the role of the agent s outside options in principal-agent relationships

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1 the role of the agent s outside options in principal-agent relationships imran rasul y university college london silvia sonderegger z university of bristol and cmpo january 2009 Abstract We consider a principal-agent model of adverse selection where, in order to trade with the principal, the agent must undertake a relationship-speci c investment which a ects his outside option to trade, i.e. the payo that he can obtain by trading with an alternative principal. This creates a distinction between the agent s ex ante (before investment) and ex post (after investment) outside options to trade. We investigate the consequences of this distinction, and show that whenever an agent s ex ante and ex post outside options di er, this may equip the principal with an additional tool for screening among di erent agent types, by randomizing over the probability with which trade occurs once the agent has undertaken the investment. In turn, this may enhance the e ciency of the optimal second-best contract. JEL Classi cation: D21, L14. Keywords: adverse selection, randomization, type-dependent outside options. We thank the associate editor and two anonymous referees for suggestions that have helped improve the paper. We have also bene tted from the comments of Fabrizio Adriani, Roman Inderst, Ian Jewitt, Bruno Jullien, Thomas Mariotti, In-Uck Park, Francesco Squintani, Max Steuer, Thomas von Ungern-Sternberg and seminar participants at LSE and the University of Lausanne. All errors are our own. y Department of Economics, University College London, Drayton ouse, 30 Gordon Street, London WC1E 6BT, United Kingdom. Tel: ; i.rasul@ucl.ac.uk. z Department of Economics, University of Bristol, 8 Woodland Road, Bristol BS8 1TN, United Kingdom. Tel: ; s.sonderegger@bristol.ac.uk. 1

2 1 Introduction In many forms of bilateral exchange, one party has to undertake relationship-speci c investments before trade can occur with their partner. An important consequence of such speci c investments is that they typically change the investing party s outside option to trade, namely the payo that he would obtain by trading with an alternative partner. For example, a rm that tailors its machinery in order to produce a speci c widget required by a certain buyer, will change its production possibilities when trading with alternative buyers whose requirements need not be the same. 1 A key distinction therefore exists between the rm s ex ante outside option, before the relationshipspeci c investment is undertaken, and their ex post outside option, after the investment has occurred. This paper investigates the consequences of this distinction in principal-agent models of adverse selection, where the agent s type is his private information, and both parties are risk neutral. We show that whenever an agent s ex ante and ex post outside options di er, this may equip the principal with an additional tool for screening among di erent agent types, by randomizing over the probability with which trade occurs once the agent has undertaken the speci c investment. In turn, this may enhance the e ciency of the optimal second-best contracts. This paper contributes to the literature on mechanism design when agents have type-dependent outside options (Lewis and Sappington 1989, Maggi and Rodriguez-Clare 1995, Jullien 2000). The literature on adverse selection identi es several cases in which the optimal mechanism can involve randomization, such as when agents have di erent levels of risk aversion (Stiglitz 1982, Arnott and Stiglitz 1988, Brito et al 1995), when the agent s type-space is multi-dimensional (Baron and Myerson 1982, Rochet 1984 and Thanassoulis 2004), or when randomization might allow non-monotonic allocation schedules to become incentive compatible (Strausz 2006). A further rationale for randomization is presented by Calzolari and Pavan (2006), who show that, in principal-agent problems with sequential contracting, randomization may be optimal, since it allows one principal to hide information from another principal. We add to the literature by considering situations where relationship-speci c investments a ect the agent s future prospects, so that his type-dependent ex ante and ex post outside options di er. This provides a novel rationale of why randomization may be optimal in principal-agent settings. The remainder of the paper is organized as follows. In Section 2 we develop the principal-agent model. Section 3 solves for the optimal second best contracts. Section 4 discusses the e ciency consequences of having both types of outside option and also addresses possible extensions. All proofs and a numerical example are in the Appendix. 2 Model Preliminaries We consider a principal-agent model with a principal P and an agent A, who contract over the production of output, q. Production is assumed to be observable and veri able. The agent s marginal cost of production,, which de nes his type, is not observed by the principal, and we assume 1 This phenomenon is not con ned to bilateral exchange between rms. Consider a traveller who wants to travel from A to B at 8pm on a given day. The traveller can choose whether to travel by train or bus. The speci c investment undertaken by the traveller in order to access a certain type of travel takes the form of him being physically present at a particular location the bus or train station at a particular time. While from an ex ante perspective the traveller s outside option to catching the 8pm bus would be to take the 8pm train, once he has made the speci c investment of arriving at the bus station prior to 8pm, his ex post outside option to catching the 8pm bus will be quite di erent. While he may for example catch the 9pm train, the 8pm train has been ruled infeasible by his earlier speci c investment. 2

3 2 f, L g, where > L > 0, and prob( = ) =. In order to trade with the principal, the agent must undertake a relationship-speci c investment, with cost normalized to zero. The agent s decision to undertake the investment is observable and veri able. A contract between the principal and the agent is denoted f; ; q; T g, where 2 f0; 1g speci es whether the agent must undertake the investment 2, 2 [0; 1] denotes the probability with which trade occurs between the parties, q 2 [0; q] denotes the output that the agent must produce in case of trade, and T 2 R + indicates the payment from the principal to the agent (independent of whether trade actually occurs or not). We assume trade can only occur if the agent has made the relationship-speci c investment so that if = 0, = 0. 3 The principal s problem consists of designing the optimal menu of contracts from which the agent makes his preferred choice. The revelation principle states this search can be con ned to the set of direct revelation mechanisms, whereby the agent is requested to report his type and is o ered a contract that is contingent upon this report. The timing of actions is then as follows. t=0 P o ers A a menu of contracts M = fm ; M L g, where M i = f i ; i ; q i ; T i g is the contract o ered to the agent when his reported type is i, i = ; L. t=0.5 If A accepts M i and M i speci es i = 1, A undertakes the relationship-speci c investment. t=1 Conditional on i = 1, trade occurs with probability i, in which case A produces q i. With probability 1 occur with certainty. i trade between A and P does not occur. If i = 0, trade between A and P does not t=1.5 Provided that he has respected the terms of the contract, A receives T i. Without loss of generality we restrict attention to contracts that always induce truthtelling and participation by the agent. Agent s Ex ante and Ex post Outside Options If the agent does not accept the principal s contract, or if his contract prescribes i = 0, then the agent does not undertake any relationship-speci c investment, and obtains a payo B i 0 from alternative trade, where i = ; L. This de nes the agent s ex ante outside option. Importantly, we allow for the possibility that ex ante outside options di er across types, so that B 6= B L. If the agent undertakes the relationship-speci c investment, but trade between the parties does not occur, then the agent obtains a payo C i < B i from alternative trade. C i captures the agent s ex post outside option, namely the value of his trading prospects with alternative principals, after having undertaken the relationship-speci c investment with the previous principal. Ex post outside options may also be type-dependent, so that C 6= C L. The expression B i C i > 0 re ects the loss in terms of the agent s alternative trading prospects from undertaking the relationship-speci c investment, which tailors his production to the principal s needs. We refer to this as the opportunity cost of randomization, since this cost is only incurred when i < 1. Payo s Both parties are assumed to be risk neutral with respect to monetary transfers and production. If a type i agent accepts a contract f i ; i ; q i ; T i g, his net expected utility is, u ( i ) = T i + i f i i q i + (1 i )C i B i g : (1) 2 Allowing the contract to specify enables us to restrict attention to contracts that are always accepted by the agent. We thank an anonymous referee for providing this suggestion. 3 By restricting attention to 2 f0; 1g we rule out the possibility of the principal randomizing over. This is done to shorten the exposition of our results. The possibility of randomization over is discussed in section 4. 3

4 The principal s expected payo is U P = i i vq i T i, where v >. Let u i denote the utility obtained by a type i agent when he truthfully declares his type. From (1), holding constant all other dimensions of the contract o ered to type i, there is a one-to-one relation between T i and u i. In what follows we will therefore characterize a contract as M i = f i ; i ; q i ; u i g. Finally, we denote L as, C C L as C, B B L as B and u u L as u. 3 Results The participation constraint for a type i agent is u i = T i + i [ i i q i + (1 i )C i B i ] 0. The incentive compatibility constraints which ensure agents nd it optimal to declare their true type are, IC : u u L + L [ L q L + (1 L )C B] : IC L : u L u + [ q (1 )C + B] : Suppose full information contracts are o ered so that i = i = 1, q i = q, and u i = 0 for i = ; L. Constraint IC becomes, 0 q B, and IC L becomes, 0 q + B. We focus on the more intuitive case in which q + B > 0 so that L types have incentives to overstate their costs and mimic types. This is embodied in assumption A1 below. 4 To ensure that under full information the optimal contract prescribes i = i = 1, q i = q for both types, assumption A2 below is required. A1: q + B > 0 A2: q (v i ) B i ; i = ; L Our rst result provides a partial characterization of type s optimal contract whenever agents are required to undertake the relationship-speci c investment. Lemma 1: It is never optimal for the principal to o er = 1 in conjunction with and q satisfying, q (1 )C + B < 0: (2) Under A1 the full information contracts would violate IC L. By o ering type agents a contract such that q (1 )C + B = 0, the principal ensures both that IC L is satis ed and that no rents are o ered to L agents. O ering agents a contract such that (2) holds would only increase the distortions of and/or q from their full information values (1 and q respectively) without generating any gain for the principal. This is essentially the rationale for Lemma 1. An implication of Lemma 1 is that the participation constraint of type L will not bind at the optimum. This is because, given type s participation, IC L implies u L u 0. In what follows, we therefore allow IC L to hold with equality, let u = 0, and ignore constraint IC. We then later verify that the solution of the relaxed problem indeed satis es IC. The principal s problem then is, max U P = q i 2[0;q], i 2[0;1], i 2f0;1g, i=;l [ q (v ) + (1 )C B ] + (1 ) L [ L q L ( ) + (1 L )C L B L ] (1 ) [ q (1 )C + B] (P) 4 For completeness, in the Appendix, we state the main results for the case in which the parameter values are such that high types have incentives to understate their type and mimic low cost types. These two cases arise because of the existence of the type-dependent ex ante outside options, B i; as has been analyzed in detail by Maggi and Rodriguez-Clare (1995). Note that in the knife-edge case where q + B = 0 the principal can o er the full information contract to both types without inducing either to mimic the other, so this is clearly her favored course of action. 4

5 subject to [ q (1 )C + B] 0: (C1) where (C1) derives from Lemma 1. We rst solve (P) ignoring (C1). If the solution satis es (C1) with strict inequality, it is the solution to the overall problem. Otherwise (C1) binds. The principal faces a standard trade-o between e ciency and informational rents. If she o ers types the e cient (full-information) contract where = = 1, q = q, then she must also o er positive rents to L types to prevent mimicking. In this case (C1) is slack. If the principal wishes to eliminate L s rents, then she must distort type s contract away from the e cient contract. 5 In this case (C1) binds so, conditional on = 1, we have, q = (1 )C B : (3) When C B > 0 i.e., the opportunity cost of randomization is higher for L than for then (3) implies =@ < 0. By lowering the principal can increase q whilst keeping L s rents at zero. A trade-o then emerges. A lower decreases the probability of trade, but it also increases q, and hence the value of trade. When the latter e ect is stronger than the former, the optimal contract (conditional on C1 binding) prescribes q = q and = (C B) = (q + C) 2 (0; 1), i.e. it prescribes randomization. 6 Proposition 1 fully describes the optimal second best contracts. 7 Proposition 1: For type L, the optimal contract always prescribes L = L = 1, q L = q. If > max C + q q ( ) C L ; ; B + q q ( ) B L ; (4) then ( C1) is slack, and the optimal contract for type has = = 1, q = q. If (4) does not hold, then ( C1) binds, and the optimal contract for type is (i) if C > C L(v ) and C B > (B C )(q+c) q(v ) C > 0: = 1, = (ii) if C < C L(v ) and B < B v < 0: = = 1 and q = B. (iii) in all the other cases: = 0. C B q+c and q = q: If prob( = ) = is su ciently high, then the principal nds it optimal to o er types the e cient contract, so as to maximize her pro t when trading with types, even if this implies that positive rents are relinquished to agents of type L. Conversely, if is su ciently low, then the principal prefers to allow (C1) to bind and so eliminate any rents to L types. Proposition 1 shows that in order for the optimal contract for to prescribe randomization, C should be positive, and su ciently large. Intuitively, C B B > 0 implies that the opportunity cost of randomization is higher for L types than for. ence, by o ering types a contract involving randomization, the principal can lower the incentives of L types to overstate their costs and mimic types. By contrast, if C B < 0, then types stand to lose more from randomization than L types, and so randomization would not help deter L from mimicking. Similarly, if C = B = 0 as is the case if both ex ante and ex post outside options are type-invariant so B = B L and C = C L then the opportunity cost of randomization is the same for both types, and again randomization is not an 5 Given the linearity of her payo, the principal would never select contracts between these extremes. 6 That (C B) = (q + C) 2 (0; 1) follows from C B > 0 and assumption A1. 7 We adopt the convention that if P is indi erent between setting i = 1 or i = 0 for i = ; L, then she selects i = 0. Similarly, if she is indi erent between all i 2 [0; 1] (resp., all q i 2 [0; q]) then P selects i = 0 (resp., q i = 0). 5

6 e ective screening tool. This clari es why type-dependent outside options are essential for randomization to be optimal. Note that, in order for randomization to be optimal, C B should not only be positive, but also su ciently large. This ensures that a small amount of randomization in the contract o ered to is su cient to deter L from mimicking, and guarantees that the principal can obtain a positive expected pro t when trading with type. What are the implications of C B > 0? From (3), we know that when (C1) binds and C B > 0 then a trade-o emerges between and q. A lower decreases the probability of trade, but it also increases q, and hence the value of trade. For randomization to be optimal, the principal must then be willing to lower the probability of trade with in order to raise q at the margin. Whether this occurs or not, depends on the precise comparison between the costs (i.e., trade with occurs less often) and the bene ts (i.e., q is higher) of randomization. To see how the former may outweigh the latter, consider the simple case where C L < C < 0, B = B L = 0. Since ex ante outside options are independent of type, if = 1, this case corresponds to the canonical model. As can be seen from (C1), leaving no rents to type L then requires q = 0. The principal s payo when dealing with type is then equal to zero. By contrast, setting < 1 allows the principal to set q = (1 ) C= > 0. ere, the cost of imposing randomization is null, since when = 1 trading with type generates no pro ts (this follows from q = 0). By contrast, if C is not too negative, i.e. C > C L (v )= ( ), then the bene t of randomization is strictly positive, since it allows the principal to obtain a strictly positive expected payo when dealing with. 8 Conditional on the principal wishing to leave no rents to type L (which, as highlighted by proposition 1, happens whenever is su ciently low), randomization is then clearly optimal. Given the linearity of her payo, if the principal nds it optimal to sacri ce in order to raise q at the margin, then she goes all the way, and sets q as high as possible in the optimal contract, i.e. q = q. From (3), is then equal to (C B) = (q + C). A Numerical Example In the Appendix, we discuss a numerical example where = 0:75, L = 0:25, q = v = 2, and the agent s ex ante and ex post outside options are B = 1:85, B L = 2:35, C = 1:75, and C L = 1:95. In that case, it is straightforward to show that, for < 0:52, the optimal contract o ered to type prescribes = 1; = 0:375 and q = q = 2: 4 Discussion E ciency Proposition 1 highlights the impact of having two (i.e., ex ante and ex post) typedependent outside options on the optimal second best contracts. Suppose that, on the contrary, C i = B i for both i = ; L, so C = B. From (3), the only way for (C1) to then bind is to set q = B=. If (4) does not hold and B B = (v ), then the optimal contract prescribes = 0, i.e. no trade between the principal and agents of type, since with q = B= the principal would never obtain a non-negative pro t when dealing with type. In contrast when B i 6= C i, trade between the principal and agents of type may occur with positive probability even if B B = (v ). 9 8 If C is very low (negative), then the transfer necessary to induce type to accept a contract involving randomization would be large, and randomization is therefore not optimal. 9 This is the case for instance in the numerical example introduced above, where B = 0:5 > B = (v ) = 0:74. 6

7 ence, in a complete contracting environment, the need for agents to undertake relationship-speci c investments ex ante that decrease the agent s outside option, can result in greater ex post e ciency, that is, at the production stage. This is because such investments enable the principal to utilize randomization as a tool to screen between agent types. To our knowledge, the earlier literature has not noted this potentially useful role for ex ante relationship-speci c investments to improve on ex post e ciency. The literature has emphasized rather, that in the presence of contractual incompleteness, investment speci city results in ex ante ine ciencies, i.e. ine ciencies at the investment stage (Grout 1984, Grossman and art 1986, art and Moore 1990). Relaxing the Linearity Assumption The restriction to linear payo functions allows us to abstract from risk-aversion considerations, and to di erentiate our results from the existing literature on randomization in mechanism design (Stiglitz 1982, Arnott and Stiglitz 1988, Brito et al 1995). owever, our results extend also to non-linear settings. To see this, suppose agents face convex production costs, so the net utility of an type i agent when accepting a contract f; ; q; T g is, T + [ i g(q) + (1 )C i B i ] : (5) where g 0 (q) > 0 and g 00 (q) > 0 for all q > 0. Suppose that, if o ered the full-information contract, a type L agent would overstate his cost and mimic type, as was the case throughout Section 3. Condition (C1) then is, [ g (q ) (1 )C + B] 0: (C1 0 ) Following the same argument as in Proposition 1, for su ciently low, the optimal contract for type agents is such that C1 0 binds. Then, conditional on = 1, we have, [(1 q = g 1 )C B] : (6) As in the linear case, when C B > 0, (6) implies =@ < 0, so that, by lowering, the principal can increase q. Clearly, this is a necessary requirement for randomization to be o ered, or else the principal would always optimally select = 1. Note that, in contrast with the linear case above, in this non-linear case the trade-o between and q may actually make U P concave in thus warranting randomization even in the absence of restrictions on feasible output quantities. To see this, let g(q) = 0:5q 2, and suppose that parameter values continue to follow the numerical example given above, but the restriction that q may not exceed q is relaxed. q Expression (6) then becomes q = 0:8 + 1:2. Conditional on = 1, the principal s expected q i payo when dealing with a type agent is U P = h2 0:8 + 1:2 0:75 0:4 + 0:6 1:75 0:1, which is concave in. The optimal contract for is = 1, = 0:78; and q = 1:53, and when dealing with type, the principal s expected pro t is 0:23. ence in this numerical example, for su ciently low the optimal contract for may again prescribe randomization, although in contrast with the linear case, the optimal q is below its rst-best value. Allowing for Randomization Over What would happen if instead of assuming 2 f0; 1g, were allowed to take any value in [0; 1]? In that case, randomization over would be possible. 7

8 owever, we argue that it would not be optimal. 10 To see why, note that, similar to what happens for randomization over, randomization over may only be optimal when (C1) binds enters the principal s payo linearly, so if (C1) is slack then optimally takes a corner value. Consider now a contract M = f ; ; q ; u g, where 2 (0; 1) and u = 0 (as discussed above, this latter condition is always satis ed at an optimum). Under this contract, the opportunity cost of randomization over for an agent of type i is equal to i q + (1 )C i B i, namely the expected net payo that the agent obtains when he undertakes the relationship-speci c investment. 11 Following the same logic as in the case of randomization over, if this opportunity cost di ers between types, then this may provide a possible rationale for randomization over to be optimal. owever, if (C1) binds, then the net payo that the agent obtains when he undertakes the relationship-speci c investment under contract M is zero for both L and. 12 ence, the opportunity cost of randomization over is equal (and null) for both types. Randomization over is therefore ine ective for screening between types. 5 Appendix 5.1 Proofs Proof of Lemma 1: We show that any menu of contracts in which = 1 and (2) holds is necessarily dominated, as P could o er a menu that, whilst violating (2), satis es both IC and IC L and yields him a strictly higher expected payo. Consider a menu M = fm ; M L g = f( ; ; q ; u ) ; ( L ; L ; q L ; u L )g such that = 1 and (2) holds. P s expected payo from M is, f [q (v ) C ] + C B u g + (1 ) L f L [q L ( ) C L ] + C L B L u L g : (7) Now consider an alternative menu c M = Under A1, c M satis es IC. It also satis es IC L provided, n cm ; c M L o, where c M = (1; b ; bq ; 0) and c M L = (1; 1; q; 0). b bq (1 b )C + B 0 (8) We now show that there exist values of b and bq which satisfy (8) with equality (i.e., violate (2)) and which are such that c M yields P a greater expected payo than M. P s expected payo from c M is, fb [bq (v ) C ] + C B g + (1 ) [q( ) B L ] : (9) A su cient condition for (9) to exceed (7) is, b [bq (v ) C ] [q (v ) C ] > 0: (10) Condition (10) ensures that P prefers M c to M. We distinguish between two cases. First, suppose that (1 )C B q. ence setting b = and bq = (1 )C B ensures (8) holds with equality. Contract M c = 1; ; (1 )C B ; 0 is feasible because, if (2) holds, then q < (1 )C B which 10 It is straightforward to see that in our framework randomization over L is also never optimal. 11 Recall that in our framework the relationship-speci c investment is necessary for trade between the principal and the agent to occur. Given a contract f, ; q; T g if the agent undertakes the relationship-speci c investment, his expected utility is T iq + (1 )C i, while if the agent does not undertake the relationship-speci c investment, his utility is T + B i: 12 This follows since, when (C1) binds, then q L + (1 )C L B L = q + (1 )C B = 0: 8

9 implies (1 )C B > 0. With b = the LS of (10) is (bq q ) (v ), which is strictly positive. ence, M c dominates M and so M c dominates M. Second, suppose (1 )C B B > q. Note that since q+b > 0 under A1, < q < (1 )C B, so C B > 0. There are then two possibilities to consider. C B In the rst case, q + C > 0. Inequality (2) can be rewritten as < q +C. By setting b = C B q +C, bq = q we ensure (8) holds with equality. The LS of (10) becomes (b ) [q (v ) C ], which is strictly positive. ence, M c C B = 1; q +C ; q ; 0 dominates M and so M c dominates M. In the second case, q + C 0. For this to hold, we require C < 0. As C B > 0, this implies B B < 0. By setting b = 1, bq = we ensure (8) holds with equality. The LS of (10) becomes B (v ) C [q (v ) C ]. Under (2), a su cient condition for this to be positive is that, C ( ) C L (v ) < 0: (11) Note however that as q + C 0 in this second case, if (11) does not hold then contract M is dominated by a contract that sets = 0. To see this, note that, by setting = 1, the extra pro t obtained by the principal is non-negative only if q u +B C (1 ) (v ). For this to be consistent with q + C 0 it is necessarily required that B C (1 ) (v ) C. In turn, this requires C ( ) C L (v ) < 0. We therefore conclude that contract M is surely dominated. Proof of Proposition 1: We divide the proof in two parts. We rst consider the case where (C1) is slack in the optimal contract. We then consider the case where (C1) binds in the optimal contract. First consider the solution of (P) ignoring (C1). It is straightforward to see the optimal M L prescribes L = L = 1, q L = q. Consider now the optimal M. Lemma 2: If condition (C1) is slack in the optimal contract, then the optimal M prescribes = = 1, q = q. Proof of Lemma 2: Ignoring (C1), the FOCs for M = q [ (v ) (1 ) ] [C + (1 ) C] = [ (v ) (1 ) ] = [ q (v ) + (1 )C B ] (1 ) [ q (1 )C + B] (14) For condition (C1) to be slack, it is necessary that = 1. ence, the LS of (14) must be positive. 13 This has implications for the optimal values of and q. If the optimal in the unconstrained problem is zero, to then have = 1 requires (C B ) (1 ) (B C) > 0. Since C B < 0, a necessary condition for this is that B C < 0. owever, when = 1 and = 0; this requirement would contradict (C1). Similarly, if the optimal q in the unconstrained problem is zero, to then have = 1 requires [(1 )C B ] (1 ) [B (1 ) C] > 0. For this to hold it is necessary that B (1 ) C < 0. owever, when = 1 and q = 0, this requirement would again contradict (C1). ence, if (C1) is slack in the optimal contract, then the optimal and the optimal q must both be strictly positive. As mentioned earlier, we adopt the convention that, if indi erent between all possible values of q 2 [0; q], the principal will select q = 0, and, similarly, if indi erent between all possible values of 2 [0; 1], the principal will select = 0. Since, as proved above, q = 0 and/or 13 Recall that, as mentioned in footnote 7, if indi erent between = 1 and = 0, P will select = 0. 9

10 = 0 are not consistent with = 1, we conclude that if (C1) is slack in the optimal contract, then the optimal q must be = q, and the optimal must be = 1. When = = 1, q = q, (C1) becomes q + B 0, which is satis ed with strict inequality by A1. It remains to verify consistency; given = = 1, q = q, the LS of (12), (13) and (14) must be strictly positive, to ensure that = = 1, q = q is indeed optimal. It is straightforward to show that this happens whenever, > max C + q q ( ) C L ; ; B + q q ( ) B L : (4) ence, when (4) holds, the optimal contracts prescribe i = i = 1, q i = q for both i = L;. Moreover, u = 0, while u L = q + B (it is straightforward to check that this satis es IC ). Condition (C1) is slack. This establishes the rst part of the proof of proposition 1. Second part: When (C1) binds, q = (1 )C B, and P s expected payo is, 14 (1 )C B U P = (v ) C + C B + (15) (1 ) L [ L q L ( ) + (1 L )C L B L ] : It is straightforward to see the optimal M L in this case also prescribes L = L = 1, q L = q. The optimal M maximizes (15) subject to q 2 [0; q]. The FOCs P C(v ) C = [C L (v ) C ( )] P (1 )C B = (v ) C + C B Two cases can arise. 15 In the rst C L (v ) C ( ) < 0, so conditional on = < 0 and P sets as low as possible. If C > < 0 and the lowest feasible solves q = (1 )C B C B, so = q+c is 1. If C < then when = 1 (C1) may only bind if q = 0. owever, = 0 is preferred by P in this case, since C. Provided B q+c (q(v ) C ) + C B > 0, the optimal > 0 and the lowest feasible solves q = 0. Similarly, if C = B it allows P to avoid having to pay a positive transfer to type to induce his participation. In the second case, C L (v ) C ( ) > 0, so conditional on = > 0 and P sets as high as possible. If B < 0, then = 1 and q = B. Provided B (v ) B > 0, it is then optimal to set = 1. If B > 0, it is then optimal to set = 0 as this is the only way to ensure (C1) binds. To see this, note that we can only be in the case C L (v ) C ( ) > 0 if C < 0 so that, if B > 0, then C < B. This implies (1 )C B < 0 for all, and therefore (C1) never binds unless = 0. Similarly, if B = 0, then when = 1 (C1) may only bind if q = 0. owever, as argued above, = 0 is then preferred by P. To complete the description of the optimal contracts, note that when (C1) binds the optimal contracts prescribe u = u L = 0. It is straightforward to check IC is satis ed in all the cases we have identi ed. 14 More precisely, q = (1 )C B must necessarily hold if = 1 and (C1) binds. If = 0, then clearly the value of q is irrelevant. The expression (15) captures P 0 s expected payo in both cases. 15 The knife-edge case where C L(v ) C () = 0 is ignored for brevity. 10

11 5.2 Assumption A1 Does Not old For completeness, we consider the case in which 0 q + B and so types have incentives to understate their costs and mimic L types. The remaining assumption A2 is assumed to still hold. The counterparts for the main results are as follows, Lemma 1B: It is never optimal for the principal to o er L = 1 in conjunction with L and q L satisfying, L q L + (1 L )C B < 0: (18) An implication is that the participation constraint of type will not bind at the optimum. The optimal contracts are now found by letting IC hold with equality, setting u L = 0, and ignoring IC L. The counterpart to (C1) is, L [ L q L + (1 L )C B] 0: (C1B) Proposition 2B: For type, the optimal contract always prescribes = = 1, q = q. If q (v L ) C L < min ; ; q (v L) B L q (v ) C v q (v ) B then ( C1B) is slack, and the optimal contract for type L has L = L = 1, q L = q. If (19) doesn t hold, then (C1B) binds, and the optimal contract for type L is, (i) if C < C L(v ) and C B < (B L C L )(q+c) q( ) C L < 0: L = 1, L = (ii) if C > C L(v ) and B < B < 0: L = L = 1 and q L = B. (iii) in all the other cases: L = A Numerical Example C B q+c and q L = q: Suppose = 0:75, L = 0:25, q = v = 2, and agent s ex ante and ex post outside options are B = 1:85, B L = 2:35, C = 1:75, and C L = 1:95. For (4) to hold we require 0:52. If < 0:52, then (C1) must bind in the optimal contract. From (3), if = 1 this implies q = , and to ensure q q = 2, we require 0:375. Conditional on = 1, the principal then selects 2 [0:375; 1] ito maximize her expected payo when dealing with a type agent, U P = h2= :25 1:75 0:1. Since U P is decreasing in, so the principal selects the lowest compatible with (C1). The optimal contract for then is, = 1; = 0:375; q = q = 2, and when dealing with type agents, the principal s expected payo is 0:18. (19) References [1] arnott.r and j.e.stiglitz (1988) Randomization with Asymmetric Information, Rand Journal of Economics 19: [2] baron.d.p and r.b.myerson (1982) Regulating a Monopolist with Unknown Costs, Econometrica 50: [3] brito.d.l, j.h.hamilton, s.m.slutzky, and j.e.stiglitz (1995) Randomization in Optimal Income Tax Schedules, Journal of Public Economics 56:

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