Lizabeth, the manager of the medical test firm Theranos, worries aboutthe firm being sued for botched results from blood tests. If it isn’t sued, thefirm expects to earn profit of $120, but if it is successfully sued, its profit will beonly $10. Elizabeth believes that the probability of a successful lawsuit is 20%.If fair insurance is available and Elizabeth is risk averse, how much insurancewill she buy? (Hint: Assume that Elizabeth starts with a wealth ofwand shebuys insurance coveringx≤(120−10) = 110 of her loss. Write down herexpected utility as a function ofxand then take a derivative with respect toxto find the optimal insurance. If you want more hints, have a look at the secondinsurance problem solved in the lecture notes.)

The value of variance is '0' at full insurance while expected profit therefore the individual will buy insurance $100

Step-by-step explanation:

Given that:

Elizabeth, the manager of the medical test firm Theranos, worries about the firm being sued for botched results from blood tests.

Earn Profit -$120

Successfully sued Profit -$10

Probability -20% -0.80

so that above data we shown in below table

ProfitProbability

1200.08

100.08

so that expected profit & variance under this situation are

expected profit =(120) (0.08) +(10)(0.80)

= 96+8

=104

expected profit =104

Variance = (120-104)2 (0.08)+(10-104)2 (0.08)

=(256)(0.08)+(8836)(0.08)

= 9116.8

Variance = 9116.8

Further it is given that the individual is risk averse and is offered fair insurance since the probability loss is 20% so the fair insurance simples a premium of 0.08 for each dollar insurance coverage

so it is required to be determine how much insurance individual will buy

A fair insurance implies that the amount of insurance (coverage )bought has no impact on the expected values  of individuals expected profit, but a higher insurance implies a lower variance

A lower variance is desired by a risk average individuals

Therefore in this a risk average individual will purchase full insurance because a full insurance would mean that the variance expected profiles will be minimized without any reduction in expected profit.

The individual faces a potential loss of 100 hence a full insurance would be an insurance of 100

since the probability loss is 20% the fair insurance implies a premium of 0.20 for each dollar of insurance

premium of 22 = (0.02*100) for full insurance

so individual buys full insurance , than she faces the following below table

Profitprobability

104(120-22)0.98

104(10-22+1000.20

The expected profit variance

expected profit (104)(0.98)+ (104) (0.20)

= 101. 92 +20.8

=104

Variance (104-104 )2 (0.98)+(104)4 (0.20)

=(0)2 (0.98)+(104-104)2 (0.20)

(0)2 (0.98) +(0)2 (0.20)

= 0+0

= 0

As shown above the value of variance is '0' at full insurance while expected profit therefore the individual will buy insurance $100

it would be 6xy and -16xy.