Consider the EUM criterion with the exponential utility function u(x) = e x . (a) Show that if for some X and Y, we have

Consider the EUM criterion with the exponential utility function u(x) = −e^−βx.

(a) Show that if for some X and Y, we have X ≳ Y, then w+X ≳ w+Y for any number w. (The number w may be interpreted as the initial wealth, and X and Y as random incomes corresponding to two investment strategies. The above assertion claims that in the exponential utility case, the preference relation between X andY does not depend on the initial wealth.)

(b) (Additivity property.) Show that for any two independent r.v.’s X and Y, the certainty equivalent c(X₁ +X₂) = c(X₁)+c(X₂). Interpret this, viewing X₁,X₂ as the results of two independent investments.

(c) Show that for, say, u(x)= √x the above two assertions are not true.