Rosario has to finish her dissertation within 10 days, that is, at time t = 1, t = 2, ..., or t = 10. It takes one day to finish the dissertation, and on the day Rosario does so, she incurs an instantaneous disutility cost equivalent to $10. Rosario is a hyperbolic discounter with β = 0.85 and δ = 1. Her (instantaneous) utility function is u(x) = x. (a) Suppose the university has a system in which it charges Rosario $1 in fees for every day she does not finish her dissertation (paid each day that it is incurred). E.g., finishing on day 2 incurs a cost of $1 paid on day 1. When does Rosario finish if she is naive? How much does she pay in penalties? (Hint, past penalties are sunk, e.g., from the perspective of t = 2 self, any penalties paid in t = 1 are sunk, and do not factor into decisions or utilities going forward.) (b) Still in the $1/day system, when does Rosario finish if she is sophisticated? (c) Now suppose that the university has a deadline system: Rosario incurs a penalty of $10 (paid on the day the dissertation is completed) if she does not finish her dissertation before day 10 (so finishing on day 9 does not trigger the penalty, but finishing on day 10 does). There are no daily penalties. When does Rosario finish in this system if she is naive? How much does she pay in penalties? (d) When does Rosario finish in the deadline system if she is sophisticated? (e) Does it make a big difference to a naive hyperbolic discounter whether she is in a day-by-day-penalty or deadline system? Explain intuitively. (f) Does it make a big difference to a sophisticated hyperbolic discounter whether she is in a day-by-day-penalty or deadline system? Explain intuitively.
Rosario has to finish her dissertation within 10 days, that is, at time t = 1, t = 2, ..., or t = 10. It takes one day to finish the dissertation, and on the day Rosario does so, she incurs an instantaneous disutility cost equivalent to $10. Rosario is a hyperbolic discounter with β = 0.85 and δ = 1. Her (instantaneous) utility function is u(x) = x.
(a) Suppose the university has a system in which it charges Rosario $1 in fees for every day she does not finish her dissertation (paid each day that it is incurred). E.g., finishing on day 2 incurs a cost of $1 paid on day 1. When does Rosario finish if she is naive? How much does she pay in penalties? (Hint, past penalties are sunk, e.g., from the perspective of t = 2 self, any penalties paid in t = 1 are sunk, and do not factor into decisions or utilities going forward.)
(b) Still in the $1/day system, when does Rosario finish if she is sophisticated?
(c) Now suppose that the university has a deadline system: Rosario incurs a penalty of $10 (paid on the day the dissertation is completed) if she does not finish her dissertation before day 10 (so finishing on day 9 does not trigger the penalty, but finishing on day 10 does). There are no daily penalties. When does Rosario finish in this system if she is naive? How much does she pay in penalties?
(d) When does Rosario finish in the deadline system if she is sophisticated?
(e) Does it make a big difference to a naive hyperbolic discounter whether she is in a day-by-day-penalty or deadline system? Explain intuitively.
(f) Does it make a big difference to a sophisticated hyperbolic discounter whether she is in a day-by-day-penalty or deadline system? Explain intuitively.
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