter on Oligopoly and Industrial Organization (see page ??). There we showed that the equilibrium probability of entry  must solve the condition of indi erence between entry (leading to the payo stated on the left{hand side) and non{entry (leading to zero payo ) NX ?1 n=0

N (n; )(n + 1) ? F = 0;

(12.1)

where N (n; ) is the probability of the event that n rivals enter, N (n; ) := N n? 1 n (1 ? )N ?1?n ; (12.2) (n) the gross pro t as a function of the number of active rms, N the number of potential competitors, and F the sunk entry cost. N is taken to be so large that (N ) < F ; in words, if all rms entered with certainty, they would su er losses. The prize question is: Does this indi erence condition have a solution  2 (0; 1), and if so, is it unique? Having mastered stochastic dominance, the answer is easy to prove. Proposition 18 (Mixed entry strategies in oligopoly) The simultaneous entry model has a unique mixed strategy equilibrium, where each potential competitor enters with positive probability less than one. Proof The proof proceeds in several steps. 1) Con rm that the left{hand side of (12.1) is positive if  = 0, and negative if  = 1. This is the easiest part. 2) Consider the probability distribution of the random number of rival entrants, n~ , as a function of  N ?1 N ? 1 m (1 ? )N ?1?m : N (n; ) := Prfn~ > ng =  m m=n+1 (12.3) By a known result one can write eq. (12.3) in the equivalent form:16  n N ?n?2 d;  N (n; ) = n!(N(N??n1)! (12.4) ? 2)! 0  (1 ?  ) !

!

X

Z

16

Take a look at rule (R16) in the brief introduction to \order statistics" in Appendix

D , page ?? f.

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from which you see immediately that  N is strict monotone increasing in . In other words, increasing  leads to FSD dominance. Since  is decreasing in n, it follows that the expected pro t from entry (the left{hand side of (12.1)) must be strict monotone decreasing in , by Proposition 1. 3) Putting pieces together, it follows that (12.1) has a unique solution  2 (0; 1). Another question concerns the impact of potential competition on the equilibrium entry strategy. Again, stochastic dominance gives the clue.

Proposition 19 (Comparative statics) The equilibrium probability of entry  is a strict monotone decreasing function of the number of potential competitors N .

Proof Follows immediately from the fact that the binomial distribution

N (n; ) is stochastically increasing (in the sense of FSD) in N as well as in .

13 Auctions Consider the English (E) and Dutch (D) auction, in the symmetric independent private values framework with N  2 bidders. Assume uniformly distributed valuations on the support [0; 1], and risk averse bidders. In Chapter ?? we show in detail that the equilibrium bid functions are bD (v) = (1 ? N1 )v, and bE (v)  v. The associated equilibrium prices are the random variables PD and PE . Their probability distribution functions (cdf's) are (see (??) and (??)

FD (x) = N N? 1 

N



xN (Dutch auction)

FE (x) = xN + N xN ?1 ? xN 



(English auction):

(13.1) (13.2)

The equilibrium price in both auctions is equal to the highest bid. In the Dutch auction it is

PD = b1(V(N ));

(13.3)

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and in the English auction

PE = b2(V(N ?1)):

(13.4)

Thereby V(N ), V(N ?1) denote the highest and second highest order statistic of a sample of N identical and independly distributed iid random valuations, respectively. An important result in auction theory is that the expected values of equilibrium prices in the Dutch and English auction are the same, E [PD ] = E [PE ]. However, PD and PD di er in their risk attributes. Indeed,

Proposition 20 (Dutch vs. English auction) The random equilibrium price of the Dutch auction, PD second order stochastically dominates that of the English auction, PE . In other words, every risk averse seller should strictly prefer the Dutch auction.

Proof To prove this stochastic dominance relationship recall that one ran-

dom variable second order stochastically dominates another, if their expected values are the same , and their probability distribution functions (cdf's) are single crossing .17 Since the seller prefers higher prices, the dominated random variable is the one that has more probability mass on both tails of the distribution. Compute the probability distribution functions (cdf's) of PD and PE , as 17

See Theorem 5.

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follows

FD(x) = PrfPD  xg = Prfb(V(N ))  xg = Prf N N? 1 V(N )  xg = PrfV(N )  N N? 1 xg N N = F N ? 1x N = N N? 1 xN ; FE (x) = PrfPE  xg = PrfV(N ?1)  xg = F (x)N + N F (x)N ?1 ? F (x)N = xN + N xN ?1 ? xN : 













Consider the di erence

(13.5) 

(13.6)

(x) := FD(x) ? FE (x) = ( N N? 1 )N + (N ? 1) xN ? NxN ?1: Evidently, (x) is an N -th polynomial with only one change in the sign of its coecients. By Descartes's Rule , it follows that (x) has at most one real positive root. At the same time, (x) must have at least one real root in (0; 1), because the expected values of PD and PE are the same. Hence, (x) has exactly one real root in (0; 1), which con rms single crossing. Finally, observe that PE has more probability mass on both tails of the distribution than PD , because FD ( N N? 1 ) = 1 = FE (1) > FE ( N N? 1 ): Therefore, PD second order stochastically dominates PE . An illustration of the stochastic dominance relationship between PD and PE is in Figure 13. 



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1 6

......... ......... ........ ......... ........ ......... ........ ............................................................................................................................................... .. . ... ........... .. ........ . ... ...... . . . .. . ...... . . . .. . ...... ..... . .. . . . . . . . . ...... . . . .. . . . . . . . . . ..... . .. . . . . . . . . . .... .. . . . ....... . .. . . . . . . . . . . .... . .. . . . .. . . . .. ....... . . . . . .. . ......... . . . . . ...... . . . . ..... . . . . . . ....... . . . . ... ... . . . . . . . . . ... .. . . . . . . . .... .. . . . . ... .. . . ... .... . . . ... .. . . . . . ... . . ... . . . . . . . .. ... . . . . . . . . .. .... . . . . . . .. ... . . . . . . . . ... .. . . . . . .. . . . . . . . . . . .. ... . . . . . .. . . . . ... ... . . . . .. . . . ... . . .. ... . . . . . . . . ... ... . . . . . . . . . . .. ... . . . . .. .. . . . . ... .... . . . . .. .... . . . . . . . .... . . ... . . . . . ... . . . . ................ . . . . . .......

FD

FE

N ?1 N

1

-

x

Figure 4: Probability distributions of PD , PE for n = 2

14 Portfolio selection II* Using the mean{variance approach it is relatively straightforward to nd out whehter it pays to mix two risky assets in a portfolio even if one asset would be strictly strictly preferable if one had to choose just one of them.18 The mean{variance approach is however not suciently general.19 Therefore, it is high time to return to this issue, using stochastic dominance in lieu of mean{variance rankings. Consider portfolios consisting of a mixture of two random prospects. We are interested in conditions under which diversi cation is optimal for all risk averse agents. Let the two prospects be described by two independent and positive random variables, X1 and X2 . Then, a portfolio is characterized by a new random variable, P () P () := X1 + (1 ? )X2 : (14.1) The portfolio selection problem is to choose a  2 R that maximizes the See the section entitled \Diversi cation and risk: a mean{variance analysis", in the Appendix to this Chapter on p. 34 . 19 Recall, variance is an unambiguous measure of risk only if the utility function is quadratic or if probability distributions are normal. 18

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investor's expected utility, E [U (P ()]. We will say that diversi cation is optimal if for all U 2 U2 there exists some  2 (0; 1) for which E [U (P ()] > maxfE [U (P (0)]; E [U (P (1)]g. A rst result, that also helps in other proofs, is

Proposition 21 Let X1 and X2 be two independent, identically distributed

random variables. Then diversi cation pays in the strong sense that E [U (P ()] > maxfE [U (P (1)]; E [U (P (0)]g, 8 2 (0; 1).

Proof See Hadar and Russell.20 But does diversi cation pay even if one of the two random prospects, say X1, SSD dominates X2 ? Surprisingly, the answer is yes | provided the dominance is not excessive, in a sense to be made precise. Before proceeding, it has to be stressed that this question makes sense only if the two prospects have the same mean: E [X1] = E [X2]. Why this? First of all, the assumption that X1 SSD dominates X2 implies E [X1]  E [X2], by Lemma 2. Therefore, if E [X1] di ered from E [X2], one would have E [X1] > E [X2]. But then 8  2 (0; 1); EP (1) = E [X1] > E [X1] + (1 ? )E [X2] =: EP (), so that | again by Lemma 2 | X1 would SSD dominate P ();  2 (0; 1), so that the assertion could not possibly hold true. Anyway, the assumption E [X1] = E [X2] is not restrictive. If assets are divisible, it is only a matter of the choice of units. As a second preliminary question we now ask: what precisely do we mean by saying that X1 dominates X2 { but not excessively so? For the purpose of formalizing this notion, we introduce yet another random variable. Let Y be independent of X2, yet identically distributed. Then, by combining X2 and Y one gets a new random variable that SSD dominates X2 (see Proposition 7). Utilizing this property, we now make the following assumption:

Assumption 1 There exists a set S  (0; 1); S 6= ;, so that 8  2 S , P~ () := X2 + (1 ? )Y SSD dominates X1. This gives precise meaning to the statement that X1 SSD dominates X2 | but not excessively so. 20

Hadar, J. and W. R. Russell [1971]. \Stochastic dominance and diversi cation", Jour-

nal of Economic Theory , 3: 288-305. (Hadar and Russell, 1971).

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With these preliminaries we come to our main \optimality of diversi cation" result.

Proposition 22 Let X1 and X2 be two independent nonnegative random variables with identical means. Suppose X1 SSD dominates X2 , and the above assumption holds. Then, for all  2 S , P () := X1 + (1 ? )X2 SSD dominates X2 as well as X1 .

Proof In the proof we make use of certain properties of the \arti cial"

portfolio described in Assumption 1, that involves a convex combination of the two random variables X2 and Y , P~ () := X2 + (1 ? )Y ,  2 S . Since X2 and Y are identically and independently distributed, the probability distributions of P () and P^ () := X1 + (1 ? )Y must be the same. Next, notice that by Proposition 7, P^ () SSD dominates P~ ().21 Combined with the other assumptions one thus arrives at the following combination of SSD relationships

P () :

=

SSD SSD SSD SSD

X1 + (1 ? )X2 X1 + (1 ? )Y =: P^ () X2 + (1 ? )Y =: P~ () X1 X2:

And the assertion follows immediately by the transitivity of stochastic dominance.

15 Income inequality* As an economist you are well trained to rank social states by their eciency, on the basis of the Pareto criterion. But how do you rank income distributions? For example, you may be asked to answer questions such as these:

 Has the distribution of income become less or more equal during the past decade?

21

In applying Proposition 7 set a =  and b = (1 ? ).

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 Is the distribution of income in France less or more equal than in Ger-

many?  Does the introduction of the value added tax in the U.S. give rise to greater or less income inequality? How would you go about to answer these questions? As a rst shot you might use the Gini{coecient, a well{known summary statistic of inequality. However, we all have some rough notion of what income inequality should mean, and given any such notion one can always nd examples of income distributions for which the ranking by the Gini{ coecient blatantly contradicts that very notion of inequality. Indeed, you may try as hard as you wish, but you cannot nd any acceptable social welfare function that consistently ranks income distributions in the same way as the Gini-coecient, as was shown rigorously by Newbery [1970].22 As early as in the 1920's, Dalton23 suggested that the study of income inequality should start with a statement of the underlying welfare norm, in the form of a social welfare function. The modern economic theory of ethics has laid out various properties that reasonable social welfare functions should share, such as symmetry and additive separability. The rationale of imposing these and other properties is the notion that ethical rankings of social states must be impartial vis a vis all individuals in society.24 In the following we adopt Dalton's advice and use a social welfare function to rank income distributions. Speci cally, we consider income distributions with equal means, , characterized by their distribution functions such as F : [0; y] ! [0; 1], and F  : [0; y] ! [0; 1], with images F (y); F (y). These are ranked according to the social welfare function W : F ! R , de ned on the set of all distribution functions F with domain [0; y]

W (F ) :=

Z

0

y

U (y)dF (y):

(15.1)

22

Newbery, D. [1970]. \A theorem on the measurement of inequality", Journal of Eco-

23

Dalton, H. [1920]. \The measurement of the inequality of incomes", Economic Jour-

nomic Theory, 2: 264-266. nal, 30.

Social welfare functions that have these properties are usually referred to as utilitarian. One of the most ardent and sophisticated defenders of modern utilitarianism is Harsanyi, and you may wish to consult one of his beautifully clear contributions on this important subject. Consult Harsanyi, J. C. [1976]. Essays on Ethics, Social Behavior, and Scienti c Explanations. Reidel-Publisher. 24

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The only additional requirement shall be that the utility function U : R ! R be an element of the class U2. We ask: under what conditions does F rank higher than F , regardless of the particular form of the utility function U 2 U2 . This leads us to the notion of Lorenz{dominance and | even though no random variables are involved here | back to SSD. You are surely familiar with the ingenious statistical representation of income distributions by Lorenz curves. The Lorenz curve plots percentages of the population, ordered by their income level from bottom to top, and their respective percentage share of total income, as depicted in the diagram 15 below. The diagonal represents the line of absolute equality. Therefore, Lorenz curves are always on or below the diagonal.25 % of income

6

... ...... ........ .... . .... ... .... ... . . . .... ... .... . ... ..... .... ... ..... . .... .... . . . . . . . .... ... .... . . .... ..... . .... . . .... ..... . .... . . . . . . . . .. .... . . ..... .... . ..... .... . . . .... ..... . .... . . . .... . . . ..... . ... . . .... .. ..... . . . ..... .... . ..... . . . . .... ..... . . . . ..... . . . . .... .... .. .... .. . . . . .... ... ..... . . . . ... .... . ... . . . . . . . . .. .... . . . . . . . .. . . . . . ..... .... .... .. . . . . . ....... ..... .... . .... . . . . . ....... .... . . . . . . . . . . . . . ...... .... .. .... . . . . . ........ ..... ... .... . . . . . ......... .... .... .. . . . . . . . . . . . . ... . .... ...... . . . . ......... ..... ..... ...... .... . . . . ........... .... ...... . .... . . . . . . . . . . ... .... . ....... . . .... ...... .... ...... . . .... ...... ..... ........ .... . . .............. . . . . ...... .... . ..................... ..... .... ............................

Line of absolute equality

Lorenz-Curve

-

% of population

Figure 5: Lorenz curve Incidentally, the Gini coecient is de ned the ratio of the area between the Lorenz curve and the line of absolute inequality, and the area under the line of full equality. 25

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Stated formally, the Lorenz curve26 is the graph of a function ' : [0; 1] ! [0; 1] that maps the images of the distribution functions F (y) and F (y) into income shares, as follows F ?1 (z) 1 y f (y)dy; F (z) (15.2) '(z; F ) :=  0 Z

'(z; F ) := 1

F ?1 (z)

Z

0

y f  (y)dy;

(15.3)

where  := yf (y)dy = yf (y)dy. Based on this familiar concept we now de ne Lorenz dominance: R

R

De nition 1 Consider two income distributions with equal mean, described by the distribution functions F and F . F \Lorenz dominates" F , F L F , if F is above F  everywhere, i.e. if 8z 2 [0; 1]; '(z ; F )  '(z ; F ). Of course, any welfare function W ranks distributions the same way as Lorenz dominance. But Lorenz dominance induces only a partial ordering of income distributions. However, as shown in the following main result, the welfare function W cannot consistently rank distributions in the same manner for all utility functions U 2 U2, unless these distributions can be ranked by Lorenz dominance. Interestingly, the proof employs SSD, which is why the present material is an application of stochastic dominance theorems.

Proposition 23 The income distribution function F ranks higher than F , W (F )  W (F ), for all utility functions U 2 U2, if and only if F Lorenz dominates F , F L F . Proof By Proposition 3, the statement W (F )  W (F ); 8U 2 U2, is equiv-

alent to the statement that F dominates F  in the sense of SSD. Therefore, one only needs to show that Lorenz dominance is equivalent to SSD. The details of the proof are spelled out in Atkinson (Atkinson, 1972).27 Not surprisingly due to Lorenz, M. O. [1905], \Methods for measuring concentration of wealth", Journal of the American Statistical Association, 9. 27 Atkinson, A. B. [1972]. \On the measurement of inequality", Journal of Economic Theory, 2:244-263. 26

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16 Supplements This section covers two supplementary topics: 1) some basics of the mean{ variance analysis of portfolio diversi cation, and 2) a brief introduction on so called \Arrow{bonds". This material is an important ingredient of the analysis of uncertainty in economics, even though it does not use stochastic dominance. This material is supplementary. It may be used independently or to motivate the stochastic dominance analysis of portfolio selection.

16.1 Diversi cation and risk: mean{variance approach Suppose you want to invest a xed sum of money. You are o ered two nancial assets, say two di erent bonds or stock. Stripped o all unimportant detail, these assets are fully characterized by their random returns, described the two random variables X and Y . Both assets promise the same average return E [X ] = E [Y ], and have the same unit price.28 But X is at least as risky and perhaps riskier than Y . You are risk averse. Would you be well advised to invest in a mixed asset, with a positive fraction of your wealth invested in X and Y , or should you rather put it all into the low risk asset Y?

Portfolios A mixed asset or portfolio is characterized by a new random

variable Z , de ned as a linear combination of the random returns of its constituent assets

Z := X + (1 ? )Y; 2 R :

(16.1)

Since the two assets' average returns are the same, the average return of the portfolio Z is una ected by the asset mix. Therefore, you can be single{minded and focus on the e ect of mixing assets on risk. Your decision problem is thus reduced to choose that particular that minimizes the risk of the resulting portfolio. 28 Assuming both identical expected values and unit prices makes this analysis a special case. Nevertheless, some of the following results, such as the convexity of the variance of a portfolio, are central to the general mean variance analysis of portfolio choice. A nice introductory text is Sharpe, W. F. Portfolio Theory and Capital Markets, McGraw{Hill 1970.

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While we are particularly interested in nding out whether the optimal is a real number between 0 and 1, which represents diversi cation, notice that we have not excluded that is negative or greater than 1. A negative indicates that the investor actually sells or issues the asset X , and > 1 that he issues Y .

Conjectures Before you continue with the formal analysis, sit back for a

moment and let your intuition guide you. As X is at least as risky as Y (and perhaps even riskier), you may think that by combining them one can only increase risk relative to that from holding only the low risk asset. Therefore, it would seem that you should stay away from the high risk asset X . On the other hand, you may recall the old adage that \one should never put all eggs into one basket" , which suggests that it is always better to diversify.

Outlook As it turns out, neither conjecture holds true without quali ca-

tion. If the covariance (or the correlation) of X and Y is suciently \low", risk is subadditivity (the total is less than the sum of its parts), and diversi cation is bene cial. But, diversi cation actually increases risk if the covariance is suciently \high". Therefore, it all depends on the degree of association between the two assets' returns, measured by their covariance or correlation. Essentially, risk is like toxic chemicals. To determine the toxic properties of a blend of chemicals it is less important how poisonous the ingredients are than how they work together. Some perfectly harmless substances combine to a deadly mixture, while other highly toxic ingredients completely neutralize each other.

Assumptions In the following exercise we develop a tight, necessary and

sucient condition for the optimality of diversi cation. In line with the \mean{variance approach" to the theory of nance, we assume that the variance of Z is a suitable measure of risk.29 As a convention, we let

x2 := Var(X )  Var(Y ) =: y2:

(16.2)

As we explained in this chapter, variance can only serve this purpose if utility functions are quadratic or if probability distributions of X and Y are normal distributions (see example 3 on page 11). These conditions are, of course, awfully restrictive. 29

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By a known result (see Rule R 3, Appendix D ), one has

z2 := Var(Z ) = 2x2 + (1 ? )2y2 + 2 (1 ? )Cov (X; Y ):

(16.3)

The question is: does z2 have a minimum for some 2 (0; 1)?

A lemma that clears the road Lemma 4 Suppose x2 > y2 or X and Y are not perfectly correlated. Then z2 reaches a minimum at

 =

y2 ? Cov (X; Y ) x2 + y2 ? 2Cov (X; Y ) :

(16.4)

Proof De ne the function  : R ! R , ( ) := z2.  is twice continuously di erentiable, and one has

0( ) = 2 x2 ? 2(1 ? )y2 + 2Cov (X; Y ) ? 4 Cov (X; Y ): q

(16.5)

q

Denote the standard deviations by x := x2, y := y2, and the correlation by  := Cov (X; Y )=(xy ). Also recall that  2 [?1; 1] (by Rule R7, Appendix D ). Therefore

00( ) = 2x2 + 2y2 ? 4Cov (X; Y ) = 2x2 + 2y2 ? 4xy  2x2 + 2y2 ? 4xy ; (by   1) = 2(x ? y )2  0 (by 16.2);

(16.6)

with at least one inequality holding strict because  < 1 or x > y . Hence,  is strictly convex, and therefore the stationary point , determined in (16.4), is a unique global minimum. (Note, the nominator of (16.4) is equal to 21 00( ) > 0, and hence does not vanish.)

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Main result It remains to be shown that  2 (0; 1), under reasonable circumstances. This question is answered by the following main result

Proposition 24 Suppose x  y . Diversi cation is optimal,  2 (0; 1), if and only if  < y =x .

Proof We need to show that  reaches a minimum at  2 (0; 1). Since  is convex (see (16.6), and (0) = Var(Y )  Var(X ) = (1), the proof is complete if 0(0) < 0, as illustrated in the diagram below. Insert = 0 into (16.5), and one obtains

0(0) = 2[Cov (X; Y ) ? y2]  2(xy ? y2) < 0 () y > x:

Example 4 Assume the joint probability distribution of two assets' random returns X and Y summarized in Table 4. Then one obtains

Pr(X; Y ) Y = 8 Y = 16  X=5 3=5 1=5 4=5 X = 30 3=20 1=20 1=5 3=4 1=4 1 

E (X ) = E (Y ) = 10; Var(X ) = 100; Var(Y ) = 12; Cov (X; Y ) = 0; and hence   0:107. Therefore, almost 11% of the entire investment should be put into the high risk security X .

Summary We have shown that the optimality of diversi cation hinges

upon the degree of association between the individual assets' returns. In particular, diversi cation reduces risk, and hence is bene cial, if either 1) both assets are equally risky (y = x), but  < 1, or 2) the correlation is

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var(Z )

6

var(X )



... ... ... ... ... ... .... ... . . . ... .... .... ... ... .... .... ... .... .... ... .... .... . . . .... .... .... ..... .... .... .... ..... .... ..... ..... .... .... . . . . ...... ..... ...... ...... ...... ....... ...... ..... ....... ...... ...... ...... . ...... . . . . . . . .......... ..................... ............................. ...........

... ...

.. ...

... ...

.. ...

. .. ..

var(Y ) 0



1

-

Figure 6: The Variance of a Mixed Portfolio negative or vanishes (  0). In turn, diversi cation actually increases risk if the two assets are highly correlated ( is positive and close to 1) and one is more risky than the other. Having obtained clear cut results, summarized by the necessary and sucient condition in Proposition 24, you may be ready to look at the generalized analysis, using a less restrictive measure of risk, in Sect. 14 of this Chapter.

16.2 The demand for \Arrow{bonds" Goods are distinguished by physical properties, time and place of delivery, and states of nature. In a complete market system there is a market for each and every good. Completeness may thus require an enormous number of markets | some of which may end up to be so thin as to make price taking utterly implausible. However, as Arrow30 showed in a justly celebrated contribution, a complete market system can be replicated by a surprisingly small number of markets. The reason is that a set of state contingent nancial securities | called \Arrow-bonds" | actually spans the full set of state contingent comArrow, K. [1953]. \Le r^ole des valeurs boursieres pour la repartition la meilleure des  risques", Econometrie , 11: 41{47; English translation in: Arrow, K. [1976]. Essays in the Theory of Risk Bearing. North{Holland, Essay 4. 30

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modity claims for goods distinguished by physical characteristics, date and place of delivery. In this section we get you acquainted with Arrow{bonds, and perform some instructive exercises on the demand for these nancial assets.31 We will not pursue their spanning property here, but return to it in the chapter on General Equilibrium Theory. So what is an Arrow{bond anyway? Suppose the economy is in either one of n mutually exclusive states of nature, denoted by i 2 f1; :::; ng, each of which occurs with probability pi > 0, pi = 1. An Arrow{bond is de ned as a nancial asset that promises to pay the bearer $ Xi > 0 if state i occurs, and nothing otherwise. Completeness of the market for Arrow{bonds means that there is one Arrow{bond for each and every state i 2 f1; : : : ; ng. In the following we assume that bond prices are linear, and units are chosen in such a way that one Arrow{bond costs $ 1. Now consider the portfolio choice of a price{taking agent who has a $ 1 endowment that s/he plans to invest in Arrow{bonds. The household is risk{averse with preferences represented by the von Neumann{Morgenstern utility function U : R + ! R U (y) := ln(y): (16.7) Denote the demand for type-i Arrow{bonds by ai  0. We distinguish two kinds of feasible portfolios, described by elements of the sets A and A, respectively

A := fa 2 R n+ j A := fa 2 R n+ j

X

X

ai  1g (feasible portfolios),

ai = 1g (feasible no{cash portfolios).

Optimal no{cash portfolios What portfolio is optimal if the household is committed to spend the entire wealth on Arrow{bonds? In order to nd the answer, you have to solve the well{behaved decision problem The analysis follows Arrow, K. [1970]. \The value and demand for information", in: McGuire, C. B. and R. Radner, eds., Decision and Organization, Chapter 6, reprinted in: Arrow, K. [1976]. Essays in the Theory of Risk Bearing. North{Holland, Essay 12. . 31

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max a

X

i

pi U (aiXi); s.t. a 2 A:

(16.8)

Denoting the Lagrangian of the constraint ai = 1 by   0, the solution is characterized by the following Kuhn{Tucker conditions pi   with = if a > 0; and a = 1: (16.9) i i ai Proposition 25 The optimal no{cash portfolio exhibits ai = pi ; 8 i = 1; : : : ; n: P

X

Proof By condition (16.9) a solution cannot exhibit ai = 0 for any i. There-

fore, a solution | if one exists | must satisfy 0 < ai = pi =, and ai = 1. Obviously, there exists precisely one solution:  = 1; ai = p: P

Remark 1 Notice that the Kuhn{Tucker conditions do indeed describe a

maximum. Also, notice that the optimal portfolio is una ected by the bond return Xi . This property is quite surprising, don't you agree?

16.2.1 When is it optimal to hold no cash? Of course, the household may choose to hold cash and not spend all wealth on Arrow{bonds. This leads us to search for necessary and sucient conditions for not holding cash in an optimal portfolio. A key role is played by the existence of what we call a sure{gain{portfolio, which is de ned as follows.

De nition 2 A feasible portfolio a 2 A is called \sure{gain{portfolio" if aiXi  1; 8 i 2 f1; : : : ; ng. Lemma 5 A sure{gain{portfolio exists if and only if 1=Xi  1. Proof 1) If a sure{gain{portfolio a 2 A exists, one has ai  1=Xi and therefore, 1  ai  1=Xi , which proves the assertion. 2) Suppose 1=Xi  1. P

P

P

P

We proceed by designing a sure{gain{portfolio. For this purpose set ai = X1 , which is obviously a sure{gain{portfolio. Therefore, it only remains to be shown that this portfolio is also an element of A. But this follows immediately from the assumption 1=Xi  1. i

P

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Proposition 26 The optimal portfolio does not include cash if and only if there exists a sure{gain{portfolio.

X  1 is necessary and sucient for not holding cash. In order to analyze the decision problem, introduce the slack{variable

Proof By the Lemma all we have to show is that b := 1 ?

X

P



1



i

ai:

Then we seek to maximize the expected utility X

pi U (aiXi + b); s.t.

X

ai + b = 1; ai  0; b  0;

the solution of which is characterized by the Kuhn{Tucker conditions

piXi U 0(aiXi + b)   with = if ai > 0; X

(16.10)

piU 0(aiXi + b)  ;

(16.11)

together with the constraint. 1) To prove necessity suppose b = 0 (no{cash portfolio). Then again, by (16.7), condition (16.10) cannot be satis ed at ai = 0, not for any i, so that (16.10) must be satis ed with equality, if a solution exists at all. Divide (16.10) by Xi , take the sum over all i, and one obtains, together with (16.2.1)



X

1=Xi =

X

pi U 0(aiXi + b)  :

Since  > 0 (recall (16.10) is satis ed with equality), one concludes 1=Xi  1, as asserted. 2) To prove suciency, we show that for any portfolio in which not all funds are invested, there is another one that is at least as good where all funds are invested. The assertion then follows immediately by the uniqueness of the optimal solution. Consider a portfolio a 2 A for which b := 1 ? ai  0, and another sure{gain{portfolio a 2 A. Now construct the portfolio, P

P

a^i := ai + bai:

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Since a is a sure{gain{portfolio one has

a^iXi = aiXi + baiXi  aiXi + b; and therefore, by the monotonicity of U X

piU (^aiXi ) 

X

piU (aiXi + b):

At the same time X

a^i =

X

ai + b

X

ai =

X

ai + b = 1;

a^i  0: Therefore, the portfolio a^ is feasible, does not include cash, and yet is at least as good as a.

17 Notes on the literature The exposition of the theory of stochastic dominance is based on Hadar and Russell (Hadar and Russell, 1969), (Hadar and Russell, 1971), Meyer (Meyer, 1989), Meyer and Ormiston (Meyer and Ormiston, 1983), and Ormiston (Ormiston, 1992). For further details the reader is referred to the collection of essays edited by Whitmore and Findlay (Whitmore and Findlay, 1982), especially the contributions by Fishburn and Vickson (Fishburn and Vickson, 1978) and Hadar and Russell (Hadar and Russell, 1982). The extension to third and higher order stochastic dominance is straightforward. The interested reader is referred to Whitmore (Whitmore, 1970). The applications on diversi cation of simple portfolios is based on Hadar and Russell (Hadar and Russell, 1982), (Hadar and Russell, 1971). If you want to know more about this topic take a look at Fisburn and Porter (Fishburn and Porter, 1976) and Hadar and Seo (Hadar and Seo, 1990). The application on the competitive rm is based on Hadar and Russell (Hadar and Russell, 1982) and Ormiston (Ormiston, 1992). If you want to know more about this topic, take a look at Sandmo's (Sandmo, 1971) seminal contribution. The distinction between the income and substitution e ect may have seem misplaced to you in the context of the theory of rm. In that case, take a look at Davis (Davis, 1989).

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The application on labor supply uses results derived in the analysis of social security risk by Prinz and Wolfstetter (Prinz and Wolfstetter, 1991). The application on mixed entry strategies in a Cournot oligopoly uses the entry model by Dixit and Shapiro (Dixit and Shapiro, 1986). The application on income inequality and stochastic dominance is based on a celebrated result by Atkinson (Atkinson, 1972). In the current literature the concept of Lorenz dominance was generalized so that it can also be applied to income distributions with di erent means, and the so called Atkinson theorem was extended correspondingly. If you are interested in these matters, take a look at Yitzhaki (Yitzhaki, 1982). For a general survey on income inequality, we also recommend Sen's (Sen, 1972) monograph very highly. In the literature there are many more applications of stochastic dominance. We mention the explanation of the optimal deductible in insurance contracts by Arrow (Arrow, 1963) and Eeckhoudt, Gollier and Schlesinger (Eeckhoudt, Gollier and Schlesinger, 1991), the theory of wage indexation by Adolph and Wolfstetter (Adolph and Wolfstetter, 1991), and Holmstrm's (Holmstrm, 1979) analysis of the principal{agent problem which will be discussed extensively in the chapter on Information and Incentives . Finally we refer the reader to the well{known ranking of risk proposed by Rothschild and Stiglitz (Rothschild and Stiglitz, 1970) (Rothschild and Stiglitz, 1971), and its critical evaluation by Landsberger and Meilijson (Landsberger and Meilijson, 1990).

References Adolph, B. and Wolfstetter, E. (1991). Wage indexation, informational externalities, and monetary policy, Oxford Economic Papers 43: 368{390. Arrow, K. J. (1963). Social Choice and Individual Values, 2 edn, Wiley, New York. Arrow, K. J. (1976a). The role of securities in the optimal allocation of risk{ bearing, in K. J. Arrow (ed.), Essays in the Theory of Risk Bearing, North{Holland, pp. 121{133.

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Arrow, K. J. (1976b). The value and demand for information, in K. J. Arrow (ed.), Essays in the Theory of Risk Bearing, North{Holland, pp. 267{ 278. Atkinson, A. B. (1972). On the measurement of inequality, Journal of Economic Theory 2: 244{263. Breen, W. and Savage, J. (1968). Portfolio distributions and tests of security selection models, Journal of Finance 23: 805{819. Dalton, H. (1920). The measurement of the inequality of incomes, Economic Journal. Davis, G. K. (1989). Income and substitution e ects for mean{preserving spreads, International Economic Review 30: 131{136. Dixit, A. and Shapiro, C. (1986). Entry dynamics with mixed strategies, in L. G. Thomas (ed.), The Economics of Strategic Planning, Lexington Books, Lexington, pp. 63{79. Eeckhoudt, L., Gollier, C. and Schlesinger, H. (1991). Increases in risk and deductibe insurance, Journal of Economic Theory 551: 435{440. Fishburn, P. C. and Porter, L. B. (1976). Optimal portfolios with one safe and one risky asset, Management Science 22: 1064{1073. Fishburn, P. C. and Vickson, R. G. (1978). Theoretical foundations of stochastic dominance, in G. A. Whitemore and M. C. Findlay (eds), Stochastic Dominance, Lexington Books, p. Kap.2. Hadar, J. and Russell, W. R. (1969). Rules for ordering uncertain prospects, American Economic Review 59: 25{34. Hadar, J. and Russell, W. R. (1971). Stochastic dominance and diversi cation, Journal of Economic Theory 3: 288{305. Hadar, J. and Russell, W. R. (1982). Applications in economic theory and analysis, in G. A. Whitmore and M. C. Findlay (eds), Stochastic Dominance, Lexington Books, pp. 295{333. Hadar, J. and Seo, T. K. (1990). The e ects of shifts in a return distribution on optimal portfolios, International Economics Review 31: 721{736.

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Harsanyi, J. C. (1976). Essays on Ethics, Social Behavior, and Scienti c Explanations, Reidel Publisher. Holmstrm, B. (1979). Moral hazard and observability, Bell Journal of Economics 10: 74{91. Landsberger, M. and Meilijson, I. (1990). A tale of two tails: An alternative characterization of comparative risk, Journal of Risk and Uncertainty 3: 65{82. Lorenz, M. O. (1905). Methods for measuring concentration of wealth, Journal of the American Statistical Association. Mandelbrot, B. (1963). The variation of certain speculative stock prices, Journal of Business 36: 394{419. Menezes, C. F. and Hanson, D. L. (1970). On the theory of risk aversion, International Economic Review 11: 481{487. Meyer, J. (1989). Stochastic dominance and transformations of random variables, in T. B. Fomby and T. K. Seo (eds), Studies in the Economics of Uncertainty: In Honor of Josef Hadar, Springer Verlag, New York| Berlin, pp. 45{57. Meyer, J. and Ormiston, M. B. (1983). The comparative statics of cumulative distribution function changes for the class of risk averse agents, Journal of Economic Theory 31: 153{169. Newbery, D. (1970). A theorem on the measurement of inequality, Journal of Economic Theory 2: 264{266. Ormiston, M. B. (1992). First and second degree transformations and comparative statics under uncertainty, International Economic Review 33: 33{44. Pratt, J. W. (1989). Utility functions, interest rates, and the demand for bonds, in T. B. Fomby and T. K. Seo (eds), Studies in the Economics of Uncertainty: In Honor of Josef Hadar, Springer Verlag, New York| Berlin, pp. 225{231. Prinz, A. and Wolfstetter, E. (1991). Unsichere Rente: wie reagieren Arbeitsangebot und Ersparnis?, Jahrbucher f. Nationalokonomie und Statistik.

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Rothschild, M. and Stiglitz, J. E. (1970). Increasing risk I: a de nition, Journal of Economic Theory 2: 225{243. Rothschild, M. and Stiglitz, J. E. (1971). Increasing risk II: its economic consequences, Journal of Economic Theory 3: 66{84. Sandmo, A. (1971). On the theory of the competitive rm under price uncertainty, American Economic Review 61: 65{73. Sen, A. (1972). On Economic Inequality, Oxford University Press. Sharpe, W. F. (1970). Portfolio Theory and Capital Markets, McGraw{Hill. Whitmore, G. A. (1970). Third{degree stochastic dominance, American Economic Review 60: 457{459. Whitmore, G. A. and Findlay, M. C. (eds) (1982). Stochastic Dominance, Lexington Books. Yitzhaki, S. (1982). Stochastic dominance, mean variance and Gini's mean di erence, American Economic Review 72: 178{185.