A decision-maker with initial wealth w faces a probability of incurring a loss. If the loss occurs, with probability p the amount of the loss is ₁ and with probability 1-p the amount of the loss is £2, where ₁>₂ > 0. The decision-maker can buy insurance against both losses at a price of q dollars per unit. If she purchases z units of insurance, she receives z dollars if either loss occurs, even if z is greater than the amount of the loss. There is no limit to the amount of insurance she can purchase. (a) First suppose that the decision-maker is a risk averse expected utility maximizer with von Neumann-Morgenstern utility u(y) over quantities of wealthy. i. (Write down the first-order condition characterizing the optimal choice of z when it is interior. Solution: The first-order condition is (1−q)ñpu² (w − l₁ + (1 − g)x)+(1−g)x(1−p)u' (w − l₂ + (1 − g)x) = g(1−z)u'(w—qx). ii. (, Is the highest price q at which the decision-maker fully insures against the larger loss ₁ greater than, equal to, or less than ? Prove your answer. Solution: Atq=, the above FOC simplifies to pu' (w-l₁ + (1-9)x) + (1 − p)u' (w − l₂ + (1 − q)x) = u(w − qr). I If z = {₁, the left-hand side of this equation is smaller than the right-hand side. Since u' is decreasing, we must have r
A decision-maker with initial wealth w faces a probability of incurring a loss. If the loss occurs, with probability p the amount of the loss is ₁ and with probability 1-p the amount of the loss is £2, where ₁>₂ > 0. The decision-maker can buy insurance against both losses at a price of q dollars per unit. If she purchases z units of insurance, she receives z dollars if either loss occurs, even if z is greater than the amount of the loss. There is no limit to the amount of insurance she can purchase. (a) First suppose that the decision-maker is a risk averse expected utility maximizer with von Neumann-Morgenstern utility u(y) over quantities of wealthy. i. (Write down the first-order condition characterizing the optimal choice of z when it is interior. Solution: The first-order condition is (1−q)ñpu² (w − l₁ + (1 − g)x)+(1−g)x(1−p)u' (w − l₂ + (1 − g)x) = g(1−z)u'(w—qx). ii. (, Is the highest price q at which the decision-maker fully insures against the larger loss ₁ greater than, equal to, or less than ? Prove your answer. Solution: Atq=, the above FOC simplifies to pu' (w-l₁ + (1-9)x) + (1 − p)u' (w − l₂ + (1 − q)x) = u(w − qr). I If z = {₁, the left-hand side of this equation is smaller than the right-hand side. Since u' is decreasing, we must have r
Chapter7: Uncertainty
Section: Chapter Questions
Problem 7.4P
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please only do: if you can teach explain steps of how to solve each part
what is the optimizatio formula that was use for foc? please solve
each part
Transcribed Image Text:4. A decision-maker with initial wealth w faces a probability of incurring a loss. If the loss
occurs, with probability p the amount of the loss is ₁ and with probability 1-p the amount
of the loss is £2, where ₁ > ₂ > 0. The decision-maker can buy insurance against both losses
at a price of q dollars per unit. If she purchases a units of insurance, she receives a dollars if
either loss occurs, even if z is greater than the amount of the loss. There is no limit
to the amount of insurance she can purchase.
(a) First suppose that the decision-maker is a risk averse expected utility maximizer with
von Neumann-Morgenstern utility u(y) over quantities of wealth y.
i. () Write down the first-order condition characterizing the optimal choice of a when
it is interior.
Solution: The first-order condition is
(1-q)πpu' (w-l₁ + (1 − q)x)+(1−q)π(1−p)u' (w − l₂ + (1 — q)x) = q (1-7)u' (w-qx).
ii. (, Is the highest price q at which the decision-maker fully insures against the larger
loss ₁ greater than, equal to, or less than ? Prove your answer.
Solution: At q = n, the above FOC simplifies to
pu' (wl₁ + (1-q)x)+ (1 − p)u' (w−l₂ + (1 − q)x) = u'(w — qx).
If x= ₁, the left-hand side of this equation is smaller than the right-hand side.
Since u' is decreasing, we must have x < ₁ for this equation to hold as decreasing
x causes the left-hand side to increase and the right-hand side to decrease. (This
argument implicitly assumes that p1; if p = 1 then this is essentially a standard
insurance problem and r = l₁ is optimal when q = n.)
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