3 Harder Suppose Warren did know about tornadoes but didnt know about insurance

3 harder suppose warren did know about tornadoes but

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3. [Harder] Suppose Warren did know about tornadoes, but didn’t know about insurance. His refer- ence lottery therefore includes the risk but not the insurance premium: r 3 = <w 0 - 1000 , 0 . 05; w 0 , 0 . 95 > (that is, a 5% chance of having ( w 0 - 1000) and a 95% chance of having w 0 ). (a) What is his utility now if he buys full insurance at a premium of $ p ? (assume p < 1000) EU ( buy | r 3 ) = E [ w ] + E [ v ( w - r 3 )] = w 0 - p + (0 . 05 v ( w 0 - p - ( w 0 - 1000)) + 0 . 95 v ( w 0 - p - ( w 0 ))) = w 0 - p + (0 . 05 v (1000 - p ) + 0 . 95 v ( - p )) = w 0 - p + (0 . 05(1000 - p ) + 0 . 95( - 2 p )) = w 0 - p + 50 - . 05 p - 1 . 9 p = w 0 + 50 - 2 . 95 p 4
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EC 310 - Behavioural Economics (b) What is the highest premium such that Warren will prefer to buy full insurance? Explain why your answer differs from part 1. EU ( don 0 t | r 3 ) = E [ w ] + E [ v ( w - r 3 )] = 0 . 05( w 0 - 1000) + 0 . 95 w 0 + 0 . 05 [0 . 05 v ( w 0 - 1000 - ( w 0 - 1000)) + 0 . 95 v ( w 0 - 1000 - ( w 0 ))] + 0 . 95 [0 . 05 v ( w 0 - ( w 0 - 1000)) + 0 . 95 v ( w 0 - ( w 0 ))] = w 0 - 50 + (0 . 05)(0 . 95) v ( - 1000) + (0 . 95)(0 . 05) v (1000) = w 0 - 50 - 95 + 47 . 5 = w 0 - 97 . 5 So he will buy if: w 0 + 50 - 2 . 95 p w 0 - 97 . 5 50 p In general, this case is going to be hard to predict. Paying the premium will feel partly like a gain and partly like a loss, while getting hit by a tornado feels only partly like a loss. With diminishing sensitivity, this will get complicated fast. With piecewise-linearity, the effects net out. It turns out that there is also a 5% independent risk that Warren’s home will be struck by an earthquake, which would cause $2000 in damage. 4. If Warren narrowly brackets his decisions, and includes each premium in his reference point when considering each insurance contract (i.e. as in part (1)), what is the total amount Warren will be willing to pay to fully insure against these two risks? Solving for earthquake insurance separately, the problem is as in (1): EU [ buy | r 1 ] = w 0 - p + v ( w 0 - p - ( w 0 - p )) = w 0 - p EU [ don 0 t | r 1 ] = E [ w ] + E [ v ( w - r 1 )] = 0 . 05( w 0 - 2000) + 0 . 95( w 0 ) + [0 . 05 v ( w 0 - 2000 - ( w 0 - p )) + 0 . 95 v ( w 0 - ( w 0 - p ))] = w 0 - 100 + [0 . 05 v ( p - 2000) + 0 . 95 v ( p )] = w 0 - 100 + (0 . 05( - 2)(2000 - p ) + 0 . 95( p )] = w 0 - 100 - 200 + . 1 p + . 95 p = w 0 - 300 + 1 . 05 p So he will buy if: w 0 - p w 0 - 300 + 1 . 05 p 146 . 34 p So in total, he will be willing to pay 146 . 34 + 73 . 17 = 219 . 51 5. If Warren were offered a single policy to cover both risks (and he incorporated the premium into 5
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EC 310 - Behavioural Economics his reference point), what is the most he would pay to insure? EU [ buy ] = w 0 - p + v ( w 0 - p - ( w 0 - p )) = w 0 - p EU [ don 0 t ] = E [ w ] + E [ v ( w - r )] = w 0 - 150 + . 05 2 v ( w 0 - 3000 - ( w 0 - p )) + ( . 05)( . 95) v ( w 0 - 1000 - ( w 0 - p )) +( . 05)( . 95) v ( w 0 - 2000 - ( w 0 - p )) + . 95 2 v ( w 0 - ( w 0 - p )) = w 0 - 150 + 1 400 v ( p - 3000) + 19 400 v ( p - 1000) + 19 400 v ( p - 2000) + 361 400 v ( p ) = w 0 - 150 + 2 400 ( p - 3000) + 38 400 ( p - 1000) + 38 400 ( p - 2000) + 361 400 p = w 0 - 150 + 2 400 + 38 400 + 38 400 + 361 400 p - 1 400 [6000 + 38000 + 76000] = w 0 - 450 + 1 . 0975 p So he will buy if: w 0 - p w 0 - 450 + 1 . 0975 p 214 . 54 p So not the biggest effect in the world, but in general when Warren broadly-brackets his risks he will be willing to pay less than if he considers them one-at-a-time. Note that diminishing sensitiv- ity would make this effect much stronger, since getting hit by both a tornado and an earthquake wouldn’t be as bad as suffering each as an independent loss. 6
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