Reference no: EM133251998

Consider a drug offender who is on probation and regularly monitored for drug use. He is deciding whether to take a hit and receive an immediate instantaneous utility of h = 1, knowing that he will be punished with certainty for violating the terms of his probation. The criminal justice system would like to deter drug use, and is contemplating two options for punishing probation violators. Option one: give violators a short jail sentence of jsdays immediately, yielding a utility of -jsimmediately. Option two: give violators a long jail sentence of jldays in the future, yielding a utility of -jlat the beginning of the sentence (i.e. all of the utility impact of the sentence is felt in a lump sum at the beginning of the sentence).

The offender's utility is zero if he neither takes a hit nor serves any jail time.

(a) Assume drug offenders are exponential discounters (i.e. β = 1) with δ ≤1. Let js= 1.01 and jl= 2, to be served a month in the future.

(i) What is the largest monthly discount factor δmsuch that the short jail sentence successfully deters drug use while the long jail sentence does not?

(ii) What is the largest yearly discount factor δywhere this condition holds? [Hint: since a month is 12 of a year, the yearly discount factor is a simple function of the monthly discount factor.]

(b) Now assume drug offenders are quasi-hyperbolic discounters with beta ≤1 and δy= 0.99. If js= 1.01 and jl= 2 is to be served a month in the future, what is the largest β such that the immediate short jail sentence successfully deters drug use while the delayed long jail sentence does not?

(c) Consider an exponential discounter with yearly discount factor, δy, found in part (a).

(i) Would a massive sentence of jl= 20,000 starting two years from now be effective at deterring drug use?

(ii) Compare the effectiveness of the massive sentence to the effectiveness of a 1.01- day sentence served immediately, which you determined in part (a). Explain the psychological intuition for the similarities or differences between the effectiveness of these two punishments.

(d) Consider the quasi-hyperbolic discounter, with δ = 0.99 and the β found in part (b).

(i) Will the massive delayed sentence deter drug use?

(ii) Compare your answer to your answer from part (c). Explain the psychological intuition for the similarities or differences between these answers.