EconS 503 Homework #8. Answer Key

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1 EonS 503 Homework #8 Answer Key Exerise #1 Damaged good strategy (Menu riing) 1. It is immediate that otimal rie is = 3 whih yields rofits of ππ = 3/ (the alternative being a rie of = 1, yielding ππ = 1).. The damaged version is sold to the low-valuation onsumers and the normal version to the high-valuation onsumers. Hene, dddddddddddddd = 5/10. To satisfy the inentive onstraint of the high tye, nnnnnnnnnnnn must be hosen suh that 6 10 dddddddddddddd = 1 10 = 3 nnnnnnnnnnnn or, equivalently, nnnnnnnnnnnn = 9/10, whih is less than 3. This results in rofits of (1/)(5/10 1/10) + (1/ )(9/10) = 33/0 > 3/. 3. Hene, the firm makes a higher rofit by introduing the damages version in site of the higher ost of rodution for this version. The high-valuation onsumers are also better off when the damages version is introdued, as they fae a lower rie. Finally, the lowvaluation onsumers obtain zero utility in both ases. The introdution of the damaged version thus results in a Pareto imrovement. Exerise # - Nonlinear riing 1. The monooly hoses to maximize ππ = ( )(1 ). The rofit-maximizing rie is easily found as UU = 1+. The orresonding rofit is ππuu = (1 ). 4. Faing a tariff (mm, ), the artiiation onstraint of a onsumer is CCCC() mm 0, where CCCC() is the onsumer surlus rie. With demand QQ() = 1, we have CCCC() = (1 ). As the monoolist s best interest is to set mm = CCCC(), its roblem is thus to set so as to maximize ππ = CCCC() + ( )(1 ) = 1 (1 )(1 + ). The first-order ondition yields = 0. Hene, the otimal two-art tariff is (mm, ) = (1 ),. The otimal two-art tariff onsists of selling the good at marginal ost, so as to generate the largest onsumer surlus, and in aturing this surlus fully through the fixed fee. The orresonding rofit is ππ TT PP = mm = (1 ). Welfare is maximized under the two-art tariff as it involves marginal ost riing (whereas uniform riing results in a deadweight loss).

2 3. Consumer surlus for agents of tye at any rie is omuted as CCSS () = ( ) 4. We reall from art () of the exerise that CCSS 1 () = (1 ) for any 1. For any rie where the two tyes of onsumer buy (i.e., 1), we hek that CCSS () CCSS 1 (). One otion for the monoolist is to make sure that onsumers of both tyes buy; for this, mm = CCSS 1 () and 1. We then have the same roblem as in art (), with = 1 : (mm, ) = 1, 1 and ππ 8 1 = 1. The alternative otion is to sell only to tye- 8 onsumers with mm = CCSS () and. The monoolist s roblem is then to hoose to maximize ππ = (1/) ( ) + ( 1/)(1 /). 4 The first-order ondition yields = 1/. As = 1/ (i.e., marginal ost riing violates the onstraint), the monoolist hooses a rie of = 1, so that mm = 1/4. The resulting rofit is ππ = 1 1 = 1. Therefore, ππ 4 > ππ 1, imlying that the monoolist refers to sell to tye- onsumers only by setting a rie above marginal ost. Homework #3 Moral Hazard a) The ontrat seifies the ayment ww ss you reeive when you deliver the beer, and the ayment ww FF you reeive when you do not deliver the beer. The two inequalities are the inentive omatibility onstraint and your friend s individual rationality onstraint: 0.9( ee 0.ww SS) + 0.1( ee 0.ww FF) ee 0.(ww FF+5) 0.8(8 ww SS ) + 0.1( ww FF ) 3 (IC) (IR) The solution to these equations (when they hold with equality) is ww SS = 4.81 and ww FF = 1.5. b) The two inequalities are the inentive omatibility onstraint and your own individual rationality onstraint: 0.9( ee 0.ww SS) + 0.1( ee 0.ww FF) ee 0.(ww FF+5) 0.9( ee 0.ww SS) + 0.1( ee 0.ww FF) 1 (IC) (IR) We make these equalities, and solve the two equations. The two imly that ee.(ww FF+5) = 1, and so (taking logs 0. (ww FF + 5) = 0. Hene, ww FF = 5. From (IR), 0.9( ee 0.ww SS) ee 0.( 5) = 1

3 ee 0.ww SS = 1 0.1ee = ww SS = log(0.809) = 0.1 ww SS = 1.06 In the first-best ontrat, it is observable whether the beer is drunken or onfisated, and so that agreement an require that the beer is delivered. The ayment is onstant beause the risk-neutral arty should bear all the risk. The ayment is the solution to your individual rationality onstraint: ee 0.(ww) = 1. Hene, w = 0. Your utility is -1 in both the first-best and seond-best ontrats, beause it is assumed here that your friend gets all the gains from trade. Your friend s utility is higher for the first-best than for the seond-best agreements: Case Utility 1st-best 0.9 (8) +0.1 (0) = (8-1.06) (0 - ( -5)) = nd-best 6.75

4 CHAPTER 4. HIDDEN ACTION, MORAL HAZARD 44 Maximizing with reset to R(x) for every ositive x leads to the following rst order 1 = (x) u 0 (W 0 + R (x) x) = (a) for 8x > (0) = 0, 1 u 0 (W 0 + R (0)) = 1 0 (a) 1 (a) where we assumed that (a) never equals to 0 or 1. In other words, we assume that it is extremely ostly for the agent to ensure there will be no loss whatsoever, but extremely hea to make sure that the loss does not our with robability 1. Given the ositive Lagrange multiliers and 0 (a) < 0, we nd for 8x > 0 1 u 0 (W 0 + R (0)) > 1 > 1 u 0 (W 0 + R (x) x), R (0) > R (x) x: So the otimal ontrat is suh that the insuree is unished when a loss ours, but the unishment is not related to the size of the loss x sine the robability of a larger loss is indeendent of the agent s e ort. Therefore, the seond-best ontrat will indue variation in the insuree s ayo only between 0 and x > 0. The exat shae of the ontrat will be fully determined by the rst order ondition with reset to e ort and the zero-ro t ondition. 4.3 Question 1 Consider a rinial-agent roblem with three exogenous states of nature, 1 ; ; and 3 ; two e ort levels, a L and a H ; and two outut levels, distributed as follows as a funtion of the state of nature and the e ort level: State of nature 1 3 Probability 0:5 0:5 0:5 Outut under a H Outut under a L The rinial is risk neutral while the agent has utility funtion w when reeiving monetary omensation w, minus the ost of e ort, whih is normalized to 0 for a L and to 0.1 for a H. The agent s reservation exeted utility is Derive the rst-best ontrat.. Derive the seond-best ontrat when only outut levels are observable. 3. Assume the rinial an buy for a rie of 0.1 an information system that allows the arties to verify whether state of nature 3 haened or not. Will the rinial buy this information system? Disuss.

5 CHAPTER 4. HIDDEN ACTION, MORAL HAZARD First-Best Contrat The robabilities are as follows: Pr (q = 18ja H ) = 0:75 Pr (q = 18ja L ) = 0:5: Thus, when the rinial an ontrat on the level of e ort, he will solve max 0:5q(a i; 1 ) + 0:5q(a i ; ) + 0:5q(a i ; 3 ) a i;w w subjet to w (ai ) 0:1 Under ation hoie a H, w H 0:1 = 0:1 ) w H = 0:04: Therefore, E P (a H ) = 18 (0:75) + 1 (0:5) 0:04 = 13:71: Under ation hoie a L, w L = 0:1 ) wl = 0:01: Therefore, E P (a L ) = 18 (0:5) + 1 (0:75) 0:01 = 5:4: Thus, the otimal ontrat sei es a = a H and w = 0: Seond-Best Contrat When only outut levels are observable, the otimal ontrat to indue the e ort level a H solves a minimization of the wage bill, subjet to min w 18;w 1 0:75w :5w 1 0:75 w :5 w 1 0:1 0:1 (IR) 0:75 w :5 w 1 0:1 0:5 w :75 w 1. (IC) We an rove that in the two outomes/two ations ase both (IR) and (IC) must be binding at the otimum. If (IR) is not binding, the exeted wage bill an be dereased by lowering w 1 whereas the (IC) will even be relaxed. If (IC) is not binding, but the (IR) is, then we an derease w 18 and inrease w 1 suh that the (IR) is still binding. In artiular, given w1 dw 1 = 3, dw 18 w18

6 CHAPTER 4. HIDDEN ACTION, MORAL HAZARD 46 this would hange the exeted wage bill by w1 0:75dw :5( 3 )dw 18 : w18 By (IC), we have w 1 < w 18. Therefore, a derease in w 18 leads to a negative hange in the wage bill, 0:75(1 r w1 w 18 )dw 18 < 0 With the binding restritions (IR) and (IC), it is easy to nd the otimal wage shedule w 1 = 0:005 w 18 = 0:065: Note that the limited liability onstraint imlied by the utility funtion w is not binding here. The exeted utility of the rinial is E P (a H ) = 13:75 0:065 0:75 0:005 0:5 = 13:705: So the exeted ro t of induing a H is greater than the exeted ro t of induing a L, regardless of whether the e ort level is observable or not Information System For this sei rie the answer is trivially no. The rinial is not willing to buy the information system, beause even with erfet information the highest ro t he an attain is 13:71. Therefore, the maximal gain of using the information system will always be smaller than the ost of imlementing the system, 13:71 13:705 = 0:0075 < 0:1: We now nd the ro ts the rinial an make when state 3 an be observed. Then we an nd for whih rie the rinial would be willing to ay for the information system. Let us denote by y the signal that takes the value 1 when 3 is realized and the value 0 otherwise. If y = 1, the rinial an still not identify the e ort level being exerted. However, when y = 0, only the low e ort level an lead to an outut level of 1. So by enalizing the agent in nitely when q = 1 and y = 0, the rst-best an be attained. Indeed, the agent will hoose the high e ort level to be sure to avoid the unishment. Given that no variation in reeived inome is needed, the high e ort an be imlemented at the lowest ost. However, given the liability onstraint imlied by the utility funtion, the rinial is restrited to ositive levels of omensation. Hene, the rinial will ay w 1 = 0 when the outut is 1 and the signal is 0: In addition, w 18 and w 3, the wages aid when q = 18 and y = 1; q = 1 resetively, are set to solve min w 18;w 3 0:75w :5w 3

7 CHAPTER 4. HIDDEN ACTION, MORAL HAZARD 47 subjet to 0:75 w :5 w 3 0:1 0:1 (IR) 0:75 w :5 w 3 0:1 0:5 w :75 3 w w3. (IC) 3 The inentive omatibility onstraint is binding and simli es to Hene, and 0:75 w 18 0:1 = 0:5 w 18 : w 18 = 0:04: 0: 0:75 0: w 3 = = 0:04: 0:5 So the rst-best an be ahieved by the ontrat w 18 = w 3 = 0:04 and w 1 = 0. Finally, from the alulation above, the rinial will imlement the information system if the ost falls below 0: Question 13 Consider the modi ed linear managerial-inentive-sheme roblem, where the manager s e ort, a a ets urrent ro ts, q 1 = a + " q1, and future ro ts, q = a+" q, where " qt are i.i.d. with normal distribution N(0; q). The manager retires at the end of the rst eriod, and the manager s omensation annot be based on q. However, her omensation an deend on the stok rie P = a + " P, where " P N(0; P ). Derive the otimal omensation ontrat t = w + fq 1 + sp. Disuss how it deends on P and on its relation with q. Comare your solution with that in the Chater CEO Comensation Note that this question uses notation that is not onsistent with the exosition in setion The rogram for this roblem is the following, max a;w;f;s E (q 1 + q t) subjet to E e [t (a)] e t (IR) a arg max E e [t (ba)] (IC) ba t = w + fq 1 + sp

8 CHAPTER 4. HIDDEN ACTION, MORAL HAZARD 48, subjet to max a a;w;f;s (w + fa + sa) w + fa + sa f q + s P + sf qp a t a arg max w + fba + sba ba f q + s P + sf qp ba : We an aly the rst-order aroah and substitute the rst order ondition a = f + s for the inentive omatibility onstraint. Introduing this e ient level of e ort and substituting the outside oortunity level lus the risk remium lus the ost of e ort for the wage, we obtain max f;s + s f f q + s (f + s) P + sf qp t: The rst order onditions with reset to f and s are resetively 4 f q + s qp s P + f qp After some rewriting, the equations beome f + s f + s = 0 = 0: and we nally nd f = s ( + qp ) 1 + q f = s ( + P 1 + qp ), f = s = q + P q + P P qp qp + ( P q qp ) q qp qp + ( P q qp ): 4.4. Comarison Comared to the examle in Chater 4, the imortane of the agent s e ort has doubled for the rinial. Among the ontratible variables, only the imat of a hange in e ort on the stok rie has doubled. Although the agent s e ort

9 CHAPTER 4. HIDDEN ACTION, MORAL HAZARD 49 now determines the outut in two eriods, the outut in the seond eriod is not ontratible. The stok rie therefore beomes a more reise indiator of e ort than the ontratible outut. If we onsider the relative size of the inentives at the otimum f s = P q qp qp and omare this to the equivalent ratio in Chater 4 1 f qp s = P, q qp we notie that the rinial uts relatively more weight on the stok ries. This weight is exatly double when both indiators are indeendent, qp = Question 14 Consider the following rinial-agent roblem. There is a rojet whose robability of suess is a (a is also the e ort made by the risk-neutral agent, at ost a ). In ase of suess the return is R, and in ase of failure the return is 0. The arameter R an take two values, X with robability and 1 with robability 1. To undertake the rojet, the agent needs to borrow an amount I from the rinial. The sequene of events is as follows: First, the rinial o ers the agent a debt ontrat, with fae value D 0. The agent aets or rejets this ontrat. Seond, nature determines the value R that would our in ase of suess. This value is observed by both rinial and agent. The rinial an then hoose to lower the debt from D 0 to D 1. The agent hooses a level of e ort a. This level is not observed by the rinial. The rojet sueeds or not. If the rojet sueeds, the agent ays the minimum of R and the fae value of debt D 1. Answer the following questions: 1. Comute the subgame-erfet equilibrium of this game as a funtion of I,, and X:. When do we have D 1 < D 0? Disuss. 1 In setion f and s and w and t are interhanged.