8. certainty equivalents Ralph is contemplating a lottery. A fair coin will be tossed. If the coin shows "heads," Ralph will be paid 100 dollars. If the coin shows

"tails," Ralph will be paid nothing. So the expected value of this lottery is .5(100) + .5(0) = 50.

(a) Initially suppose Ralph’s utility for wealth is given by U(w) = √w and that Ralph’s initial wealth is zero. Determine Ralph’s certainty equivalent and risk premium for this lottery. Why is CE < 50? Also, why is CE > 0?

(b) Using the same root utility, now determine Ralph’s risk premium for initial wealth wi ∈ {0, 5, 10, 25, 50, 100, 500, 1,000}.

Interpret your finding.

(c) Now let U(w) = −exp(−ρw), with ρ = .01. Repeat the construction in

(b) above. Interpret your finding.

(d) Again let U(w) = −exp(−ρw). Determine Ralph’s risk premium for ρ ∈ {.0005, .001, .01, .06, .1, 1}. Interpret your finding.

What happened to initial wealth in your construction?