For each of the following utility functions over wealth w, compute the following:

i) the expected utility and the certainty equivalent of the lottery: 10% chance of ending up with $100 and 90% chance of ending up with $10,000;

ii) x, where if you start with $10,000, you would be indifferent between gaining $x for sure and a free lottery ticket giving you a 1-in-1,000,000 chance of gaining $990,000 (and otherwise gaining nothing, thus staying at $10,000).

a) w^0.5

b) log₁₀(w)

c) w

d) w²