Suppose there are two investors, the first having constant relative risk aversion > 0 and the

Suppose there are two investors, the first having constant relative risk aversion ρ > 0 and the second having constant relative risk aversion 2ρ.

(a) Show that the Pareto-optimal sharing rules are w˜ 1 = ˜wm +η − 
η2 +2ηw˜ m , and w˜ 2 = 
η2 +2ηw˜ m − η, for η > 0. Hint: Use the first-order condition and the quadratic formula.
Because η is arbitrary in (0,∞), there are many equivalent ways to write the sharing rules.

(b) Suppose the market is complete and satisfies the law of one price. Show that the SDF in a competitive equilibrium is m˜ = γ
*
η2 + 2ηw˜ m − η
+−2ρ
for positive constants γ and η.