METODOLOGÍA

The probability measure over states of risk is

ν ∈∆(S),

and the utility function of the individual is

U =Eνu.

Assumption 3

(2) prob{rj ≥ 0} =1, prob{rj = 0} 6= 1, rj is linearly independent with

return vectors of other assets, andErl_j <+∞for all natural number

l.

Assumption 30

(1) u isC2 on _R++, is strictly increasing and strictly concave;

(2) prob{rj ≥ 0} =1, prob{rj = 0} 6= 1, rj is linearly independent with

return vectors of other assets, and Erl

j <+∞ for l= 1,2. Assumption 300

(1) uisC∞on_R++, is strictly increasing and strictly concave, andu(n)=

∂nu

∂xn 6= 0, at somex∈(0,∞),n = 1, ...;

(2) prob{rj ≥ 0} =1, prob{rj = 0} 6= 1, rj is linearly independent with

return vectors of other assets, and Erl

j <+∞ for l= 1,2.

At prices of assets p∈Po, the optimization problem of the individual is

max

y∈RJ

Eνu(Ry)

s.t. py = 1.

The demands for assets satisfies necessary and sufficient first order condi- tions

Eνu

0

(Ry)R=λp, λ >0;

py = 1.

As a consequence, the demand for assets identifies the family of marginal rates of substitution mjk(y) = Eνu 0 (Ry)rj Eνu 0 (Ry)rk >0.

Lemma 2 (Green, Lau and Polemarchakis, 1979). Suppose that

(1) the utility index u, and asset return rj for j = 1, ..., J satisfy As- sumption 3;

Then, the demand for assets identifies the cardinal index for risk u up to a positive affine transformation.

Proof. Without loss of generality, we normalize u(0) = 0, and u0(0) = 1. Differentiation of the functional equation

Eνu

0

(Ry)rj =mjk(y)Eνu

0

(Ry)rk, (3.1)

with respect to yk yields

Eνu 00 (Ry)rjrk= ∂mjk(y) ∂yk Eνu 0 (Ry)rk+mjk(y)Eνu 00 (Ry)r2_k. (3.2) At y=(0,...,0), we have u00(0)(Erjrk−Er2k) = ∂mjk(y) ∂yk Eνrk. (3.3)

Repeat the above differentiation with respect to yj, and also evaluate

aty = (0, ...,0), we have

u00(0)(Erjrk−Er2j) =−

∂mjk(y)

∂yj

Eνrk. (3.4)

Hold’s inequality can be used to show that Erjrk − Er2k = 0 and

Erjrk−Er2j = 0 cannot hold at the same time, so we can uniquely recover

u00(0). Higher-order differentiations will recover higher-order derivatives of

u at 0, which will recover index ugiven the assumptions on u.

Remark 10. Green, Lau, and Polemarchakis (1979) do not require a risk- free asset. Instead, the cardinal risk index is analytic at x= 0.

Lemma 3 (Dybvig and Polemarchakis, 1981). Suppose that

(1) the utility index u, and asset return rj for j = 1, ..., J satisfy As- sumption 30;

(2) there is an asset that is risk-free: r1 = 1 across states of risk;

(3) the probability measure over states of risk, ν ∈∆(S), is known. Then, the demand for assets identifies the cardinal index for risk u up to a positive affine transformation.

Proof. Differentiation of the functional equation Eνu 0 (Ry)rj =mjk(y)Eνu 0 (Ry)rk, (3.5)

with respect to yk yields

Eνu 00 (Ry)rjrk= ∂mjk(y) ∂yk Eνu 0 (Ry)rk+mjk(y)Eνu 00 (Ry)r2_k. (3.6) For j = 1, Eνu 00 (Ry)rk = ∂mjk(y) ∂yk Eνu 0 (Ry)rk+mjk(y)Eνu 00 (Ry)r2_k, (3.7) which, evaluated at the portfolio y= (x,0, ...,0), yields

u00(x) = ∂mjk(x,0, ...,0) ∂yk u0(x) +mjk(x,0, ...,0)u 00 (x)Eνr2k. (3.8) Since mjk(x,0, ...,0)u 00

(x) < 0, this identifies the risk-aversion of the individual, −u 00 (x) u0(x) = ∂mjk(x,0,...,0) ∂yk mjk(x,0, ...,0)Eνr2_k−1 , x∈(0,∞), (3.9)

or, equivalently, the cardinal risk index, u, up to an affine transformation. To see this, notice that the right hand side is observable, and we integrate both sides to get lnu

0

(x)

+ lnB, where B is an integration constant. Expo- nentiate and integrate again, we get Bu(x) +C,where C is an integration constant, which is positive affine transformation of u(x).

Remark 11. The argument does not require full knowledge of the distribu- tion of payoff, (R,ν), only of the second moment of the payoff of a risky asset.

Lemma 4 (Polemarchakis,1983). Suppose that

(1) the utility index u, and asset return rj for j = 1, ..., J satisfy As- sumption 300;

(3) the cardinal risk index u is known.

Then, the demand for assets identifies all moments of the distribution of payoffs of assets.

Proof. Without loss of generality,u is smooth and un_{(1)6= 0 at x= 1.}

As previously, only, here, with u(1)_{(x) =u}0_{(x) and u}(2)_{(x) =u}00_(x),

Eνu(2)(Ry)rjrk =

∂mjk(y)

∂yk

Eνu(1)(Ry)rk+mjk(y)Eνu(2)(Ry)r2k, (3.10)

which, evaluated at y= (1,0, ...,0), and with u(1) _{= 1, yields}

Eνu(2)(1)rjrk =

∂mjk(y)

∂yk

+mjk(y)u(2)(1)Eνr2k. (3.11)

Since mjk(y)>0, while u2(1)6= 0, this identifies

Eνr2k and Eνrjrk,

the second moments of the distribution of payoffs of assets. Higher order derivatives identify higher moments; at each step the coefficients of the moments to be identified do not vanish.

Remark 12. Instead of the cardinal risk index, u, it suffices to knowEνr2k,

the second moment of the distribution of returns of a risky asset.

Remark 13. Knowing second moment of the return of one risky asset is necessary for above recovery argument. One example is this: an investor with a CARA cardinal utility index, u(x) = −e−ρx _{demands a risky asset}

with normally distributed payoffs, r2 ∼ N(µ, σ2) against a risk-free asset

with payoffr1 =raccording toy2 = µ

−r

ρσ2; it follows that simultaneous iden- tification of the cardinal risk aversion index and the distribution of payoffs assets, without any, even partial, knowledge of either, is not possible.