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Security Market Microstructure
Lecture 1: Expected Utility Theory, Pareto Efficiency, WalrasianEquilibrium and Radner Equilibrium
Instructor: Chyi-Mei Chen(Tel) 3366-1086(Email) [email protected](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 of
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.
1
all feasible lotteries; that is, P contains all probability distributionsover the Z. We call P the investment opportunity set. From now on,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).
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
Here comes our first main result.
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-t
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).
3
consumption, 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 investorseeks to maximize E0[
∑Tt=0 δ
tu(ct)], where δ ∈ (0, 1) is the discrete-timediscount 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.
4
• 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 issaid 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
Lemma 1 (Jensen Inequality) If f : ℜ → ℜ is concave (respec-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 R
uR(x) are referred to respectively
as 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?)
8
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 wealthW0 = 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 constantW0 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
11
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
],
[h0Xh
]),
and assuming that each traded asset has unit supply, we have the ag-
16
gregate consumption being
C =
[g0 + h0X1
].
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 =
π1π2...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 two
investors. 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’s
utility-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’s
utility-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∗0Xh∗
])
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 theevent 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) = u11(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)
30
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
31
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
21. Let us give an example for Pareto optimal sharing rules. Suppose thatthe economy has one single consumption good, I = 2, and the aggregateconsumption C has a compact support [0, 1]. Agent j has von NeumannMorgenstern utility function
uj(c) = −e−ajc, j = 1, 2,
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.
32
where aj > 0 are constants. We are looking for two increasing functionss1(C) and s2(C) = C − s1(C) such that for some positive constantπ ∈ [0, 1], s1(·) is the solution to
maxs(·)
J(s(·)) ≡ πE[−e−a1s(C)] + (1− π)E[−e−a2[C−s(C)]].
Note that for an arbitrary function t(·),
maxa∈R
H(a) ≡ J(s1(·) + at(·))
assumes maximum at a = 0, as s1(·) is Pareto efficient. Thus, in caseH(·) is smooth enough,
H ′(0) = 0.
Expanding, we see that s1(C) = α + βC for some constants α andβ > 0. That is, for CARA people, the optimal sharing rules are linear.Note that to implement a linear sharing rule by competitive marketssystem, we only need to open two markets, one for a risky asset thatpays dividends proportional to C, and the other a riskless asset. Thissays that two fund separation holds for CARA people.28
22. Now we review the concept of Radner equilibrium in a 2-period econ-omy with incomplete markets. There are two dates, 0 and 1, andconsumption takes place only at date 1. We call date 0 ex-ante anddate 1 ex-post. There are S possible states at date 1 and there areL consumption goods (where recall that we have assumed L = 1 insection 2). There are I agents maximizing expected utilities, which arederived from their date-1 consumptions. Agents trade assets at date 0and may be able to trade in the date-1 spot markets for consumptiongoods. First suppose that markets are Debreu complete: there are SLcontingent assets, where the (s, l)-th asset delivers 1 unit of good l ifstate s occurs at date 1 and nothing otherwise. Assume that agent i’svon Neumann-Morgenstern utility function vi : RL
+ → R is continu-ously differentiable, increasing, and strictly concave, her endowments
28Optimal sharing rules are linear if and only if the I agents have linear risk tolerencesand identical risk cautiousness; i.e. their von Neumann-Morgenstern utility functions arein the HARA class. See for example Huang and Litzenberger, chapter 5. Interested readersshould refer to Cass and Stiglitz (1970).
33
are (wsl), and she holds the beliefs that state s may occur with prob.πis.
Definition 5 An Arrow-Debreu contingent market equilibrium29 is apair (psl), (xsl)i ≡ xi such that(i) ∀i = 1, 2, · · · , I, xi = argmaxui(xi) ≡ ∑S
s=1 πisv
i(xis) subject to∑s
∑l
psl(xisl − wi
sl) ≤ 0;
(ii)∑
i(xi − wi) = 0.
As an earlier theorem shows, the contingent market equilibrium alloca-tion is Pareto efficient. This is referred to as the ex-ante efficiency. Itimplies the following ex-post efficiency: for all s, the equilibrium in thedate-1 spot market for consumption goods is such that (x1s, x
2s, · · · , xIs)
is Pareto efficient relative to the set of utility functions (v1, v2, · · · , vI).It can be shown that ex-ante efficiency implies ex-post efficiency. Theconverse is not true in general.
23. Now assume that only K < SL markets are open (K assets in zerosupply are traded) at date 0, but spot markets (for goods) are all openat date 1. Thus the market structure at date 0 is incomplete. This iswhere the concept of Radner equilibrium arises.
First we define notation. An asset is denoted by a matrix
A = [asl] = [a1, a2, · · · , aS] ,29If there is a complete set of Arrow-Debreu commodities traded at date 0, we say
markets are Debreu-complete and the pure exchange economy is referred to as the Arrow-Debreu contingent markets economy. If at date 0 we have a complete set of traded con-tingent commodities, and we have rational expectations (agreeing upon all relevant mea-surable functions in the sense of Roy Radner (1972)), then no trade would occur even ifmarkets reopen. Things are different if we do not have a complete set of Arrow-Debreucommodities traded at date 0. In the latter case, trades may occur when markets reopenafter date 0, and in this case markets may or may not be dynamically complete. It is clearthat price expectations are important elements in the latter market structure; see Radner(1972, Econometrica).
34
where as is an L-vector specifying the amount of good l to be deliveredat date 1 in case state s occurs.30 Let qk, k = 1, 2, · · · , K, be thedate-0 price of asset k. Let zik be agent i’s position in asset k. Wecall zi the portfolio of agent i. The timing is as follows. At date 0,agents first trade assets and form portfolios. At date 1, the assets firstpay (liquidation) dividends (in goods; see the above definition). Thesedividends are added to endowments and then agents trade in the spotmarkets. Define
wis = wi
s +K∑k=1
zikaks ,
which will be agent i’s pre-trade endowments in state s at date 1.
Definition 6 A Radner equilibrium is a tuple
q∗ ∈ RK+ ; p
∗s ∈ RL
+ (∀s); (z∗1, z∗2, · · · , z∗I); (x∗1, x∗2, · · · , x∗I)
such that(i) ∀i, (z∗i, x∗i) maximizes ui(xi) subject to(i-1) q∗ · z∗i ≤ 0;(i-2) p∗s · x∗is ≤ p∗s · w∗i
s = p∗s · (w∗is +
∑k z
ikaks), ∀s;(ii)
∑i z
∗i = 0;(iii) ∀s, ∑
i x∗is =
∑iw
is.
Note that agents have perfect foresight in a Radner equilibrium, in thesense that at date 0 when they make portfolio dicisions they correctlypredict the measurable functions pls = pl(s) even if they may hold dif-ferent prob. measures. (Agent i’s information structure is representedby the prob. space (S = 1, 2, · · · , S, 2S , πi).)31
24. With incomplete markets, Radner equilibrium allocations are usuallyPareto inefficient. For example, let L = 1, S = 3, K = 2. Let a1 =(1 + r, 1 + r, 1 + r) and a2 = (0, 0, 1). Comparing the two equilibrium
30For example, for the (s, l) Arrow-Debreu contingent good (Arrow-Debreu security),as′ = el ·1[s′=s] where el is the l-th standard basis vector in ℜL. A bond in good l′ is suchthat asl = 1[l=l′].
31Note that the the contingent market equilibrium is a special case of Radner equilib-rium: simply let q∗ = p∗, z∗i = x∗i − wi.
35
definitions, we see that inefficiency arises in the Radner equilibriumbecause with incomplete markets an agent is faced with S + 1 budgetconstraints rather than one single constraint as in the case of completemarkets.
25. Unlike in the Arrow-Debreu economy where there will be no trade evenif markets reopen after date 0, trades will generally take place at bothdates 0 and 1 in a Radner equilibrium. At date 0, agents have perfectforesight in the sense that they correctly expect the terms of trade atdate 1 and based on these price expectations, they carry out part ofthe trading process at date 0 and postpone the rest till date 1. Itfollows that for a Radner equilibrium to implement the Pareto efficientallocation of a contingent market equilibrium, we only need to ensurethat each agent gets the right amount of pre-trade wealth in states: since the ex-ante efficient contingent market equilibrium allocationimplies that the allocation is ex-post state-s efficient for all s, and eachex-post efficient allocation is sustained by a competitive equilibrium(actually at the contigent market equilibrium prices), we only need tomake sure that the date-0 asset structure is such that in every states the optimal portfolio for each agent generates an additional wealthwhich is exactly the difference between the value of the agent’s initialstate-s endowments and the value of her contingent market equilibriumconsumption bundle. This motivates the idea of Arrow completeness.
26. Arrow proposes that Pareto efficiency is ensured if at date 0 there areproperly chosen S asset markets and all L spot markets are availableat date 1. When markets meet Arrow’s requirement, we say that theyare Arrow complete; see Arrow’s papers listed in the references.32 Theidea is as follows. Take good 1 as numeraire and rewrite the definitionfor Radner equilibrium. Let (p∗, (x∗i)) be an Arrow-Debreu contingentmarket equilibrium were we to have a complete set of traded contingent
32Arrow’s idea is further extended by Harrison and Kreps (1979) into a notion commonlyreferred to as dynamic completeness; see Lecture 5 of my Investments Course. Roughly, ifthe markets are complete over any date-t-date-(t+1) period, then markets are dynamicallycomplete, and Pareto efficiency is attained in the Radner equilibrium. In this case, aversion of the representative agent theorem also holds. See Duffie and Huang (1985) foran extension of dynamic completeness to a continuous-time economy, where the so-calledmartingale multiplicity determines whether the markets are dynamically complete.
36
goods. Now, create S assets such that asset s (the contingent moneyin state s) pays 1 unit of good 1 in state s and nothing otherwise.Consider the price expectations ps = p∗s
p∗s1for date-1 spot markets and
the date-0 asset prices qs = p∗s1. Given these prices, agent i’s portfoliodecision is such that z∗is = ps(x
∗is − wi
s). It can now be verified thattogether with x∗i this forms a Radner equilibrium.33
27. Does a Radner equilibrium exist always?
Theorem 11 Suppose that agent i’s consumption plan must be con-tained in X i ⊂ RSL, which is closed, convex, and bounded from below(implying that the set ∑i x
i ≤ ∑iw
i is compact). Suppose that ui iscontinuous and concave. Suppose that for all i and all xi, there exists[xi]′ differing from xi only in one state such that ui([xi]′) > ui(xi).Suppose that for all i, there exists at least one xi0 ∈ X i such thatxi0 ≪ wi. Finally, suppose there exists D > 0 such that for all i andall k, zik ≥ −D (so that z|∑i z
i is compact). Then there exists aRadner equilibrium.
The lower bound D represents one type of short sale constraint. With-out this assumption, Radner equilibrium need not exist. For example,suppose S = K = 2 ≤ L. Hart (1975; Journal of Economic Theory)gives an example where for all (p1, p2) = (p′1, p
′2) (subscripts for states),
(p1 · a1, p2 · a2) is linearly independent of (p1 · b1, p2 · b2) (where a, bstand for the two assets), but the two vectors are linearly dependent at(p′1, p
′2). In this case, except at the (p′1, p
′2), the attainable consumption
plans span R2 (without the bound D!) so that markets are complete,but markets are incomplete at (p′1, p
′2). It follows that the budget cor-
respondence is not upper hemi-continuous, and in Hart’s example afixed point fails to exist. If L = 1, however, this is no problem, becausethe bounds on X i translate into bounds on zi, and hence an equilib-rium can exist; this is Peter Diamond’s stock market economy (1967).Or else, if assets, instead of being real, express their returns in units
33Note that in each state at date 1, each agent will use up her pre-trade wealth (becausevi is strictly increasing). It follows that essentially each agent is facing one single budgetconstraint just like in the Arrow-Debreu economy.
37
of account, or they only deliver some numeraire good, then the non-existence problem disappears. In addition to these special cases, Duffieand Shafer (1986) show that Hart’s example is still non-generic.34
28. We now give three examples of Radner economy.
29. (Example 1: Commodity Futures) A farmer plants wheat at date0 and harvests at date 1. The farmer has the production function
q =√x,
where x ≥ 0 is the amount of input (taken as numeraire) and q theamount of the output. The date-1 wheat price is a random variablep ∼ N(µp, σ
2). The farmer can sell the output in the date-0 forwardmarket at the forward price pf (to be endogenously determined be-low). Let f be the units of the forward contract sold by the farmer. Inthe date-0 forward market, there is another trader called the urbanite.The farmer and the urbanite are both price-takers and expected util-ity maximizers with their von Neumann-Morgenstern utility functionsbeing respectively
Uϕ(w) = −e−ϕw, Uu(w) = −e−uw, ϕ, u > 0.
Being price-takers with rational expectations, the farmer and the ur-banite both know pf and p ∼ N(µp, σ
2), and they take these as given.In particular, in choosing x and f , the farmer ignores the fact that hischoice of x could affect the distribution of p.(i) Given the farmer’s beliefs about p and pf , find his optimal date-0choices for x and f . Show that, in the presence of the forward market,the farmer in choosing x behaves as if all his output were hedged in theforward market (i.e. as if f = q(x)). Consequently, f can be decom-posed into a fully hedged component plus a bet.(ii) Find the urbanite’s demand for the forward contract. Find the
34Although the non-existence problem is non-generic, a Radner equilibrium, when itexists, is generically constrained inefficient. This means that conditional on the fact ofmissing markets, the allocation in equilibrium is still less than Pareto efficient. It is shownthat monetary policy may be used to improve efficiency in this world. It all depends onwhether assets pay real dividends (or in units of account). If assets are all real, there is norole for money. On the other hand, monetary policy has real effects if assets are nominal.
38
date-0 forward market equilibrium (pf , f) using the market clearingcondition. Suppose that as a true relationship, µp = 1− q(x), or equiv-alently,
p = 1− q(x)− ϵ, ϵ ∼ N(0, σ2).
Now impose rational expectations to remove the dependence of equi-librium variables on µp. What happens to the forward price pf if σ2
(or ϕ or u) tends to zero?(iii) What is the role of the urbanite? Does there exist a level of σ2
most preferred by the urbanite?(iv) In this exercise does the existence of a forward market stabilize thedate-1 spot price for wheat?Solution. Consider part (i). The farmer’s random wealth is, givenx, f, pf and p,
p[q(x)− f ] + pff − x = pfq(x)− x+ (p− pf )[q(x)− f ].
The first order condition of the farmer’s problem gives
E[U ′ϕ · (pq′ − 1)] = 0 = E[U ′
ϕ · (−p+ pf )],
implying that1
q′(x)= pf ,
which means that x depends only on pf but not on p. This result canbe alternatively obtained by solving
maxx
pfq(x)− x;
that is, x is chosen as if the farmer will sell all the output in the currentforward market!
It is straightforward to show that
f =pf2
+pf − µp
ϕσ2
= q(x) +pf − µp
ϕσ2,
39
so that the short position the farmer takes in the forward market is theoutput plus a bet (the above second term). In particular, the bet is ashort (resp. a long) if and only if pf ≥ (resp. ≤) µp.
In part (ii), it is easy to show that the urbanite’s position isµp−pfuσ2 .
Markets clearing and rational expectations then imply that
pf =2(ϕ+ u)
3(ϕ+ u) + ϕuσ2, x = [
ϕ+ u
3(ϕ+ u) + ϕuσ2]2.
If one among ϕ, u, and σ tends to zero, then pf is asymptotically inde-pendent of all these three variables because concerns about risk becomeasymptotically unimportant at least for one of the trading parties.
In part (iii), the urbanite’s role is to provide insurance to the farmer,and being risk averse herself, the urbanite needs to be compensated(so that pf < µp, a phenomenon known as normal backwardation).35
As to the urbanite’s preferences over σ2, let Wu denote the urbanite’sterminal wealth at date 1, and we have
Wu =µp − pfuσ2
(p− pf ) ∼ N((µp − pf )
2
uσ2,(µp − pf )
2
u2σ2).
SinceE[−e−uWu ] = −e−u(E[Wu]−u
2var[Wu])
The urbanite’s welfare can be measured by
E[Wu]−u
2var[Wu]
=(µp − pf )
2
uσ2− u
2
(µp − pf )2
u2σ2
=(µp − pf )
2
2uσ2
35Note that
µp = 1−√x =
2(ϕ+ u) + ϕuσ2
3(ϕ+ u) + ϕuσ2>
2(ϕ+ u)
3(ϕ+ u) + ϕuσ2= pf .
40
=1
2uσ2[1− 3(ϕ+ u)
3(ϕ+ u) + ϕuσ2]2
=ϕ2
2uσ2
[3(ϕ+ u) + ϕuσ2]2
=ϕ2
2u
[3(ϕ+u)σ
+ ϕuσ]2
and hence the urbanite’s welfare is maximized at
σ2 =3(ϕ+ u)
ϕu.
Intuitively, a too high σ2 hurts the urbanite because the urbanite isrisk-averse, but a too low σ2 reduces the urbanite’s mean return forproviding insurance to the farmer.
Finally, the answer to part (iv) is obviously negative given the linearspecification in (ii), where x affects only the drift term of p.
30. (Example 2: Pareto inferior trade) There are n identical farmersand m identical consumers, all price-takers. Farmers can plant either asafe crop (s) or a risky crop (r). His problem is, at date 0, to determinea fraction x of his land to be planted with r, and 1 − x to be plantedwith s. Assume one unit of land for each farmer. Each unit of landgenerates 1 unit of safe crop, or alternatively, z units of risky crop,where z ∼ N(1, σ2). Each consumer has an indirect utility function
V (I, p, q) = ln(I)− aln(p)− bln(q),
where a, b are strictly positive constants and p, q are respectively thespot prices for r and s after harvest at date 1.(i) Using Roy’s identity to show that the demands for r and s arerespectively aI
pand bI
q.
(ii) Show that, given the equilibrium x, the date-1 spot prices for r ands are respectively
p =ay
xz, q =
by
1− x,
41
where y = mIn.
(iii) Show that farmer’s problem at date 0 is
maxx
E[U(xpz + q(1− x))]
with U ′ > 0 > U ′′ and q and the distribution of p given. (Simplybecause farmers are price-takers.)(iv) Find the first-order condition for the farmer at date 0. Now imposerational expectations: the optimal x solving the first-order conditionmust be such that it gives rise to the future prices q and p.(v) Conclude that in equilibrium, a farmer’s welfare is U((a+ b)y) anda consumer’s is ln(I)− (a+ b)ln((a+ b)y) + aE[ln(z)].Solution. Consider part (i). Using Roy’s identity, we have
dr = −∂V∂p∂V∂I
=aI
p.
Similarly, ds =bIq. From this, we have the aggregate demands
Qr = mdr, Qs = mds.
The aggregate supplies are price-inelastic (simply because the produc-tion decisions are made before farmers see the equilibrium spot prices):
Sr = nxz, Ss = n(1− x).
It is straightforward to get the equilibrium prices p and q. This is part(ii). Note that the price and quantity of the risky crop are inverselyrelated! More important, the revenue from the risky crop is sure tobe ay = amI
nfor each farmer! The revenue from the safe crop for
each farmer is also nonrandom, which is by = bmIn. This gives clues
regarding why opening new markets may reduce farmers’ welfare: ifopening new markets prevents farmers from fully insuring themselves,then as U ′′ < 0, farmers will suffer from excessive risk bearing.
Now, in part (iii), a price-taking farmer is to choose x given the con-stant q and the random variables p and z. The first-order conditionis necessary and sufficient for the farmer’s maximization problem: x∗
solvesE[U ′(xpz + q(1− x))(pz − q)] = 0.
42
The rational price expectations condition can now be imposed. Werequire that x∗, q, p be mutually consistent in the sense that (i) givenp and q, x∗ satisfies the above first-order condition; and (ii) given x∗,p and q respectively satisfy
p =ay
x∗z, q =
by
1− x∗.
This implies thatpz = q.
Hence we havex∗ =
a
a+ b, q = (a+ b)y.
These are parts (iii) and (iv). Part (v) is obvious.
Now imagine that there is another economy E ′ identical to the aboveeconomy E except that the random variable z in E is replaced by therandom variable z′ in E ′, and z, z′ are such that z+z′ = 2 for sure. Nowif trade is allowed, there will be trade (why?). We look for a symmetricfree trade equilibrium where x is the fraction of land planted with therisky crop. Now we have Sr = 2nx, Ss = 2n(1 − x) for sure. Onecan verify that now a farmer in economy E (respectively, E ′) has profit(az+ b)y (respectively, (az′+ b)y, which is a mean-preserving spread ofthe farmer’s profit without trade. Assume now that each consumer hasa CRRA utility function with R the measure of relative risk aversion.It can be shown that, if the farmers do not change their productionplans, then
∆U
u′=u([a+ b]y)− E[u([aθ + b]y)]
u′=Rya2σ2
2(a+ b).
However, farmers do have different production plans in the presenceof both E and E ′, and although a consumer faces no risk in the newscenario, she is hurt by the change in x (which is made by the farmersbecause of an increase in the risk of planting the risky crop). For R > σ,one can show that consumers are also worse off. Hence the opening ofa new market hurts all farmers and consumers in this example.
31. Example 3. Does the presence of speculators (those who trade be-cause they believe that they possess superior pre-trade information)
43
stabilize the stock market? Milton Friedman (1953) made this asser-tion. His argument is that, in order to make speculative profits, aspeculator buys a stock when the price is low and sells it when theprice is high, and the resulting price path is less volatile (thanks to thetrading activities of the speculators). Hart and Kreps (1986, JPE) re-futed this point. They pointed out that speculators may not buy whenthe price is low and sell when the price is high; instead, they buy whenthe price is likely to rise and sell when it is likely to fall. They gavean explicit example which shows that the presence of speculators mayresult in excessive price volatility.
Consider the following discrete-time (t = · · · ,−1, 0, 1, 2, · · ·) economywith two consumption goods X and Y and three types of people, con-sumers, speculators, and producers. Good Y is taken as the numeraire.Producers produce and supply 100 units of good X at each date. Pro-ducers do not have any storage capacity and they are price-takers.Producers consume only good Y.
Consumers are short-lived and they consume only good X. The inten-sity of date-t consumers’ demand for good X at date t is representedby θt. Assume that θt ∈ 0, 1 is i.i.d. over time, with θt = 1 withprobability 0.01, and its realization is public information at date t. Incase θt = 1, then the date-t consumers’ demand for good X is perfectlyelastic at the price $1000. If instead, θt = 0, then the date-t consumers’demand for good X is perfectly elastic at the price $10 for the first 100units, and slopes downward to $9 at 130 units, and decreases asym-potically to the price $8. At each date t, a public signal ξt ∈ 0, 1 isrealized which is correlated with θt+1. In particular, if θt+1 = 1, thenξt = 1; and if θt+1 = 0, then ξt = 1 with prob. 1
11and ξt = 0 with prob.
1011.
Finally, there are overlapping generations of competitive speculators.The date-t speculators are endowed with good Y at date t, and theycan store up to 50 units of good X for one day, and they will onlyconsume good Y at date t + 1 before they leave the economy. Thus ifdate-t speculators do purchase and store some of good X at date t, thenthey will definitely need to clear their positions at date t+ 1. Assumethat for every 5 units of good X stored at date t, only 3 units will stillbe in good condition at date t+ 1. Moreover, assume that speculators
44
are wealthy enough so that they can purchase the 50 units of good Xeven if the price of X is $1000.(i) First assume the absence of speculators. Show that for all t, thecompetitive price of good X (denoted pt) is $1000 if θt = 1 and it is$10 if θt = 0.(ii) Now assume the presence of speculators. Find the speculators’equilibrium trading strategy (by adaptedness, which must be contin-gent on the realization of θt and ξt). Show that in equilibrium, theprice of good X at some date t may be $1000, $10, or $9. Determine ifthe presence of speculators stabilizes the price.Solution. Part (i) is obvious, and we consider part (ii). At date t, sup-pose θt = 1. In this case, speculators would never buy, for if they did,they could never sell the good at any price higher than $1000. There-fore suppose that θt = 0. There are two possibilities. First, supposeξt = 0. This says that θt+1 = 0 with probability one! Considering thedecay of the good occurring in the storage process, speculators wouldnever buy in this case either: The price today can be no lower than $8,but the price tomorrow can be no higher than $10. Finally, we are leftwith the case where θt = 0 and ξt = 1. In this case, note that
pro.(θt+1 = 1|ξt = 1) =pro.(θt+1 = ξt = 1)
pro.(θt+1 = ξt = 1) + pro.(θt+1 = 0, ξt = 1)
=0.01× 1
0.01× 1 + 0.99× 111
= 0.1.
Note that the supply is inelastic at 100 + 3qt−1
5units, where qt−1 is
the quantity purchased by the date-(t− 1) speculators. As the currentdemand is low, even if the date-t speculators behave as aggressively asthey possibly can, the price will be no higher than $10. Tomorrow, theprice can be no lower than $8 in the bad state (which occurs with prob.0.9) and the price will be $1000 in the good state. For an individualspeculator who purchases q units of good X at date t, the expectedprofit will then be greater than or equal to
−q · 10 + [0.1 · 1000 + 0.9 · 8] · 3q5,
which is positive and increasing in q. We conclude that the speculatorsin total will purchase 50 units of good X in case θt = 0 and ξt = 1.
45
So, what prices are possible to emerge in equilibrium?
• In case at date t, θt = 1, then pt = $1000.
• In case θt = 0 and ξt = 1, then the date-t speculators will buy50 units, and this ensures that the net supply of good X left forconsumers can never exceed 80 units, implying that pt = $10.
• In case θt = ξt = 0, then pt depends on the state at date t− 1.
(a) If ξt−1 = 0, then date-(t − 1) speculators did not store thegood, and therefore pt is simply the low-state price in the ab-sence of speculators, which is $10.
(b) If ξt−1 = 1 = θt−1, it is obvious that date-(t − 1) speculatorsdid not store the good, and pt is still $10.
(c) If ξt−1 = 1 but θt−1 = 0, then the date-(t− 1) speculators didstore the good (and they must dump the good to the marketat date t) but the date-t speculators do not want to store thegood. In this case, the supply is inelastic at 130 units, whichleads to a price of $9.
Although the presence of speculators changes the equilibrium alloca-tions, it does not change the equilibrium price for X except in the abovecase (c). Thus our conclusion is that the presence of speculators resultsin excessive price volatility. The idea is that with a positive probabilitythe time-(t− 1) speculators may purchase 50 units of X in anticipationof a price rise at time t, but at time t the bad news arrives so that thetime-t speculators do not do the same. In this event, there is then anincrease in the net supply of X at time t, causing the price to drop to$9. In this example, the presence of the speculators increases the equi-librium consumer surplus, and it reduces the producers’ revenue whenthe aforementioned event occurs. Of course, the speculators themselvesmust benefit from trading.
32. Now we describe a general T -period Radner economy, and then offer aseries of examples with T = 3.
46
33. Assume that there are I investors endowed with some positions in Ntraded assets at date 0, where the assets can be traded at all dates0, 1, 2, · · · , T (so that they are called long-lived assets) and they gen-erate consumption (or cash flows) only at date T . Investors share thesame information structure, which is denoted by a filtered probabilityspace (Ω,FT , P,F); see sections 17-25 of Lecture 0. More precisely,each element ω ∈ Ω gives a complete description of the economy fromdate 0 to date T , and we refer to it as a state of nature; Ω is the setof all possible states of the economy; Ft is the collection of events inΩ that investors can tell whether have taken place at date t; and P isthe probability measure that assigns a probability to each date-t event.Recall that F = Ft; t = 0, 1, 2, · · · , T, with for all t ≥ 1, Ft−1 ⊂ Ft;that is, the information is accumulated over time.
34. An investor i’s endowed positions in the N traded assets at date 0 isdenoted by θiN×1(0). Let θ
i(t,Ft) denote the positions that an investortakes from date t − 1 into date t, for all t = 1, 2, · · · , T . Asset n isdenoted by the cash earnings that it generates at date T , denoted bydn(ω). The date-t cum-dividend price of asset n is denoted by Sn(t, ω).Note that Sn(T, ω) = dn(ω) for all n = 1, 2, · · · , N and all ω ∈ Ω.
35. It can be shown that if Ω is finite, then each σ-algebra F contains asubset P , which is a partition of Ω, such that every element of F isa (possibly empty) union of the elements of P. Conversely, for aeachpartition P of Ω, there exists a smallest σ-algebra F that contains P.Thus there is a one-to-one correspondence between the σ-algebras andthe partitions on Ω. The bottom line is that, if Ω is a finite set, thenwe can represent F by a sequence of finer and finer partitions of Ω,denoted by F = Pt; t = 0, 1, 2, · · · , T.
36. A natural information constraint on the asset prices is that for all t,for all Et ∈ Pt, and for all ω, ω′ ∈ Et, we must have
Sn(t, ω) = Sn(t, ω′),
and we denote that common outcome by Sn(t, Et). This happens be-cause at time t, investors know whether event Et occurs, but if Et isknown to have taken place, investors cannot tell which element in Et
47
is the true state. In this case, we say that the the asset price processSn(·, ·) is adapted to the filtration F.
37. At date 0, given θi, investor i seeks to maximize
E[T∑t=0
ρtui(ci(t, ω))],
by choosing the optimal budget-feasible investment policy
θi(t+ 1, Et); ∀Et ∈ Pt;∀t = 0, 1, 2, · · · , T − 1
and consumption plan
ci(t, Et); ∀Et ∈ Pt;∀t = 0, 1, 2, · · · , T,
where we have assumed naturally that ci(·, ·) is adapted to F, andnote that ρ ∈ (0, 1) is a discount factor measuring investor i’s timepreference. The investment policy θi is slightly different: θin(t) is thenumber of shares of asset n that investor i carries from date t− 1 intodate t, and so its value is determined using the information of Pt−1,not Pt.
Note also that the budget constraints facing investor i in the aboveoptimization problem are
N∑n=1
θin(1, E0)Sn(0, E0) + ci(0, E0) =N∑
n=1
θin(0)Sn(0, E0),
∀t = 1, 2, · · · , T, ∀Et ∈ Pt,N∑
n=1
θin(t+1, Et)Sn(t, Et)+ci(t, Et) =
N∑n=1
θin(t, Et−1)Sn(t, Et),
whereθi(T + 1, ET ) = 0, ∀n = 1, 2, · · · , N, ∀ET ∈ PT .
Here we adopt the Radner equilibrium concept: in forming the budgetconstraints, investor i rationally expects the vector stochastic process
S1(t, ω)S2(t, ω)
...SN(t, ω)
; ∀t = 0, 1, 2, · · · , T ; ∀ω ∈ Ω,
48
in the sense that, when in state ω the economy evolves to date t, themarkets-clearing prices of the N traded assets are exactly
S1(t, ω)S2(t, ω)
...SN(t, ω)
.
38. Now we consider a series of examples with T = 3.
Example 4. Consider an economy that extends for three dates (t =0, 1, 2). There are two consumers A and B, each endowed with one unitof asset X and one unit of asset Y. There are 4 possible states of nature,ω1, ω2, ω3, ω4. The two consumers’ common information structure is asfollows. At time 0, they know that the true state is among ω1, ω2, ω3, ω4.At time 1, they know whether the event E ≡ ω1, ω2 has or has notoccurred. At time 2, they learn the true state. The assets pay dividendsonly at time 2. The two investors’ subjective probability measuresand the dividends per unit (dpu, denominated in a single consumptiongood) for the two assets are summarized in the following table.
prob.’s; dpu’s/states ω1 ω2 ω3 ω4
A’s prob. 110
210
310
410
B’s prob. 410
310
210
110
X’s dpu 1 2 3 4Y’s dpu 3 4 5 6
A and B only want to consume at time 2. They have a commonvon Neumann-Morgenstern utility function for time-2 consumption:u(c) = log(c).(i) Is there a Radner equilibrium for this economy? If you think thatthere is one, find (1) the equilibrium price processes for assets X andY; and (2) A’s equilibrium trading strategy.(ii) Suppose that we add one more agent C into the economy. Suppose
49
that C has the same beliefs and preferences as A, but unlike A, C isendowed with no assets. Rather, C is endowed with e(t, at) ≥ 0 unitsof the consumption good at date t when event at occurs. You are toldthat in the equilibrium of this three-agent economy, the processes ofprices of X and Y are the same as in part (i). Find one non-zero posi-tive stochastic process e(t, at) that is consistent with this information.Solution. First consider part (i). It is clear that the markets arecomplete at time 1 whether the event E occurs or not. Let us con-jecture that the markets are dynamically complete, and then solve theequivalent Arrow-Debreu equilibrium. We shall verify our conjectureafterwards.
Thus, imagine an equivalent economy where at time 0 there is a com-plete set of traded Arrow-Debreu securities. Let ai and bi be respec-tively A’s and B’s holdings of the Arrow-Debreu security paying oneunit of consumption if and only if the time-2 state is ωi. Let pi be thetime-0 price of the Arrow-Debreu security corresponding to the time-2state ωi. Let us choose the normalization
4∑j=1
pj = 1;
that is, the time-2 consumption good is taken as numeraire. A seeks to
maxai∈ℜ;i=1,2,3,4
4∑i=1
i
10log(ai)
subject to4∑
i=1
aipi ≤4∑
j=1
2(1 + j)pj.
Similarly, B seeks to
maxbi∈ℜ;i=1,2,3,4
4∑i=1
5− i
10log(bi)
subject to4∑
i=1
bipi ≤4∑
j=1
2(1 + j)pj.
50
Solving the two maximization problems given p1, p2, p3, p4, and thensolving for p1, p2, p3, p4 using the markets clearing condition aj + bj =4(1 + j), j = 1, 2, 3, 4, we have
a1 =8
5, a2 =
24
5, a3 =
48
5, a4 = 16,
b1 =32
5, a2 =
36
5, a3 =
32
5, a4 = 4,
and
p1 =30
77, p2 =
20
77, p3 =
15
77, p4 =
12
77.
Next, imagine that markets reopen at time 1 in event E. Let q1 andq2 be the prices of the Arrow-Debreu securities corresponding to statesω1 and ω2 respectively. A’s problem is to
maxA1,A2∈ℜ
110
110
+ 210
log(A1) +210
110
+ 210
log(A2)
subject toA1q1 + A2q2 ≤ a1q1 + a2q2,
where A1 and A2 are the date-2 consumption for A in respectivelystates ω1 and ω2. Recall that if there is a complete set of traded Arrow-Debreu securities at date 0, and if traders have rational expectations,then Ai = ai, for i = 1, 2. This implies that (using q1 + q2 = 1),
q1 =3
5, q2 =
2
5.
Similarly, imagine that markets reopen at time 1 in event Ec and letq3 and q4 be the prices of the Arrow-Debreu securities correspondingto states ω3 and ω4 respectively. A’s problem is to
maxA3,A4∈ℜ
310
310
+ 410
log(A3) +410
310
+ 410
log(A4)
subject toA3q3 + A4q4 ≤ a3q3 + a4q4,
51
where A3 and A4 are the date-2 consumption for A in respectivelystates ω3 and ω4. After imposing Aj = aj, j = 3, 4, we have (usingq3 + q4 = 1),
q3 =5
9, q4 =
4
9.
Now, we consider the original Radner equilibrium. Let Px(t, at) andPy(t, at) be the equilibrium prices of assets X and Y at time t in eventat. Now we have
Px(2, ω1) = 1, Px(2, ω2) = 2, Px(2, ω3) = 3, Px(2, ω4) = 4,
Px(1, E) = Px(2, ω1)× q1 + Px(2, ω2)× q2 =7
5,
Px(1, Ec) = Px(2, ω3)× q3 + Px(2, ω4)× q4 =
31
9,
and
Px(0,Ω) =163
77.
Similarly, we have
Py(2, ω1) = 3, Py(2, ω2) = 4, Py(2, ω3) = 5, Py(2, ω4) = 6,
Py(1, E) = Py(2, ω1)× q1 + Py(2, ω2)× q2 =17
5,
Py(1, Ec) = Py(2, ω3)× q3 + Py(2, ω4)× q4 =
49
9,
and
Py(0,Ω) =317
77.
Note that the two vectors[Px(1, E)Px(1, E
c)
],
[Py(1, E)Py(1, E
c)
]
are linearly independent, and hence in the Radner equilibrium the mar-kets are dynamically complete as we conjectured.
52
Next, we consider A’s trading strategy in the Radner equilibrium. Thisis simply the trading strategy that finances the date-2 consumptionvector
a1
a2
a3
a4
=
85
245
485
16
.
Let θx(t+1, at) and θy(t+1, at) be respectively investor A’s equilibriumholdings of assets X and Y carried into time t+1 when event at occursat time t. We must have 1 3
2 4
θx(2, E)
θy(2, E)
=
85
245
,and hence we have θx(2, E)
θy(2, E)
=
4
−45
.Similarly, we have 3 5
4 6
θx(2, E
c)
θy(2, Ec)
=
485
16
,and hence we have θx(2, E
c)
θy(2, Ec)
=
565
−245
.It follows that the value of A’s trading strategy at time 1 is
Px(1, E)× θx(2, E) + Py(1, E)× θy(2, E) =72
25
53
if event E occurs, and it is
Px(1, Ec)× θx(2, E
c) + Py(1, Ec)× θy(2, E
c) =112
9
if event E does not occur.
Now, we must have Px(1, E) Py(1, E)
Px(1, Ec) Py(1, E
c)
θx(1)
θy(1)
=
7225
1129
,and hence we have θx(1)
θy(1)
=
749115
−211115
.The value of A’s trading strategy at time 0 can easily be verified toequal A’s date-0 wealth
Px(0,Ω) + Py(0,Ω).
Finally, consider part (ii). Apparently, one possible solution is that
e(0,Ω) = e(1, E) = e(1, Ec) = 0,
and
e(2, ω1)
e(2, ω2)
e(2, ω3)
e(2, ω4)
=
a1
a2
a3
a4
=
85
245
485
16
.
Example 5. Consider an economy that extends for three dates(t = 0, 1, 2). There is one single perishable consumption good, whichcannot be stored at any time. There are two consumers A and B, each
54
endowed with one unit of stock X and 480 units of the consumptiongood at time 0. There are 4 possible states of nature, ω1, ω2, ω3, ω4.The two consumers’ common information structure is as follows. Attime 0, they know that the true state is among ω1, ω2, ω3, ω4. At time1, they know whether the event E ≡ ω1, ω2 has or has not occurred.At time 2, they both learn the true state. The stock X pays a one-timeliquidation dividend at time 2. The two investors’ subjective proba-bility measures and the dividends per unit (dpu, denominated in thesingle consumption good) for stock X are summarized in the followingtable.
prob.’s; dpu’s/states ω1 ω2 ω3 ω4
A’s prob. 110
210
310
410
B’s prob. 410
310
210
110
X’s dpu 4 6 8 10
In addition to stock X, A and B are free to issue or purchase oneparticular European call option written on one unit of X which hasexercise price K and will expire at time 2. For all t = 0, 1, 2, A andB have a common von Neumann-Morgenstern utility function ut(·) fortime-t consumption: u0(z) = u2(z) = log(z), and u1(z) ≡ 0. Assumeno discounting, so that at time 0 agent i seeks to maximize
u0(ci0) + Ei[u1(c
i1) + u2(c
i2)|F0],
where cit is agent i’s time-t consumption, and Ei[·|F0] is agent i’s expec-tation operator (using his subjective probabilities) based the commontime-0 information F0.(i) Suppose that K ∈ (0, 4). Show that there is a Radner equilibriumand find the equilibrium price process for the above call option, or showthat no Radner equilibrium can exist.Solution. Consider part (i). For K ∈ (0, 4), the call option will al-ways be exercised at time 2, which together with stock X produces ariskless asset for the economy. Hence we conjecture that markets aredynamically complete with the stock and the option. Thus, imagine anequivalent economy where at time 0 there is a complete set of tradedArrow-Debreu securities. Let ai and bi be respectively A’s and B’sholdings of the Arrow-Debreu security paying one unit of consumption
55
if and only if the time-2 state is ωi. Let pi be the time-0 price of theArrow-Debreu security corresponding to the time-2 state ωi. Let uschoose the normalization
4∑j=1
pj = 1;
that is, the time-2 consumption good is taken as numeraire. Let p0 bethe time-0 price of spot consumption. A seeks to
maxcA0 ,ai∈ℜ;i=1,2,3,4
log(cA0 ) +4∑
i=1
i
10log(ai)
subject to
p0cA0 +
4∑i=1
aipi ≤4∑
j=1
2(1 + j)pj + 480p0.
Similarly, B seeks to
maxcB0 ,bi∈ℜ;i=1,2,3,4
log(cB0 ) +4∑
i=1
5− i
10log(bi)
subject to
p0cB0 +
4∑i=1
bipi ≤4∑
j=1
2(1 + j)pj + 480p0.
Note that it would be incorrect to claim that cA0 = cB0 = 480 becausethe consumption good is perishable.
Solving the above two maximization problems given p0, p1, p2, p3, p4,and then solving for p0, p1, p2, p3, p4 using the markets clearing condi-tions
cA0 + cB0 = 960, aj + bj = 4(1 + j), ∀j = 1, 2, 3, 4,
we havep0 = 1,
cA0 = cB0 = 480,
a1 =8
5, a2 =
24
5, a3 =
48
5, a4 = 16,
b1 =32
5, a2 =
36
5, a3 =
32
5, a4 = 4,
56
and
p1 =30
77, p2 =
20
77, p3 =
15
77, p4 =
12
77.
Thus for both agents, the time-0 consumption equals the time-0 en-dowment. This implies that the equilibrium asset allocation and assetprices will not change if we assume instead that the agents do not havetime-0 endowments. Note that the latter economy was studied in Ex-ample 1 above, where in that dynamically complete economy the twoagents A and B are endowed with time-2 state-dependent endowments,which are exactly the same as in the current problem (stock X in thecurrent problem is equivalent to the portfolio consisting of one unit ofeach traded asset in the economy studied in Example 1). Hence theequilibrium price processes of the 4 Arrow-Debreu securities in the cur-rent equilibrium are the same as those obtained in Example 1. Thisimplies that at all times the price of stock X in the Radner equilib-rium of the current economy is equal to the sum of the prices of thetwo traded assets in the Radner equilibrium of the economy studied inExample 1.36 Hence the price process of stock X is
Px(2, ω1) = 4, Px(2, ω2) = 6, Px(2, ω3) = 8, Px(2, ω4) = 10,
Px(1, E) =24
5,
Px(1, Ec) =
80
9,
and
Px(0,Ω) =480
77.
Since the call option is simply a portfolio consisting of one unit of stockX and shorting K units of the pure discount bond maturing at time2 with face value equal to 1 (unit of the consumption good), the priceprocess of the option is simply
Pc(2, ω1) = 4−K, Pc(2, ω2) = 6−K, Pc(2, ω3) = 8−K, Pc(2, ω4) = 10−K,36This is no accident. The endowment 480 was chosen deliberately to make sure that
those who have understood Example 1 can spare the need of going through all the compu-tations again. Of course, if the endowed time-0 consumption is not 480 for each agent, thena Radner equilibrium can still exist in general, but we have to go through the computationsall over again.
57
Pc(1, E) =24
5−K,
Pc(1, Ec) =
80
9−K,
and
Pc(0,Ω) =480
77−K.
Note that with these prices, markets are indeed dynamically complete,verifying our conjecture in the first place. This concludes part (i).
Example 6. Consider an economy that extends for three dates (t =0, 1, 2). There are two consumers A and B, each endowed with one unitof asset X and one unit of asset Y. There are 4 possible states of nature,ω1, ω2, ω3, ω4. The two consumers’ common information structure is asfollows. At time 0, they know that the true state is among ω1, ω2, ω3, ω4.At time 1, they know whether the event E ≡ ω1, ω2 has or has notoccurred. At time 2, they learn the true state. The assets pay dividendsonly at time 2. The two investors’ subjective probability measuresand the dividends per unit (dpu, denominated in a single consumptiongood) for the two assets are summarized in the following table.
prob.’s; dpu’s/states ω1 ω2 ω3 ω4
A’s prob. 110
210
310
410
B’s prob. 410
310
210
110
X’s dpu 3 0 1 2Y’s dpu 0 3 2 1
A and B only want to consume at time 2. They have a commonvon Neumann-Morgenstern utility function for time-2 consumption:u(c) = log(c). Is there a Radner equilibrium for this economy? Ifyou think that there is one, find it.Solution. There is no Radner equilibrium for this economy. To shownon-existence, first assume that a Radner equilibrium exists with dy-namically complete markets. Take the consumption good as the nu-meraire, so that
Px(1, E) + Py(1, E) = Px(1, Ec) + Py(1, E
c) = Px(0,Ω) + Py(0,Ω) = 3.
58
Now the date-1 price matrix
P ≡[Px(1, E) Py(1, E)Px(1, E
c) Py(1, Ec)
]
is either non-singular or singular. First assume that P is non-singular,so that in the Radner equilibrium markets are dynamically complete.Following the same procedure as in Problem 1, we can solve for the twoagents’ equilibrium consumption plans
a1
a2
a3
a4
=
65
125
185
245
,
b1
b2
b3
b4
=
245
185
125
65
,
and the asset prices
Px(1, E) = Py(1, E) = Px(1, Ec) = Py(1, E
c) = Px(0,Ω) = Py(0,Ω) =3
2,
apparently a contradiction to the assumption that P is non-singular.
Thus we are left with the possibility that P is singular whenever aRadner equilibrium exists. For some scalar p, we have
P ≡[p 3− pp 3− p
],
which implies that both asset X and asset Y are riskless during thedate-0-date-1 period. Thus regardless of whether event E occurs, A’sdate-1 wealth is 3.
Note that at date 1 markets become complete, whether or not event Ehas occurred. Let q1 and q2 be the date-1 prices of the Arrow-Debreusecurities corresponding to states ω1 and ω2 respectively when event Eis known to have occurred. Let q3 and q4 be the date-1 prices of theArrow-Debreu securities corresponding to states ω3 and ω4 respectively
59
when event Ec is known to have occurred. Then a1 and a2 must solvethe maximization problem
(M1) maxA1,A2
1
3log(A1) +
2
3log(A2)
subject toq1A1 + q2A2 = 3;
b1 and b2 must solve the maximization problem
(M2) maxB1,B2
4
7log(B1) +
3
7log(B2)
subject toq1B1 + q2B2 = 3;
a3 and a4 must solve the maximization problem
(M3) maxA3,A4
3
7log(A3) +
4
7log(A4)
subject toq3A3 + q4A4 = 3;
and b3 and b4 must solve the maximization problem
(M4) maxB3,B4
2
3log(B3) +
1
3log(B4)
subject toq3B3 + q4B4 = 3.
Comparing the first two maximization problems (M1) and (M2) to therest two maximization problems (M3) and (M4), we conclude that
q1 = q4, q2 = q3,
which then implies, by Px(1, E) = Px(1, Ec) = p, that
3q1 = Px(1, E) = Px(1, Ec) = q3+2q4 = q1+2q2 ⇒ q1 = q2 ⇒ q1 = q2 = q3 = q4.
60
However, by solving (M1) and (M2) directly, we have
a1 =1
q1, a2 =
2
q2, b1 =
12
7q1, b2 =
9
7q2,
so that by the markets clearing conditions,
(1 +12
7)1
q1= 3 = (2 +
9
7)1
q2⇒ q1 = q2,
a contradiction! Hence we conclude that P cannot be singular either,and no Radner equilibrium can exist for this economy.
Example 7. Consider an economy that extends for three dates (t =0, 1, 2). There are two consumers A and B, each endowed with one unitof asset X and one unit of asset Y. There are 4 possible states of nature,ω1, ω2, ω3, ω4. The two consumers’ common information structure is asfollows. At time 0, they know that the true state is among ω1, ω2, ω3, ω4.At time 1, they know whether the event E ≡ ω1, ω2 has or has notoccurred. At time 2, they learn the true state. The assets pay dividendsonly at time 2. The two investors’ subjective probability measuresand the dividends per unit (dpu, denominated in a single consumptiongood) for the two assets are summarized in the following table.
prob.’s; dpu’s/states ω1 ω2 ω3 ω4
A’s prob. 14
14
14
14
B’s prob. 12
16
16
16
X’s dpu 1 2 3 4Y’s dpu 4 3 2 1
A and B only want to consume at time 2. They have a commonvon Neumann-Morgenstern utility function for time-2 consumption:u(c) = log(c).(i) Find the price processes for assets X and Y in a Radner equilibrium.(ii) Find B’s equilibrium trading strategy.Solution. First consider part (i). It is clear that the markets arecomplete at time 1 whether the event E occurs or not. Let us con-jecture that the markets are dynamically complete, and then solve the
61
equivalent Arrow-Debreu equilibrium. We shall verify our conjectureafterwards.
Thus, imagine an equivalent economy where at time 0 there is a com-plete set of traded Arrow-Debreu securities. Let ai and bi be respec-tively A’s and B’s holdings of the Arrow-Debreu security paying oneunit of consumption if and only if the time-2 state is ωi. Let pi be thetime-0 price of the Arrow-Debreu security corresponding to the time-2state ωi. Let pE and pEc be the time-0 price of the Arrow-Debreu se-curity that pays one unit of consumption at time 1 when respectivelyevent E and event Ec occur. A seeks to
maxai∈ℜ;i=1,2,3,4
1
4
4∑i=1
log(ai)
subject to4∑
i=1
aipi ≤ 54∑
j=1
pj.
Solving, we have
ai =5∑4
j=1 pj
4pi.
Similarly, B seeks to
maxbi∈ℜ;i=1,2,3,4
1
2log(b1) +
1
6
4∑i=2
log(bi)
subject to4∑
i=1
bipi ≤ 54∑
j=1
pj.
Solving, we have
b1 =5∑4
j=1 pj
2p1, bi =
5∑4
j=1 pj
6pi, ∀i = 2, 3, 4.
Let us choose the normalization
4∑j=1
pj = 1;
62
that is, the time-2 consumption good is taken as numeraire. The mar-kets clearing condition now implies that
p1 =9
24, p2 = p3 = p4 =
5
24.
It follows that the equilibrium consumption bundles for A and B arerespectively
a1a2a3a4
=
103
666
,and
b1b2b3b4
=
203
444
.
Next, imagine that markets reopen at time 1 in event E. Let q1 andq2 be the prices of the Arrow-Debreu securities corresponding to statesω1 and ω2 respectively. A’s problem is to
maxA1,A2∈ℜ
1
2
2∑i=1
log(Ai)
subject to
A1q1 + A2q2 ≤10
3q1 + 6q2,
so that we have
A1 =103q1 + 6q22q1
, A2 =103q1 + 6q22q2
.
For A1 =103and A2 = 6, we have (normalizing q1 + q2 = 1),
q1 =9
14, q2 =
5
14.
63
Similarly, imagine that markets reopen at time 1 in event Ec and letq3 and q4 be the prices of the Arrow-Debreu securities correspondingto states ω3 and ω4 respectively. A’s problem is to
maxA3,A4∈ℜ
1
2
2∑i=1
log(Ai)
subject toA3q3 + A4q4 ≤ 6q3 + 6q4,
so that we have
A3 =6q3 + 6q4
2q3, A4 =
6q3 + 6q42q4
.
For A3 = A4 = 6, we have (normalizing q3 + q4 = 1),
q3 =1
2, q4 =
1
2.
Now, we consider the original Radner equilibrium. Let Px(t, at) andPy(t, at) be the equilibrium prices of assets X and Y at time t in eventat. Now we have
Px(2, ω1) = 1, Px(2, ω2) = 2, Px(2, ω3) = 3, Px(2, ω4) = 4,
Px(1, E) = Px(2, ω1)× q1 + Px(2, ω2)× q2 =19
14,
Px(1, Ec) = Px(2, ω3)× q3 + Px(2, ω4)× q4 =
7
2,
and
Px(0,Ω) =54
24.
Similarly, we have
Py(2, ω1) = 4, Py(2, ω2) = 3, Py(2, ω3) = 2, Py(2, ω4) = 1,
Py(1, E) = Py(2, ω1)× q1 + Py(2, ω2)× q2 =51
14,
Py(1, Ec) = Py(2, ω3)× q3 + Py(2, ω4)× q4 =
3
2,
64
and
Py(0,Ω) =66
24.
From here, we deduce pE = 1424
and pEc = 1024.
Note that the two vectors[Px(1, E)Px(1, E
c)
],
[Py(1, E)Py(1, E
c)
]
are linearly independent, and hence in the Radner equilibrium the mar-kets are dynamically complete as we conjectured.
Next, we consider B’s trading strategy in the Radner equilibrium. Thisis simply the trading strategy that finances the date-2 consumptionvector
b1b2b3b4
=
203
444
.
Let θx(t+1, at) and θy(t+1, at) be respectively B’s equilibrium holdingsof assets X and Y carried into time t+ 1 when event at occurs at timet. We must have 1 4
2 3
θx(2, E)
θy(2, E)
=
203
4
,and hence we have θx(2, E)
θy(2, E)
=
−45
2815
.Similarly, we have 3 2
4 1
θx(2, E
c)
θy(2, Ec)
=
4
4
,
65
and hence we have θx(2, Ec)
θy(2, Ec)
=
45
45
.It follows that the value of B’s trading strategy at time 1 is
Px(1, E)× θx(2, E)+Py(1, E)× θy(2, E) =19
14× (
−4
5)+
51
14× 28
15=
40
7
if event E occurs, and it is
Px(1, Ec)× θx(2, E
c) + Py(1, Ec)× θy(2, E
c) =7
2× 4
5+
3
2× 4
5= 4
if event E does not occur.
Now, we must have1914
5114
72
32
θx(1)
θy(1)
=
407
4
,and hence we have θx(1)
θy(1)
=
2850
6850
.The value of B’s trading strategy at time 0 is thus
54
24× x(1) +
66
24× y(1) = 5.
39. If an economy with incomplete markets would like to open new mar-kets, what kind of assets should be chosen so that new markets can beopened for them? The recent literature on security design and financialinnovation intends to answer this question. Here we give a brief reviewfor the results that are applicable to our current two-period economywith perfect but incomplete financial markets.
Recall the simple economy with two investors U and V that we de-scribed in section 1. First observe that to attain Pareto efficiency it
66
suffices for the economy to have a traded Arrow-Debreu security foreach aggregate consumption level. This follows from Corollary 1: if instates j and k, Cj = Ck, then a Pareto efficient allocation must pre-scribe the same consumption allocation for U and V in states j andk; that is, aj = ak and bj = bk. Thus we do not need to have onetraded Arrow-Debreu security for each state of nature, unless of courseCj = Ck for all j = k.
What if at date 0 we do not have a complete set of traded Arrow-Debreusecurities for distinct date-1 aggregate consumption levels? Ross (1976)demonstrates how we can use financial options to complete the market.To demonstrate Ross’s idea, suppose that there are three likely date-1events, s1, s2, and s3, corresponding to three likely levels of date-1 ag-gregate consumption. Suppose that there is only one traded (primary,or underlying) asset X1 at date 0, of which the date-1 payoff vector is
s1 s2 s3X1 4 3 1
so that markets are incomplete. Consider introducing at date 0 a (Eu-ropean) call option C(X1, q) written on 1 unit of X1 with exercise priceq and expiry equal to date 1. It is clear that markets become completeif we can introduce two options C(X1, 1) and C(X1, 2), for the threepayoff (row) vectors below are linearly independent:
s1 s2 s3X1 4 3 1
C(X1, 1) 3 2 0C(X1, 2) 2 1 0
However, let us not get over-optimistic at this point. Introducing fi-nancial options on X1 can complete the market in this example becausethere exists a one-to-one correspondence between the set of likely date-1 payoffs of asset X1 and the set of likely levels of date-1 aggregateconsumption. If, for example, the date-1 vector of X1 were
s1 s2 s3X1 2 2 1
67
then it is impossible to complete the market by introducing financialoptions written on X1. Hence introducing financial options can com-plete the market only if the set of traded primary assets has date-1payoff vectors that can fully separate the levels of date-1 aggregateconsumption. Ross emphasizes that this is only a necessary conditionfor adding ordinary financial options to be able to help complete themarket, and is generally insufficient. Consider the following examplewhere there are 4 likely levels of date-1 aggregate consumption:
s1 s2 s3 s4X1 1 1 2 2X2 1 2 1 2
Note that the date-1 payoffs of X1 and X2 fully separate the 4 eventss1, s2, s3, and s4. After introducing C(X1, 1) and C(X2, 1), we have
s1 s2 s3 s4X1 1 1 2 2X2 1 2 1 2
C(X1, 1) 0 0 1 1C(X2, 1) 0 1 0 1
but markets are still incomplete! What we need is a general option thatcan be written on either a portfolio on X1 and X2, or that has a payoffdepending on the date-1 payoffs of both X1 and X2. For example,consider the portfolio Xp that consists of 2 units of X1 and 1 unit ofX2. The date-1 payoff of Xp is
s1 s2 s3 s4Xp 3 4 5 6
Now, introducing two ordinary financial options written on Xp com-pletes the market:
s1 s2 s3 s4X1 1 1 2 2X2 1 2 1 2
C(Xp, 3) 0 1 2 3C(Xp, 4) 0 0 1 2
68
Since creating financial options is much cheaper than creating a primaryasset (like issuing a common stock), Ross’s theory emphasizes the roleof financial options in helping financial markets attain Pareto efficiency.
40. We have thus far assumed that investors U and V have homogeneousbeliefs; that is, they hold the same probabilities for the n date-1 statesof nature. When investors have homogeneous beliefs, financial marketsserve the sole function of facilitating risk sharing. We have shown thatfor that purpose, we need to have one Arrow-Debreu security for eachdistinct level of date-1 aggregate consumption in order to guaranteePareto efficient competitive equilibrium.
In reality, investors generally have different subjective beliefs. In thiscase, besides the risk-sharing motive, investors may want to trade as-sets for speculative reasons, and to attain Pareto efficiency there mustbe sufficiently many traded assets that provide one gambling opportu-nity for each event for which investors disagree about the conditionalprobabilities for the states contained in that event. This is the maintheme of this section.
The first useful insight in this aspect is due to Hakansson (1977). Todemonstrate his idea, assume that n = 5, with
C(ω1) = C(ω2) = C(ω4) = 3, C(ω3) = C(ω5) = 6,
and defines1 = ω1, ω2, ω4, s2 = ω3, ω5.
Let πU(ωk) and πV (ωk) be the probabilities that U and V respectivelyassign to state ωk.
Hakansson’s theorem says that if
πU(ωk|s1) = πV (ωk|s1), ∀k = 1, 2, 4,
andπU(ωj|s2) = πV (ωj|s2), ∀k = 3, 5,
then financial markets only play the role of facilitating risk sharing; thatis, if U and V can trade two Arrow-Debreu securities, one for s1 andthe other for s2, then they will not have speculative demands although
69
the two sets of probabilities πU(·) and πV (·) differ. The following is anexample.
ω1 ω2 ω3 ω4 ω5
πU(·) 112
16
13
14
16
πV (·) 19
29
29
13
19
Note that U and V have the same posterior beliefs conditional on eithers1 or s2:
ω1 ω2 ω3 ω4 ω5
πU(·|s1) 16
13
0 12
0πV (·|s1) 1
613
0 12
0πU(·|s2) 0 0 2
30 1
3
πV (·|s2) 0 0 23
0 13
Hakansson’s result tells us when the speculative motives of asset tradingwill disappear so that we only need to have enough financial markets toensure efficient risk sharing, but he does not tell us how many marketswe need to have if the speculative trading motives still exist given theset of Arrow-Debreu securities tracking all the likely date-1 aggregateconsumption levels. The answer to this question is provided by Amerish(1985).
Let Ω denote the set of likely date-1 states of nature. We need to define3 partitions of Ω in order to explain Amerish’s theory. First, let Γ bethe partition that contains all the sj’s. That is, events contained inΓ correspond to the likely realized levels of date-1 aggregate consump-tion. The number of elements of Γ is the number of Arrow-Debreusecurities that we need to provide to the investors in order to ensureefficient risk-sharing. Second, we need to define another partition Λ,of which the number of elements is the number of financial assets thatwe need to provide to investors in order to allow them to resolve thedifferences between their subjective beliefs. That is, this partition Λtells us U’s and V’s speculative trading motives. From Γ and Λ, wecan then construct a partition which is finer than both Γ and Λ, andis the coarsest partition that has this property. We shall denote thispartition by Θ. Amerish’s theorem says that to ensure Pareto efficientcompetitive equilibrium, we need to have one Arrow-Debreu securityfor each distinct element contained in Θ.
70
Let us explain Amerish’s theory by looking at the following example.Assume that n = 5, with
C(ω1) = C(ω2) = 3, C(ω3) = 2, C(ω4) = C(ω5) = 4,
and defineΓ = s1, s2, s3,
withs1 = ω1, ω2, s2 = ω3, s3 = ω4, ω5.
The role of Γ should be self-evident.
Now, to define Λ, we need to specify the two investors’ subjective be-liefs. Assume that
ω1 ω2 ω3 ω4 ω5
πU(·) 13
18
724
112
16
πV (·) 320
320
110
15
25
The partition Λ is the coarsest partition on Ω that possesses the prop-erty that conditional on each element contained in Λ, U and V sharethe same posterior beliefs. Amerish provides a procedure to find sucha partition. At first, for each ωj, define
ψU(ωj) ≡πU(ωj)
πU (ωj)+πV (ωj)
2
,
and
ψV (ωj) ≡πV (ωj)
πU (ωj)+πV (ωj)
2
.
Then Λ is defined by the equivalence relation that for each partitionset s ∈ Λ,
ωj, ωk ∈ s ∈ Λ ⇔ ψU(ωj) = ψU(ωk) and ψV (ωj) = ψV (ωk).
In the current example, we have
ω1 ω2 ω3 ω4 ω5
ψU(·) 4029
1011
7047
1017
1017
ψV (·) 1829
1211
2447
2417
2417
71
so that we have
Λ = ω1, ω2, ω3, ω4, ω5.
That is, to resolve the difference between the two investors’ subjectivebeliefs, or to allow asset trading based on speculative motives, we needto have at least 4 Arrow-Debreu securities.
Now, to satisfy both the risk-sharing demands represented by Γ andthe speculative demands represented by Λ, we must find a partitionthat is at least as fine as both Γ and Λ. In fact, we only need to findthe coarsest partition that has this property. In the current example,we have
Θ = Λ,
and so we need 4 Arrow-Debreu securities that, respectively, pay 1 unitof consumption at date 1 if and only if exactly one element containedin Λ occurs at date 1. Note that having 3 Arrow-Debreu securitiesassociated with respectively the events s1, s2 and s3 is not enough toensure Pareto efficiency, exactly because U and V cannot resolve theirdifference in subjective beliefs by trading only these 3 securities. Moreprecisely, in trading the Arrow-Debreu security associated with s1, U ismore optimistic about the likelihood of ω1 and more pessimistic aboutthe likelihood of ω2 than V, and it would be more efficient if we cansplit this Arrow-Debreu security into two securities associated with ω1
and ω2 respectively. This will allow U and V to engage in a side betthat Pareto improves their welfare.
As another example, consider the following U’s and V’s subjective be-liefs:
ω1 ω2 ω3 ω4 ω5
πU(·) 13
18
724
18
18
πV (·) 320
320
110
15
25
Show that in this case
Θ = Λ = ω1, ω2, ω3, ω4, ω5,
so that we need to have one Arrow-Debreu security for each state ofnature in order to attain Pareto efficiency.
72
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