Finance TheoryExpected Utility Theory, Pareto Efficiency and Walrasian Equilibrium

Instructor: Chyi-Mei Chen(Tel) 3366-1086(Email) cchen@ccms.ntu.edu.tw(Website) http://www.fin.ntu.edu.tw/∼cchen/

1. There are three dimensions of an investor that matter in asset trading:the investor’s initial endowments, her preferences, and her beliefs (orsubjective probabilities) about the uncertain future. In a two-periodeconomy where an investor trades assets at date 0 and receives returnsat date 1, the investor’s endowments are usually represented by herdate-0 initial wealth W0 > 0, and her preferences represented by avon Neumann-Morgenstern utility function u(·) for her date-1 randomwealth W . It has been a standard assumption in finance theory thatan investor in making her date-0 investment decisions seeks to maxi-mize E[u(W )], where the expectation E[·] is taken using the investor’ssubjective probability distribution for W .

2. A decision maker facing a set A of feasible alternatives is said to berational if she is endowed with a weak preference relation ≽ on A.1 Con-sider a date-0 investment environment where all investment projects (orlotteries) generate cash flows only at date 1, and there are N possiblelevels of date-1 cash flows, denoted by z1 < z2 < · · · < zN . DefineZ ≡ z1 < z2 < · · · < zN. In this case, an investment project (or alottery) p is a probability distribution over the set Z, which generatesconsumption level z ∈ Z with probability p(z). Let P be the set ofall feasible lotteries; that is, P contains all probability distributionsover the Z. We call P the investment opportunity set. From now on,

1A binary relation ≽ on A is complete if for all a, b ∈ A, either a ≽ b or b ≽ a (or both),and it is transitive if for all a, b, c ∈ A, a ≽ c whenever a ≽ b and b ≽ c. A binary relation≽ on A is a weak preference on A if it is both complete and transitive. From a weakpreference ≽ on A, we can derive the strict preference ≻ on A as follows. For all a, b ∈ A,a ≻ b if it is not true that b ≽ a. Similarly, we can derive the indifference relation ∼ on Afrom the weak preference ≽ on A: for all a, b ∈ A, we have a ∼ b if both a ≽ b and b ≽ a.

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an investor facing P is called rational if she is endowed with a weakpreference ≽ on P.

3. Our first main result is that, a rational investor’s weak preference onP satisfies the following three behavioral assumptions if and only if sheis an EU-maximizer.

Axiom 1 (Axiom of Reduction) For all a ∈ [0, 1] and for allp, r ∈ P, the investor feels indifferent about the simple lottery ap +(1 − a)r (which yields the consumption level z ∈ Z with probabilityap(z) + (1− a)r(z)) and the compound lottery2 that with probability ahe gets to take the lottery p and with probability 1− a he gets to takethe lottery r.3

Axiom 2 (Independence Axiom) For all p, q, r ∈ P and a ∈ (0, 1],p ≻ q ⇒ ap+ (1− a)r ≻ aq + (1− a)r.

Axiom 3 (Continuity Axiom) For all p, q, r ∈ P, p ≻ q ≻ r ⇒ap+ (1− a)r ≻ q ≻ bp+ (1− b)r for some a, b ∈ (0, 1).

Here comes our first main result.

2While a simple lottery is a probability distribution on Z, a (two-stage) compoundlottery is a probability distribution on P. In other words, the outcome of a simple lotteryis a cash flow z ∈ Z, but the outcome of a compound lottery may be an opportunity to playanother lottery p ∈ P. Note also that the above assumed date-0 investment environmenthas ruled out investment projects involving cash outflows at date 0. Moreover, if thelottery p represents one unit of a traded security, holding 0.5 units of that security maynot be a feasible investment project at date 0. To see this, suppose that p generates zNwith probability one. Then 0.5 units of p will generate 0.5zN for sure, but it may happenthat zj = 0.5zN for all j = 1, 2, · · · , N .

3Suppose that N = 2 and zj = j, for all j = 1, 2. Suppose that p, q, r ∈ P are suchthat

p =

13

23

, q =

23

13

, r =

12

12

.

Axiom 1 says that a compound lottery that allows the investor to take p and q both withprobability 1

2 is regarded as equivalent to r by the investor.

2

Theorem 1 A weak preference ≽ on P has an expected utility functionrepresentation if and only if it satisfies Axioms 1-3. That is, there existsa utility function u : Z → ℜ such that

∀p, q ∈ P, p ≻ q

⇔ H(p) ≡∑z∈Z

p(z)u(z) ≡ Ep[u] > Eq[u] ≡∑z∈Z

q(z)u(z) ≡ H(q),

if and only if Axioms 1,2, and 3 hold.

Moreover, if there exist p, q ∈ P such that p ≻ q, then the utilityfunction u is unique up to a positive affine transformation in the sensethat if v : Z → ℜ is such that

∀p, q ∈ P,∑z∈Z

p(z)v(z) >∑z∈Z

q(z)v(z) ⇔ p ≻ q,

then there exist some a ∈ ℜ and b ∈ ℜ++ such that for all z ∈ Z,

u(z) = a+ bv(z).

4. The above u(·) is referred to as a von Neumann-Morgenstern (VNM)utility function (cf. von Neumann and Morgenstern (1953)), whichis defined on Z. Note that u is a cardinal utility function. We usu-ally take Z to be ℜ+, representing the set of (non-negative) consump-tion or wealth levels.4 In a dynamic setting where an investor facesa stream of lotteries, we usually assume that the investor maximizesa discounted sum of temporal VNM utility functions. (We say thatthe investor’s utility function is time-additive or time-separable.) Incontinuous-time models, for example, we usually let T = [0, T ] de-note the time span, and the investor’s objective function is representedas E0[

∫t∈T e−ρtu(ct)dt], where E0[·] is the expectation operator condi-

tional on the time-0 information, ct is the investor’s random time-tconsumption, and ρ and e−ρt are referred to as respectively the dis-count rate and the discount factor. In discrete-time models, we usuallylet T = 0, 1, 2, · · · , t, t + 1, · · · , T, and we assume that the investor

4However, for the case where Z = ℜ+, the three axioms specified above are no longersufficient for the preference to be representable by an expected utility function; a fourthaxiom called “the sure thing principle” is now needed. See Kreps (1988).

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seeks to maximize E0[∑T

t=0 δtu(ct)], where δ ∈ (0, 1) is the discrete-time

discount factor.5

5. From now on, investors are assumed to maximize expected utility whenmaking investment decisions. An investor is represented by a pair(W0, u), where W0 is the investor’s (non-random) initial wealth, whichconsists of cash, and u(·) is the investor’s VNM utility function forterminal wealth. We shall consider the investment problem facing aninvestor endowed with VNM utility function u : ℜ+ → ℜ. A fair gam-ble is a lottery that generates zero expected profits. An investor withVNM utility function u is risk neutral (respectively, risk averse andrisk seeking) if given any initial wealth W0 and facing any fair gamblez, she would feel indifferent about (respectively, become worse off, andbecome better off) taking the fair gamble z:

E[u(W0 + z)] = (respectively, ≤ and ≥)u(W0).

6. We shall need to know some properties of a concave function. A func-tion f : ℜn → ℜ is concave (respectively, strictly concave) if for allx,y ∈ ℜn and for all λ ∈ [0, 1],

f(λx+ (1− λ)y) ≥ (respectively, >)λf(x) + (1− λ)f(y).

A function f is convex (respectively, strictly convex) if −f is concave(respectively, strictly concave). A function f is affine, if both f and−f are concave. An affine function is linear if f(0) = 0. A twice-differentiable function f : ℜn → ℜ is concave (respectively, strictlyconcave) if and only if its Hessian is everywhere negative semi-definite(respectively, negative definite). This implies that a twice-differentiablefunction f : ℜ → ℜ is concave (respectively, strictly concave) if andonly if f ′′ ≤ 0 (respectively, f ′′ < 0).6

We shall frequently need to use the following lemma.

5This time-additive utility function gives rise to the so-called equity premium puzzle infinance literature, which we shall explain in a subsequent lecture.

6Recall the following definitions.

• A symmetric matrix An×n is said to be positive definite (or PD), if for all xn×1 =0n×1, the quadratic form xTAx > 0. (The quadratic form is a second-degreepolynomial in x. With x given, it becomes a scalar.) A symmetric matrix An×n is

4

Lemma 1 (Jensen Inequality) If f : ℜ → ℜ is concave (respec-

said to be negative definite (or ND), if −A is positive definite. For example, thematrix

A2×2 =

2 0

0 1

is PD, and the matrix

B2×2 =

−2 0

0 −3

is ND.

• A symmetric matrix An×n is said to be positive semi-definite (or PSD), if for allxn×1 ∈ Rn, the quadratic form xTAx ≥ 0. A symmetric matrix An×n is said tobe negative semi-definite (or NSD), if −A is positive semi-definite. For example,the matrix

C2×2 =

2 0

0 0

is PSD, and the matrix

D2×2 =

0 0

0 −3

is NSD.

• Consider a twice differentiable function f : ℜn → ℜ. Let the Df : ℜn → ℜn be thevector function

Df =

∂f∂x1∂f∂x2

...∂f∂xn

,

which will be referred to as the gradient of f . Let D2f : ℜn → ℜn2

be the matrixfunction

D2f =

∂2f

∂x1∂x1

∂2f∂x1∂x2

· · · ∂2f∂x1∂xn

∂2f∂x2∂x1

∂2f∂x2∂x2

· · · ∂2f∂x2∂xn

......

......

∂2f∂xn∂x1

∂2f∂xn∂x2

· · · ∂2f∂xn∂xn

,

which will be referred to as the Hessian of f .

Note that when n = 1, the gradient of f becomes f ′ and the Hessian of f becomes f ′′.

5

tively, convex), and x is a random variable with finite E[x], then7

E[f(x)] ≤ (respectively, ≥)f(E[x]).

The preceding lemma implies the following result.

Theorem 2 An investor endowed with a VNM utility function u(·) isrisk averse if and only if u(·) is concave.

7. Given a VNM utility function u : ℜ+ → ℜ with u′(x) > 0 > u′′(x) forall x > 0, we can define two new functions Ru

A and RuR as follows.

RuA(x) = −u′′(x)

u′(x), Ru

R(x) = −xu′′(x)

u′(x), ∀x > 0.

The two functional values RuA(x) and Ru

R(x) are referred to respectivelyas the Arrow-Pratt measures for absolute and for relative risk aversionat the wealth level x; see Arrow (1970) and Pratt (1964). Observe thatif Ru

A(·) = ρ > 0 is a constant function, then u(x) = −e−ρx,8and inthis case, we refer to u(·) as a CARA (constant absolute risk aversion)utility function with ρ being the associated coefficient of absolute riskaversion. (If ρ = 0, then apparently u(z) = z, so that the investor isrisk neutral.) Similarly, u(·) is a CRRA (constant relative risk aversion)utility function if Ru

R(·) is a constant function. In this case eitheru(x) = log(x) or u(x) = xp for some p ∈ (0, 1) (up to a positive affinetransform).9 If Ru

A(·) is an increasing (decreasing) function, then u(·)7Readers interested in other properties of concave functions can look up Tiel (1984).8Note that

ρ = RuA(x) = −u′′(x)

u′(x)= −d log(u′(x))

dx⇒ −ρx+ k = log(u′(x))

⇒ e−ρx+k = u′(x) ⇒ u(x) = a− be−ρx

for some k, a,∈ ℜ, b ∈ ℜ++. Since a VNM utility function is unique up to a positive affinetransform, we can, for example, pick a = 0, b = 1.

9Note that if RuR(x) = ρ > 0 for some constant ρ, then we have

ρ

x= Ru

A(x) = −u′′(x)

u′(x)= −d log(u′(x))

dx⇒ −ρ log(x) + k = log(u′(x)),

6

is referred to as an IARA (DARA) utility function. Some evidence hassuggested that most investors have DARA preferences, and in this caseif u′′′ exists, then one can show that u′′′ ≥ 0.

8. In a static setting, investors trade assets at date 0 and the assets gener-ate cash flows at date 1. Let p and x be the date-0 price of an asset andthe date-1 (random) cash flow generated by the asset.10 We refer to x

p,

xp− 1 and E[ x

p− 1] as respectively the dollar return, the rate of return,

and the expected rate of return on the asset. The asset is referred toas riskless if x

pis non-random, and in this case we denote E[ x

p− 1] by

rf . In the presence of a riskless asset, we refer to xp− (1 + rf ) and

E[ xp] − (1 + rf ) as respectively the excess rate of return and the risk

premium on the asset.

9. Suppose that an investor with initial wealth W0 can trade a risklessasset and a risky asset at date 0, and let r be the rate of return on therisky asset. Let a be the amount of her initial wealth to be invested inthe risky asset. Define f(a) = E[u((W0 − a)(1 + rf ) + a(1 + r))]. Notethat f(a) is the expected utility that the investor can obtain from thedate-1 wealth

W ≡ (W0 − a)(1 + rf ) + a(1 + r),

which is partially determined by the choice of a.11 Assume u′ > 0 > u′′.

and we have u′(x) = ekx−ρ, so that if ρ = 1, u(x) = a + b log(x); and if ρ = 1, thenu(x) = a + bx1−ρ, for some k, a,∈ ℜ, b ∈ ℜ++. Again we can pick a = 0 and b = 1. Tomake sure that u′ > 0 > u′′, we assume that ρ ∈ (0, 1).

10In a pure exchange economy where consumers are endowed with consumption goodsbut they cannot produce any of those goods (an unrealistic assumption; I know), x is anexogenous variable, but p, which is determined in the equilibrium of financial markets, isan endogenous variable. In a production economy where people can produce new goods(or greater amounts of existing goods) using existing goods, x must be determined en-dogenously also. However, it is standard in financial engineering that we take the pricesof some traded assets (called underlying assets) as exogeneous and then determine endoge-nously the prices of other assets (called derivative assets) accordingly. This procedureis subject to criticism because economic equilibrium should impose some restrictions onthe prices of underlying assets, and arbitrarily assuming the behavior of those prices mayimply inconsistency with investors’ rationality, or with the markets clearing condition.

11If a < 0, then the investor is selling the risky asset short; and if a > W0, then theinvestor is selling the riskless asset short, or is borrowing.

7

Under some mild conditions,12 we have

f ′(a) = E[∂

∂au((W0 − a)(1 + rf ) + a(1 + r))]

= E[u′(W0(1 + rf ) + a(r − rf ))(r − rf )],

and

f ′′(a) = E[∂2

(∂a)2u((W0 − a)(1 + rf ) + a(1 + r))]

= E[u′′(W0(1 + rf ) + a(r − rf ))(r − rf )2],

where note that u′′(W0(1+rf )+a(r−rf )) is a negative random variableand (r − rf )

2 is a positive random variable, so that f ′′(a) < 0. Thisshows that f(a) is strictly concave in a. The investor seeks to

maxa∈ℜ

f(a);

that is, the investor would like to find the optimal amount of money a∗

that should be spent on the risky asset, taking the asset returns (r, rf )as given, where optimality means that the investor’s expected utilityis maximized.13 Since f(·) is concave, our mathematical review showsthat the optimal amount a∗ to be invested in the risky asset can befound by solving the first-order condition f ′(a∗) = 0 if f ′(+∞) < 0;one necessary condition for the existence of a solution to f ′(a∗) = 0 isthat the probabilities of the two events r > rf and r < rf are bothpositive, which we shall assume to hold from now on.

Lemma 2 If u(·) is twice-differentiable with u′ > 0 > u′′, then a∗ > 0as long as E[r] > rf . That is, no matter how risk-averse the investormay be, she will take a long position in the risky asset.

12In general, what is needed is a uniform integrability condition; see Section 4.5 of Chung(1974).

13Hence we have assumed that markets are perfect. In particular, the investor is assumedto be a price-taker. (Where?)

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Theorem 3 Assume that E[r] > rf . Other things being equal, anincrease in W0 leads to an increase (repsectively, a decrease) in a∗ ifRu

A is a decreasing (respectively, increasing) function. Other thingsbeing equal, an increase in W0 leads to an increase (repsectively, adecrease) in a

W0(which is the portfolio weight for the risky asset) if Ru

R

is a decreasing (respectively, increasing) function.

Example 1 Suppose that Mrs. A’s VNM utility function is u(x) =√x. Suppose that her initial wealth is W0 = 100, 000, and that the

riskless lending and borrowing rate is rf = 0. The rate of return on therisky asset, r, is equally likely to be −0.1 and 0.5. How much shouldshe borrow or lend? (Show that she should borrow 700,000.)

Example 2 Suppose that Mrs. B’s VNM utility function is u(x) =−e−x, and the riskless rate is rf = 0.1. The rate of return on the riskyasset, r, is a normal random variable with mean 0.3 and variance 0.04.Mrs. B has initial wealth W0 = 1, 000, 000. How much should she spendon the risky asset?14

14Hint: Given a random variable z, its moment generating function is defined as

Mz(t) ≡ E[etz], ∀t > 0.

If z ∼N(µ, σ2), then

Mz(t) = etµ+t2σ2

2 .

Now, consider an investor with von Neumann-Morgenstern utility function u(x) = −e−Ax,where A > 0 is the investor’s measure of absolute risk aversion. The investor seeks to

maxE[u(W )] = −E[e−AW ],

where the terminal wealth W is normally distributed

W ∼ N(µW , σ2W ).

Note that−AW ∼ N(−AµW , A2σ2

W ),

and hence the investor’s objective function can be re-written as

E[u(W )] = −M−AW (1) = −e−AµW+ 12A

2σ2W

= −e−A[µW− 12Aσ2

W ] = u(µW − 1

2Aσ2

W ),

9

10. Suppose that at time 0 two agents U and V are faced with 3 possibletime-1 states, ω1, ω2, ω3. Let us call Ω ≡ ω1, ω2, ω3 the sample space.Let P and Q be U’s and V’s subjective probabilities regarding the 3possible time-1 states. A (real-valued) random variable (r.v.) is roughlya function mapping from Ω into ℜ. Consider the probability measuresP and Q and the 6 random variables described in the table below.

prob. and r.v./state ω1 ω2 ω3

P 12

12

0Q 0 1

434

x 1 2 3y 1 2 3z 1 2 4s 5 2 4t 4 6 7w 6 4 1

The table says that U assigns zero probability to the event ω3 and Vassigns zero probability to the event ω1. Apparently, x : Ω → ℜ andy : Ω → ℜ are the same function, and hence we say the two randomvariables are equal, and we write x = y. Although x : Ω → ℜ andz : Ω → ℜ are not the same function, but from U’s perspective, theevent

ω ∈ Ω : x(ω) = z(ω) = ω3

may occur only with probability zero. Hence we say that x = z P -almost surely. Similarly, from V’s perspective, z and s are nearly the

where µW − 12Aσ2

W is referred to as the certainty equivalent of W for the investor.Since u′(·) > 0, the solution that solves

maxE[u(W )] = u(µW − 1

2Aσ2

W )

must be identical to the solution that solves

maxµW − 1

2Aσ2

W .

For this reason, the investor essentially has a mean-variance utility function.

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same (the two functions differ only on an event that, from V’s perspec-tive, may occur with zero probability), and hence we say that z = sQ-almost surely. Finally, neither U nor V would regard t and w as thesame function, but note that from U’s perspective, these two randomvariables share the same distribution function; that is, for each andevery r ∈ ℜ, we have

P (ω ∈ Ω : t(ω) ≤ r) = P (ω ∈ Ω : w(ω) ≤ r),

and hence we say that from U’s perspective, t and w are equal in distri-bution. Note that given any constant W0 and given any von Neumann-Morgenstern utility function u(·), we have E[u(W0+ t)] = E[u(W0+w)]as long as t and w are equal in distribution. Finally, observe that x = yimplies that x = y P -almost surely for any probability measure P de-fined on Ω; and if x = y P -almost surely under some probabilitymeasure P , then under P , x and y must share the same distributionfunction.

11. Sometimes we need a criterion to rank the riskiness of investmentprojects. Here we go over two prevalent criteria. The first criterionis called first-degree or first-order stochastic dominance, and it is de-fined as follows. Consider two risky assets h and g of which the ratesof return rh and rg take values in a common support, say the unit in-terval [0, 1]. (The compact-support assumption is made only to easethe exposition, and the following main results hold for general distri-butions.) Let the distribution functions of rh and rg be respectively Hand G, where H and G are such that H(z) = G(z) = 0 if z < 0 andH(z) = G(z) = 1 if z ≥ 1.

Now imagine that we are able to invite all investors whose VNM utilityfunctions’ first derivatives are non-negative to rank these two distri-bution functions. Assume that all investors are endowed with a non-random initial wealth, say 1 dollar. (Again, the one-dollar assumptionis made only for ease of demonstration.) Let U1 be this set of VNMutility functions. We say that H (stochastically) dominates G in thefirst degree (or in the first order), written as H ≥FSD G, if and only ifevery investor contained in U1 (weakly) prefers H to G; that is, if and

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only if ∫[0,1]

u(1 + z)dH(z) ≥∫[0,1]

u(1 + z)dG(z), ∀u ∈ U1.

Theorem 4 The following statements are equivalent.(i) The distribution functions H,G are such that H(z) ≤ G(z) for allz ∈ ℜ.(ii) H ≥FSD G.(iii) There exists a random variable e ≥ 0 such that rh and rg + e havethe same distribution function; that is, rh and rg+e are equal in distribution.15

Now, imagine that the reference group consists instead of all investorswhose VNM utility functions are weakly concave. The correspondingcriterion with this new reference group is called the second-degree orsecond-order stochastic dominance. More precisely, let U2 be the setof weakly concave VNM utility functions; i.e. those u(·) with u′′ ≤ 0almost everywhere on (1, 2). We say that H (stochastically) dominatesG in the second degree (or in the second order), denoted by H ≥SSD G,if and only if every investor with her VNM utility function u containedin U2 (weakly) prefers H to G; that is, if and only if∫

[0,1]u(1 + z)dH(z) ≥

∫[0,1]

u(1 + z)dG(z), ∀u ∈ U2.

Theorem 5 The following statements are equivalent.(i) The distribution functions H,G are such that (A) S(y) ≡

∫ y0 H(z)−

G(z)dz ≤ 0 for all y ∈ ℜ; and (B) rh and rg have the same expectedvalue.(ii) H ≥SSD G.(iii) There exists a random variable e such that for each and everyrealization of rh, E[e|rh] = 0, and such that rg and rh + e have thesame distribution function.

Proposition 1 If x ≥SSD y, then E[x] = E[y] and var[x] ≤var[y].

15

12

For more information about the first-degree and second-degree stochas-tic dominance, see Hadar and Russell (1969).16 Note that for a risk-averse investor endowed with non-random initial wealth and two mutu-ally exclusive investment projects H and G, G should never be chosenover H if H ≥SSD G.17 Based on this idea, empirical methods havebeen developed to investigate whether there coexist two traded assets ofwhich the rates of return satisfy the relation of second-degree stochas-tic dominance.18 If such a pair of traded assets can be found, it willbe taken as evidence that financial markets are less than fully efficient.Note that second-degree stochastic dominance really does not renderthe latter implication, for it is only natural that investors also takepositions in other risky assets besides the pair of assets under compari-son, and in that case an investor may rationally hold the stochasticallydominated asset for hedging reasons.

12. Now we go over the notion of Pareto efficiency, and relate Walrasian

16Hadar, J., and W. Russell, 1969, Rules for ordering uncertain prospects, AmericanEconomic Review, 59, 25-34.

17This follows from statement (iii) in the preceding theorem. Without loss of generality,assume that the initial wealth is one dollar, and the VNM utility function u is such that0 > u′′. We have

E[u(1 + rg)] = E[u(1 + rh + e)] = E[E[u(1 + rh + e)|rh]]

≤ E[u(E[1 + rh + e|rh])] = E[u(1 + rh)],

where the first equality follows from the fact that E[u(w)] = E[u(z)] whenever w and zare equal in distribution, the second equality follows from the law of iterated expectations,and the inequality follows from Jensen’s inequality for conditional expectations. Note thatwhen the initial wealth is random, the conclusion is no longer true. For example, let kbe a positive constant, rg = k − z, rh = k, and let z, a fair gamble, be the investor’srandom initial wealth. In this case, H ≥SSD G obviously, but the expected utility fromtaking project G is u(k), and the expected utility from taking project H is E[u(k + z)] <u(E[k + z]) = u(k), where the inequality follows from Jensen’s inequality. Thus projectG should be chosen over H for the risk-averse investor whose random initial wealth isz. The idea is that, a (more) risky project may turn out to be a better instrument forhedging, when a risk-averse investor is already suffering from an endowment risk (we callit a background risk).

18See for example Kaur, A., B. Rao, and H. Singh, 1994, Testing for Second-OrderStochastic Dominance of Two Distributions, Econometric Theory, 10, 849-866. See alsothe references therein.

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equilibrium to Pareto efficiency. For ease of exposition, we shall restrictattention to an economy consisting of two investors, U and V, that canconsume only one single consumption good at dates 0 and 1. Thereare n possible date-1 states, and we assume that U and V agree thatstate j may occur with probability πj > 0. Moreover, we assume thatthe aggregate amount of consumption is C0 > 0 at date 0 and Cj instate j at date 1. Let a0 ≥ 0 and aj ≥ 0 denote respectively U’samounts of date-0 and date-1-state-j consumption. Let b0 ≥ 0 andbj ≥ 0 denote respectively V’s amounts of date-0 and date-1-state-jconsumption. Define

C(n+1)×1 =

C0

C1...Cn

,

a(n+1)×1 =

a0a1...an

,and

b(n+1)×1 =

b0b1...bn

.U and V are then assumed to maximize respectively

U(a) ≡ u0(a0) +n∑

j=1

πju1(aj)

and

V (b) ≡ v0(a0) +n∑

j=1

πjv1(aj),

where the four functions u0, u1, v0, v1 : ℜ+ → ℜ are strictly increasingand strictly concave.

14

Definition 1 Given the aggregate consumption vector C, a,b is a

feasible allocation if a,b ∈ ℜ(n+1)+ and a + b = C. A feasible allo-

cation a,b is Pareto efficient or Pareto optimal if there does not

exist another feasible allocation a, b such that U(a) ≥ U(a) andV (b) ≥ V (b) with at least one of these inequalities being strict.

Theorem 6 A feasible allocation a,b is Pareto efficient if and onlyif there exist two real numbers α, β ≥ 0, with α2 + β2 = 0, such thata,b solves the following maximization problem:

maxa,b∈A

αU(a) + βV (b),

where A denotes the set of feasible allocations.

Theorem 7 A feasible allocation a,b is Pareto efficient if and onlyif one of the following three situations occur:(i) a = C;(ii) b = C;(iii) there exists a constant µ > 0 such that

u′0(a0)

v′0(b0)=

u′1(aj)

v′1(bj)= µ, ∀j = 1, 2, · · · , n;

that is, the Borch rule holds.

Corollary 1 Suppose that a,b is a Pareto efficient allocation suchthat a = 0 = b. Then for all j, k ∈ 1, 2, · · · , n, j = k, if bj > bk,then aj > ak, so that Cj > Ck also.

13. To attain a Pareto efficient allocation, U and V can sign a risk-sharingcontract. However, for an economy with a lot of people, it may beinfeasible for those people to attain Pareto efficient allocations via pri-vate contracting. In this case one is curious about how securities trad-ing may replace the role of private contracting and help people attain

15

Pareto efficient allocations. It turns out that the competitive equilib-rium allocations of a securities markets economy are Pareto efficient ifsome conditions are met. In particular, this will be true if (i) marketsare complete (see below for a formal definition); or (ii) if a riskless assetand a risky asset tracking the date-1 aggregate consumption are tradedand all the people have linear risk tolerance with identical risk cautious-ness; or (iii) equilibrium securities returns are elliptically distributed.The conditions stated in either (ii) or (iii) imply that two-fund separa-tion holds; see a more detailed discussion my notes for the Investmentscourse.

14. Continue to assume that the economy has only two investors U andV. Suppose that there are m assets traded in date-0 perfect financialmarkets, which generate returns (in amounts of the single consumptiongood) at date 1. More precisely, suppose that there are n possiblestates at date 1, and for all j = 1, 2, · · · ,m, one share of asset j willgenerate xkj units of consumption in state k, for all k = 1, 2, · · · , n.Let Xn×m be the matrix of which the (k, j)-th element is xkj. We shalldenote asset j by the (n × 1)-vector xj, which is the j-th column ofmatrix X. Correspondingly, let pj be the date-0 market-clearing priceof asset j, and let the (m × 1)-vector p be such that its j-th elementis pj. The pair (X,p) will be referred to as a price system, where notethat X and p are respectively exogenous and endogenous variables,and their elements represent respectively date-1 and date-2 cash flows(units of consumption).

15. Assume that U is endowed with g0 > 0 units of date-0 consumptionand gj > 0 units of asset j, and let gm×1 be the vector whose j-thelement is gj. Assume that V is endowed with h0 > 0 units of date-0consumption and hj > 0 units of asset j, and let hm×1 be the vectorwhose j-th element is hj. Then the pre-trade allocation is

(

[g0Xg

],

[h0

Xh

]),

and assuming that each traded asset has unit supply, we have the ag-

16

gregate consumption being

C =

[g0 + h0

X1

].

16. The above 2-period 2-people m-asset n-state competitive economy canbe concisely written as

E = u0, u1, v0, v1, π, g0,g, h0,h,X,

in which u0, u1, v0, v1 stand for the investors’ preferences, the vector

π =

π1

π2...πn

stands for the investors’ homogeneous beliefs, g0,g, h0,h stand for theinvestors’ endowments, and X fully describes the m traded assets.These are all premitives or data or exogenous parameters of the com-petitive economy.

Our task is to characterize the equilibrium prices p and the equilibriumconsumption and portfolio policies (g∗0,g

∗) and (h∗0,h

∗) for the twoinvestors. These three pieces together define a Walrasian equilibriumas the following definition states.

Definition 2 A competitive equilibrium (or a Walrasian equilibrium)of economy E is a triple (p, (g∗0,g

∗), (h∗0,h

∗)) such that(i) given p, the consumption and portfolio policy (g∗0,g

∗) solves U’sutility-maximization problem

maxa0,dm×1

U(a)

subject toa0 + p′d ≤ g0 + p′g,

a =

[a0Xd

]∈ ℜn+1

+ ;

17

(ii) given p, the consumption and portfolio policy (h∗0,h

∗) solves V’sutility-maximization problem

maxb0,tm×1

V (b)

subject tob0 + p′t ≤ h0 + p′h,

b =

[b0Xt

]∈ ℜn+1

+ ;

and(iii) the date-0 markets for the consumption good and for the m assetmarkets all clear given p; that is,

g∗ + h∗ = g + h = 1,

andg∗0 + h∗

0 = g0 + h0.

Definition 3 An asset is represented as a vector z ∈ ℜn, such thatits k-th element, zk, stands for the amount of consumption that onecan get in state k at date 1 if holding 1 unit of asset z from date 0 todate 1. The date-0 asset markets are complete if for each conceivableasset z ∈ ℜn, there correspondingly exists some portfolio ym×1 of the mtraded assets such that z = Xy. We say that the date-0 asset marketsare incomplete if they are not complete.

Definition 4 A type-0 arbitrage opportunity is a portfolio q0 of the mtraded assets such that p′q0 < 0 and Xq0 ≥ 0n×1.

19 A type-1 arbitrageopportunity is a portfolio q1 such that p′q1 ≤ 0 and Xq1 > 0n×1. If theprice system (X,p) admits no type-0 or type-1 arbitrage opportunities,then the price system is arbitrage free.

19We write zn×1 ≥ 0n×1 if for all j = 1, 2, · · · , n, the j-th element zj of z is greaterthan or equal to zero. We write z > 0, if z ≥ 0 and z = 0. We write z >> 0 if for allj = 1, 2, · · · , n, the j-th element zj of z is greater than zero.

18

In plain words, a type-0 arbitrage opportunity is a portfolio strategythat allows an investor to make money at date 0, and to never losemoney at date 1; and a type-1 arbitrage opportunity is a portfoliostrategy that allows an investor to not lose money at date 0, and tomake money in at least one state at date 1.

Theorem 8 Suppose that (p, (g∗0,g∗), (h∗

0,h∗)) is a competitive equilib-

rium for economy E. Then the price system (X,p) admits no arbitrageopportunities.

The preceding theorem says that one necessary condition for competi-tive equilibrium is that the equilibrium prices p together with X shouldadmit no arbitrage opportunities. Our next theorem gives an equiva-lent condition for the absence of arbitrage opportunities from the pricesystem (X,p).

Theorem 9 The price system (X,p) is arbitrage free if and only ifthere exists a vector f ∈ Rn

++ such that p = X′f , or equivalently, forall j = 1, 2, · · · ,m,

pj =n∑

k=1

fkxkj.

The vector f is called a state price vector.20

An immediate consequence of the preceding theorem is that followingcorollary.

Corollary 2 An arbitrage-free price system (X,p) admits a uniquestate price vector if and only if the date-0 asset markets are complete;that is, if and only if the rank of X is

ρ(X) = n.

20Its elements are nothing but the date-0 prices of the n Arrow-Debreu securities.

19

Theorem 10 Suppose that a competitive equilibrium (p, (g∗0,g∗), (h∗

0,h∗))

exists. If ρ(X) = n, then the equilibrium allocation

(

[g∗0Xg∗

],

[h∗0

Xh∗

])

is Pareto efficient.

17. Now we review the martingale pricing theorem. Suppose in a two-period (dates 0 and 1) economy there is one single consumption good(i.e., money) and one single investor with date-0 endowment e0 (herinitial wealth) and date-1 endowment e1. Assume that there are npossible date-1 states, denoted by ω1, ω2, · · · , ωn. Let Ω be the setcontaining these n states. The outcome or realization of e1 in state ωi

is written e1(ωi), or simply e1i. At date 0, financial markets open, andassets are available for trading. First we consider an ideal situationwhere there are n assets traded in the date-0 markets, where one unitof the j-th asset will generate 1 dollar in state ωj but nothing in otherstates, and it is called the state-ωj Arrow-Debreu security. We saythat the date-0 markets are complete because all the n Arrow-Debreusecurities are available for trading at date 0. The date-0 price forthe state-ωj Arrow-Debreu security is denoted by ϕωj

, or simply ϕj.The single investor seeks to maximize the sum of her expected utilitiesobtained at dates 0 and 1:

maxc0,c1(ωj)

u0(c0) +n∑

j=1

πju1(c1(ωj))],

subject to

c0 +n∑

j=1

ϕjc1(ωj) = e0 +n∑

j=1

ϕje1(ωj),

where (i) for t = 0, 1, ut(·) is the investor’s date-t VNM utility function,with u′

t > 0 ≥ u′′t ; (ii) ∀j = 1, 2, · · · , n, πj > 0 is the probability for the

event that the date-1 state of the world is ωj; (iii) c0 is the amount ofmoney that the investor decides to spend at date 0; and (iv) c1(ωj), orsimply c1j, is the amount of money she decides to spend at date 1 whenthe date-1 state is ωj. Note that, since 1 unit of the j-th Arrow-Debreu

20

security pays 1 dollar in and only in state ωj, the investor must holdc1(ωj) units of the j-th Arrow-Debreu security from date 0 to date 1in order to get c1(ωj) dollars at date 1 when state ωj occurs. Note alsothat the investor is a price-taker: she believes that ϕj; j = 1, 2, · · · , nwill not be affected by her buying and selling those n securities.21

Our task here is to characterize the set of equilibrium prices ϕj; j =1, 2, · · · , n. By equilibrium, we mean competitive equilibrium or Wal-rasian equilibrium, in which (i) given the prices ϕj; j = 1, 2, · · · , n,the amounts that the investor chooses to spend at date 0 and date 1,c0, c1j; j = 1, 2, · · · , n, are expected-utility-maximizing; and (ii) given

21The presence of a complete set of Arrow-Debreu securities allows an investor to transferpurchasing power back and forth between t = 0 and t = 1, and between any two states att = 1. Let me give an example. Suppose that n = 2, and ϕ1 = 0.8, ϕ2 = 1. Suppose thatyour endowed income is as follows:

e0 = 2000, e1(ω1) = 1000, e1(ω2) = 100.

(i) Suppose that you only want to consume in state ω1 at t = 1. How much can you spendin state ω1 at t = 1? The answer is

1

ϕ1[e0 + e1(ω2)× ϕ2] + e1(ω1) = 3625.

You have to carry out the following time-0 transactions in order to make this consumptionplan possible.

• At first, you have to transfer your time-1 ω2-state income back to time 0. This canbe done by short selling 100 units of the 2nd Arrow-Debreu security: borrowing 100units of the 2nd Arrow-Debreu security and selling at t = 0, you will immediatelyget 100ϕ2 at t = 0, but you will have to return 100 in (and only in) state ω2 at t = 1,which leaves you with e1(ω2)− 100 = 0 in state ω2 at time 1.

• Next, you can spend e0 + 100ϕ2 on the first Arrow-Debreu security. That is, youcan purchase e0+100ϕ2

ϕ1units of the first Arrow-Debreu security at t = 0, and carry

them into time 1. Then, you will have no money left at t = 0, implying that yourconsumption at time 0 is zero, and you will also have zero consumption in stateω2 at time 1. However, you can consume more than e1(ω1) in state ω1 at time 1.Indeed, the total consumption in state ω1 at t = 1 becomes

e0 + 100ϕ2

ϕ1+ e1(ω1) = 3625.

Now, as another exercise, determine how much you can spend at time 0, if you onlywant to consume at time 0. Detail the needed transactions.

21

the prices ϕj; j = 1, 2, · · · , n, the demands for the n assets, which areagain c0, c1j; j = 1, 2, · · · , n, equal the supplies of the n assets;22 thatis, the n asset markets clear under the prices ϕj; j = 1, 2, · · · , n.Replacing c0 by e0 +

∑nj=1 ϕj[e1(ωj) − c1(ωj)] into the objective func-

tion u0(c0)+∑n

j=1 πju1(c1(ωj))], we have a maximization problem witha concave objective function, for which the first-order conditions arenecessary and sufficient for the optimal solution:

πju′1(c1(ωj))

u′0(c0)

= ϕj, ∀j = 1, 2, · · · , n.

Now, the markets clearing condition requires that

c0 = e0, c1(ωj) = e1(ωj), ∀j = 1, 2, · · · , n.

(Since there are no other people in this economy, markets clear if andonly if the prices of the n assets adjust in such a way that the investorfinds it optimal to consume her endowments at each date in each state.)It follows that

πju′1(e1(ωj))

u′0(e0)

= ϕj, ∀j = 1, 2, · · · , n.

Note that every conceivable asset generating cash flows at date 1 canbe represented by a vector x ∈ ℜn, where xj, the j-th element of x,denotes the cash flow generated by one share of asset x in state ωj

at date 1. (We have assumed that investors care about money andnothing else.) Now observe an important fact: every conceivable assetthat generates cash flows at date 1 is a portfolio of the n Arrow-Debreusecurities. For example, an asset that pays a per-unit cash flow x(ωj)in state ωj can be regarded as a portfolio that consists of x(ω1) unitsof the first Arrow-Debreu security, x(ω2) units of the second Arrow-Debreu security, and so on. For example, suppose that n = 3. Observe

22What is the supply of the j-th Arrow-Debreu security? The answer is e1(ωj). To seethis, note that there is only one investor in this economy, and holding e1(ωj) units of thej-th Arrow-Debreu security is the same as being endowed with e1(ωj) dollars in state ωj

at date 1.

22

that x(ω1)x(ω2)x(ω3)

= x(ω1)

100

+ x(ω2)

010

+ x(ω3)

001

.Now let Px denote the price of the asset on the above left-hand side.(The right-hand side is a portfolio of the three traded Arrow-Debreusecurities.) Note that in equilibrium Px must equal the date-0 valueof the portfolio on the above right-hand side: if not, then the investorwould still want to alter her equilibrium positions in the n traded assetsbecause an arbitrage opportunity has shown up, which contradicts theassumption that the economy has reached an equilibrium. It followsthat

Px =n∑

j=1

ϕjx(ωj) =n∑

j=1

πju′1(e1(ωj))

u′0(e0)

x(ωj)

= E[u′1(e1)

u′0(e0)

x].

Note that the above pricing formula holds for any conceivable assetxn×1. In particular, a pure discount bond with face value equal to onedollar is an asset that promises to pay x(ωj) = 1, ∀j = 1, 2, · · · , n. Thisis obviously a riskless asset. Its date-0 price, according to the abovepricing formula, is

E[u′1(e1)

u′0(e0)

].

Now, recall that rf is the rate of return on any riskless asset. Since thepure discount bond is a riskless asset, we must have

1

1 + rf= E[

u′1(e1)

u′0(e0)

] ⇒ rf =1

E[u′1(e1)

u′0(e0)

]− 1.

Thus, for an asset that pays a per-unit cash flow x(ωj) in state ωj, itsdate-0 price can be further written as

Px = E[u′1(e1)

u′0(e0)

x]×E[

u′1(e1)

u′0(e0)

]

E[u′1(e1)

u′0(e0)

]

23

=

E[

u′1(e1)u′0(e0)

E[u′1(e1)

u′0(e0)

]x]

1 + rf=

E[ξx]

1 + rf,

where the random variable

ξ ≡u′1(e1)

u′0(e0)

E[u′1(e1)

u′0(e0)

].

Because u′0, u

′1 > 0, the realizations of ξ are all positive; that is, ξ is a

positive random variable.

Recall that for all j = 1, 2, · · · , n, πj is the real probability for state ωj.We can define a set of martingale probabilities as follows.

∀ωj ∈ Ω, π∗j = πjξ(ωj).

We claim that (A) ∀j = 1, 2, · · · , n, π∗j > 0; and (B)

∑nj=1 π

∗j = 1. Hence

π∗j ; j = 1, 2, · · · , n are indeed well-defined probabilities. To see that

(A) is true, note that π∗j = πjξ(ωj), with πj > 0 and ξ(ωj) > 0. To see

that (B) is true, note that

n∑j=1

π∗j =

n∑j=1

πjξ(ωj) =1

E[u′1(e1)

u′0(e0)

]

n∑j=1

πju′1(e1(ωj))

u′0(e0)

=E[

u′1(e1)

u′0(e0)

]

E[u′1(e1)

u′0(e0)

]= 1.

Now, recall that for an asset that pays a per-unit cash flow x(ωj) instate ωj, its date-0 price can be further written as

Px =E[ξx]

1 + rf=

1

1 + rfE[ξx] =

1

1 + rf

n∑j=1

πj ξ(ωj)x(ωj)

=1

1 + rf

n∑j=1

π∗j x(ωj) =

1

1 + rfE∗[x] =

E∗[x]

1 + rf,

24

where E∗[·] is the expectation that is taken using the new probabilitiesπ∗

j ; j = 1, 2, · · · , n. The formula

Px =E∗[x]

1 + rf

is called the martingale pricing formula for any asset x.

If you ask someone what would be the fair date-0 price for an asset thatpays x at date 1, you would probably get the answer E[x]. Financetheory will tell you that two things may go wrong with this answer:first, one has to pay the price at date 0, and then wait to get the cashflow x at date 1, and there is a time value for money; and second, theprice that one pays at date 0 is a sure amount of money, and in returnthe investor will get an uncertain cash flow x at date 1. The abovepricing formula tells us that, to take care of the time value of money(that is, 1 dollar today is usually worth more than 1 dollar tomor-row), the expected date-1 cash flow must be discounted (i.e., dividedby (1 + rf )) before it transforms into the date-0 price; and moreover,the original probabilities πj; j = 1, 2, · · · , n must be replaced by aset of new probabilities π∗

j ; j = 1, 2, · · · , n in order to reflect the factthat u′′

t ≤ 0 (that is, an investor typically hates uncertainty). In fact,if u′′

t = 0 so that the investor is risk-neutral and no longer botheredby the presence of uncertainty, then the old and the new probabilitieswill coincide with each other. In that case, the date-0 price of an assetis equal to its expected date-1 cash flow divided by (1 + rf ). If theinvestor is risk-neutral and has no time preferences (i.e., she feels indif-ferent about 1 dollar today and 1 dollar tomorrow), then discountingcan also be spared, and the fair date-0 price of asset x for this investoris indeed E[x]!23

18. Our next example will derive the Black-Scholes option pricing formula.Suppose that I = 1, so that the Walrasian economy has a single in-vestor, but there is an infinite number of possible states at date 1,

23The random variable ξ is a proability density function commonly referred to as theRadon-Nikodym derivative in probability theory and the theory of asset pricing. Verifythat ξ = 1 if in the above analysis the single investor is risk neutral; that is, if u′′

1 = u′′0 = 0.

25

so that ek; k = 1, 2, · · · , n is now replaced by a continuous randomvariable e1, with

log(e1) ∼ N(µ, σ2).

Suppose that e0 = 1, and u10(x) = u1

1(x) = u(x) = x1−ρ

1−ρ, where 0 <

ρ < 1. Suppose that there are only one risky asset and one risklessasset available for trading at date 0, both in zero net supply, where theriskless rate is rf . The risky asset will pay x = max(e1 − k, 0) at date1, and will be referred to as a call option with exercise price k. Let pbe the date-0 equilibrium price of the call option. Let us find p.

Since assets are in zero net supply, the asset prices must adjust in such amanner that investor 1 chooses not to trade either asset in equilibrium.So, in equilibrium, the investor’s welfare is measured by

u(e0) + E[u(e1)] =1

1− ρ+ E[

e1−ρ1

1− ρ].

In equilibrium, the following function

g(ϵ) ≡ u(e0 − ϵp) + E[u(e1 + ϵx)]

should have a maximum at ϵ = 0. Hence we have

−u′(e0)p+ E[u′(e1)x] = 0 ⇒ p = E[u′(e1)

u′(e0)x].

Hence the equilibrium price of the option, or simply, the option pre-mium, is equal to

p =∫ +∞

−∞e−ρz max(ez − k, 0)f(z)dz,

where

f(z) =1√2πσ

e−(z−µ)2

2σ2 , −∞ < z < +∞

is the density function for a random variable distributed as N(µ, σ2).Hence we have

p =∫ +∞

z=log(k)e(1−ρ)zf(z)dz − k

∫ +∞

z=log(k)e−ρzf(z)dz.

This is the famous Black-Scholes Option-Pricing Formula, which wasoriginally derived in a continuous-time dynamically complete economy.

26

19. Our next example considers stock trading with incomplete markets.Suppose that there are two investors in the two-period frictionless econ-omy, where for i = 1, 2, investor i has von Neumann-Morgenstern utilityfunction

u(z) = −e−ρiz, ∀z ∈ ℜ,

whereρ2 > ρ1 > 0.

There are one risky asset and one riskless asset available for trading atdate 0. The riskless rate of return is rf . The risky asset is a commonstock, which has 2 shares outstanding, with the two investors eachholding one share before trading starts at date 0. Nobody is endowedwith the riskless asset, so that the riskless asset is in zero net supply.Let x ∼N(µ, σ2) be the date-1 cash flow generated by one share of thecommon stock, where

µ >2σ2

1ρ1

+ 1ρ2

> 0.

(i) Let P be the date-0 stock price. Let Di(P ) be investor i’s demandfor the common stock at date 0, given that the stock price is P . FindD1(·) and D2(·).(ii) Write down the markets-clearing condition, and obtain the equilib-rium stock price P ∗.(iii) Plug P ∗ into D1(·) and D2(·) and determine which investor is buy-ing the stock, and which investor is selling the stock in equilibrium atdate 0.(iv) Which one between the two investors is borrowing in equilibriumat date 0? Which one is lending? Why?

Solution. Consider part (i). Investor i’s problem is

maxDi∈ℜ

E[−e−ρi[Dix+(1−Di)P (1+rf )]] = e−ρi[(1−Di)P (1+rf )]E[−e−ρiDix].

Note that the random variable

−ρiDix ∼ N(−ρiDiµ, ρ2iD

2i σ

2),

27

so that, by the formula of moment generating function for a Gaussianrandom variable,24 we have

E[−e−ρiDix] = −e−ρiDiµ+12ρ2iD

2i σ

2

.

Thus investor i seeks to

maxDi∈ℜ

−e−ρi[Diµ+(1−Di)P (1+rf )− 12ρiD

2i σ

2]

= u(Diµ+ (1−Di)P (1 + rf )−1

2ρiD

2i σ

2),

where recall that u(·) is strictly increasing. Thus in searching for theutility-maximizing Di, the investor can simply solve the following max-imization problem:

maxDi∈ℜ

W (Di) ≡ Diµ+ (1−Di)P (1 + rf )−1

2ρiD

2i σ

2.

In finance literature, W (Di) is referred to as the certainty equivalentof W for the investor, induced by the investment strategy Di.

The last maximization problem involves only a quadratic objectivefunction, and hence is easy to solve. The necessary and sufficient first-order condition gives

Di(P ) =µ− P (1 + rf )

ρiσ2, i = 1, 2.

This demand function exhibits several interesting properties. At first,investor i takes a long position (i.e., Di > 0) if and only if the expectedrate of return on the risky asset is greater than rf , and that positionshrinks when the investor becomes more risk-averse (ρi gets higher) orthe riskiness of the risky asset rises (σ2 gets higher). This finishes part(i).

24It says that for z ∼N(e, V ),

Mz(t) ≡ E[etz] = ete+t2V2 .

28

Now, for part (ii), the markets-clearing condition requires that at theequilibrium price P ∗,

D1(P∗) +D2(P

∗) = 2 ⇒ P ∗ =µ− ρσ2

1 + rf,

where

ρ =2

1ρ1

+ 1ρ2

is the harmonic mean of ρ1 and ρ2. Note that P ∗ equals µ1+rf

(the

martingale pricing formula for risk neutral investors!) if the riskinessof the common stock, measured by σ2, vanishes, or if ρ vanishes (whichhappens if either ρ1 or ρ2 is equal to zero). Thus the term ρσ2 representsa risk discount: the two investors are actually risk averse, not riskneutral. Note that our condition

µ >2σ2

1ρ1

+ 1ρ2

ensures that both investors will take long positions in stock.

Next, consider part (iii). It is easy to see that

Di(P∗) =

ρ

ρi, i = 1, 2,

and henceD1(P

∗) > 1 > D2(P∗).

This is a very intuitive result. Investor 2 is more risk averse thaninvestor 1, since ρ2 > ρ1, and hence in equilibrium investor 1 must holdmore of the risky common stock than investor 2 does. Note also thatthis result verifies our earlier conjecture that i−Di(P

∗) = 0 for i = 1, 2.

For part (iv), recall that the two investors start with the same initialwealth, and so the investor holding more of the risk common stock mustalso be the one that is borrowing in equilibrium. Hence investor 1 isborrowing and investor 2 is lending in equilibrium.

29

20. Our next example concerns the design of Pareto efficient sharing rules.Consider two partners U and V facing the set Ω (defined in our firstapplication above) of date-1 uncertain states. Their endowed date-1incomes are respectively aj and bj in state ωj, and we define Cj =aj + bj to be the aggregate consumption in state ωj for the two people.Their date-1 VNM utility functions are respectively u(·) and v(·), whereu′, v′ > 0 ≥ u′′, v′′. Our task here is to find the optimal way for the twopeople to share their aggregate consumption in each and every date-1 state. Here “optimal” means Pareto optimality: an arrangementbetween the two people is not Pareto optimal or efficient, if there isanother arrangement that can make both of them weakly happier, andat least one of them strictly happier. When the latter does not happen,then the original arrangement is Pareto optimal (or Pareto efficient).

Suppose that U and V sign a contract, which specifies that U shouldget sj in state ωj. This contract sj; j = 1, 2, · · · , n is feasible if andonly if both U and V are happy to sign it. In other words, it is feasibleif and only if

n∑j=1

πju(sj) ≥n∑

j=1

πju(aj) ≡ u0;

andn∑

j=1

πjv(Cj − sj) ≥n∑

j=1

πjv(bj) ≡ v0.

The latter two inequalities are referred to as respectively U’s and V’sindividual rationality constraints. We call u0 and v0 respectively U’sand V’s reservation utilities. Our question here is, if there is at leastone feasible contract for U and V, which feasible contract is Paretoefficient? What does it look like?

Suppose that U has the right to design the contract; i.e., U has theright to choose sj; j = 1, 2, · · · , n. Then U seeks to

maxsj ;j=1,2,···,n

f(s1, s2, · · · , sn) ≡n∑

j=1

πju(sj)

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subject to

g(s1, s2, · · · , sn) ≡ v0 −n∑

j=1

πjv(Cj − sj) ≤ 0.

It can be verified that f and g are respectively concave and convexfunctions of (s1, s2, · · · , sn), and hence we can apply the Kuhn-Tuckertheorem to solve the problem. Let µ ≥ 0 be the Lagrange multiplierfor the constraint, and the Kuhn-Tucker condition requires that

∀j = 1, 2, · · · , n, ∂f

∂sj(s∗1, s

∗2, · · · , s∗n) = µ

∂g

∂sj(s∗1, s

∗2, · · · , s∗n).

Using the definitions of f and g, we obtain the following Borch rule forPareto optimal risk sharing:

∀j = 1, 2, · · · , n, u′(sj) = µv′(Cj − sj).

Two interesting cases arise: (i) u′′ = 0 > v′′; and (ii) u′′ < 0 = v′′. Incase (i), u′ is a constant, and the Borch rule implies that v′(Cj − sj)must be independent of j! Since v′(·) is a strictly decreasing function,this can happen only when

C1 − s1 = C2 − s2 = · · · = Cn − sn;

that is, V’s date-1 income becomes riskless under the Pareto optimalrisk sharing! The idea is quite simple: if U is risk neutral and V isrisk averse, then it is efficient to let U bear all the risk in aggregateconsumption. Case (ii) is similar. In fact, given that v′ is a constant,u′(sj) has to be independent of j, and since u′(·) is strictly decreasing,this implies that

s1 = s2 = · · · = sn.

Again, the risk-averse party must receive a fixed income at date 1!

The above analysis has implications about how two partners of a privatefirm should share profits. If exactly one of them is risk-averse, the risk-neutral guy should become the shareholder, who issues a riskless debtto her partner. The above analysis also tells us something about the

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design of an insurance policy. If the insurance company is risk-neutralwhile the insuree is risk-averse, then there should be no deductibles inan insurance policy: the insuree should have non-random wealth afterthe date-1 reimbursements are made.25 Let us extend the analysis abit further.

Suppose that the above case (i) holds. Suppose further that U is aninsurance company, and V is an insuree. Suppose that there are twoequally likely date-1 states (n = 2), with b1 = 0, b2 = 4, and v(z) =

√z.

Now, an insurance contract is defined as a pair (p, r), where V mustpay U p in both states at date 1 (p is the insurance premium), and Uwill reimburse r to V in and only in state 1. We will see how U’s andV’s bargaining power may affect p and r.(1) Assume that U can choose (p, r), which V can only accept or reject.Compute p and r.26

(2) Assume that V can choose (p, r), which U can only accept or reject.Compute p and r.27

References

1. Arrow, K., and G. Debreu, 1954, Existence of equilibrium for a com-petitive economy, Econometrica, 22, 265-290.

2. Arrow, K., 1964, The role of securities in the optimal allocation of riskbearing, Review of Economic Studies, 31, 91-96.

25Two questions arise at this point. First, why are there always deductibles in the real-world insurance contracts? Second, do the above results depend on who has the right todesign sj ; j = 1, 2, · · · , n?

26Hint: The optimal (p, r) must make V feel indifferent about buying and not buyingthe insurance, and it must also satisfy the Borch rule. Hence we have

12

√0 + r − p+ 1

2

√4− p = 1

2

√0 + 1

2

√4,

0 + r − p = 4− p⇒ r = 4, p = 3.

In this case, U’s expected profit is p− 12r = 1, and V’s expected utility is 1.

27Hint: The optimal (p, r) must make U feel indifferent about selling and not sellingthe insurance, and it must also satisfy the Borch rule. Hence we have

12 (−r + p) + 1

2 · p = 0,0 + r − p = 4− p

⇒ r = 4, p = 2.

In this case, U’s expected profit is zero, and V’s expected utility is√2.

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4. Borch, K., 1962, Equilibrium in a Reinsurance Market, Econometrica,30, 424-444.

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14. Grossman, S., and O. Hart, 1983, An analysis of the principal-agentproblem, Econometrica, 51, 7-45.

15. Harrison, M., and D. Kreps, 1979, Martingales and arbitrage in multi-period securities markets, Journal of Economic Theory, 20, 381-408.

16. Hart, O., 1975, On the optimality of equilibrium when the marketstructure is incomplete, Journal of Economic Theory, 11, 418-443.

17. Huang, C., and R. Litzenberger, 1988, Foundations for Financial Eco-nomics, New York: Elsevier Science.

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23. Magill, M., and W. Shafer, 1991, Incomplete markets, in W. Hilden-brand and H. Sonnenschein (eds.) Handbook of Mathematical Eco-nomics, Volume IV, Chapter 30, New York: Elsevier Science.

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