Show that any monotone LRT utility function is a monotone affine transform of one of the five utility functions: negative exponential, log, power, shifted log, or shifted power. Hint: Consider first the special cases (i) risk tolerance = A and (ii) risk tolerance = Bw. In case (i) use the fact that u(w)

u

(w) = dlogu

(w)

dw and in case (ii) use the fact that wu(w)

u

(w) = dlog u

(w)

dlogw to derive formulas for logu

(w) and hence u

(w) and hence u(w). For the case A = 0 and B = 0, define v(w) = u

w−A B



, show that the risk tolerance of v is Bw, apply the results from case (ii) to v, and then derive the form of u.