What does it mean to say that an insurance contract is

Question

Recall that in an expected utility model with a good g and bad b state of nature, preferences over state-contingent consumption bundles (c_g, cb) are represented by the utility function

EU (c_g, c_b) = (1 - p)U(cg) pU (c_b) 

for some function U, where p is the probability of the bad state occurring. Suppose that Bob has preferences given by (1) with U(c) = _N/E and that in the absence of insurance his consumption would be (c_g, cb) = (14,5).

Suppose that p = 1/5 and that an insurance company offers Bob arbitrarily many units of the state-contingent consumption bundle (-1, 4). That is, Bob can choose to add (-t, 3t) of consumption to his current consumption for any t > 0.

1. What does it mean to say that an insurance contract is actuarially fair? Is the above contract actuarially fair?

2. How much of the above contract will Bob purchase (i.e. what is the optimal level of t from his perspective)?

3. What will his state-contingent consumption be after he purchases t units of this contract?

4. Now suppose that the an insurance company offers Bob arbitrarily many units of the state-contingent consumption bundle (-2, 4). How much of the above contract will Bob purchase and what will be ensuing state-contingent consumption?