problem code : 140609MICRO

[vishruti]

Consider a strictly risk averse decision maker who has an initial wealth of W but who runs a risk of loos of Rs. D. The probability of loss is "p". It is possible for the agent to buy insurance.One unit of insurance costs Rs. q and pays Re. 1 if loss occurs. If x units of insurance are bought then what is the wealth of the individual if no loss occurs? What is the wealth if loss occurs? Also find the expected wealth. What is the optimal level of x if U(W) = W ? Is the insurance fair?

(source: mas collel)

[Smriti]

if no loss occurs then the individual's wealth is: w-xq
if the loss occurs then the individuals wealth is: w-d+x-xq
therefore expected wealth is:p(w-d+x-xq)+(1-p)(w-xq)
optimal level of insurance is derived from: expected wealth=w
so, x=pd/p-q
i am not sure about the fairness bit

[vishruti]

the expected wealth is (1-p) ( w - xq) + p ( w -xq - d +x)

we have to maximize expected wealth wrt x

then we have p = q from first order condition

the condition is reduced to u'(w - d + x*- x*) = u'(w - x*)

because at x* First order condition is -q(1-p)u'(w-x*q) + p(1-q)u'(w-D + x*(1-p)) = 0

now since u'(.) is a decreasing function this implies

w - d +x*(1-p) = w - x*p

hence x*  = d , the optimum level of insurance

and it is fair because the final wealth is w - pd , weather loss occurs or not and the insurer is insured completely