Question

1. A consulting firm must decide now whether to hire a young engineer. The main purpose of this engineer would be to work on a contract that the firm hopes to win. This new contract will begin in 2 months and last for 4 months. If the firm hires the engineer and wins the contract, the firm will make a profit of $6000 on the new contract (this figure allows for the cost of the new engineer during the 4 months of the contract). During the new employee’s first 2 months (before the new contract begins), he can be assigned to various projects, reducing overtime costs for other employees by $1200 per month. The cost of employing this engineer (salary, taxes, and benefits) is $2,000 per month, so it is clear that the firm will lose money on him for his first 2 months. If the firm hires the engineer but does not win the contract, it cannot fire him until he has completed 3 months of work. During the third month he will not really be needed, and the work he will do will be worth only $400 to the firm. If the firm does not hire the engineer but wins the contract, it will be forced to hire somebody in a rush. There is only a 45% chance that it would be able to find another young qualified engineer on such notice for $2000/month, and if it is unable to find one it will have to pay $3,000/month for an experienced engineer to work 4 months on the project.

a) The president of the firm feels the firm has a 60% chance of winning the contract. Should it hire the engineer now or wait until it knows whether it won the contract?

b) What is the “indifference probability” of winning the contract, i.e. the probability at which the firm is indifferent between hiring the engineer now and waiting until it knows whether it won the contract?

2. On the first of the month, you must decide whether or not to buy a monthly bus pass for $25. You usually drive to work, but your car must go in for service, and you will be forced to ride the bus. Over the phone, the mechanic estimated an 80% chance that the car’s problem is major, requiring 10 working days of service. However, if the mechanic finds only a “minor” problem, you will be without the car for only 5 working days. The bus fare is $3.00 per day. Should you buy the bus pass? Base your decision on the expected cost of riding the bus for the month.

Answer 1

1. the firm has 2 options

1. Hire the engineer
   1. The cost of the engineer for the 1st 2 months is (2000-1200)*2=$1600
   2. Win the contract with a probability of P(W)=0.60: The profit is $6000 which covers the cost of the engineer. But he has costed $1600 in the first 2 months. Hence the net profit is 6000-1600=$4400
   3. Do not win contract, with a probability of P(NW)=0.40: The profit is zero. The engineer costs (2000-1200) for each of first 2 months and the third month he costs 2000-400=$1600. The total cost of hiring the engineer is (2000-1200)*2+1600=$3200. The net profit is -$3200
2. Do not hire the engineer (before winning the contract)
   1. Win the contract, with a probability of 0.60: The profit is $6000.
      1. Hire an engineer at $2000/month, with a probability of 0.45: the cost of the engineer for 4 months is covers in the profit. Net profit is $6000
      2. Hire an engineer at $3000/month, with a probability of 0.55: the cost of the engineer for 4 months covered in the profit is $2000/month, giving a shortfall of $1000/month.  Net profit is $6000-1000*4=$2000
   2. Do not win the contract, with a probability of 0.40: Net profit =$0

the following decision tree shows these values

Moving from the right to right

Chance node 4:

The expected payoff is

Chance node 2:

the expected payoff is

chance node 3:

The expected payoff is

Decision node 1:

the firm has to decide between 2 options

1. Hire an engineer at an expected payoff $1360
2. Do not hire an engineer, at an expected payoff $2,280

The firm needs to wait until it knows whether it won the contract.

b) Let p be the probability of winning the contract.

The expected payoff at node 2 is

The expected payoff at node 3 is

the firm is indifferent between hiring the engineer now and waiting until it knows whether it won the contract if these 2 expected values are the same

the probability at which the firm is indifferent between hiring the engineer now and waiting until it knows whether it won the contract is 0.8421

2. You have to choose from 2 options

1. Buy a monthly pass at $25.
2. Do not buy the month pass
   1. Major problem with a probability of 0.80. The cost of bus fare for 10 days = 3*10=$30
   2. Minor problem with a probability of 0.20. The cost of bus fare for 5 days = 3*5=$15
   3. the expected cost is

You have to choose between buying a monthly pass at $25 vs not buying a pass at an expected cost of $27.

You should buy the bus pass