Suppose you and I are the only two individuals in the world and we both face individual risk in the following way: I get more consumption in state 1 than in state 2 while you get more consumption in state 2 than in state 1.
a. Suppose we are both risk averse and our tastes are state-independent. Will we fully insure one another in a competitive equilibrium?
b. How does your answer change if there is aggregate risk in the sense that overall consumption is higher in state 2 than in state 1?
c. Is it possible that we insure each other if our tastes are risk neutral and state-independent? If so, are the terms actuarily fair?
d. Suppose that there is no aggregate risk but our tastes are state-dependent. How might we fully insure each other if our beliefs about the probability of each state differ?
What will be an ideal response?
a. Yes. (See panel (a) of Graph 17.10 in the text.)
b. We can no longer fully insure each other. (See Graph 17.11 in the text.)
c. Risk neutrality implies that our indifference curves are straight lines. This implies they cannot be tangent in the Edgeworth Box -- unless they lie right on top of one another. But this is precisely the case as our indifference curves have the same slope (given state-independence) as long as we have common beliefs about probabilities. That slope is the same as the slope of the contract line that incorporates actuarily fair terms of trade. Thus, any insurance will be actuarily fair -- although none of it will make either party better (or worse) off.
d. State-dependence of tastes will cause individuals to no longer want to fully insure when terms are actuarily fair (as they would under state-independent tastes and no aggregate risk). This is because the slopes if indifference curves along the 45-degree line no are no longer equal to actuarily fair slopes of contract lines. But if beliefs differ, the slopes of our indifference curves might still be equal along the 45 degree line -- making it possible that we might fully insure each other.
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