The utility of a consumer given a U= W^1/2 with an initial endowment of 900 birr is determined by the consumer's willingness to pay to avoid risk. If the consumer is involved in a game of fair gamble which could lead to a win/loss of 500 birr, then the consumer is willing to pay an amount equal to the expected utility of the win/loss.
In order to calculate the amount the consumer is willing to pay to avoid risk, we need to first calculate the expected utility of the win/loss. The expected utility is equal to the probability of winning (1/2) multiplied by the utility of winning (900 birr) plus the probability of losing (1/2) multiplied by the utility of losing (400 birr).
Therefore, the expected utility of the win/loss is 500 birr. Since the consumer is willing to pay an amount equal to the expected utility of the win/loss, the consumer is willing to pay 500 birr to avoid risk.