Question

Consider the case of a high school graduate considering his career options. He can take up a life of crime that will pay him a salary of $100,000 if he is not caught. However, there is a 10% chance that we will be caught, and go to prison where he will earn $200 in the prison laundry. He can also work in a non-criminal job for a guaranteed $65,000. His utility over money is U(M) = √M, and he does not lose utility from going to prison other than the low earnings.
a) What is his expected income if he chooses a life of crime?
b) What is his expected utility if he chooses a life of crime?
c) Which career do we expect him to choose?
d) Suppose the local government is considering two programs of equal cost. One would increase police funding which would raise the apprehension rate to 25%, and another would fund a job-placement program that would increase the salary of the non-criminal job to $75,000. Which one of these would reduce crime more?

Answer 1

a) expected income if he chooses life of crime

expected income if he does not get caught + expected income if he lands in prison

= prob(he does not get caught)*(Income from crime) + Prob(he gets caught)*(income from prison)

= (.9)(100,000) + (.1)(200)

= 90,000+20 = $90,020

b) Expected utility from life of crime

utility (expected income from life of crime) = (90020)^(1/2) = 300

c) Gauranteed income in a non-criminal job = $65,000

Utility from non-criminal job = (65,000)^(1/2) = 255

Since, expected utility from life of crime (300) > expected utility from non-criminal job(255)

Therefore, he will choose life of crime.

d) if apprehension rate that is probability of getting caught increases from 10% to 25% and thus probability of not getting caught would fall to 75%

This changes Expected Income from life of crime

= (.75)(100,000) +(.25)(200) = 75,050

Expected utility from life of crime = (75,050)^(1/2) = 273.95

Also, since income from non-criminal job increases to $75,000 then

Expected utility from non-criminal job is = (75,000)^(1/2) = 274.86

therefore, both the schemes would be reducing crime equally.