Problem Set 1 for EC 219ADmitry TaubinskyDue date: September 21, by 12:00pmNote: Remember that it is okay (and even encouraged) to work with other people onthis problem set. But each person must separately write up and turn in the solutions,and must provide the names of all of the people he or she worked with on the problems.Problem 1Angie has trouble waking up in the morning.Her goal is to wake up at 6am, and she sets heralarm clock for that time.From her evening self’s perspective, sleeping in generates a benefitbthat morning, but a costc > bthat she incurs later throughout the day. Her morning self, however,chooses to sleep in ifb > βc, whereβ≤1 is her present bias. Angie may not be fully sophisticatedabout her present bias, however, and believes that her present bias isˆβ≥β.Part a.What is the highest value ofβfor which Angie will end up sleeping in in the morning?Part b.Suppose that Angie can purchase Clocky, a loud and obnoxious alarm clock that runsaway from her in the morning, and makes sleeping in less pleasant. The price of Clocky isp < c-b,and what it does is that it lowers the benefit of sleeping in tobclocky< b. Angie purchases Clockywhen her perception of its benefit (eitherc-bor 0, depending on whether she believes it willchange her behavior) is greater than or equal to the pricep >0. Show that whenˆβ= 1, Angie willnot purchase Clocky.Part c.Show that whenβ= 0, Clocky will not be effective in changing Angie’s behavior.Part d.What is the range ofβfor which Clocky will be effective in changing Angie’s morningbehavior?In other words, compute the valuesβand¯β– expressed in terms of the parametersb, c, bclocky– such that Clocky will change Angie’s morning behavior if and only ifβ< β <¯β.Part e.What is the range ofˆβfor which Angie will choose to purchase Clocky?Part f.What is the range of (β,ˆβ) for which Angie will choose to purchase Clocky, but by doingso will make herself worse off since Clocky will actually be ineffective? Assume thatbclocky>0 forthis part.1

Part g.The point of this story about Angie and Clocky is to get you to think about how peoplechoose commitment devices, and whether people are actually always made better off by their choiceof commitment devices. Please explain in words the broad principles about when offering peoplecommitment devices can make people better off, and when this can backfire and make people worseoff.Problem 2Constantine likes his smokes, but a little too much because of his self-control problems. He chooseshis consumption of cigarettesxaccording to the utility functionU(x) = (Z-px) +u(x)-βhx,wherepis the price,his the marginal health cost of cigarette consumption that he incurs in thefuture,u(x) is his immediate pleasure of smoking that he experiences in the present,Z-pxis theleftover money that he spends on other consumption, andβ≤1 is his present-bias. Constantine’slong-run utility isV(x) = (Z-px) +u(x)-hx.That is,Vis the utility function Constantine