Consider a decision maker that who is considering whether or not to make an investment for D dollars, that has probability p1 of returning R1 dollars, or p2 of returning R2 dollars. Assume that R1 = 0; R2 = 10; p1 = p2 = 0.5 and D = 5.

The celebrated von-Neumann-Morgenstern expected utility maximization theorem says that under a reasonable set of mathematical conditions, there exists a utility function U (x) over the value of the outcome x, such that the decision maker chooses the decision that maximizes their expected utility E[U(x)].

In the case of the investment example, the decision maker compares p1* U(R1 D) + p2* U(R2 D) (the expected utility of making the investment) to zero (the expected utility of passing, i.e., not making investment), and chooses the decision that yields the higher value.

We'll assume U is differentiable at least two times, with U (0) = 0 and U (x) > 0 (i.e., U (x) is strictly increasing).

Suppose that U(x) < 0 everywhere. Does the investor prefer to invest or pass? Explain your answer.

Suppose that U(x) > 0 everywhere. Does the investor prefer to invest or pass? Explain your answer.