Question

A worker has a utility function over income U(I) = √ I. She is picking between two jobs for the following year. Job X pays $40,000; she can get this job for sure. She is also considering Job Y; there are other candidates for this job, and the probability of not getting it during the hiring process is 0.2 (she would get no income at all if that happened, and the process concludes after Job X’s deadline). If she gets the job, she would make $Z. Aside from the probabilities of being hired and the salaries, the jobs are exactly the same. Which will be her decision for the year if Z is (a) 36,100? (b) 40,000? (c) 60,025? (d) 62,500? (e) 67,600? Show your work in each case (if you do not need to do any utility calculations to know the answer, state so). Additionally, calculate the expected value of her income for Job X and Job Y in situation c) above. How do you reconcile these figures with her actual decision?

Answer 1

We do not need the utility calculations since it is only dependent on I. Higher the I, higher the utility. Therefore, the

a)

Income from job X = $40,000

Income from job Y = $36,100

Since income of job X is higher, the worker will prefer job X ( Since utility would be higher for job X because of higher income.

b)

Income from job X = $40,000

Income from job Y = $40,000

Since income of job X is same as income from job Y, the worker would be indifferent between the jobs. ( Since utility would be same for both because of equal incomes.

c)

Income from job X = $40,000

Income from job Y = $60,025

Since income of job Y is higher, the worker will prefer job Y ( Since utility would be higher for job Y because of higher income.

d)

Income from job X = $40,000

Income from job Y = $62,500

Since income of job Y is higher, the worker will prefer job Y ( Since utility would be higher for job Y because of higher income.

e)

Income from job X = $40,000

Income from job Y = $67,600

Since income of job Y is higher, the worker will prefer job Y ( Since utility would be higher for job Y because of higher income.

Probability of getting job X = 100% ; Salary = $40,000

Expected income (X) = $40000 ( 40000 x 1 )

Probability of job getting job Y = 80% (1 - probability of not getting job Y = 1 - 0.2 ) ; Salary = $60,025

Expected income (Y) = 0.8 x 60025 = $48,020

Since expected income is higher for job Y, worker should prefer job Y. It is same decision which the worker would have taken based on utility as well.