Explain what type of utility function this means we canuse to represent her preferences

If June's preferences over state-contingent wealth bundles conform to the theory of Subjective Expected Utility, this means that she evaluates each bundle based on its expected utility, which is a weighted average of the utilities of the wealth levels in each state, with the weights being her subjective probabilities of the states.

This implies that we can represent her preferences using a von Neumann-Morgenstern utility function. This type of utility function assigns a real number (the utility) to each possible outcome in a way that reflects the individual's preferences over the outcomes.

The utility function is typically assumed to be increasing (more wealth is preferred to less) and concave (reflecting risk aversion). A common form of the utility function is u(x) = ln(x), but other forms can also be used, depending on the specific characteristics of the individual's preferences.

In June's case, if she is risk averse, her utility function would be concave. This means that she prefers a certain outcome to a risky one with the same expected wealth. If she is risk neutral, her utility function would be linear. And if she is risk loving, her utility function would be convex.

The exact form of the utility function would depend on the specific shape of her indifference curves, which represent the combinations of wealth in the two states that give her the same level of utility. These indifference curves can be derived from the utility function by setting the expected utility equal to a constant and solving for one of the wealth levels in terms of the other.