Problem 4. Choice Under Uncertainty (short problems). 1. Under certainty, any increasing monotone transformation of a utility function is also a utility function representing the same preferences. Under uncertainty, we must re- strict this statement to linear transformations if we are to keep the same preference representation. Check this with the following example. Assume an initial utility function attributes the following values to three possible outcomes: B u(B) = 100 M u(M) = 10 P u(P) = 50 (a) Check that with this initial utility function, the lottery L1 = (0.5, 0.5,0), which pays B with probability 0.5 and M with probability 0.5, is strictly preferred to L2 = (0, 0, 1), which pays P with probability 1, that is L1 = (0.5, 0.5, 0) > L2 = (0, 0, 1). (b) Suppose we have the following transformations: f(u) = a+bu, a20,b >0, g(u) = In(u). Check that under f, L1 > L2, and under g, L2 > L1. 2. Find the reduced lottery corresponding to the compound lottery L = (L1, L2; 1, 1 - #) where L1 = (1,0), L2 = (7, 1- T)