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A risk adverse agent

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In state of nature 1 the individual has income w, whereas in state of nature 2 the individual's income is y < w. 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 &#8719;s in state 1. Prove that a risk-averse von Neumann-Morgenstern individual will over-insure, fully-insure, or under-insure according as the insurance is available at a price &#8719; lower than, equal to, or higher than the actuarially fair price.

&#8719; - symbol pi

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In state of nature 1 the individual has income w, whereas in state of nature 2 the individual's income is y < w. 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 &#8719;s in state 1. Prove that a risk-averse von Neumann-Morgenstern individual will over-insure, fully-insure, or under-insure according as the insurance is available at a price &#8719; lower than, equal to, or higher than the actuarially fair price.

&#8719; - symbol pi
Part 1: A risk adverse agent offered actuarially fair insurance choose to insure fully

Suppose the total loss to the property in the event of the loss is T and the value of the property insured is q (where q<=T). The probability of loss is p.
T = w-y (change in worth if the loss event takes place)
Let the premium rate is &#8719;
Thus, premium = &#8719;q
The ...

Solution Summary

A risk adverse agent is assessed.

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Important Note:
Please try to use mathematical notation as much as you can to demonstrate your answer, but don't forget to carefully define each step you make.

Question
(a) Define "risk averse".
(b) Why does a risk averse agent offered actuarially fair insurance choose to insure fully?
(c) What does the agent choose if the terms are worse than actuarially fair?
(d) Show there are either one or two kinds of equilibrium in competitive insurance markets, depending on how equilibrium is defined.
(e) Explain, in the context of competitive insurance markets, what is meant by: "Separating equilibrium" and "Pooling equilibrium". What assumptions are required for each of these types of equilibrium, in turn, to exist?

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