Consider the insurance problem from class in which an agent with wealth w dollars faces a probability € (0,1) of incurring a loss of dollars, where 0 < l< w. He can purchase insurance at a price q € (0, 1) per unit that pays 1 dollar per unit purchased if the loss occurs. Suppose the agent is an expected utility maximizer with von Neumann-Morgenstern utility u(x) = ln(x + 1). You may assume throughout this question that the parameters are such that the Ai b. Jution nel antilid stomi

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please  only do: if you can teach explain steps of how to solve each part: b 

what is the optimization to use for foc? 

 


please solve for x

also the other part

Consider the insurance problem from class in which an agent with wealth w dollars faces a
probability € (0,1) of incurring a loss of dollars, where 0 < l< w. He can purchase
insurance at a price q € (0,1) per unit that pays 1 dollar per unit purchased if the loss occurs.
Suppose the agent is an expected utility maximizer with von Neumann-Morgenstern utility
u(x) = ln(x + 1). You may assume throughout this question that the parameters are such
that the agent's expected utility maximization problem has an interior solution.
(a) Find the certainty equivalent of the lottery corresponding to no insurance.
Solution: We have (1-7) ln(w+1) + π ln(w −l + 1) = ln(CE+1) and hence CE =
(w+1) ¹ (w − l + 1)" - 1.
(b) Suppose insurance is actuarially fair. What is the effect on demand for insurance of a
marginal increase in l? That is, if x is the quantity of insurance demanded, what is the
value of əx/al?
Solution: Since the DM fully insures, the quantity of insurance increases by exactly the
amount of the increase in l.
(c) Suppose q>. What is the effect on demand for insurance of a marginal increase in l?
Solution: The FOC is
which is equivalent to
Hence
-q(1-π)
(1-q)π
(w-qx+1) (w+(1-q)x-l+1)
+
= 0,
(w+(1-q)x-l+1)q(1 − π) = (1 − q)n(w− qx + 1).
dx
q(1 — q)(1 − n) — q(1 — π) = −(1 — q) q
-
дг
de
Transcribed Image Text:Consider the insurance problem from class in which an agent with wealth w dollars faces a probability € (0,1) of incurring a loss of dollars, where 0 < l< w. He can purchase insurance at a price q € (0,1) per unit that pays 1 dollar per unit purchased if the loss occurs. Suppose the agent is an expected utility maximizer with von Neumann-Morgenstern utility u(x) = ln(x + 1). You may assume throughout this question that the parameters are such that the agent's expected utility maximization problem has an interior solution. (a) Find the certainty equivalent of the lottery corresponding to no insurance. Solution: We have (1-7) ln(w+1) + π ln(w −l + 1) = ln(CE+1) and hence CE = (w+1) ¹ (w − l + 1)" - 1. (b) Suppose insurance is actuarially fair. What is the effect on demand for insurance of a marginal increase in l? That is, if x is the quantity of insurance demanded, what is the value of əx/al? Solution: Since the DM fully insures, the quantity of insurance increases by exactly the amount of the increase in l. (c) Suppose q>. What is the effect on demand for insurance of a marginal increase in l? Solution: The FOC is which is equivalent to Hence -q(1-π) (1-q)π (w-qx+1) (w+(1-q)x-l+1) + = 0, (w+(1-q)x-l+1)q(1 − π) = (1 − q)n(w− qx + 1). dx q(1 — q)(1 − n) — q(1 — π) = −(1 — q) q - дг de
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