Question

2. A consumer's utility of income is given by the function U(.). The consumer has initial income $Y, but risks losing $L with probability p. If a company offers insurance, the contract consists of coverage $q at price $a per dollar of coverage. (a) Using the notation above, what is the consumer's expected utility with insurance? (b) Define actuarially fair insurance. What are the sufficient assumptions for insurance to be actuarially fair? (c) Prove that, if insurance is actuarially fair, the consumer will demand full insurance coverage: q* = L. (d) Prove that, if insurance charges a loading factor $t, such that a = p + t, the consumer will demand less than full insurance: q* <L.

Answer 1

Giveni- - A consumon's vitility of income funchon UC). is given by the Heore:- Probility P, coverageq, utility funn u, income y, Loss, Premiyon (inzuorance) | $ = a. above, wheet is the coryumes's Gepeche utility a) using the nation with trywance ? Here is Eu (with insurance) -SG - Y-M, CQay-M-172, = N-3 Premium (total) payment to cover a amount loss in Bad state a. since Premium Mis paid before state to would is known So Income h uniformly by min both periods. i EU = (1-P) u[y-M] tpu [y-M-L+q] Since Maqa , they EU = (1-P) a[y-aq] tpu [y-L+q- aq] Ary,

b) Actuaouilly paint Insurance the Insurance that never change the porchogey eepeeled congh CEC), it Simply consih in some staty a reduce in others, so that ret Expected pay off is zero Sufficient condition - The peor dollock premium (for benefit preceived) equals the prot of Bad state [a=p] for os : . . Het expected Loss = net expected gain. . By paying Premium M benfit to quis recind if Losy occuy & Met Gain (9-4) So (1-P)M= P(q-m) M-ph= pq -PM 1. M = pa = Fence pa teoce.- (c) To prove q=L, Actually faiar Insurance is always a Eell Injurane scheme. as Eu = (1-P) u[y-aq]+ Pu [y-Ltq (1-al] Now Max Eu wat Bonefit q. EU - CI-P) au Cy-aq) + P (1-0) u' [y-L+q (l-a)] aq put a Eu ao aq

. Then 8 (14) u'[y-L= q (1-ar] = (14) au [y-aq] Now. in favor ynsurance asp They person being Risk Amorse, utility funch is concave, So ay dimirüstung mie of cong", yo. y - L+2 (1-a) = - az => q-qa-L= nag This [2 =2]

29 d) Now q = ptt on fair insurance, asp Nowo JEU = -(1-P) a u' Cy-aq) +P C1-a) u' Cy4+2 (10] 80. GD u'Cy-42] + (1-2) Pul [y-Ltq (1-2)] Now as asp, &(1-a) <(10 9. SCO concoue i we need, u'ęcy <u'[ca], for concame utility function than Gula u ule hunction t ca alger they u [y-ag] P(1-2) >> u' [y-L+q (1-01] alle as P(1-9) 21,8o.u [y-aq] <ul [y-L+q (1-2)] a(1-P) a so X-gq3t-L+q-qa [39] 0