(a) Explain carefully what is meant by the wealth. Branwell is worried that his
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(a) Explain carefully what is meant by the wealth. Branwell is worried that his precaces the prospe and destroyed by fire. He works out that he faces the pent wealth 0.95, 0.05]. Explain how this relates to state con illustrate this prospect on a diagram. t his precious book collection may be ates to state contingent wealth and An insurance company offers Branwell insurance on the terms: in return for a premium of £1 they will pay that his books are destroyed. Branwell can (b) buy as much insurance as he likes on these terms. Define precisely what is meant o prospect. Does the insurance company offer Branwe actuarially fair terms? y a fair ll insurance on On a new diagram draw the budget constraint that Branwell faces, as described in part (b). Branwell is a risk averse expected utility (c) maximiser. What does this mean? Show on your diagram hovwExplanation / Answer
State Contingent Commodity -
Insurance is a ‘state contingent commodity,’a good that you buy now but only consume if a specific state of the world arises. This insurance is purchased before the state of the world is knownand you can only make the claim for the payout if the relevant state arises.
Consider an individual who has to decide how much insurance cover to buy. Formally, she maximizes her expected utility by choosing the optimal cover or indemnity C: His state dependent utility function would look like -
E[U] = (1 )u(W P(C)) + u(W P(C) L + C).
Assumption - Inndividual is risk averse (von Neumann–Morgenstern utility function is increasing and strictly concave)
is the probability that a loss of size L occurs
W is her wealth in the event of no loss.
P is the insurance premium paid, which can in general be thought of as a function of C, the cover
Our problem here is slightly less complex than how the above function looks like. Let us see our prospect which is [200,40,0.95,0.05]
State of World - It corresponds to the amount of the loss incurred by the insurance buyer. State 1 is when there is no loss and for each possible loss there is an additional state (here one again)
We define the buyer’s wealth in each state of the world, W, as her state contingent wealth. Before entering into an insurance contract, the consumer has given endowments of state contingent wealth, W0 if no loss occurs, and W0 L given the occurrence of loss L > 0.
W0 = 200
L = 40
W0 - L = 160
If he buys insurance he gets a cover C= of 10 for which he pays P = 1 pound
We assume the buyer solves the problem
max
C0
u¯ = (1 )u(W0 P) + u(W0 L P + C)
An insurance that is actuarially fair when the premium is equal to expected claims:
Premium = p · A
where p is the expected probability of a claim,
and A is the amount that the insurance company will pay in the event of an accident
here p = 0.05 and A = 10 which implie p.A = 0.5
Therefore the insurance compancy is not giving a fair prospect as it is charging a higher premium = 1.
How much insurance will a risk averse person buy?
Consider a person with an initial endowment consisting of three things:.
A level of wealth w0 = 200 ;
a probability of an accident of p = 0.05;
and the amount of the loss, L = 40
Expected utility if uninsured is:
E(U|I = 0) = (1 p)U(w0) + pU(wo L) = 0.95*200 + 0.05*160 = 198
Expected utility if insured is:
E(U|I = 1) = (1 p)U(w0 pA) + pU(wo L + A pA)
This needs to be maximised with respect to A.
Solvinf the FOC gives A = L.
Therefore he can buy 40/10 = 4 insurance.
Consumers can use their endowment (equivalent to budget set) to shift wealth across states of the world via insurance, just like budget set can be used to shift consumption across goods X, Y.
Let’s say that this consumer can buy actuarially fair insurance. What will it sell for?
• If you want $1.00 in Good state, this will sell of $0.95 prior to the state being revealed.
• If you want $1.00 in Bad state, this will sell for $0.05 prior to the state being revealed.
The price ration required for our budget constraint will be Pg/Pb = 0.95/0/05 = 19
E(w) = 0.95 * 200 + 0.05 * 160 = 198
So, I would need to give you more than 200 in the good state to compensate for this risk. — Bearing risk is psychically costly —must be compensated. (Understanding this would be helpful to design our indifference curves which would be convex to the origin)
EQUILIBRIUM - indifference curve will be tangent to the budget set at exactly the point where wealth is equated across states. (Draw on your own)
Lastly, if the probability to loss is lower than 0.05, Branwell would be willing to purchase lesser insurance. (Calculation cannot be done becuse exact figure for new p is not mentioned)
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