3. Disability Insurance Assume that everybody earns a wage of $100. Individuals face a probability q of getting disabled. If they are disabled their wage becomes $0. Individuals can purchase insurance from private firms. It provides them with a benefit b if they get disabled. The price of insurance (the insurance premium) is p. Assume in all questions that the market for insurance is perfectly competitive (firms earn zero profit) so that insurance plans are actuarially fair (unless explicitly stated otherwise). Assume initially that there are two types of people that differ in the probability of getting disabled q and their utility function over consumption c: . Type L: qL = 5% and u(c) = Vc . Type H: qH = 35% and u(c) = Vc There are 50 people of each type. Assume ONLY IN (a) that there is perfect information, i.e. the insurance company knows the type of each consumer and can offer a different plan to each type (i.e. some plan (bL, PL) to Type L; some plan (bH, PH) to Type H.) (a) Assume that the firms offer full insurance. Calculate the benefit and price offered to each type, ((bL, PL); (bH, PH)). Due to perfectly competitive Mrk, -9b, and because firmsoffer full insurance, b=100 $ forboth group. 7 PL= bix9= 100x 0.05= 5 (- - b = income depreciation in padscafe PH= bHXEH = (20 *035= 35 Do (bull = ( (0o. 5 ) ( by. PM ) = ( 100, 35 )Assume for the rest of the problem that firms cannot observe types. (b) Assume that there is only one insurance company (so it may earn a profit) and it offers only one insurance plan. It offers a plan with a benefit of b = 100 for a price of p = 50. Call this insurance plan Plan #1. Which types will choose to buy it? (Hint: Calculate utilities rather than the maximize utilities) alulothyinsurance: (19/ 5up + 9 5. pub = chelscoop + 85 760-P = Stop Elulofxby insurance : ( 12 ) JW =(1 91 5100 = (01(-2 ) willingness topay - 10o-P 2 10(1-8) = 100-P 2(-0(1-8)2 PS 100-cock-al 7 P=loo(1- (12809)) # p=(--(29-92) 410 4 (2-2 ) For Low -) PS 100 (0 05) (195 ) 7 1= 9.95 Fight P= (00 ( 0.25 ) (165) = ) P5 5275 Therefore only flight group will choose to buy plan # 1 (c) Will the insurance company still be able to offer Plan #1 in a perfectly competitive equilibrium? Because now insurance company cannotdifferentiate types, andina perfectly competitive equilibrium, the insurance company cannot make a profit, so it wouldn't be able to offer plan $1 if itspriceis higher than acevarially fair price 7 actuaring fair price = 09stars x 100 = 50 *0. 4: 20 There fore, plant'sprice 50720, insurance Comparing is not able to offerplant, and more. (d) A new firm decides to offer a second plan so that there are now two plans offered in the economy: Plan #1 and the new plan called Plan #2. Plan #2 has benefit b = 2.5. What is the price for Plan #2 if the new firm expects both types to buy it and it sets the price actuarially fairly? plan #2's new actuarially fair price will be ous-fors 2 - X 2.5 = 0 272.5 = 0.3(e) At this price for Plan #2, which types will buy Plan #2, if any? Which types will buy Plan #1, if any? Which will buy no plan at all, if any? (Hint: You've computed several of the necessary utilities in (b)) J At this price Lowtype will choose planta because nemprice 0.5 isLowerthan 9.15 Feightype have to choosebetween $ 1 1$ 2, Elulotbuy insurance is ( Fal w-P + 9 Jonpb f(2 = ) ( kal Jco-o.s + 9) 0.5+25 = (1-a) Ja95 +912=6.98 7. Hightypewill choose FID (f) Entrepreneurs learn that the new firm was making a profit under Plan #2. Those en- trepreneurs start lots of new firms so that any firm offering a plan with benefit b = 2.5 has to set the premium equal to actual average costs. As a result, Plan #2 disappears, but there is a new plan called Plan #3 with b = 2.5 and premium p equal to the plan's actual average costs. Assume that the types who used to buy Plan #2 (computed in the previous problem) buy Plan #3 and are the only ones who buy it. What premium will Plan #3 have?(g) Only Plan #1 and Plan #3 are offered in the economy. Which types (if any) will buy Plan #1? Which types (if any) will buy Plan #37 Which types (if any) will buy no plan? (h) Let social welfare equal the sum of all individuals' expected utility. Under which economic scenario is social welfare maximized: the economic scenario of part (a) or the economic scenario of (g)? (Note: You may but definitely do not have to use algebra in your answer.)