Priority pricing by a durable goods monopolist

Transcription

1 Priority ricing by a durable goods monoolist João Correia-da-Silva February 15 th, 17. Abstract. A durable goods monoolist faces buyers with rivately observed valuations in two eriods, being unable to commit in the first eriod on the mechanism that she will roose in the second eriod. The monoolist is able to use a mediation device that revents her from discriminating buyers in the second eriod based on the choices they made in the first eriod. Moreover, the monoolist is able to revent buyers from retrading among themselves. It is shown that, although there is a continuum of ossible valuations, the otimal first-eriod mechanism is a menu with at most two ossibilities: a relatively high rice with no rationing; and a relatively low rice subject to rationing. In the second eriod, the monoolist otimally osts an intermediate rice. Keywords: Sequential mechanism design, Mediated mechanisms, Coase conjecture. JEL Classification Numbers: D8 VERY PRELIMINARY I am extremely grateful to Daniel Garrett for very useful conversations at an early stage of this roject. I also thank Armin Schmutzler, Ben Golub, Bruno Jullien, Doh-Shin Jeon, François Salanié, Misha Drugov, Renato Gomes and Takuro Yamashita. The usual disclaimer alies. This research has been funded by the Euroean Commission through the Marie Sk lodowska Curie Fellowshi H- MSCA-IF ; and by FEDER (through COMPETE) and Portuguese ublic funds (through FCT) in the framework of rojects PTDC/IIM-ECO/594/1 and PEst-OE/EGE/UI415/14. 1

2 1 Introduction The market ower of a monoolist who sells durable goods over time is mitigated by her inability to commit to future rices (Coase, 197). In each moment in time, the buyers that are most willing to trade are those with the highest valuations for the durable good. As a result, in the following eriod, the seller faces buyers with lower valuations and thus has incentives to set a lower rice. Anticiating this, buyers become unwilling to ay a rice that is much higher than the rice they exect to be charged in future eriods. In some sense, the monoolist faces cometition from her future selves. This classical argument is known as the Coase conjecture. Research on the Coase conjecture has mostly focused on the case in which the seller osts rices. Skreta (6) showed that this restriction is inconsequential because the otimal mechanism under sequential rationality constraints is actually a sequence of osted rices. In a model with a finite number of eriods where, in each eriod, the seller rooses a sequentially otimal mechanism to a buyer with a rivately observed valuation, the seller does not benefit from the ossibility of offering various combinations of ayment and robability of trade. This kind of second degree rice discrimination brings no additional revenue to the seller. In contrast, considering a model with two eriods where the seller cannot discriminate buyers in the second eriod based on the choices they made in the first eriod, Denicolò and Garella (1999) showed that rationing could alleviate the commitment roblem of the monoolist by reducing her incentives to lower rices in future eriods. This is the case if buyers are sufficiently atient. Rationing is beneficial because it alters the comosition of subsequent demand in a favourable way, by keeing some of the buyers with high valuations unserved. As a result, the sequentially otimal second-eriod rice becomes higher, which means that rationing countervails the roblem of limited commitment. The contrast between the results of Skreta (6) and Denicolò and Garella (1999) means that the ability of the seller to discriminate, in the second eriod, between buyers

3 that did not get the object due to rationing and buyers that did not wish to trade is detrimental to the seller. If discrimination is ossible, buyers that are rationed suffer a ratchet effect because their relatively high valuations are revealed. If discrimination is not ossible, the willingness to ay of those buyers is higher because those that end u being rationed will be offered better terms in the second eriod, because they are ooled with buyers with lower valuations that rejected trade in the first eriod. In this aer, it is shown that the otimal first-eriod mechanism may be either a relatively high osted rice, a relatively low rice with rationing, or a combination of the two. More comlicated mechanisms do not increase the seller s revenue. In an examle with linear demand (uniformly distributed valuations), we show that, for a given discount factor of the seller: a osted rice is otimal if the discount factor of the buyer is low; a rice with rationing is otimal if the discount factor of the buyer is high; a combination of the two is otimal if the discount factor of the buyer is intermediate. 1 Priority ricing has been studied, among others, by: Harris and Raviv (1981), Chao and Wilson (1987), Wilson (1989), Nocke and Peitz (7), Desai et al. (7) and Su (1), in models with uncertain demand or uncertain suly; and by Su (7), in a continuous-time finite-horizon model where buyers with heterogeneous discount factors arrive over time. Rationing has also been the subject of investigations by: Gilbert and Klemerer (), in a model with entry costs; Bulow and Klemerer (), in a common-value environment; Liu and Van Ryzin (8), in a model with commitment; and by Deb and Said (15) and Courty and Nasiry (16) in models where consumers are uncertain about their valuations. In other related work: McAfee and Wiseman (8) showed that caacity costs counter the Coase conjecture, while Board and Pycia (14) introduced outside otions and showed that these also counter the Coase conjecture. 1 With a common discount factor: if agents are relatively imatient, it is otimal for the seller to ost a rice; otherwise, it is otimal to offer a menu comosed by a relatively high rice guaranteeing delivery and a relatively low rice with rationing. 3

4 Model A seller of an indivisible durable good (worth nothing to her) faces a buyer who has a rivately known valuation for the good, v [, 1], drawn according to a cumulative robability distribution, F, which has continuous and strictly ositive density, f, and is such that v is strictly increasing. 3 Trade may occur in eriod 1 or eriod. In each eriod, the seller rooses a menu comosed by airs (ˆr, ˆt), where ˆr [, 1] is a robability of trade and ˆt IR is a transfer (from buyer to seller). The buyer chooses a air from the menu. We assume that the air (, ) is always in the menu to reflect the fact that the buyer can reject to contract. The seller cannot make commitments in eriod 1 regarding her roosal in eriod. Moreover, the seller cannot observe the buyer s choice from the menu. The only thing that the seller observes is whether or not trade takes lace. Therefore, the seller is not able to condition her eriod roosal on the buyer s eriod 1 choice from the menu. 4 Buyer and seller are risk-neutral and have ossibly different discount factors, δ b [, 1] and δ s [, 1]. 5 The seller wishes to maximize the exected discounted value of the transfer, while the buyer wishes to maximize the exected discounted value of the difference between the value derived from the good and the transfer made to the seller. 6 Equivalently, we may think of a continuum of buyers with valuations distributed in [, 1]. 3 This assumtion, known as regularity or monotonicity of virtual utility, is standard and weaker than weakly-increasing hazard rate,. 4 Skreta (6) studied the case in which the seller observes the buyer s choice, and showed that the otimal mechanism is a decreasing sequence of osted rices. Such a mechanism is also feasible without the ability of observing the buyer s choice, therefore, it is not detrimental for the seller to be unable to observe the buyer s choice. We will find that not observing the buyer s choice is actually beneficial if the layers are sufficiently atient. In a sense, it allows the seller to commit not to use the information revealed by the buyer s choice. 5 Without atience (δ b = δ s = ), it is otimal for the seller to set the monooly rice in eriod 1. Without imatience (δ b = δ s = 1), it is otimal for the seller to set a rice higher or equal to 1 in eriod 1, skiing eriod 1, and then set the monooly rice in eriod. 6 The case in which there is a different seller in the second eriod can be catured by setting δ s =. More generally, the case in which the eriod 1 seller gets a fraction τ [, 1] of eriod revenues, and δ [, 1] is her discount factor, can be catured by setting δ s = τδ. 4

5 In the second eriod, the otimal strategy for the seller is to ost a rice: 7 { argmax } f (v) dv, where f (v) is the osterior robability belief of the seller about the valuation of the buyer after conditioning on trade not having been executed in eriod 1. 7 See Börgers (15, Section.). 5

6 3 Analysis Our aroach to the seller s roblem is to start by taking as given and solve the one-shot mechanism design roblem of choosing the eriod 1 menu under the additional restriction that the given is otimal in eriod. Then, find the otimal value of. In the light of this aroach, the revelation rincile holds because lack of commitment is transformed into an exogenous restriction on the allocation in eriod 1. Restricting to incentive comatible direct mechanisms, let r : [, 1] [, 1] and t : [, 1] IR secify, resectively, the robability of trade and the transfer (from buyer to seller) as a function of the valuation announced by the buyer. The roblem of the seller can be written as: 8 { max { max r,t s.t. t(v) dv + δ s } [1 r(v)] dv v argmax {r(v )v t(v ) + 1 v δ b (v ) [1 r(v )]} v { } argmax [1 r(v)] dv. (1) Lemma 1. Truthful revelation imlies that r is weakly-increasing. Proof. To verify that r must be weakly-increasing, consider simultaneously two incentive constraints: a buyer with valuation v does not want to announce valuation v and vice-versa: r(v)v t(v) + 1 v δ b (v ) [1 r(v)] r(v )v t(v ) + 1 v δ b (v ) [ 1 r(v ) ] r(v )v t(v ) + 1 v δ b (v ) [ 1 r(v ) ] r(v)v t(v) + 1 v δ b (v ) [1 r(v)]. Adding the two constraints and simlifying, we obtain: [ r(v) r(v ) ] [ v v + 1 v δ b (v ) 1 v δ b (v ) ]. 8 Whenever the maximization is over an emty set, consider a default value of zero. 6

7 If v > v, the exression inside the second arenthesis is strictly ositive: (i) if v, it is v v ; (ii) if v, it is (1 δ b )(v v ); (iii) if v < < v, it is (1 δ b )v + δ b v. Hence, r(v) r(v ) cannot be strictly negative. It is convenient to incororate the truth-telling constraint in the objective function. Lemma. The embedded roblem in (1) is equivalent to: { [ max r(v) v r + (1 δ b ) s.t. ] dv r(v) r is weakly-increasing { argmax [ ] } v + δ b δ s 1 δ b dv + δ s [1 F ( )] } [1 r(v)] dv. () Proof. From the truth-telling condition in roblem (1), U(v) is the maximum of a collection of increasing and convex functions of v, where each function corresonds to a given value of v. This imlies that U(v) is increasing and convex. Since U(v) is convex: it is differentiable with the ossible excetion of countably many oints, and is equal to the integral of its derivative. 9 With truthful revelation, the buyer s ayoff is: U(v) = r(v)v t(v)+1 v δ b (v ) [1 r(v)]; which imlies that: U (v) = r (v)v + r(v) t (v) + 1 v δ b [1 r(v)] + 1 v δ b (v ) [ r (v)]. Using the first-order incentive constraint, r (v)v t (v) + 1 v δ b (v ) [ r (v)] =, we obtain: U (v) = r(v) + 1 v δ b [1 r(v)]. Hence: U(v) U() = r(ṽ) dṽ + (1 δ b ) r(ṽ) dṽ + δ b (v ), if v r(ṽ) dṽ, if v <. 9 See Börgers (15, Lemmas. and.3) or Royden and Fitzatrick (1, Chater 6). 7

8 From Lemma 1, it is necessary that r is weakly-increasing. We now check that (together with the first-order incentive constraint) this condition is also sufficient. (i) Suosing that v < v : U(v) r(v )v t(v ) U(v) U(v ) r(v )(v v ) v r(ṽ) dṽ v r(v ) dṽ. (ii) Suosing that v < v: U(v) r(v )v t(v ) + δ b (v ) [ 1 r(v ) ] U(v) U(v ) r(v )(v v ) + δ b (v v ) [ 1 r(v ) ] (1 δ b ) r(ṽ) + δ b dṽ (1 δ b ) r(v ) + δ b dṽ. v v (iii) Finally, suosing that v < < v: U(v) r(v )v t(v ) + δ b (v ) [ 1 r(v ) ] U(v) U(v ) r(v )(v v ) + δ b (v ) [ 1 r(v ) ] r(ṽ)dṽ + (1 δ b ) r(ṽ) + δ b dṽ r(v )dṽ + (1 δ b ) r(v ) + δ b dṽ. v v In all cases (i)-(iii), the conditions hold if r is weakly-increasing. Hence, using the fact that t(v) = r(v)v + 1 v δ b (v ) [1 r(v)] U(v), the embedded roblem in (1) becomes: { max {r(v)v + 1 v δ b (v ) [1 r(v)] U(v)} dv + δ s r,u s.t. r is weakly-increasing { argmax } [1 r(v)] dv. } [1 r(v)] dv Maniulating the objective function, and setting to zero the ayoff of a buyer with zero valuation, we obtain: 8

9 = { r(v)v { r(v)v dv [1 F ( )] } r(ṽ) dṽ dv + r(ṽ) dṽ + (1 δ b ) + (δ b δ s ) {r(v)v + δ b (v ) [1 r(v)]} dv } r(ṽ) dṽ + δ b (v ) dv + δ s [1 r(v)] dv r(v) dv (1 δ b ) r(ṽ) dṽ dv + (1 δ b ) r(ṽ) dṽ dv r(v)v dv r(v) dv + δ s [1 F ( )]. (3) Using the fact that: r(ṽ) dṽ dv = ṽ r(ṽ) dv dṽ = [F ( ) F (ṽ)] r(ṽ) dṽ, and that r(ṽ) dṽ dv = the objective function becomes: = r(v)v dv [1 F ( )] r(v) dv (1 δ b ) + (δ b δ s ) [ v + δ s [1 F ( )], ṽ r(ṽ) dv dṽ = [F ( ) F (v)] r(v) dv + (1 δ b ) [1 F (v)] r(v) dv r(v) dv + δ s [1 F ( )] ] r(v) dv + (1 δ b ) [1 F (ṽ)] r(ṽ) dṽ, r(v)v dv [ ] v + δ b δ s 1 δ b r(v) dv which is the objective function in the statement of this lemma. We roceed by showing that the otimal eriod 1 menu includes at most two alternatives besides the null contract: one is to ay a relatively high rice to obtain the good with 1% robability; the other is to ay a relatively low rice to obtain the good with a robability that is lower than 1%. 9

10 Lemma 3. The solution of roblem () satisfies: r(v) = 1, if v (v, 1) ˆr, if v (v, v), if v (, v), where v [, ], v [, 1], and ˆr [, 1). Proof. This is a corollary of Lemma 6 for the articular case in which G(1) =. Buyers with high valuations (v > v) refer to ay the high rice to guarantee delivery; while buyers with intermediate valuations (v < v < v) refer to ay the low rice desite the risk of being rationed. Buyers with low valuations (v < v) refer the null contract. All buyers that ay to obtain the good in eriod have been rationed in eriod 1. Those that refer the null contract in eriod 1 also refer the null contract in eriod. Among the buyers that ay the low rice and risk rationing, those with relatively high valuations trade in eriod. Notice that the seller may roose a menu with a single alternative besides the null contract. If v = 1, all tyes of buyer are rationed; if v = v or ˆr =, there is no rationing. 1

11 4 Increasing roortional hazard rate To solve the reduced roblem (), we only need to find the otimal values of v [, ], v [, 1], and ˆr [, 1). To solve the comlete roblem (1), we also need to find the otimal value of, which is taken as given in the reduced roblem (). In this section, we restrict F to be such that v is strictly increasing.1 This assumtion of increasing roortional hazard rate guarantees that the single-eriod rofit function, [1 F ()], is single-eaked. For simlicity of exosition, we also restrict F to be twice-differentiable. From Lemma 3, the otimal eriod 1 mechanism is defined by the trile (v, v, ˆr). The following lemma establishes that a sequentially otimal rice osted in eriod cannot be higher than the single-eriod monooly rice, m argmax { [1 F ()]}. The lemma also gives the value of v as a function of. Lemma 4. If (v, v, ˆr, ) is a solution of roblem (1), then: m and v >. Moreover: F (v) = F ( ) + f( ). Proof. From Lemma 3, we know that [v, v]. If a ossibly different rice [v, v] is osted in eriod, the resulting rofit is (1 ˆr) dv = (1 ˆr) [F (v) F ()]. Therefore, the first-order condition for sequential otimality is: [ ] F (v) F () f() = [1 F ()] F (v) F () 1 F () f() 1 F () =. It is clear that the exression inside the second arenthesis is strictly decreasing in, because it is the difference between a decreasing function and a strictly increasing function. Therefore: there is a single that satisfies this condition, and is a strictly increasing function of v. For to be sequentially otimal, we must have: F (v) = F ( ) + f( ). If = m, we obtain v = 1. If > m, we obtain a contradiction because v 1. 1 Like regularity, this assumtion is weaker than weakly-increasing hazard rate,. 11

12 The eriod 1 mechanism can be equivalently characterized by a trile ( 1, r, ˆr), where 1 is the rice aid to get the object with certainty, and r is the conditional rice aid to get the object with robability ˆr. 11 Truth-telling imlies that r = v, and that 1 is such that: v 1 = ˆr(v r ) + (1 ˆr)δ b (v ) 1 = (1 δ b )(1 ˆr)v + ˆrv + (1 ˆr)δ b, which allows us to write the resulting rofit of the seller as: 1 [1 F (v)] + rˆr [F (v) F (v)] + δ s (1 ˆr) [F (v) F ( )] = (1 ˆr) [(1 δ b )v + δ b ] [1 F (v)] + ˆrv [1 F (v)] + δ s (1 ˆr) [F (v) F ( )]. The only candidate deviation from is to the unique rice d (, v) that satisfies the first-order condition in that domain. If ˆr =, the value of v is irrelevant. Lemma 5. If (v, v, ˆr, ) is a solution of roblem (1), then: v (, ). Moreover, v is defined as a function of and ˆr by: F ( d )+ d f( d ) = ˆrF (v)+(1 ˆr) [F ( ) + f( ) ]; where d is defined by: d f( d) = (1 ˆr) f( ). Proof. Let m and consider r defined as in Lemma 3, with ˆr < 1 and v determined by Lemma 4. If we could ignore the sequential otimality restriction in roblem (), the otimal v would be the uer bound, v =, because virtual utility is negative for v [, ]. However, as we verify below, if v =, a downward deviation from would be rofitable. For [, v], rofit in eriod is dv + (1 ˆr) dv = [F (v) F ()] + (1 ˆr) [F (v) F (v)]. The derivative of rofit with resect to, for (, v), is: v F (v) F () f() + (1 ˆr) [F (v) F (v)] =. 11 From risk-neutrality, it is equivalent to ay rˆr unconditionally (with robability one) or to ay r only if trade is executed (with robability ˆr). 1

13 Therefore, if = v = the (left) derivative of rofit with resect to is negative: f( ) + (1 ˆr) f( ) <, which imlies that v < is necessary for sequential otimality of. Of course, if v =, sequential otimality of would be guaranteed because, by construction of v in Lemma 4, = is the single rice that satisfies the first-order condition for (v, v). Since the objective function in roblem () is strictly increasing in v, we want to find the maximum v (, ) subject to sequential otimality of. The first-order condition for local rofit maximization with (, v) is: F (v) + (1 ˆr) [F (v) F (v)] F () f() = ˆrF (v)+(1 ˆr)F (v) F () 1 F () f() 1 F () =. The exression on the left-hand side is strictly decreasing in, because it is the difference between a decreasing function and a strictly increasing function. There is, therefore, a single d that can satisfy the first-order condition, and d is a strictly increasing function of v and a decreasing function of ˆr (strictly whenever v < v). The rice d, which is the candidate deviation from, is imlicitly defined by: ˆrF (v) + (1 ˆr) F (v) = F ( d ) + d f( d ). To verify that there is a threshold value of v below which is sequential otimal and above which it is not, notice that the difference between the rofit for = (v, v) and the rofit for = d (, v) can be written as: π( ) π( d ) = = d π ()d + v π ()d d [(1 ˆr) F (v) + ˆrF (v) F () f()] d + (1 ˆr) v [F (v) F () f()] d, where the second integral is clearly decreasing in v (the argument is ositive), while the first 13

14 integral is decreasing in v if and only if: (1 ˆr) F (v) + ˆrF (v) F (v) v + ˆrF (v) ˆrF ( d ) < (1 ˆr) F (v) + ˆrF (v) F ( d ) d f( d ) < (1 ˆr) F (v) + v (1 ˆr) F ( d ) d f( d ), which is true because the left-hand side is zero while the right-hand side is ositive. So that it is not beneficial to deviate from to d, we must have d v, and: d {F (v) F ( d ) + (1 ˆr) d [F (v) F (v)]} (1 ˆr) [F (v) F ( )] d f( d) (1 ˆr) f( ). We now verify that f() is a strictly increasing function of (, m ). Its derivative is ositive if and only if: f() + f () >. We know that the derivative of ositive, therefore, f() + f () f() + f() only if f() 1 F () f() 1 F () is strictly f() 1 F () >. Notice that f() + 1 F () < if and 1 F () < 1, which is equivalent to f() <. The latter is true for < m. This imlies that f() + f () >. Loosely, for given ˆr, we must have sufficiently low v, so that d is sufficiently low to satisfy d f( d) (1 ˆr) f(). By insection of the objective function in roblem (), it is clear that r(v) should be as low as ossible for v [, ], because the virtual valuation is negative. Keeing ˆr constant, this imlies that it is better to have v as high as ossible. Hence, the above restriction is binding. Given and ˆr, the value of v must be such that: d f( d) = (1 ˆr) f(), where d is imlicitly defined by: F ( d ) + d f( d ) = ˆrF (v) + (1 ˆr) F (v). 14

15 5 Uniform distribution To be able to comute the otimal mechanism, ( 1, r, ˆr, ), we now roceed under the assumtion that the valuation of the buyer is drawn from a uniform distribution: F (v) = v, v [, 1]. We say that the otimal eriod 1 mechanism is: (i) a osted rice without rationing whenever ˆr = ; (ii) a menu consisting of a higher rice without rationing and a lower rice with rationing whenever ˆr (, 1) and v < v < 1; (iii) a osted rice with rationing whenever ˆr (, 1) and v < v = 1. Proosition 1. The otimal eriod 1 mechanism is: (i) a osted rice without rationing if and only if: δ s 4+8δ b+δb δ3 b δ b ; +δb (ii) a menu consisting of a higher rice without rationing and a lower rice with rationing if and only if: δ b +6δ b 4 4(1 δ b ) 3 δ b < δ s < 4+8δ b+δb δ3 b δ b ; +δb (iii) a osted rice with rationing if and only if: δ s δ b +6δ b 4 4(1 δ b ) 3 δ b. Proof. The single-eriod monooly rice is m = 1, thus 1. From Lemma??, v =. Incentive comatibility imlies that r = v, and that: v 1 = r(v r ) + (1 r)δ b (v ) 1 = (1 δ b ) (1 r) v + rv. The resulting rofit of the seller can be written as: 1 (1 v) + r r (v v) + δ s (1 r)(v ) ) ] = [(1 δ b (1 r) v + rv (1 v) + rv(v v) + δ s (1 r) v 4 ) ( ) = (1 r) [(1 δ b v 1 δ b δs 4 v ] + r ( v v ). To check otimality of = v, it is sufficient to check that it is not rofitable to deviate to d = 1 [(1 r)v + rv], which is the single alternative satisfying the first-order condition: 15

16 (1 r)(v ) (1 r)(v v) d + (v d ) d 1 4 (1 r)v 1 4 [(1 r)v + rv] r (v v)v (v v). Our roblem is thus reduced to: { ) ( ) max (1 r) [(1 δ b v,v,r v 1 δ b δs 4 v ] + r ( v v )} s.t. r (v v)v. (v v) Since the objective function is an affine function of r, the otimum is attained either with r = or with r = (v v)v (v v). With r =, that is, with a osted rice in eriod 1, the maximum rofit is: attained with v = ) ( ) π P = max {(1 δ b v,v v 1 δ b δs 4 v } = ( δ b) 4(4 δ b δ, (4) s) δ b 4 δ b δ s. Since the value of v is irrelevant, we can ick v = v =, in which case r = = (v v)v (v v). Hence, the otimum is always attained with r = (v v)v (v v). Let x [, 1 π(x, v) = ] be the ratio between v and v. With r = (v v)v (v v) [ ] [( 1 1 x (1 x) = x v (1 x) [1 δ b ) ( 1 δ b v ( 1 δ b δs 4 1 δ b δs 4 ) ] v + xv(1 x) (1 x) = xv 4(1 x) [4 4x δ b x (8 8x δ b δ s )xv]. = 1 x, rofit is given by: (1 x) ) v ] + 1 x ( (1 x) xv x v ) (1 xv) Of course, setting x = 1, we obtain r = and the same exression for rofit as in (4). This is the rofit that results from an otimally chosen osted rice. Setting x = is never otimal because the resulting rofit is zero. For x (, 1 ), rofit π(x, ) is strictly concave as a function of v. The first-order condition with resect to v yields: [4 4x δ b x (8 8x δ b δ s )xv 1 ] xv 1 (8 8x δ b δ s ) = v 1 = x δ bx x(8 8x δ b δ s). 16

17 It is straightforward to check that v 1 is a strictly ositive, continuous, and decreasing function of x. Since v 1 > 1 for x and v 1 < 1 for x 1, there is a threshold value of x, x 1 = 1 δ b δ s (1 δ b δ s) (, ) 1, below which v1 > 1, and above which v 1 < 1. For x (, x 1 ], the first-order condition yields v 1 1, thus we have a corner solution, with v = 1 and rofit given by: π(x, 1) = x(4 1x+8x +δ sx) 4(1 x). For x ( x 1, 1 ), the first-order condition yields v1 < 1, thus we have an interior solution, with v = x δ bx x(8 8x δ b δ s) and rofit given by: π (x) = max v π(x, v) = ( x δ b x) 4(1 x) (8 8x δ b δ s). Attainable rofit can thus be written as a function of x, which is continuously differentiable (even at x 1 ): x(4 1x+8x +δ sx), if x (, x 4(1 x) π(x) = 1 ] ( x δ b x) 4(1 x) (8 8x δ b δ, if x ( (5) s) x 1, 1 ). The function in the second branch of (5), π (x), is an uer bound to the exression in the first branch of (5), π(x, 1). Therefore, if the maximum of π (x) is attained with x ( x 1, 1 ), it is the global solution. Maximizing π (x), the first-order condition yields three solutions: x a = +δ b, which is outside the domain; x b = 4 δ b+ 5δb δ3 b δ bδ s δb δs (+δ b ), which is also outside the domain because 4 δ b (+δ b ) 1 ; and x c = 4 δ b 5δb δ3 b δ bδ s δb δs (+δ b ), the lowest of the three, which may be inside the domain. Since π (x) is increasing at x =, x c is a local maximizer (whenever it is well-defined). If x c [ 1, + ) or if x c is not well-defined, the maximizer is x = 1, which means that the otimal mechanism is a osted rice. If x c ( x 1, 1 ), then it maximizes π(x). If xc (, x 1 ), the maximizer is in the first branch of (5). The third candidate (whenever it exists) belongs to [ 1, + ) if and only if: 4 δ b 5δb δ3 b δ bδ s δb δs (+δ b ) 1 δ b 5δb δ3 b δ bδ s δb δ s 4 8δ b δ b + δ3 b + δ bδ s + δ b δ s δ s 4+8δ b+δb δ3 b, (6) δ b +δb 17

18 which is surely the case if δ b 1.1 Observe that if δ s < 4+8δ b+δb δ3 b, then x δ b +δb c is well-defined and is lower than 1, which means that x c ɛ, for small ɛ > imroves on x c. It can be verified that x c (, x 1 ) if and only if: δ s < δ b + 6δ b 4 4(1 δ b ) 3 δ b. (7) We have, therefore, x c ( x 1, 1 ) if and only if: δ b + 6δ b 4 4(1 δ b ) 3 δ b < δ s < 4+8δb+δ b δ3 b. (8) δ b +δb When x c (, x 1 ), the maximum is in the first branch of (5). The exression of π(x, 1) does not deend on δ b. As a function of x, π(, 1), is concave in the relevant domain and increasing at x =. We are sure that the maximum is interior because: π(x, 1) x = dπ (x) x=x1 dx <. x=x1 1 Because the right-hand side of (6) is strictly ositive if and only if δ b > 1. 18

19 6 Additional buyers arriving in the second eriod We now consider that the seller faces a continuum of buyers that arrive in two eriods. In eriod 1, there is still a unit mass of buyers with valuations distributed according to F. But now, in eriod, there is an additional mass G(1) of buyers, with valuations distributed in v [, 1] according to a cumulative function G, with density g, which is also such that v G(1) G(v) g(v) and vg(v) G(1) G(v) are strictly increasing.13 In eriod, the seller cannot discriminate buyers as a function of the eriod in which they arrive. Otherwise, buyers arriving in eriod would be irrelevant for the choice of the mechanism to be roosed in eriod 1. The roblem of the seller becomes: { max { max t(v) dv + δ s [1 r(v)] dv + δ s r,t s.t. } g(v) dv v argmax {r(v )v t(v ) + 1 v δ b (v ) [1 r(v )]} v { } argmax [1 r(v)] dv + g(v) dv. (9) Observe that the objective function of the embedded roblem in (9) only differs from that in (1) by the revenue obtained from the buyers that arrive in the second eriod, which is a constant, that is, does not deend on (r, t). Looking at the restrictions in (9), while the truth-telling condition is unchanged relatively to those in (1), the sequential otimality condition also differs by a term that corresonds to the revenue obtained from the buyers that arrive in the second eriod. It is clear that Lemma 1 alies, because only the truth-telling condition is involved. Lemma also alies, once we include the above-mentioned extra terms terms. The 13 It should be clear that the cumulative distribution function is G(v) g(v) G(1), with density G(1). 19

20 embedded roblem in (9) is thus equivalent to: { [ max r(v) v r s.t. ] [ ] dv + (1 δ b ) r(v) v + δ b δ s 1 δ b dv + δ s [1 F ( )] + δ s [G(1) G( )]} r is weakly-increasing argmax { } [1 r(v)] dv + [G(1) G()]. (1) Finally, following the stes of the roof of Lemma 3, we obtain the same characterization we had obtained in the scenario where no additional buyers arrived in the second eriod. Lemma 6. The solution of roblem (1) satisfies: r(v) = 1, if v (v, 1) ˆr, if v (v, v), if v (, v), where v [, ], v [, 1], and ˆr [, 1). Proof. Assume, by way of contradiction, that r is a solution of roblem () that does not satisfy the restrictions of the lemma. Let D ( r) [1 r(v)] dv and D c ( r) [1 r(v)] dv. Construct r by letting: 1, if v (v, 1] r(v) = r( ), if v [v, v], if v [, v), where v is such that D c(r) [1 r(v)] dv = F (v) + v [1 r( )] dv = D c( r), and v is such that D (r) [1 r(v)] dv = [1 r( )] dv = D ( r). This means that the thresholds are such that the masses of buyers with valuations above and with

21 valuations below that trade in eriod 1 remain constant when r is relaced by r. Notice that v [, ] and that v [, 1]. Notice also that r( ) < 1, otherwise could never be otimal under r. Observe that r imroves on r in terms of the objective function of roblem (1), because the integrals in the objective function can be seen as averages of virtual utility, and r laces more weight on higher virtual utilities (and less weight on lower virtual utilities) than r. More formally, since [r(v) r(v)] dv = : = = + [ [r(v) r(v)] v ] dv [ [r(v) r(v)] v v + [ r(v) v v [ r( ) r(v)] v + ] dv ] dv + [ v v + ] dv, where non-negativity results from the argument of the first integral being the roduct of two non-ositive terms, while that of the second integral is the roduct of two non-negative terms. Using a similar argument, since [r(v) r(v)] dv = : [r(v) r(v)] [ ] v + δ b δ s 1 δ b dv. Since virtual utility, v set, the imrovement is strict., is strictly increasing and r differs from r in a non-negligible It is clear that r is non-increasing, therefore, to finish the roof, we only need to verify that r satisfies the sequential otimality condition of roblem (1). Setting yields the same rofit in eriod under r and under r. We will conclude the roof by showing that if, demand (and thus rofit) in eriod cannot be higher under r than under r. We can ignore buyers who arrive in eriod because, for any given, their demand is the same under r and under r. If > v, eriod demand from buyers who arrived in eriod 1 would be zero under r. 1

22 If (, v): [1 r(v)] dv = = = [1 r(v)] dv [1 r(v)] dv [1 r(v)] dv [1 r(v)] dv. [1 r(v)] dv [1 r(v)] dv [1 r(v)] dv Similarly, if (v, ): [1 r(v)] dv = = = [1 r(v)] dv + [1 r(v)] dv + [1 r(v)] dv + [1 r(v)] dv [1 r(v)] dv [1 r(v)] dv [1 r(v)] dv Finally, if (, v): [1 r(v)] dv = = = [1 r(v)] dv [1 r(v)] dv + [1 r(v)] dv [1 r(v)] dv [1 r(v)] dv [1 r(v)] dv [1 r(v)] dv

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