In most cases, it is illegal to buy alcohol for minors, but we imagine that the size
Question:
In most cases, it is illegal to buy alcohol for minors, but we imagine that the size of the penalty affects how compliant a person is with the law. That is, if someone risks being fined more money, they are less likely to agree to buy alcohol for someone who is underage. The state tests compliance by sending out undercover agents who pretend to be minors and who then solicit alcohol from individuals who are heading to a liquor store. They subsequently record how many are willing to buy for the “minors”. When the fine was $350, Oregon saw an 81% compliance with the law. In 2019, they announced that they hoped to increase compliance to 90% by doubling the fine. Furthermore, let’s suppose that it is impossible to get 100% compliance (in the sense that it would require an infinitely large fine). With these facts in mind, a model for the fine required (in thousands of dollars) to ensure c% compliance with the law could be of the form P(c) = Ac + B K − c for positive constants A, B, and K.
(a) Use the provided information to find the values of constants A, B, and K.
(b) What does the model predict compliance would be if there was no fine? Does that make sense?
(c) Find the long-term behavior of P(c). Is this behavior meaningful in the context of the model?
Differential Equations and Linear Algebra
ISBN: 978-0131860612
2nd edition
Authors: Jerry Farlow, James E. Hall, Jean Marie McDill, Beverly H. West