Question

1. The owner of a chicken processing plant makes $1,000,000 per year from his chicken processing plant, and $1,000,000 per year from other sources. He notices that a total of $30,000 worth of chickens per year are being taken home by his 100 employees to supplement their minimum wage living standards (this works out to less than $6 per week per employee). These chickens are being taken out through the fire safety doors, so the owner is considering blocking the doors permanently. This is against the law because any reasonable person knows that sealing fire doors kills people in the event of a fire. The probability of a fire is 0.01 per year (one percent per year). If a fire breaks out it will be discovered that the owner sealed the safety doors, and the owner will have to pay $F = 700, 000 worth of fines and legal fees. The owner’s von Neumann- Morgenstern utility function is u(x) = √x where x is the owner’s income in a given year. The owner cares only about x.

   1. Give the owner’s expected utility if he leaves the fire doors unsealed.

   2. Give the owner’s expected utility if he permanently seals the doors.

   3. Give an algebraic expression for the owner’s expected utility as a function of the fine and fees F if he decides to seal the doors.

   4. Using the expression from the previous problem and the owner’s expected utility if he leaves the fire doors unsealed, solve for the minimal fines and legal fees that will make the owner decide not to seal the doors.

   5. If the owner contributes $C to state politicians, his fines and fees in the event of a fire will be reduced to $700, 000 − 100C. In other words, each dollar contributed to a politician reduces fees paid by $100, and the politicians are, in effect, selling insurance to the plant owner. Find the C that gives the plant owner complete coverage and check that the owner wants to make this contribution.

Answer 1

1) Owner's income if he leaves the doors unsealed: 1,000,000 (From other sources)+1,000,000-30,000 (from chicken processing plant)
x=1,970,000

Therefore expected utility if he leaves the doors unsealed: 1,970,000^0.5=1403.567

2) The Owner's expected income if he seals the door: 1,000,000 (From other sources)+1,000,000-700,000*0.01 (from chicken processing plant after accounting for the expected loss of income from the chance the fire breaks out)

x= 1,993,000

Expected utility if he leaves the doors sealed: 0.99*2,000,000^0.5+0.01*1,300,000^0.05=1411.473

3) The owner's expected income with a fine F= 1,000,000 (From other sources)+1,000,000-F*0.01 (From chicken processing plant) = 2,000,000-0.01F

The algebraic expression for the owner's expected utility if he seals the door with a fine F= 0.99*2,000,000^0.5+0.01*(2,000,000-F)^0.05

4) Therefore the owner will decide not to seal the door if his expected utility from sealing becomes just equal to the expected utility from not sealing the door=1403.567 as calculated above

0.99*2,000,000^0.5+0.01*(2,000,000-F)^0.05=1403.567

(2,000,000-F)^0.05=100(3.49857)= 349.857

F=2,000,000-349.857²=1,877,600

Therefore if the Fines and legal fees become equal to or greater than F=1,877,600 the owner will decide not to seal the doors.

5) If the owner contributes $C to state politicians

His expected utility will be 0.99*(2,000,000-C)^0.5+0.01*(1,300,000+100C)^0.05

The plant owner gets complete coverage when 100C=700,000

C=7000$

The owner will want to make this contribution if the expected utility

0.99*(2,000,000-C)^0.5+0.01*(1,300,000+100C)^0.05 is higher than 0.99*2,000,000^0.5+0.01*1,300,000^0.05=1411.473

putting C=7000, we get

0.99*(1,993,000)^0.5+0.01*(2,000,000)^0.05=1411.761

Therefore the owner wants to make this contribution as it increases his expected utility.

Hope it's clear. Do ask for any clarifications required.