Web Appendices of Selling to Overcon dent Consumers


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1 Web Appenices of Selling to Overcon ent Consumers Michael D. Grubb A Option Pricing Intuition This appenix provies aitional intuition base on option pricing for the result in Proposition 2. Consier the case of monopoly. At time one, the monopolist is selling a series of call options, or equivalently units bunle with put options, rather than units themselves. The marginal price charge for a unit q at time two is simply the strike price of the option sol on unit q at time one. The series of call options being sol are interrelate; a call option for unit q can t be exercise unless the call option for unit q 1 has alreay been exercise. However, it is useful to consier the market for each option inepenently. Accoring to Proposition 2 (equation 8), when there is no pooling an the satiation constraint is not bining, the optimal marginal price for a unit q is: ^P q (q) = C q (q) + V q (q; ^ (q)) F (^ (q)) f(^ (q)) F (^ (q)) (14) It turns out that this is exactly the strike price that maximizes the net value of a call or put option on unit q given the i erence in priors between the two parties. 4 To show this explicitly, write the net value NV of a call option on minute q as the i erence between the consumers value of the option CV an the rm s cost of proviing that option, F V. The option will be exercise whenever the consumer values unit q more than the strike price p, that is whenever V q (q; ) p. Let (p) enote the minimum type who exercises the call option, characterize by the equality V q (q; (p)) = p. The consumers value for the option is their expecte value receive upon exercise, less the expecte strike price pai, where expectations are base on the consumers prior F (): CV (p) = Z (p) V q (q; ) f () [1 F ( (p))] p 4 This parallels Mussa an Rosen s (1978) ning in their static screening moel, that the optimal marginal price for unit q is ientical to the optimal monopoly price for unit q if the market for unit q were treate inepenently of all other units. 1
2 The rm s cost of proviing the option is the probability of exercise base on the rm s prior F () times the i erence between the cost of unit q an the strike price receive: F V (p) = [1 F ( (p))] (c p) Putting these two pieces together, the net value of the call option is equal to the consumers expecte value of consumption less the rms expecte cost of prouction plus an aitional term ue to the gap in perceptions: NV (p) = Z (p) V q (q; ) f () [1 F ( (p))] c + [F ( (p)) F ( (p))] p The aitional term [F ( (p)) F ( (p))] p represents the i erence between the exercise payment the rm expects to receive an the consumer expects to pay. The term [F ( (p)) F ( (p))] represents the isagreement between the parties about the probability of exercise. As a monopolist selling call options on unit q earns the net value NV (p) of the call option by charging the consumer CV (p) upfront, a monopolist shoul set the strike price p to maximize N V (p). By the implicit function theorem, p (p) = 1 V q (q;(p)), so the rst orer conition which characterizes the optimal strike price is: f ( (p)) V q (q; (p)) [p C q (q)] = [F ( (p)) F ( (p))] (15) As claime earlier, this is ientical to the characterization of the optimal marginal price ^P q (q) for the complete nonlinear pricing problem when monotonicity an satiation constraints are not bining (equation 14). Showing that the optimal marginal price for unit q is given by the optimal strike price for a call option on unit q is useful, because the rst orer conition q (q; ) = 0 can be interprete in the option pricing framework. Consier the choice of exercise price p for an option on unit q. A small change in the exercise price has two e ects. First, if a consumer is on the margin, it will change the consumers exercise ecision. Secon, it changes the payment mae upon exercise by all inframarginal consumers. In a commonprior moel, the inframarginal e ect woul net to zero as the payment is a transfer between the two parties. This is not the case here, however, as the two parties isagree on the likelihoo of exercise by [F ( (p)) F ( (p))]. term Consier the rst orer conition as given above in equation (15). On the left han sie, the f((p)) V q (q;(p)) represents the probability that the consumer is on the margin an that a marginal increase in the strike price p woul stop the consumer exercising. The term [p C q (q)] is the cost 2
3 to the rm if the consumer is on the margin an no longer exercises. There is no change in the consumer s value of the option by a change in exercise behavior at the margin, as the margin is precisely where the consumer is ini erent to exercise (V q (q; (p)) = p). On the right han sie, the term [F ( (p)) F ( (p))] is the rm s gain on inframarginal consumers from charging a slightly higher exercise price. This is because consumers believe they will pay [1 F ( (p))] more in exercise fees, an therefore are willing to pay [1 F ( (p))] less upfront for the option. However the rm believes they will actually pay [1 F ( (p))] more in exercise fees, an the i erence [F ( (p)) F ( (p))] is the rm s perceive gain. The rst orer conition requires that at the optimal strike price p, the cost of losing marginal consumers f((p)) V q (q;(p)) [p C q (q)] is exactly o set by the "perception arbitrage" gain on inframarginal consumers [F ( (p)) F ( (p))]. Setting the strike price above or below marginal cost is always costly because it reuces e ciency. In the iscussion above, referring to [F ( (p)) F ( (p))] as a "gain" to the rm for a marginal increase in strike price implies that the term [F ( (p)) F ( (p))] is positive. This is the case for (p) >, when consumers unerestimate their probability of exercise. In this case, from the rm s perspective, raising the strike price above marginal cost increases pro ts on inframarginal consumers, thereby e ectively exploiting the perception gap. On the other han, for (p) <, the term [F ( (p)) F ( (p))] is negative an consumers overestimate their probability of exercise. In this case reucing the strike price below marginal cost exploits the perception gap between consumers an the rm. Fixing an the rm s prior F (), the absolute value of the perception gap is largest when the consumer s prior is at either of two extremes, F () = 1 or F () = 0. When F () = 1, the optimal marginal price reuces to the monopoly price for unit q where the market for minute q is inepenent of all other units. This is because consumers believe there is zero probability that they will want to exercise a call option for unit q. The rm cannot charge anything for an option at time one; essentially the rm must wait to charge the monopoly price until time two when consumers realize their true value. Similarly, when F () = 0, the optimal marginal price reuces to the monopsony price for unit q. Now rather than thinking of a call option, think of the monopolist as selling a bunle unit an put option at time one. In this case consumers believe they will consume the unit for sure an exercise the put option with zero probability. This means that the rm cannot charge anything for the put option upfront, an must wait until time two when consumers learn their true values an buy units back from them at the monopsony price. The rm s ability to o so is of course limite by free isposal which means the rm coul not buy back units for a negative price. 3
4 Marginal price can therefore be compare to three benchmarks. For all quantities q, the marginal price will lie somewhere between the monopoly price p ml (q) an the maximum of the monopsony price p ms (q) an zero, hitting either extreme when F () = 1 or F () = 0, respectively. When F () = F (), marginal price is equal to marginal cost. To illustrate this point, the equilibrium marginal price for the running example with positive marginal cost c = $0:035 an low overcon ence = 0:25 previously shown in Figure 4, panel C is replotte with the monopoly an monopsony prices for comparison in Figure 10. $ Optimal MP Monopoly Monopsony MC quantity Figure 10: Equilibrium pricing for Example 1 given c = $0:035 an = 0:25: Marginal price is plotte along with benchmarks: (1) marginal cost, (2) ex post monopoly price  the upper boun, an (3) ex post monopsony price  the lower boun. B Pooling As it was omitte from the main text, a characterization of pooling quantities when the monotonicity constraint bins is provie below in Lemma 4. This is useful because it facilitates the calculation of pooling quantities in numerical examples. Lemma 4 On any interval [ 1 ; 2 ] such that monotonicity (but not nonnegativity) is bining insie, but not just outsie the interval, the equilibrium allocation is constant at some level ^q > 0 for all 2 [ 1 ; 2 ]. Further, the pooling quantity ^q an bouns of the pooling interval [ 1 ; 2 ] must maintain continuity of ^q () an satisfy the rst orer conition on average: R 2 1 q (^q; ) = 0. Nonnegativity bins insie, but not just above, the interval [; 2 ] only if R 2 q (0; ) 0 an q R ( 2 ) = 0. Proof. Given the result in Lemma 2, the proof is omitte as it closely follows ironing results for the stanar screening moel. The result follows from the application of stanar results in 4
5 optimal control theory (Seiersta an Sysæter 1987). See for example the analogous proof given in Fuenberg an Tirole (1991), appenix to chapter 7. Note that because the virtual surplus function is strictly quasiconcave, but not necessarily strictly concave, optimal control results yiel necessary rather than su cient conitions for the optimal allocation. For the special case V qq = 0, virtual surplus is strictlyconcave an applying the ironing algorithm suggeste by Fuenberg an Tirole (1991) using the conitions in Lemmas 2 an 4 is su cient to ientify the uniquely optimal allocation. Proposition 2 characterizes marginal pricing at quantities for which there is no pooling, an states that marginal price will jump iscretely upwars at quantities where there is pooling. It is therefore interesting to know when q R () will be locally ecreasing, so that the equilibrium allocation ^q () involves pooling. First, a preliminary result is helpful. Proposition 5 compares the relaxe allocation to the rst best allocation, showing that the relaxe allocation is above rst best whenever F () > F () an is below rst best whenever F () < F (): Proposition 5 Given maintaine assumptions: 8 >< q F B () q R () = q F B () >: < q F B () F () > F () F () = F () F () < F () () strict i C q q F B () > 0 Proof. See Section B.1 at the en of this appenix. Given that F () crosses F () once from below, the relationship between the relaxe allocation an the (strictly increasing) rst best allocation given in Proposition 5 leas to the conclusion that q R () is strictly increasing near the bottom an near the top. More can be sai about pooling when consumers either have nearly correct beliefs, or are extremely overcon ent. When consumers prior is close to that of the rm, the relaxe solution is close to rst best, an like rst best is strictly increasing. In this case the equilibrium an relaxe allocations are ientical. When consumers are extremely overcon ent such that their prior is close to the belief that = with probability one, the relaxe solution is strictly ecreasing at or just above. In this case ironing will be require an an interval of types aroun pool at the same quantity ^q ( ). The intuition is that when the consumers prior is exactly the belief that = with probability one, [F () F ()] falls iscontinuously below zero at. Thus, by Proposition 5, the relaxe solution must rop iscontinuously from weakly above rst best just below to strictly below rst best at. These results an the notion of "closeness" are mae precise in Proposition 6. 5
6 Proposition 6 (1) There exists " > 0 such that if jf () j < " for all then q R () is strictly increasing for all. (2) There exists a nite constant, such that if f ( ) > an f () is continuous at then q R () is strictly ecreasing just above. Proof. If neither nonnegativity nor satiation constraints bin at, Lemma 2 an the implicit function theorem imply that qr q(q () = R ();). In this case the sign of the cross partial qq(q R ();) erivative q etermines whether q R () is increasing or ecreasing. For any at which F is su ciently close to F in both level an slope, q > 0, but when f () is su ciently large q < 0. See the proof at the en of this appenix for etails. B.1 Pooling Appenix Proofs B.1.1 Small Lemma 5 Lemma 5 Satiation q S () an rst best q F B () quantities are continuously i erentiable, strictly positive, an strictly increasing. Satiation quantity is higher than rst best quantity, an strictly so when marginal costs are strictly positive at q F B. Proof. Satiation an rst best quantities are strictly positive because by assumption: V q (0; ) > C q (0) 0. Therefore given maintaine assumptions, q S () an q F B () exist, an are continuous functions characterize by the rst orer conitions V q q S () ; = 0 an V q q F B () ; = C q q F B () respectively. Zero marginal cost at q F B () implies q S () = q F B (). When C q q F B () > 0, V qq (q; ) < 0 implies that q S () > q F B (). The implicit function theorem implies qs = V q 0 an qf B = V q V qq C qq > 0. V qq > B.1.2 Proof of Proposition 5 Proof. The relaxe allocation maximizes virtual surplus (q; ) within the constraint set 0; q S () an (q; ) is strictly quasiconcave in q (See proof of Lemma 2). Moreover, q F B 2 (0; q S ()] an C q q q F B () ; = V q q F B () ; F () F () f() since the rst best allocation satis es V q q F B () ; = q F B () (Lemma 5). Therefore there are three cases to consier: 1. F () = F (): In this case virtual surplus an true surplus are equal so q R () = q F B (). 2. F () > F (): In this case q q F B () ; > 0 an therefore q R () q F B (). When C q q F B () = 0, Lemma 5 shows that q S () = q F B () an therefore the satiation constraint bins: q R () = q F B () = q S (). When C q q F B () > 0, Lemma 5 shows that satiation is not bining at rst best, an therefore the comparison is strict: q R () > q F B (). 6
7 3. F () < F (): In this case q q F B () ; < 0 an therefore q R () < q F B () since, by Lemma 5, nonnegativity is not bining at rst best (q F B () > 0). B.1.3 Proof of Proposition 6 Proof. If, over the interval ( 1 ; 2 ), neither nonnegativity nor satiation constraints bin an q (q; ) is continuously i erentiable, then in the same interval q q R ; = 0 (Lemma 2) an the implicit function theorem implies qr q(q () = R ();). (Given maintaine assumptions, qq(q R ();) q (q; ) is continuously i erentiable at where f () is continuous.) Therefore, in the same interval q R () will be strictly increasing if q q R () ; > 0 an strictly ecreasing if q q R () ; < 0. Part (1): As the nonnegativity constraint is not bining at (q R () = q F B () > 0), the upper boun q S () is strictly increasing, an q R () is continuous, q R () will be strictly increasing for all if the cross partial erivative q (q; ) is strictly positive for all (q; ) 2 0; q S ;. De ne ' (q; ) an ": (Note that " is well e ne since 0; q S ; is compact an F 2 C 2 an V (q; ) 2 C 3 imply ' (q; ) is continuous.) ' (q; ) 1 + ( ) jv q (q; )j V q (q; ) 1 " min (q;)2[0;q S ( )][; ] ' (q; ) +! > 0 By i erentiation: q (q; ) = V q (q; ) F () F () + V q (q; ) 1 + f () f 2 [F () () F ()]! As > 0 an V q (q; ) > 0: q (q; ) jv q (q; )j jf () F ()j +V q (q; ) 1 j f ()j f 2 jf () () F ()j! The assumption j f ()j < " implies that jf () F ()j < "( ) an therefore that: q (q; ) > V q (q; ) (1 "' (q; )) By the e nition of ", this implies q (q; ) > 0 for all (q; ) 2 0; q S ;. 7
8 Part (2): The rst step is to show that q (q; ) < 0. De ne : max q2[0;q S ( )] jv q (q; )j V q (q; ) + 2f ( ) + f ( )! f ( ) As > 0 an V q (q; ) > 0: jf () q (q; ) jv q (q; )j F ()j + V q (q; ) 2 f () + f 2 jf () () F ()j! since jf () F ()j 1: q (q; ) V q (q; ) jv q (q; )j V q (q; ) f ()! By e nition of an f ( ) >, it follows that for all q 2 0; q S : q (q; ) < 0. Given continuity of f () at, q (q; ) is also continuous at. Therefore for some 1 > 0, q (q; ) < 0 just above in the interval 2 [ ; + 1 ). The secon step is to show that for some 2 > 0 neither satiation nor nonnegativity constraints are bining in the interval ( ; + 2 ). First, satiation is not bining just above as q R is below rst best here (Proposition 5), which is always below the satiation boun (Lemma 5). Secon, nonnegativity is not bining just to the right of because q R ( ) = q F B ( ) > 0 (Proposition 5 an Lemma 5) an q R () is continuous (Lemma 2). Steps one an two imply that q R () is strictly ecreasing in the interval ( ; + min f 1 ; 2 g) just above. Therefore, q R () is either strictly ecreasing at or has a kink at an is strictly ecreasing just above. In either case, monotonicity is violate at an the equilibrium allocation will involve pooling at. C Monopoly MultiTari Menu Extension In this appenix, I exten the single tari moel explore in etail in the main paper to a multitari monopoly moel. The moel is escribe in Section 5 of the paper. (Note that I replace equation (1) by the stricter assumption V qq = 0.) The moel an solution methos are closely relate to Courty an Li (2000). To work with the new problem, rst e ne: De nition 2 (1) Let q c (s; ; 0 ) minfq(s; 0 ); q S ()g be the consumption quantity of a consumer who chooses tari s, is of type, an reports type 0. Consumption quantity of a consumer who honestly reports true type is q c (s; ) q c (s; ; ). (2) u(s; ; 0 ) V (q c (s; ; 0 ); ) P (s; 0 ) 8
9 is the utility of a consumer who chooses tari s, is of type, an reports type 0. (3) u (s; ) u (s; ; ) is the utility of a consumer who chooses a tari s, an honestly reports true type. (4) U (s; s 0 ) R u (s0 ; ) f (js) is the true expecte utility of a consumer who receives signal s, chooses tari s 0, an later reports honestly. U (s; s 0 ) R u (s0 ; ) f (js) is the analogous perceive expecte utility. (5) U (s) U (s; s) is the expecte utility of a consumer who honestly chooses the intene tari s given signal s, an later reports honestly. U (s) U (s; s) is the analogous perceive expecte utility. Invoking the revelation principle, the monopolist s problem may then be written as: max q(s;)0 P (s;) such that " Z # E (P (s; ) C (q (s; ))) f (js) 1. Global IC2 u (s; ; ) u(s; ; 0 ) 8s 2 S; 8; Global IC1 U (s; s) U (s; s 0 ) 8s; s 0 2 S 3. Participation U (s) 0 8s 2 S As the signal s oes not enter the consumers value function irectly, the secon perio incentive compatibility constraints may be hanle just as they are in the single tari moel (See part 1 of the proof of Proposition 1). In particular, secon perio local incentive u (s; ) = V (q c (s; ) ; )) an qc (s; ) 0) are necessary an su cient for secon perio global incentive compatibility. Moreover, Lemma 1 naturally extens to the multitari setting. Applying the same satiation re nement q (s; ) q S (), the istinction between q c (s; ) an q (s; ) may be roppe. The next step is to express payments in terms of consumer utility, an substitute in the secon perio local incentive constraint, just as was one in the singletari moel (See parts 2 an 3 of the proof of Proposition 1). This yiels analogs of equations (3) an (5): U s; s 0 = u s 0 ; + E V q s 0 ; ; 1 F (js) f (js) s U (s) U (s) = E V (q (s; ) ; ) F (js) F (js) f (js) s P (s; ) = V (q (s; ) ; ) Z (16) (17) V (q (s; z) ; z) z u (s; ) (18) In the single tari moel, the utility of the lowest type was etermine by the bining participation constraint. To pin own the utilities u (s; ) in the multitari moel requires consieration 9
10 of both participation an rstperio incentive constraints given an assume orering of the signal space S. I will consier signal spaces S which are orere either by FOSD, or a more general reverse secon orer stochastic ominance (RSOSD). 5 Note that I assume the orering, either FOSD or RSOSD, applies to consumer beliefs F (js). In the stanar single perio screening moel the participation constraint will bin for the lowest type, an this guarantees that it is satis e for all higher types. The same is true here given the assume orering of S, as state in Lemma 6. Lemma 6 If S is orere by FOSD then the participation constraint bins at the bottom (U (s) = 0). This couple with rst perio incentive constraints are su cient for participation to hol for all higher types s >s. The same is true if S is orere by RSOSD an V 0. Proof. It is su cient to show that U (s; s 0 ) is nonecreasing in s. If this is true, then IC1 an U (s) 0 imply participation is satis e: U (s; s) U (s; s) 0. Hence if U (s) 0 were not bining, pro ts coul be increase by raising xe fees of all tari s. Now by e nition, U (s; s 0 ) E [u (s 0 ; ) js]. By local IC2 an V 0 (which follows from V (0; ) = 0 an V q > 0), it is clear that conitional on tari choice s 0, consumers utility is nonecreasing in realize : u s 0 ; = V q s 0 ; ; 0 Given a FOSD orering, this is su cient for U (s; s 0 ) to be nonecreasing in s (Haar an Russell 1969, Hanoch an Levy 1969). Taking a secon erivative of consumer secon perio utility shows that conitional on tari choice s 0, consumers utility is convex in if V 0. V q > 0, monotonicity q (s 0 ; ) 0, an local IC2: u s 0 ; = V q q s 0 ; ; q s 0 ; {z } {z } (+) (+) by IC2 This follows from increasing i erences + V q s 0 ; ; 0 {z } 0 (assumption) Uner RSOSD orering, this implies that U (s) is increasing in s. (This result is analogous to the stanar result that if X secon orer stochastically ominates Y then E [h (X)] E [h (Y )] for any concave utility h (Rothschil an Stiglitz 1970, Haar an Russell 1969, Hanoch an Levy 1969). The proof is similar an hence omitte.) 5 Courty an Li (2000) restrict attention primarily to orerings by rst orer stochastic ominance or mean preserving sprea. However, in their two type moel, they o mention that orerings can be constructe from the combination of the two which essentially allows for the more general reverse secon orer stochastic ominance orerings I consier here. 10
11 Given Lemma 6 an the preceing iscussion, the monopolist s problem can be simpli e as escribe in Lemma 7. Lemma 7 Given a FOSD orering of S, or a RSOSD orering of S an V 0, the monopolist s problem reuces to the following constraine maximization over allocations q (s; ) an utilities u (s; ) for U (s) = U (s; s) an U (s; s 0 ) given by equation (16): max q(s;)2[0;q S ()] u(s;) such that E [ (s; q (s; ) ; )] E [U (s)] 1. Monotonicity q (s; ) nonecreasing in 2. Global IC1 U (s; s) U (s; s 0 ) 8s; s 0 2 S 3. Participation U (s) = 0 (s; q; ) V (q; ) C (q) + V (q; ) F (js) F (js) f (js) Payments P (s; ) are given as a function of the allocation q (s; ) an utilities u (s; ) by equation (18). Secon perio local incentive compatibility ( u (q; ) = V (q (s; ) ; )) always hols for the escribe payments. Proof. Firm pro ts can be reexpresse as shown in (equation 19). E [ (s; )] = E [S (s; )] + E [U (s) U (s)] E [U (s)] (19) Substituting equation (17) for ctional surplus gives the objective function. The participation constraint follows from Lemma 6. For hanling of the satiation constraint an secon perio incentive compatibility, refer to iscussion in the text an proofs of Lemma 1 an Proposition 1. Now, assume that there are two possible rstperio signals s 2 fl; Hg, an that the probability of signal H is. There are two rstperio incentive constraints. Given Lemma 6, the ownwar incentive constraint U (H; H) U (H; L) (ICH) must be bining. Otherwise the monopolist coul raise the xe fee P (H; ), an increase pro ts. Together with equation (16) an the bining participation constraint U (L) = 0 (IRL), this pins own U (H) as a function of the allocation q (L; ). Substituting the bining IRL an ICH constraints for u (L; ) an u (H; ) into both the monopolist s objective function escribe in Lemma 7 an the remaining upwar incentive constraint U (L; L) U (L; H) (ICL) elivers a nal simpli cation of the monopolist s problem in Proposition 7. 11
12 Proposition 7 De ne a new virtual surplus (q L ; q H ; ) an function (q L ; q H ; ) by equations (20) an (21) respectively. (q L ; q H ; ) (H; q H; ) f H () + (1 ) (L; q L ; ) f L () V (q L ; ) (FL () F H ()) (20) (q L ; q H ; ) [F L () F H ()] [V (q L ; ) V (q H ; )] (21) Given either a FOSD orering of S, or a RSOSD orering an V 0: 1. A monopolist s optimal twotari menu solves the reuce problem: max q L ();q H ()2[0;q S ()] such that IC2 ICL Z (q L () ; q H () ; ) q L () an q H () nonecreasing in R (q L () ; q H () ; ) 0 2. Payments are given by equations (18), an (2223): u (L; ) = E V (q L () ; ) 1 u (H; ) = E F L () f L () L [V (q L () ; ) V (q H () ; )] 1 F H () f H () (22) H + u (L; ) (23) Proof. Beginning with the result in Lemma 7, the simpli cation is as follows: Equation (16) an U (L) = 0 imply equation (22). Equation (16) an U (H; H) = U (H; L) imply equation (23). By equation (16) an equations (2223) U (H) is R V (q L () ; ) [FL () F H ()]. As U (L) = 0, this means E [U (s)] = R V (q L () ; ) [FL () F H ()], which leas to the new term in the revise virtual surplus function. Further, as U (L) = 0, the upwar incentive constraint (ICL) is U (L; H) 0. By equation (16) an equations (2223) U (L; H) is R (q L () ; q H () ; ) which gives the new expression for the upwar incentive constraint. Given the problem escribe by Proposition 7, a stanar approach woul be to solve a relaxe problem which ignores the upwar incentive constraint, an then check that the constraint is satis e. Given a FOSD orering, a su cient conition for the resulting relaxe allocation to solve the full problem is for q (s; ) to be nonecreasing in s. (Su ciency follows from ICH bining an V q > 0.) This is not the most prouctive approach in this case, because the su cient conition is likely to fail with high levels of overcon ence. An alternative approach is to irectly incorporate the upwar incentive constraint into the maximization problem using optimal control techniques. 12
13 Proposition 8 Given either a FOSD orering or a RSOSD orering an V 0: The equilibrium allocations ^q L () an ^q H () are continuous an piecewise smooth. For xe 0, e ne relaxe allocations (which ignore monotonicity constraints): ql R () = arg max q L 2[0;q S ()] qh R () = arg max q H 2[0;q S ()] + (L; q L ; ) 1 V (q L ; ) F L () F H () f L () (H; q H ; ) + V (q H ; ) F L () F H () f H () The relaxe allocations ql R () an qr H () are continuous an piecewise smooth functions characterize by their respective rst orer conitions except where satiation or nonnegativity constraints bin. Moreover, there exists a nonnegative constant 0 such that: (24) (25) 1. On any interval over which a monotonicity constraint is not bining, the corresponing equilibrium allocation is equal to the relaxe allocation: ^q s () = qs R (). 2. On any interval [ 1 ; 2 ] such that the monotonicity constraint is bining insie, but not just outsie the interval for tari s, the equilibrium allocation is constant at some level ^q s () = q 0 for all 2 [ 1 ; 2 ]. Further, the pooling quantity q 0 an bouns of the pooling interval [ 1 ; 2 ] must satisfy qs R ( 1 ) = qs R ( 2 ) = q 0 an meet the rst orer conition from the relaxe problem in expectation over the interval. For instance, for q L () this secon conition is: Z 2 1 q (L; ^q; ) + 1 V q (^q; ) F L () F H () f L () = 0 f L () 3. Complementary slackness: = 0 or ICL bins with equality ( R (^q L () ; ^q H () ; ) = 0). Proof. I apply Seiersta an Sysæter (1987) Chapter 6 Theorem 13, which gives su cient conitions for a solution. To apply the theorem, I rst restate the problem escribe by Proposition 7 in the optimal control framework. This inclues translating the upwar incentive constraint into a state variable k () = R (q L (z) ; q H (z) ; z) z with enpoint constraints k () = 0 (automatically satis e) an k( ) 0. Remaining state variables are q L () an q H (), which have free enpoints. Control variables are c L (), c H (), an c k (), an the control set is R 3. Costate variables are L (), H (), an k (). Z max (q L () ; q H () ; ) q L ;q H 13
14 State Control Costate Control  State Relation q L () c L () L () _q L () = c L () q H () c H () H () _q H () = c H () k () c k () k () k _ () = (ql ; q H ; ) En Point Constraints Lagrangian Multipliers (by e nition) k () = 0 ICL k( ) 0 Control Constraints Lagrangian Multipliers Monotonicity c L () 0 L () c H () 0 H () State Constraints Lagrangian Multipliers Nonnegativity q L () 0 L (), L, L Satiation q H () 0 H (), H, H q S () q L () 0 L (), L, L q S () q H () 0 H (), H, H The Hamiltonian an Lagrangian for this problem are: H = (q L ; q H ; ) + L () c L () + H () c H () k () (q L ; q H ; ) L = H + L () c L () + H () c H () + L () q L () + H () q H () + L () q S () q L () + H () q S () q H () The following 7 conitions are su cient conitions for a solution: 1. Control an state constraints are quasiconcave in states q L, q H, an k, as well as controls c L, c H, an c k for each. Enpoint constraints are concave in state variables. This is satis e because all constraints are linear. 2. The Hamiltonian is concave in states q L, q H, an k, as well as controls c L, c H, an c k for each. This is satis e because H is constant in k an c k, linear in c L an c H, strictly concave in q L an q H (equations 2627), an all other cross partials are zero. This relies on V qq = 0 14
15 an S qq (q; ) < = (1 ) S qq (q L ; ) < 0 (26) = S qq (q H ; ) < 0 (27) 3. State an costates are continuous an piecewise i erentiable in. Lagrangian multiplier functions as well as controls are piecewise continuous in. All constraints are satis e. ^L ^L ^L = k 5. L L ; H H ; k 6. Complementary Slackness conitions (a) For all : L ; H ; L ; H ; L ; H 0 L c L = H c H = L q L = H q H = L q S q L = H q S q H = 0 (b) For L ; H ; L ; H 0 L q L ( ) = H q H ( ) = L q S ( ) q L ( ) = H q S ( ) q H ( ) = 0 (c) For L ; H ; L ; H 0 L q L () = H q H () = L q S () q L () = H q S () q H () = 0 () For ICL 0 k () = k( ) = 0 7. Transversality Conitions L ( ) = L L ; H ( ) = H H ; k ( ) = L () = L + L ; H () = H + H k () = 15
16 Fix 0. Let k () = = =. As k () is constant, it is continuous an i erentiable, an k = 0. Neither the state k () nor the control c k () enter the Lagrangian, k = 0. For any allocation, k () = 0 by e nition, so k () = 0. Putting asie for now the constraint k( ) 0 an the complementary slackness conition k( ) = 0, all other conitions concerning state k () are satis e. Moreover, the Hamiltonian an Lagrangian are both aitively separable in q L an q H. Hence the remaining conitions are ientical to those for two inepenent maximization problems. Namely: max q L 2[0;q S ()] q L0 max q H 2[0;q S ()] q H0 (L; q L ; ) + 1 V (q L ; ) F L () F H () f L () (H; q H ; ) + V (q H ; ) F L () F H () f H () For xe 0, the solution to these problems exists, satis es Proposition 8 parts 12, an meets all the relevant conitions in 37 above. The proof for this statement is omitte, because the solution for each subproblem given any xe 0, is similar to the singletari case, an closely parallels stanar screening results. The iea is that for regions ( 1 ; 2 ) where monotonicity is not bining for allocation ^q s (), the ^q s () subproblem conitions are the KuhnTucker conitions for the relaxe solution q R s (). For regions in which monotonicity is bining, the characterization closely parallels Fuenberg an Tirole s (1991) treatment of ironing in the stanar screening moel. For the nice properties of the relaxe solution state in the proposition, refer to the analogous proof of Proposition 2 part 1. All that remains to show is that there exists a 0, such that k( ) 0 an k( ) = 0 given q L () an q H () that solve the respective subproblems for that. If k( ) 0 for = 0 there is no problem. If k( ) < 0 for = 0, the result follows from the intermeiate value theorem an two observations: (1) k( ) varies continuously with, an (2) for su ciently large k( ) > 0. The latter point follows because each subproblem is a maximization of E [ (q L ; q H ; )] + k( ). As increases, the weight place on k( ) in the objective increases, an k( ) moves towars its maximum. Given FOSD, k( ) is maximize (given nonnegativity, satiation, an monotonicity) at fq L () ; q H ()g = 0; q S (), for which q H > q L an therefore, by bining ICH an V q > 0, ICL must be strictly satis e. The argument is more involve, but the constraine maximum of k( ) is also strictly positive given RSOSD. 16
17 D Aitional Tables Tables 67 replicate Tables 34 base on customerweighte rather than billweighte average mistake sizes. Table 6: Frequency an size of ex post "mistakes" (fall 2002 menu). Plan 0 Customers Plan 1 Customers Plan 2 Customers Customers 393 (62%) 92 (15%) 124 (20%) Bills 5, ,185 Alternative Consiere Plan 1, 2, or 3 Plan 0 Plan 0 Alternative Lower Cost Ex Post 5% 65% 49% Conitional Avg. Saving y 25%** 68%** 54%** Unconitional Avg. Saving y NA 39%** 5% ycustomerweighte average per month, as a percentage of Plan 1 monthly xefee. ** 99% con ence. Table 7: Unerusage versus overusage for customers who coul have save on plan 0. Plan 1 Customers (60) Plan 2 Customers (61) Bills (523) Potential Saving y Bills (578) Potential Saving y Unerusage 56% 15% 59% 31% Intermeiate 28% (7%) 33% (13%) Overusage 16% 60% 9% 35% Total 100% 68% 100% 53% ycustomerweighte average per month as a percentage of Plan 1 monthly xefee. 17
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