Given the probabilities, Prove the von Neumann-Morgenstern model of risk-averse.

In a state of nature 1 the individual has income w, whereas in the state of nature 2 the individual's income is y > . The probabilities that these states will occur are (1-p) and p, respectively. The individual can purchase insurance before the state of nature is known; an increase in income of s in state 2 can be purchased by a reduction in income of (pi sign) s in state 1. Prove that a risk-averse von Neumann-Morgenstern perticular will over-insure, fully insure, or under insure as per as the insurance is available at a price (pi sign) lower than, equal to, or higher than the actuarially fair price.