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Contracts with Endogenous Information No 780 WARWICK ECONOMIC RESEARCH PAPERS DEPARTMENT OF ECONOMICS Contracts with Endogenous Information Dezsö Szalayy University of Warwick June 23, 2005, revised: December 14, 2006 This paper is a much revised version of a chapter of my doctoral thesis submitted 2001 in Mannheim. I thank Jacques Crémer, Bhaskar Dutta, Martin Hellwig, Philippe Jehiel, Ian Jewitt, James Malcomson, Meg Meyer, Benny Moldovanu, Marco Ottaviani, Tiago Ribeiro, Peter Sorensen, Thomas von Ungern-Sternberg, Juuso Välimäki, seminar participants at ESSET 2004, an anonymous associate editor and two referees for their valuable comments. All errors are my own. y Department of Economics, University of Warwick, Dezso.Szalay@warwick.ac.uk 1 Gibbet Hill Road, CV4 7AL, Coventry UK, Abstract I study covert information acquisition and reporting in a principal agent problem allowing for general technologies of information acquisition. When posteriors satisfy local versions of the standard First Order Stochastic Dominance and Concavity/Convexity of the Distribution Function conditions, a …rst-order approach is justi…ed. Under the same conditions, informativeness and riskiness of reports are equivalent. High powered contracts, that make the agent’s informational rents more risky, are used to increase incentives for information acquisition, insensitive contracts are used to reduce incentives for information gathering. The value of information to the agent is always positive. The value of information to the principal is ambiguous. JEL Classi…cation: D82, D83, L51 Keywords: Asymmetric Information, Mechanism Design, Information Acquisition, Stochastic Ordering, Value of Information 2 1 Introduction A vast literature on contracting and mechanism design has investigated the consequences of asymmetric information on the e¢ ciency and distributive properties of allocations. In most of this literature the model’s primitive is an information structure. However, in some economic problems it is reasonable to assume that economic agents do only possess information because they expect to make use of it. Moreover, their e¤ort to gather information is often unobservable to others. Thus, an information acquisition technology rather than the information structure itself should be taken as the model’s primitive, and contracts serve the double role of motivating the acquisition of information and ensuring its truthful revelation. How does this second role a¤ect the nature of optimal screening contracts? Since Demski and Sappington (1987) have raised this question, many investigations have followed. Notably, a prominent literature has investigated how optimal supply arrangements in procurement should be changed to account for costs of acquiring information about cost-of-production conditions (see, e.g., Crémer and Khalil (1992), Crémer, Khalil, and Rochet (1998a,b), Lewis and Sappington (1997), Sobel (1993), and La¤ont and Martimort (2002) for a survey of these models). More recently, I myself (Szalay (2005)) have analyzed how decision-making in an advisor-advisee relationship should be structured to guarantee high quality advice. The …ndings of this literature are as follows. If the buyer in the procurement context wants to make sure the seller is well informed, then he should o¤er “high powered” incentive contracts. Compared to a supply arrangement with a seller who is already well informed about his costs, the seller will bene…t from an unusually high order if his marginal costs are lower than expected, but he will also receive an exceptionally low order if his costs are higher than ex ante expected. As a result, the quantity supplied is discontinuous and drops sharply when the seller’s cost is higher than ex ante expected. If the buyer does not want the seller to become informed, then the supply arrangement should be rigid and should make little use of the seller’s information. Both cases can occur, depending on the cost of information acquisition and the timing of events.1 The structure of decision-making in Szalay’s (2005) model of advice displays an exaggeration property that is akin to a high powered incentive contract. If the advisor recommends an action that is higher than the 1 This result depends on the absence of competition. Compte and Jehiel (2002) reinvestigate the case studied by Crémer et al. (1998b) allowing many agents to compete. While Crémer et al. (1998b) showed that information acquisition is socially wasteful, Compte and Jehiel (2002) show that it may become desirable again when agents compete. 3 ex ante expected action then the advisee takes an action that is even higher than the recommended one; if the advisor’s proposed action is lower than the ex ante expected one, then the advisee takes an even lower action. Similar to the procurement case, the decision schedule is discontinuous and increases sharply at the prior mean. Information acquisition in all these papers is of an all-or-nothing nature, where the person who acquires information is in equilibrium either completely informed or does not receive additional information at all. I raise a simple question: how do the insights of this literature depend on this simpli…cation? I …nd that super powered incentive contracts and exaggeration are general features of contracts with endogenous information, discontinuities are not. To demonstrate these …ndings, I develop a general but still tractable model of information acquisition. Since the techniques I use can be applied to a wide class of problems with endogenous information, the model is of interest well beyond the context of procurement and the speci…c question I raise. I study the procurement problem that Crémer et al. (1998a) have analyzed. A buyer wishes to obtain parts from a seller. Neither the seller nor the buyer knows ex ante how costly it is to produce these parts, say because they both engage in this particular kind of activity for the …rst time. The buyer begins by o¤ering a menu of contracts to the seller. Before the seller has to accept or reject o¤ers he can acquire information about his costs. In contrast to Crémer et al. (1998a), the seller can exert a continuous choice of e¤ort and receive a continuum of noisy signals. An increase in the seller’s e¤ort improves the quality of the signal he receives stochastically. Both the seller’s choice of e¤ort and the signal he receives are known only to him but not to the buyer. After the seller has observed a signal he either accepts one of the contracts or walks away without further sanction. The seller learns the true cost of production only when he produces. Allowing for a continuous quality of noisy information introduces considerable technical di¢ culties, and one of the contributions of this paper is to demonstrate an elegant way over these hurdles. A rich model of information acquisition leads naturally into a problem of multi-dimensional screening. Ex post, when the seller has acquired a noisy signal, his entire posterior, a multi-dimensional object, may be relevant for contracting. Thus, the buyer faces a problem of multi-dimensional screening, which is potentially quite nasty to solve2 . However, when the seller’s utility is linear in his information variable (e.g., his constant marginal costs), then the seller’s preference over 2 See McAfee and McMillan (1988) for a screening problem where types have more dimensions than the principal has screening instruments available. See also Armstrong and Rochet (1999) and Rochet and Stole (2003) for overviews of multidimensional screening problems. 4 contracts depends e¤ectively only on the mean calculated from the posterior distribution. Since this is a one dimensional statistic, the problem at the reporting stage is reduced to the well known one-dimensional screening problem. To understand the seller’s ex ante problem of how much e¤ort to invest in information acquisition, one has to study the dependence of the ex ante distribution of the conditional expectation on e¤ort. One can resort to standard di¤erentiability methods to describe the optimal amount of e¤ort spent on information acquisition only if the seller’s e¤ort in‡uences the ex ante distribution of the conditional mean in a particular way. The seller’s optimal choice of information acquisition is adequately described by a …rst-order condition for any contract that ensures truthful communication of information, if and only if the seller’s e¤ort increases the riskiness of the ex ante distribution of the posterior expectation at a decreasing rate, where riskiness is understood in the sense of Rothschild and Stiglitz (1970).3 The second contribution of this paper is to provide statistical foundations for increasing risk in the distribution of conditional expectation in terms of the primitives of the experiment structures. I obtain an in‡uence of the desired sort when I impose two assumptions. First, the marginal distributions of signals and true costs are given and the sellers e¤ort in‡uences only the joint distribution of these two variables.4 Second, an increase in e¤ort increases the posterior for signals above the prior expected signal value in the sense of First Order Stochastic Dominance (FOSD), and the posterior satis…es the Convexity of the Distribution Function Condition (CDFC). For signals below the prior expected signal, an increase in e¤ort decreases the posterior in the sense of FOSD and the posterior satis…es the Concavity of the Distribution Function Condition. It is interesting to contrast these conditions with those used to justify the traditional “…rst-order approach” in problems of pure moral hazard (Rogerson (1985) and Jewitt (1988)). My conditions are local versions of the standard FOSD and CDFC conditions. I impose local rather than the usual global conditions, because the latter imply changing means (Milgrom (1981)), which is a rather undesirable feature of a model of information acquisition; the law of iterated expectations requires that the means be independent of the amount of information acquisition. My conditions 3 Note that this notion of riskiness is somewhat di¤erent from Blackwell’s, which states that one information structure is Blackwell-better than another if it gives rise to a more risky distribution of the posterior. Riskiness of the posterior expectation is a less restricting condition. Heuristically, while Blackwell requires the distribution of all moments to be more risky, the present concept requires only that the distribution of the …rst moment is more risky. The di¤erence arises because I impose restrictions on the seller’s utility function, while Blackwell’s criterion orders information structures for all decision makers whose utility function belongs to a class. For more recent approaches that order information structures, see Karlin and Rubin (1956), Lehmann (1988), and Athey and Levin (2001). 4 A statistical structure of essentially this type is called a copula (see, e.g. Nelsen (2006))). 5 are less restrictive than the ones used to justify the traditional …rst order approach. In problems of pure moral hazard one has to ensure the monotonicity of contracts by imposing in addition the Monotone Likelihood Ratio Property (MLRP), which makes the speci…cation overall rather restrictive. In contrast, there is no need to ensure the monotonicity of contracts when there is adverse selection, because monotonicity of contracts is a necessary condition for implementability (Guesnerie and La¤ont (1984)). Therefore, it is fair to say that the …rst-order approach goes through more easily than in a problem of pure moral hazard. A second statistical model that delivers the same reduced form is a stochastic experiment structure that is similar in nature to the spanning condition studied in Grossman and Hart (1983). In that speci…cation, an experiment is the realization of two independent random variables; a signal which follows a given marginal distribution and an informativeness parameter whose distribution depends on the agent’s e¤ort. The posterior satis…es a local version of MLRP; for signals above the mean, a posterior arising from a relatively more informative experiment places relatively more weight on the high realization of costs, for signals below the mean, it places relatively more weight on the low realizations of costs. Finally, an increase in e¤ort makes it more likely to observe a more informative experiment in the sense of FOSD, and the distribution of informativeness satis…es a CDFC condition. The main insight arising from this analysis is that informativeness and risk are equivalent in any tractable model. It is in fact this equivalence result that explains the …ndings of the literature on the value of information and the structure of optimal contracts. The value of information depends on the seller’s and the buyer’s attitudes towards risk, that is, the shape of their indirect utility functions. It is well known that only convex indirect utility pro…les of the seller are implementable (see Rochet (1985)). Thus, incentive compatibility makes the seller a quasi-risk lover so that he always likes to have more information. In contrast, the shape of the buyer’s indirect utility function is a more complex issue. It depends both on his direct utility function and the distribution of types. More information can either be a blessing or a curse to the buyer5 , and I provide su¢ cient conditions for both cases. Similarly, the structure of the optimal supply arrangement is more risky than its exogenous information counterpart when the buyer provides the seller with extra incentives for information acquisition, and is less risky when the buyer reduces the seller’s incentives to acquire information. In the former case, when the seller’s expected cost is surprisingly low he is rewarded by an extra increase in production that increases his informational rent at the margin, and punished 5 This con…rms results of Green and Stokey (1981), who do, however, not relate their results to risk. 6 if his expected cost is surprisingly high. These results con…rm and generalize those of Crémer et al. (1998a) and eliminate the undesirable discontinuity in their supply arrangement due to the all-or-nothing nature of information acquisition. But the analysis is of use beyond that context and can be applied to any model that relies on a linear environment. Ordering better information by riskiness in the distribution of conditional expectations is an extremely useful concept. In contemporaneous work Dai and Lewis (2005) have studied a model of sequential screening with two possible levels of precision of information that obey this ordering. They show that experts with di¤erentially precise information can be screened by the extent of decision authority embodied in contracts. As in the present paper, the value of information to the principal is ambiguous. However, they show that this ambiguity can be overcome by varying the timing structure of the interaction between the principal and the expert. Dai and Lewis (2005) and the present paper complement each other. While their aim is to develop a model that is easily tractable, the current paper provides general statistical foundations for the reduced form they employ and thereby con…rms the generality of their …ndings. Moreover, the justi…cation of the …rst-order approach in terms of the primitives of the experiment structure is a novelty of my model. More recently, Shi (2006) has studied information acquisition in optimal auctions showing how the optimal reserve price is a¤ected by the fact that information is endogenous. Shi studies information structures that are “rotation ordered”, a concept that Johnson and Myatt (2006) have used to study general transformations of demand. The information structures used in this paper satisfy the rotation order. In contrast to Shi (2006), this paper derives more general statistical foundations in terms of experiment structures that induce the desired ordering in the ex ante distribution of conditional expectations. Closest related to the present paper in terms of its aim to uncover the general principles of information acquisition are Gromb and Martimort (2004) and Malcomson (2004). Gromb and Martimort (2004) establish the Principle of Incentives for Expertise, according to which an agent should be rewarded when his advice is con…rmed either by the facts or by the advice of other agents. There are two main di¤erences to the present paper. First, their setup is simpler on the informational side but richer on the organizational side, in that they allow for multiple agents. Second, they allow for contracting contingent on advice and ex post realizations whereas I focus on the case where the agent’s information is not veri…able ex post. Malcomson (2004) analyzes the standard principal agent problem, where the agent not only exerts some e¤ort but also makes a decision. The main di¤erence to the present paper is the role of communication. I allow for com- 7 munication while Malcomson considers the case where the principal commits to a single contract in advance. Moreover, Malcomson’s main interest is in characterizing conditions under which the addition of the agent acquiring a signal makes the problem and its solution any di¤erent from the standard principal agent problem, and its solution, respectively. In contrast, the present approach allows for a complete characterization of the optimal mechanism. Bergemann and Välimäki (2002) analyze incentives for information acquisition in ex post e¢ cient mechanisms. They show that incentives for information acquisition in a private value environment are related to supermodularity in the agents’payo¤ functions.6 In contrast to the present paper contracts are only proposed after information has been acquired. As a result, information acquisition may be either excessive or insu¢ cient although the seller’s payo¤ function in the present model is submodular in the state and the contracting variable. The information structures used in the present paper connect the contracting literature to a literature on the value of information in decision problems, a line of research that has been initiated by Blackwell (1951), and Karlin and Rubin (1956), and further pursued by Lehmann (1988), and most recently by Athey and Levin (2001). The combination of these two literatures delivers a powerful approach, that should prove useful to study further applications, because the predictions of the model are robust within a large class of information gathering technologies. One such application, already pursued by Shi (2006), is the study of optimal auctions with endogenous information (see Myerson (1981) for the case of exogenous information). His approach nicely complements the literature on auctions with endogenous information that has restricted attention to a class of mechanisms, e.g., …rst versus second price auctions (see Tan (1992), Hausch and Li (1993), Stegemann (1996), and more recently Persico (2000) on this).7 The paper is organized as follows. Sections 2 to 4 contain the main theory. In section 2 I spell out the main model. Section 3 contains the main result on the validity of the …rst-order approach. Section 4 derives the statistical foundation of the second order stochastic dominance relation in the distribution of the conditional expectation. Sections 5 and 6 contain the main implications of the theory. Section 5 derives some results on the value of information, section 6 discusses the form of optimal contracts. Section 7 derives two alternative formulations of experiments. In the …rst variation, I allow for moving supports, and show that the …rst-order approach is typically not valid in this framework but would deliver - if valid- essentially the same structural predictions except 6 They note that e¢ cient mechanisms in the linear environment can be based on conditional expectations. result that the auction format with the higher risk sensitivity induces more information acquisition 7 Persico’s corresponds to the result that the marginal value of information for the agent is positive. 8 for distortions at the top. The second variation provides a particularly useful simpli…cation of the main model which I term stochastic experiment structures. Section 8 concludes. Long proofs are in the appendix. 2 The Model The model is a variant of the Baron and Myerson (1982) model where I allow for general, endogenous information structures. A buyer (henceforth the principal) contracts with a seller (henceforth the agent) for the production of a good. The good is divisible, so output can be produced in any quantity, q: q is observable and contractible. The agent receives a monetary transfer t from the principal and has costs of producing the quantity q equal to q. Both parties are risk neu- tral with respect to transfers. The principal derives gross surplus V (q) from consumption; where V (q) is de…ned on [0; 1) and satis…es the conditions8 Vq (q) > 0; Vqq (q) < 0; limq!0 Vq (q) = 1; limq!1 Vq (q) = 0: Thus the principal’s net utility is V (q) t The agent’s payo¤ from receiving the transfer t and producing the amount q is given as t q Ex ante the principal and the agent do not know the precise value of prior about it, which is supported on ; with cdf P ( ) ; where ; but share a common > 0: Once the principal has committed himself to the terms of the contract but before production takes place, the agent may acquire additional information about : Information acquisition is modeled as a costly choice of e¤ort e, that in‡uences the informativeness of certain experiments. An experiment is a joint distribution of and and a random variable depends on the agent’s e¤ort. The marginal distributions of and : This distribution are both independent of e; so e¤ort in‡uences only the joint distribution of the two variables (so roughly speaking the correlation between the two variables) but not their marginal distributions9 . The random variable has typical realization 2 [ ; ] ; and follows a distribution with an arbitrary density k ( ) > 08 and cdf K ( ) : Since the distribution of has full support, K ( ), contains the same information 8 Throughout 9 The the paper subscripts will denote derivatives of functions with respect to their argument. assumption that the marginal of is independent of e will be important for the results in sections 4 through 6, but is not needed for the results in section 3. Since the changes to incorporate the case where the marginal of depends on e are minor, I leave it to the reader to explore this extension. 9 as does itself, but is much more convenient to work with. So, I denote the random variable S = K ( ) as the signal. As is well known, S is distributed on a support [s; s] = [0; 1] and follows a uniform distribution, regardless of the function K ( ).10 I let H ( j s; e) denote the resulting posterior cdf and let h ( j s; e) denote the density of the posterior distribution, and assume that this density is di¤erentiable in s and e to the order needed. Experiments can be ordered in the sense that high values of s indicate high costs in the sense of First Order Stochastic Dominance 1 < Hs ( js; e ) < 08e (1) implies that R (1) dH ( j s; e) is increasing in s with a bounded rate of change: Below I will also introduce a precise sense in which higher e¤ort corresponds to more informative experiments. For the time being this is not important and the only restriction I impose on the in‡uence of e¤ort on H ( j s; e) is He ( j s; e) = He ( j s; e) = 0 (2) (1) and (2) imply that there is a lowest and a highest estimate of costs conditional on the agent’s information and these bounds are both independent of the level of e¤ort the agent exerts. ForR R mally, dHe ( j s; e) = dHe ( j s; e) = 0: This property is convenient because the relevant contracting variable will have a …xed support. The cost of e¤ort is g (e) ; a strictly convex function, that satis…es ge (e) > 0 for e > 0; gee (e) > 0 for all e; ge (0) = 0; and lime!e ge (e) = 1; where e is an upper bound on e that can be taken as in…nite most of the time, except for some speci…c examples. The game has the following time structure: + + P o¤ers + + + s is realized A accepts and observed a contract and delivers by A or refuses and learns the A produces A exerts a menu of e¤ort e contracts to participate true costs only when producing 1 0 This approach to model dependence among random variables is closely related to the notion of a copula, de…ned as the distribution function C (P ( ) ; K ( ) ; ) on [0; 1]2 : The marginal distributions of P and K are uniform on [0; 1], regardless of the functions P ( ) and K ( ) themselves. The function C ( ) embodies the correlation structure between the random variables. In the present context, it is more convenient to specify the joint distribution over and K ( ) : Otherwise the structure is the same. 10 First, the principal o¤ers a menu of contracts. Then the agent chooses an e¤ort level, e; that determines the informativeness of the experiment. The experiment is realized and observed by the agent. Given this information he decides whether or not to participate, and, contingent on participating, also which contract to accept. If the agent refuses to participate the game ends. If the agent agreed to participate, production and transfers take place according to the contract the agent has chosen. Notice that the agent learns the true cost only at the time when he produces, not before. In particular, he does not know the true cost when he selects any of the o¤ered contracts or his outside option. I assume that the agent’s choice of e¤ort is not observable to the principal and that the value of the signal is the agent’s private knowledge. 3 Justifying a First Order Approach As is customary, I will characterize optimal solutions to the contracting problem taking as given that the principal whishes to implement a given level of e¤ort, and will say very little about the optimal choice of e¤ort to implement11 . I think of contracting in terms of mechanism design. A mechanism is a tuple fq( ); t( )g which speci…es quantities of production and transfers to the agent as a function of a (vector valued) message m; the agent sends to the principal. Invoking the Revelation Principle I can restrict attention to direct, incentive compatible mechanisms, fq( ); t( )g that depend only on a reported tuple of signal realization and value of e¤ort (^ s; e^) : Hence, one can write the principal’s problem as follows: max q( ; ); t( ; ) Z s (V (q (s; e)) t (s; e)) ds (3) s s.t. 8s; e : Z (t (s; e) Z e 2 arg max e Z q (s; e)) dH ( js; e ) (t (s; e) (Z s s Z (t (^ s; e^) q (s; e)) dH ( js; e ) (t (s; e) q (^ s; e^)) dH ( js; e ) 8^ s; e^ 0 8s; e ! q (s; e)) dH ( js; e ) ds (4) (5) ) g (e) (6) (4) requires that the agent …nds it optimal to report the true signal value and the true signal informativeness. (5) ensures that the agent …nds it optimal to participate for all possible realizations 1 1 As is well known from the problem of pure moral hazard, the problem of determining the optimal choice of e¤ort has almost no regularity structure. 11 of signal and informativeness. (6) imposes that the agent’s choice of how much e¤ort to acquire is optimal given the contract the principal o¤ers. Observe that the agent’s ex ante expected utility net of costs of information acquisition is always nonnegative. Notice that I impose (5) for all values of s and e; not only the equilibrium choice of e¤ort. This involves no loss of generality under the non-moving support assumption. Extensions to the case of moving supports will be studied below. The screening problem is multi-dimensional, and therefore potentially extremely complicated. However, due to the fact that the agent’s utility is linear in ; and linearity is preserved under expecR tations, the agent’s utility depends e¤ectively only on the one-dimensional statistic dH ( js; e ) (and the agent’s reported type). For this reason, similar to Biais et al. (2000) in a di¤erent context, we can observe that non-stochastic mechanisms can only make use of this one dimensional statistic of the type instead of the two-dimensional type itself.12 Since the agent’s conditional expectation is the relevant contracting variable it is important to understand the properties of this variable. Denote the function (s; e) = Suppose that (s; e) = Z dH ( js; e ) for some real number : Given that (s; e) is increasing in s; the function is invertible and the signal that generated a value of the conditional expectation equal to s = 1 ( ; e) : Ex ante, i.e., before s is realized, the value of the conditional expectation is a random variable itself, say. Using the fact that the distribution of s is uniform, the cdf of given e is F ( ; e) = 8 > > > > < > > > > : Due to condition (2), the support of = satis…es (s; e) for all e and = 0 1 f or < (s; e) ( ; e) f or (s; e) 1 > is the interval ; for (s; e) (7) (s; e) ; independent of e¤ort. Formally, I have (s; e) for all e: Together with the law of iterated expectations, the non-moving support property places some restrictions on the in‡uence of e on F ( ; e) : De…ne EX as the expectation operator when the expectation is taken with respect to X. The law of iterated expectations requires that ES [E [ j s; e]] = E [ ] : Changing variables and integrating by parts, I can write Es [E [ j s; e]] = 1 2 Bergemann Z F ( ; e) d and Välimäki (2002) have noted that this is also the relevant contracting variable in ex post e¢ cient mechanisms in the linear environment, since e¢ cient mechanisms are non-stochastic. 12 This property must hold for any e: Since E [ ] is independent of e; it follows that Z Fe ( ; e) d = 0 (8) (8) is a condition that any model with …xed supports must ful…l. If (8) fails to hold, then an increase in e¤ort changes the ex ante mean of the distribution, which implies that e¤ort is not purely a measure of informativeness but also of something else. It is obvious that the same conditions imply also that Z Fee ( ; e) d = 0 (9) I now use this change of variables to state (3) s.t. (4) ; (5) and (6) ; equivalently as a message game with messages ^ 2 ; about “perceived costs”. In this formulation, the principal’s problem is max q( ); t( ) Z (V (q ( )) t ( )) dF ( ; e) s:t: t( ) q( ) t ^ q ^ 8 ;^ t( ) q( ) 08 8 <Z e 2 arg max (t ( ) q ( )) dF ( ; e) e : 9 = g (e) ; In order to solve this problem I need to be able to replace the …nal constraint by a …rst-order condition. Proposition 1 The principal’s problem (3) s.t. (4) ; (5) and (6) is equivalent to the following problem max q( ) Z V (q ( )) + Z + F ( ; e) f ( ; e) Fe ( ; e) q ( ) d s:t:q ( ) for some Lagrange multiplier (10) ! ge (e) 0 if and only if Z and (8) ; and q ( ) f ( ; e) d Z y Fe ( ; e) d 08y (11) Fee ( ; e) d 08y (12) y 13 and (9) : That is, if and only if an increase in e induces a mean preserving spread in the sense of Rothschild and Stiglitz (1970), at a decreasing rate. It is well known13 that the set of implementable contracts satis…es t( ) = q( ) + q ( ) R q( )d and 0: Substituting out transfers and integrating by parts one obtains the principal’s objective function: the principal maximizes expected surplus net of the agent’s virtual surplus (Myerson (1981)). Proceeding likewise for the agent’s expected utility one obtains the expression in the constraint to problem (10) : After an integration by parts the agent’s …rst-order condition can then be expressed as Z Z Fe ( ; e) d q ( ) d ge (e) = 0 (13) From (13) it is obvious that (11) renders the agent’s expected gross utility (gross of costs of information acquisition) non-decreasing in e for any non-increasing quantity schedule; (12) renders the agent’s expected gross utility concave in e. The agent is in fact a quasi-risk lover because his indirect utility under any implementable contract is a convex function of (Rochet (1985)). Therefore he likes increases in risk in the distribution of types in the sense of Rothschild and Stiglitz (1970)14 . Moreover, since the …rst-order condition must be a valid description for any non-increasing quantity schedule, conditions (11) and (12) are also necessary. The complete proof is in the appendix. The upshot of proposition 1 is that one can complement the Mirrlees approach to reporting by a …rst-order approach to information acquisition, which yields a fairly easily tractable problem. Before I proceed to apply the approach to the speci…c context of procurement, I characterize su¢ cient conditions on the Bayesian updating process that induce second order stochastic dominance shifts in the distribution of 4 : On the Informativeness of Experiments In this section I study the properties of the distribution of the conditional expectation. I obtain su¢ cient conditions on the conditional distribution of given s and e such that the distributions of the conditional expectation for di¤erent levels of e can be ordered by Second Order Stochastic Dominance. 1 3 For convenience of the reader the derivation is reproduced in the appendix. A more detailed treatment is found in Fudenberg and Tirole (1991), chap 7. 1 4 See also Dai and Lewis (2005), who have observed this independently in a two experiment model. 14 Recall that a high signal indicates a high naturally if, e.g., in the sense of (1): This sort of dependence arises and s are a¢ liated. Consider now the dependence on e: Let s~ ES S denote the expected value of the signal s: I impose the following two conditions. First, a local FOSD condition, that I shall denote LFOSD henceforth: He ( js; e ) > 0 for s 2 (s; s~) and He ( js; e ) < 0 for s 2 (~ s; s) (14) Second, a local concavity/convexity condition of the distribution function, that I shall denote LCDFC henceforth Hee ( js; e ) 0 for s 2 (s; s~) and Hee ( js; e ) 0 for s 2 (~ s; s) (15) The reason I impose these conditions in a local rather than the usual global sense is because a global version of (14) would imply that for each s an increase in e increases the posterior. But since the distribution of the signal is …xed, this would imply an increase in the ex ante mean, which is inconsistent with the law of iterated expectations. Similarly, note that the assumption that supports are non-moving, (2), directly implies that Hee ( j s; e) = Hee ( j s; e) = 0: Therefore, I have to impose local restrictions on the concavity/convexity of the distribution function, again because the usual global version would be inconsistent with the law of iterated expectations. Let ~ E : Experiments that satisfy these conditions have the desired properties: Proposition 2 Assume that experiments satisfy (1) ; (2) ; (14) ; and (15) : Then, i) F ( ; e) has a non-moving support; for all e F ( ; e) = 0 and F ; e = 1; ii) F ( ; e) satis…es Fe ( ; e) = Fe ~; e = Fe Fe ( ; e) > 0 f or 2 ;e = 0 ;~ Fe ( ; e) < 0 f or 2 ~; and thus condition (11) ; and iii) F ( ; e) satis…es Fee ( ; e) = Fee ~; e = Fee ;~ Fee ( ; e) 0 f or 2 Fee ( ; e) 0 f or 2 ~; and thus condition (12) : 15 ;e = 0 We know from (7) that the properties of the distribution function F ( ; e) simply correspond to the properties of the inverse of the conditional expectation function. Relative to the prior mean the agent revises his posterior expectation upwards if he receives a signal higher than ex ante expected, and downwards if he receives a downward surprise. If he receives the expected signal, s = s~; no revision takes place. The upward (downward) revision for surprisingly high (low) signals is the larger the higher is e: As a consequence the conditional expectation functions for di¤erent e all cross at = (s; e) ; at = (s; e) ; and at the prior mean ~: Hence, all distributions of F ( ; e) for di¤erent levels of e¤ort satisfy a triple crossing property, and cross at the bounds of the support and at the prior mean ~: The distribution of conditional distribution of inherits the concavity/convexity properties of the given s and e: Before I illustrate these results with an example, I give alternative su¢ cient conditions on the posteriors that justify conditions (11) and (12) in proposition 1. Although these conditions are more restrictive, they may prove useful in other applications, because they imply more structure. In particular, one may impose a local version of the Monotone Likelihood Ratio Property: @ @ he ( js; e ) h ( js; e ) < 0 for s 2 (s; s~) and @ @ he ( js; e ) h ( js; e ) > 0 for s 2 (~ s; s) If the conditional distribution satis…es this condition and the agent receives a signal which is higher (lower) than ex ante expected, then it is relatively more likely that indeed the state is high (low) for a higher level of e: In this sense the signal is more informative when e¤ort is higher. Non-moving supports and di¤erentiability in s then require then that @ @ he ( js;e ) h( js;e ) =0 for s 2 fs; s~; sg : Building on the proofs of Milgrom (1981) it is straightforward to show that these local version of the MLRP condition imply (14) : Moreover, one can also show that under these assumptions the distribution of has fe ( ;e) f ( ;e) @ @ T 0 for inherits the Local Monotone Likelihood Ratio Property, i.e., one T ~:15 However, as is well known (Jewitt (1988)), joint conditions on the likelihood ratios and the convexity properties of the distribution function are rather restrictive. Therefore, I use the weaker condition (14) : The following simple example illustrates the properties. With a slight departure of our notation let the marginal cost be = B+ for some B > 1 and suppose the marginal of is uniform on [ 1; 1] : The marginal of s is uniform on [0; 1] and the posterior density is h( j s; e) = 1+ (a+bx(s)+ey(s)) ; 2 x (1) = 1 and y (s) = s s 1 2 (1 where x (s) is an increasing function satisfying x (0) = 0 and s) : h ( j s; e) satis…es conditions (14) and (15) : In fact it also satis…es the local monotone likelihood ratio property: Computing the posterior expectation, I 1 5 This last statement follows directly from Milgrom’s (1981) proposition 3. 16 (s; e) = B + 13 (a + bx (s) + ey (s)) : Sensible parameter restrictions are a = obtain in which case the support of : For various reasons e; the (s; e) coincides with the support of upper bound on e; must not be too large. Provided this is the case, h ( everywhere and 3 and b = 6; j s; e) is strictly positive (s; e) is strictly increasing in s: The function y (s) embodies the important assumptions that I have made sofar. y (s) takes a value of zero for s 2 0; 21 ; 1 ; it is negative for s 2 0; 21 and positive for s 2 1 2; 1 : Therefore, an increase in e decreases the conditional expectation for low signal values and increases it for high Z1 values. Since y (s) ds = 0, the law of iterated expectations is respected for all e: 0 In addition to these conditions that ensure the regularity properties of my problem with respect to the agent’s choice of e¤ort, it will also be convenient to have conditions that guarantee that the monotonicity constraint in problem (10) is non binding at the optimum. Without such regularity conditions, one may encounter problems of bunching that are well known and do not add much to the present discussion. Proposition 3 If H ( js; e ) satis…es condition (1) ; then the distribution of no atoms for all e: The distribution of satis…es in addition s The distribution satis…es s= 1 @ 2 F ( ;e) @ 2 f ( ;e) (s; e) s (s; e) ss 0 ( 0) at @ F ( ;e) @ f ( ;e) has full support and 0 if and only if 18s (16) if and only if @ @s s ss (s;e) s (s;e) 0 ( 0) at ( ; e) : distribution, condition (16) is equivalent to the condition 2In terms of the conditional 3 Z 7 d 6 Hs ( js; e ) d 5 0; but that is hardly more informative than condition (16) ; which says s 4 ds that the distribution of has a non-decreasing inverse hazard rate if and only if the conditional expectation function is not too concave in the sense of a standard curvature measure. I will henceforth assume that the posterior expectation satis…es (16) for all values of e; since this avoids unnecessary technicalities. In the example given above the condition is satis…ed for e not too large if the function x satis…es sxss (s) xs (s) > 1 for all s: For completeness I state also the convexity properties of the inverse hazard rate at this point. This will prove useful for the analysis of contracts below. I show in the appendix that for the speci…c example where x (s) = small; the distribution of x (s) = 81 19 + 1 19 9 5 1 5 (s 2 3) and e is su¢ ciently has a concave inverse hazard rate; and for the speci…c case where 2 (s + 9) and e is su¢ ciently small, the inverse hazard rate is convex. 17 In the remainder of this paper I apply the …rst-order approach to study the speci…c problem of procurement. The …rst step is to sign the multiplier : The second is to characterize the structure of optimal contracts. 5 The Value of Information In this section I establish two results. First, I show that it is optimal to implement a strictly positive amount of information acquisition, that is, that the optimal level of e¤ort is strictly positive. I conclude from this result that the value of information to the principal is positive. Second, I show that the level of e¤ort can be either too small or too large relative to the amount of e¤ort that maximizes the expected surplus. In particular I will show that whether there is too much or too little information acquisition depends on the principal’s quasi-attitudes toward risk, that is, on the shape of his indirect utility function. Consider …rst the value of information to the principal, which I de…ne as the di¤erence in expected utility when he implements a positive amount of e¤ort and zero e¤ort. Implementing e = 0 requires that information has no value to the agent, neither for his decision what type to report conditional on participating, nor on his decision whether or not to participate. This means that production must be independent of the agent’s announced type and that the transfer is so high that even type breaks even ex post. This is very expensive from the principal’s perspective. To show this is suboptimal, I have to show that there exist contracts that give the principal a higher utility. It is hard to show this directly, because the level of the principal’s utility depends on the shadow cost of implementing e¤ort at the optimal level of e¤ort: Therefore I establish my result in an indirect way, showing that there exist (possibly suboptimal) contracts that implement a positive level of e¤ort at a zero shadow cost and that give the principal a higher utility than any contract that implements e = 0: Since the principal will be able to do even better if he is allowed to implement any level of e¤ort, this argument shows that implementing e = 0 can’t be optimal, or in other words, that information has a strictly positive value to the principal. To make this argument I denote q ( ; e) an optimal quantity schedule contingent on the e¤ort level e: Suppose the principal o¤ers a contract that implements a level of e¤ort e at zero shadow cost; that is the value of the multiplier ; associated to the problem of implementing e¤ort e is zero. Then, we know from Baron and Myerson (1982) that the optimal quantity schedule satis…es 18 the condition Vq q BM ( ; e) = + F ( ; e) f ( ; e) To see this, maximize (10) point-wise with respect to q for (17) = 0: Conversely consistency with = 0 requires that the agent be willing to choose the e¤ort level e that the quantity schedule q BM ( ; e) is conditioned on. Let e^ denote the level of e¤ort that the agent …nds optimal to exert when he is o¤ered a contract with associated quantity schedule q BM ( ; e) : e^ satis…es the …rst-order condition Z Fe ( ; e^) q BM ( ; e) d ge (^ e) = 0 (18) The solution of (18) ; when viewed as a function of e; de…nes a best reply for the agent, e^ = r q BM ( ; e) : Contract o¤er and e¤ort choice are in simultaneous equilibrium if e = r q BM ( ; e) (19) Let e denote the (possibly empty) set of solutions to (19) : If e is non-empty, then the principal can implement any e¤ort level in e by o¤ering the associated Baron-Myerson quantity schedule de…ned by (17). O¤ering such a contract, the principal extracts some rent, and therefore he does better than under the contract where the agent is always paid as if he had costs equal to . Proposition 4 It is optimal to implement a positive level of e¤ ort. Formally, the set e; de…ned by (17) ; (18), and (19) ; is non-empty. To ease notation again in what follows I will drop the dependence of the optimal quantity schedule on e where this can be done without creating confusion. Consider a locally optimal choice of e¤ort to implement, and denote such a locally optimal value of e by e ; and the associated multiplier by Z ((V (q ( )) : Such a choice satis…es the …rst-order condition q ( ) fe ( ; e )) Fe ( ; e ) q ( )) d + Z Fee ( ; e ) q ( ) d ! gee (e ) where I have used the envelope theorem to conclude that all indirect e¤ects through e and =0 on q ( ; e) are zero around an optimum.16 Rearranging the …rst-order condition, and substituting from the …rst-order condition with respect to the agent’s e¤ort choice, I can write = 1 6 Notice R (V (q ( )) q ( ) fe ( ; e )) d ge (e ) R Fee ( ; e ) q ( ) d gee (e ) also that the envelope theorem, and therefore the statement of the …rst-order condition, applies regardless of whether or not the contract is strictly monotonic. 19 The term inside the brackets of the denominator is the second-order condition of the agent’s e¤ort choice. Hence, the sign of is equal to the sign of the numerator. If the increase in the social surplus due to an increase in e exceeds the marginal cost of acquiring information, then positive; if the two terms are just equal then optimum. I will now argue that is is zero; otherwise the multiplier is negative at the can be of either sign at a stationary point of the principal’s problem, and will give su¢ cient conditions for each case to occur. I use the following chain of reasoning. Let e~ denote an element of e; de…ned by (19) ; and let e~ denote the smallest element in e and let e~ denote the largest element in e: By de…nition (~ e) = e~ = 0: Since an increase in makes contracts more risky in the sense of a mean preserving spread, and the agent is a quasi-risk lover - because incentive compatible indirect utility pro…les are convex - we must have < 0 for any e < e~ and > 0 for any e > e~. To establish my result, it su¢ ces to give su¢ cient conditions that render the principal’s utility i) locally decreasing around e = e~ and ii) locally increasing around e = e~: By implication the principal’s utility will be locally decreasing around e~ in the former case and will be locally increasing around e~ in the latter case, which implies the desired result. An increase in the agent’s e¤ort increases the likelihood of more extreme cost perceptions. The principal bene…ts ex post if the agent’s signal is better than expected but is harmed if the agent perceives his cost as being higher. Whether the principal likes to consume such a lottery depends on the shape of his indirect utility function. In turn the shape of the indirect utility function depends on the curvature of the direct utility function and on properties of the family of distributions fF ( ; e)ge 0 : De…ne (q) = V 00 (q) V 0 (q) (q) is the Arrow-Pratt measure of absolute risk aversion with respect to production shocks in the function V (q) : Proposition 5 i) If q (q) 0 and F ( ;e) f ( ;e) is convex in point to the principal’s problem of choosing e where ii) Suppose that q (q) 0 and F ( ;e) f ( ;e) is concave in for all e; then there exists a stationary < 0: for all e: Then there exists z > 0 such that z implies that there exists a stationary point to the principal’s problem of choosing e where > 0: The examples given after proposition 3 illustrate the conditions on the inverse hazard rates. Non-decreasing absolute risk aversion in the direct utility function V (q) plus a convex inverse 20 hazard rate are su¢ cient to render the principal’s indirect utility function concave everywhere. Therefore he behaves as a quasi-risk averter and is harmed by a small increase in e¤ort. However, if V (q) has the more natural property of non-increasing absolute risk aversion and the inverse hazard rate is concave then the opposite may happen. However, this case is somewhat more subtle because it is impossible to render the principal’s indirect utility function convex everywhere. Note that these arguments show the existence of local maximizers with the property that is positive or negative, respectively; of course, the proposition does not say anything about the optimal level of e¤ort to implement. Since it is well known that the problem of choosing an optimal level of e¤ort to implement has almost no structure, I shall not dwell on this here. Instead I will proceed to characterize optimal contracts for both constellations where the shadow cost of e¤ort is positive or negative. 6 The Structure of Contracts Let fq ( ) ; t ( )g 8 denote a menu of contracts that optimally implements a given amount of e¤ort in a truth-telling equilibrium. I shall characterize such contracts, taking their existence for granted.17 The main obstacle to this analysis is that value of the multiplier is unknown. A global treatment necessitates the use of dynamic optimization and delivers little additional insights. Therefore it is useful to characterize the solution for e¤ort levels thatsare easy to implement in the s F ( ;e) F ( ;e) V ar ( fe( ;e) ) V ar ( fe( ;e) j E ) following sense. De…ne the measures and : HeurisF ( ;e) F ( ;e) V ar ( + f ( ;e) ) V ar ( + f ( ;e) j E ) tically, the higher is ; the “easier” is the inference about the unobserved e¤ort from observing relative the variation of the agent’s virtual surplus. subinterval of is considered. Lemma 1 Suppose that 0 for measures an analogous ratio when only a @ @ + F ( ;e) f ( ;e) ~: Then, the multiplier satis…es 1 Fe ( ;e) f ( ;e) 1 0 for all 1 and that @ @ + F ( ;e) f ( ;e) + 1 Fe ( ;e) f ( ;e) : j j measures the utility loss due to the need to give extra (less) incentives for information gathering when marginal costs of information gathering, evaluated at a given e¤ort level, increase by a small amount. One way to place a bound on this loss is to …nd a simple contract that continues to implement a given level of e¤ort when marginal cost of e¤ort increase (decrease) by a small amount. One di¢ culty is again to avoid the need to invoke control theory to make this point. 1 7 Conditions for existence of solutions for exogenous type distributions can be found in Guesnerie and La¤ont (1984). With a suitable adjustment for the endogeneity of information their results could be carried over. 21 The monotonicity conditions in the statement of the lemma are imposed to this end. Then, starting from a strictly monotonic contract, the principal can shock the amount of production by adding Z Fe ( ;e) Fe ( ;e) " f ( ;e) to the original quantity schedule. Since f ( ;e) f ( ; e) d = 0; this shock constitutes a mean-preserving spread, which has two e¤ects. On the one hand, it gives the agent an additional incentive to acquire information. On the other hand it reduces the principal’s payo¤ by making his consumption more risky and by increasing the expected payments to the agent by an amount that is proportional to the covariance between the agent’s virtual surplus and the measure Fe ( ;e) f ( ;e) : In turn, the covariance of these two measures is bounded above by the product of their standard deviations. Finally, ", the size of the shock needed to undo the increase in marginal costs, is computed from the agent’s incentive constraint for the choice of e; " is inversely proportional to the variance of Fe ( ;e) f ( ;e) : Taken together, this reasoning shows that the incremental cost of implementing the original level of e¤ort with this contract is at most 1 : Similarly, when the marginal cost of e¤ort at the current level of e¤ort decreases by a small amount, the principal can prevent the agent from increasing his choice of e¤ort by raising production by ;e) " Ffe(( ;e) for ~: Since an increase in the agent’s ~; so that e¤ort would make extreme cost perceptions more likely, Fe ( ; e) is non-positive for the agent has less of an incentive to acquire information. Using the same reasoning as for the case where the marginal cost increases, I show that the utility loss to the principal resulting from this change of contracts is at most 1 : In terms of the example given above, these conditions are satis…ed, e.g., if the posterior density takes the form h ( j s; e) = that the variation in variation in Fe ( ;e) f ( ;e) = 1+ (a+bx(s)+e y(s)) 2 when is su¢ ciently small. The reason is is then largely exogenous and depends on the function x (s) ; whereas the e 1 ( ; e) s ( e ( ; e) ; e) depends on the level of . If is small, then the e¤ect of an increase in e on the conditional expectation is small for all signals that the agent may receive. Therefore the variance of this measure is small as well. Thus, if the agent receives information that does have a large impact on his information in costs, but the informative content of this information is largely independent of the agent’s e¤ort, then the value of the multiplier is bounded. Conversely, the value of the multiplier may become large, when an increase in the agent’s e¤ort changes the informative content of the signal dramatically. I abstain from a discussion of the latter case, because the advantage of bounding the absolute value of the multiplier is that one can characterize the solution to the contracting problem without recourse to control techniques: Proposition 6 Suppose that @ @ + F ( ;e) f ( ;e) 1 Fe ( ;e) f ( ;e) 22 0 for all and that @ @ + F ( ;e) f ( ;e) + 1 Fe ( ;e) f ( ;e) 0 for all : Then the optimal quantity schedule is characterized by Vq (q ( )) = + F ( ; e) f ( ; e) Fe ( ; e) f ( ; e) (20) The formal proof of this proposition is omitted, since it follows straightforwardly from the previous results. The production schedule coincides with the Baron Myerson schedule at the top, at the prior mean, and at the bottom. Otherwise, there is an additional distortion. The direction of the extra distortion depends on whether the principal wants to give the agent more or less of an incentive to acquire information relative to the Baron Myerson contract. In the former case production is increased for surprisingly low cost perceptions and decreased for surprisingly bad cost assessments. The sensitivity of the production scheme with respect to the agent’s information is increased to provide extra incentives for information acquisition. In the latter case, the reverse happens and production is more equalized in order to dampen the agent’s interest in additional information. The size of the additional distortion depends on how informative a given message is about the agent’s unobserved e¤ort choice.18 In the remainder of this article I study how these results are a¤ected by changes in the underlying structure of experiments. 7 Alternative Experiment Structures 7.1 Moving Supports and Distortions at the top Sofar, I have characterized solutions to the contracting problem when the support of the agent’s conditional expectation is …xed. This is analytically very convenient, but moving supports may easily arise. To see this, modify the example19 to the case where a = b = 0 and y (s) = s 12 ; so 1+e(s 21 ) that the posterior density becomes h ( j s; e) = : Again this posterior satis…es (14) 2 (in fact, also the local monotone likelihood ratio property) and (15) : One veri…es that (s; e) = e(s 1 ) B + 3 2 : The bounds of the support are (s; e) = B 6e and (s; e) = B + 6e : A sensible upper bound on e is e = 6; in which case the support of otherwise, the support of 1 8 The term Fe ( ;e) f ( ;e) for e = e coincides with the support of is a subset of the support of and the upper bound is increasing in e has an interpretation in terms of hypothesis testing. Write Fe ( ;e) f ( ;e) F ( ;e) = : Fe( ;e) F ( ;e) F ( ;e) derivative of the log-likelihood if the statistician observes only if the values in a sample are smaller than to compute the optimal value of e: This measure is important in the contract because the production at f ( ;e) F ( ;e) is the and wants changes normalizes by the conditional density. the rent of all types who are at least as e¢ cient as : Division by 1 9 This speci…cation of the example is adapted from Ottaviani and Sorensen (2001). 23 ; and the lower bound is decreasing in e: For all values of e; the distribution of is uniform. Thus it is natural to wonder how the analysis is a¤ected by the possibility of moving supports. I will show in this section that there are some problems with the …rst-order approach; it is not possible to justify such an approach in general. However, whenever such an approach is valid, then the main qualitative features of contracts remain unchanged. However, one notable exception is that there is now a distortion at the top. There are some essential di¤erences in the agent’s problem. I will stick to the following notation in this section. I let (e) and (e) denote the upper and the lower bound of the support of the conditional mean, respectively. I assume that the upper bound is increasing in e and that the lower bound is decreasing in e: In addition, I let and denote the bounds of the support associated to the e¤ort level that the principal wishes to implement. Notice that these are independent of the agent’s actual actions. Obviously the principal’s contract o¤er satis…es the participation constraint of type with equality. Suppose the agent chooses an e¤ort level that is higher than the one the principal wishes to implement. If the agent receives a high signal, then his participation constraint is violated for all 2 ; (e) : So, the agent refuses to participate and obtains zero rent in this case. Suppose after choosing an e¤ort level that is too high, the agent receives a very low signal. In that case, for 2 [ (e) ; ] the agent will announce to have costs equal to : Suppose on the other hand, that the agent chooses an e¤ort level which is too low. In that case we have (e) < ; Z which implies that type (e) receives a strictly positive rent equal to q ( ) d : It follows from (e) these considerations that I can always write the agent’s indirect utility, u ( ) ; for any given e¤ort choice and any e¤ort (and support) that the principal wishes to implement as 8 9 > > < Z = u ( ) = max 0; q ( ) d > > : ; (21) Finally, consider the probability distribution. It has the properties that F ( (e) ; e) = 0 and F (e) ; e = 1. Moreover, it satis…es 8 > > > > 0 dF ( ; e) < = f ( ; e) > 0 > d > > > : 0 for < (e) for 2 (e) ; (e) (22) > (e) I can now derive the agent’s ex ante expected utility from (21) and (22) : This is somewhat tedious but straightforward, so I relegate the derivation of the following result to the appendix. 24 Result 1 With moving supports the agent’s ex ante expected indirect utility satis…es Z E [u ( )] = F ( ; e) q ( ; e) d At …rst sight it is puzzling that there seems to be no di¤erence to the case of non-moving supports. There are di¤erences, but the fact is that (21) and (22) go together so nicely that the di¤erences add up to zero. However, there is a crucial di¤erence at the ex ante stage when the agent chooses the level of e¤ort. An incentive compatible choice of e¤ort must satisfy the condition 8 9 > > <Z = e = arg max q ( ) F ( ; e^) d g (^ e) > e^ > : ; (23) Unfortunately, (23) cannot simply be replaced by the …rst-order condition Z q ( ) Fe ( ; e) d ge (e) = 0 for any arbitrary, incentive compatible quantity schedule q ( ) : Even if I impose the same conditions as before, namely that the law of iterated expectations holds, and that an increase in e¤ort induces a local …rst order stochastic dominance shift, and that the distribution satis…es the local concavity/convexity conditions, it is no longer true that the agent prefers to have more information (at the same cost). To see this, integrate by parts to obtain Z q ( ) Fe ( ; e) d = q Under my assumptions q where < (e) and Fe Fe ;e ;e q ( ) Fe ( ; e) q ( ) Fe ( ; e) Z q ( ) Z Fe ( ; e) d d 0; and this inequality is strict for the case > (e) : Hence, one can …nd monotonic quantity schedules where the agent does not value additional information. It is also no longer true that the agent’s expected indirect utility (gross of e¤ort costs) is concave in e¤ort, since Z and q q ( ) Fee ( ; e) d = q Fee ;e q ( ) Fee ( ; e) Fee ;e q ( ) Fee ( ; e) Z q ( ) 0 with a strict inequality when Z Fee ( ; e) d d < (e) and > (e) : Hence, the same caveat applies here. However, whenever the …rst-order condition adequately describes the solution to the agent’s problem, I have the following result. Proposition 7 If the …rst-order approach is valid, and the conditions in lemma 1 and proposition 6 hold, then an optimal quantity schedule satis…es the condition Vq (q ( )) = + F ( ; e) f ( ; e) 25 Fe ( ; e) f ( ; e) For the case where > 0 ( < 0) the level of production at the top is higher (smaller) than the Baron Myerson quantity at = ; the level of production at = is lower (higher) than the Baron Myerson quantity. The rationale for this result is simple. With moving supports, an increase of the agent’s e¤ort does have an impact on the mass at the bounds of the support that the principal wishes to implement; at the lower bound the agent’s e¤ort increases the value of the distribution function at the margin, at the upper bound of the support his e¤ort decreases the mass at the margin. Hence, there are additional distortions to consider relative to the case with a …xed distribution of types. 7.2 Stochastic Experiments I end this article with a discussion of a class of updating processes that gives rise to a particularly tractable model. Suppose e¤ort does not in‡uence the posterior distribution directly, but rather in‡uences only the likelihood of obtaining di¤erent posteriors that are independent of e¤ort. I show in this section that the …rst-order approach is rather easy to justify in that case. In addition, all the qualitative insights developed for the more general model are still valid. Suppose an experiment is the realization of two random variables, S and I; and a resulting posterior with cdf H ( j s; i) : The variable S is still the signal, I is an informativeness parameter. Typical realizations of these variables are s 2 [s; s] = [0; 1] and i 2 [0; 1] ; respectively. The marginal distributions of s and i are independent of each other and fully supported with densities k (s) = 1 for s 2 [0; 1] (and zero otherwise) and l (i; e) ; respectively. Let L (i; e) denote the cdf of the random variable i20 . Assume that l (i; e) > 0 for all i and all e: Denote the conditional R dH ( js; i ). The interpretation of the random variable is expectation function as (s; i) = unchanged. Provided that we can write s = 2 0 An 1 (s; i) is strictly increasing in s for all i, the function is invertible and ( ; i) for the value of s that generates the conditional expected value : The intuitive example of this experiment structure -although discrete instead of continuous- would take the signal s as a red light on a junction, with signal realizations fred, orange, greeng and the informativeness of the signal as fgood,badg : The informativeness refers to whether I expect to have priority if I receive a green realization. Assume that informativeness depends only on whether the junction is in Napels or in Zürich, say. From observing the colour of the signal I cannot infer whether I am in Napels or in Zürich. Knowing that I am in Zürich does not help me to infer whether the signal should be red or green. Hence s and i are independent. Moreover, the frequency with which the signal changes coulours is (probably) the same at junctions in Napels and Zürich. However, if I do know that I am in Zürich this changes my posterior belief relative to the one I would have in Napels whether I will receive priority on the junction when the signal is green. 26 cdf of conditional on i is F i ( ; i) = 8 > > > > < 0 1 > > > > : f or < (s; i) ( ; i) f or (s; i) 1 (s; i) > (s; i) Let F ( ; e) denote the unconditional cdf of : I have F ( ; e) = Z 1 F i ( ; i) dL (i; e) (24) 0 By construction, ; is independent of e¤ort and its distribution is fully supported on an interval ; independent of e¤ort where = mini (s; i) and = maxi (s; i) : To order experiments, I assume that the posterior density satis…es the local monotone likelihood ratio property, formally, I assume that @ @ hi ( js; i ) h ( js; i ) < 0 for s 2 (s; s~) and @ @ hi ( js; i ) h ( js; i ) > 0 for s 2 (~ s; s) (25) and @ @ hi ( js; i ) h ( js; i ) = 0 for s 2 fs; s~; sg (26) As I have explained in section 3, (25) implies that higher values of i correspond to more informative experiments. In particular, this implies again that conditional on a signal above (below) the mean, the posterior distribution conditional on a given informativeness i is the higher (lower) in the sense of FOSD the higher is i: In addition let Le (i; e) 0 and Lee (i; e) 0 (27) Then, an increase in e¤ort makes it more likely to perform a more informative experiment; and the marginal impact of e¤ort on the distribution of experiments is decreasing in e: Within this structure, I have the following result: Proposition 8 Given conditions (25) ; (26) ; and (27) ; the distribution F ( ; e) satis…es conditions (11) ; (8) ; (12) ; and (9) ; and hence the …rst-order approach is valid. Under the monotonicity conditions in proposition 6, the optimal quantity schedule satis…es the condition Vq (q ( )) = + F ( ; e) f ( ; e) Fe ( ; e) f ( ; e) Thus, it is easy to justify a …rst-order approach if we think of the agent’s e¤ort as of “spanning” the possible posteriors. Moreover, this model is appealing because it comprises much of the 27 existing literature and therefore generalizes the …ndings of this literature. All-or-nothing information acquisition corresponds to the case where there are just two distributions of the conditional expectation conditional on i, F 0 ( ; 0) and F 1 ( ; 1) ; the distribution F 0 ( ; 0) has mass one at E =E and the distribution F 1 ( ; 1) corresponds to the distribution P ( ) : In the current setup I assume that the distribution F 0 ( ; 0) has no atoms, but of course it can be close to a mass-point at E : This assumption eliminates the discontinuities found in the earlier literature. Moreover, I allow for a continuum of levels of informativeness, i; that are (heuristically) ordered the way that the distributions F i ( ; i) are the closer to P ( ) the higher is i21 : Since this model is particularly easy to handle it should prove useful in further applications. 8 Conclusion The main result of the paper is information and risk are equivalent in a wide class of reporting games with endogenous information. It is justi…ed to describe the amount of information acquisition by the solution of a …rst-order condition for any incentive compatible contract, if and only if the agent’s information gathering increases risk in the ex ante distribution of the conditional expectation in the sense of Rothschild and Stiglitz (1970). Su¢ cient conditions on experiment structures are provided that generate such an ordering. The robust results that follow from the approach are that contracts that provide the agent with extra incentives for information acquisition are more sensitive to the agent’s information relative to their …xed information counterparts. The reverse is true when incentives for information acquisition are reduced. Results beyond these depend on the speci…c information structure and are therefore not robust. The paper has derived a tractable modeling of information acquisition and a reduced form which is relatively easy to handle. It can be used to address any problem of mechanism design in the single agent case and extends easily to multi-agent mechanism design problems in the linear, private values environment. 21 I thank an anonymous referee for suggesting this interpretation. 28 9 Appendix Truth-telling: For convenience I summarize the Proof of proposition 1, preliminaries. known features of the contract. For a more extensive treatment, see Fudenberg and Tirole (1991). Let u ;^ = t ^ q ^ and u ( ) = max^ t ^ q ^ : In a truth-telling equilibrium ^ = : By the envelope theorem, u ( ) = q ( ) : Moreover, the least e¢ cient type ; is indi¤erent R R u ( )d = q( )d : The …rst order = 0: Hence u ( ) = between participating and not, u condition t^ ^ q^ ^ ^= = 0 holds almost everywhere. Hence q^ ^ d = 0; a.e., so that q^ ^ t^ ^ ^ q^ ^ ^ d^ 0 is necessary for truth-telling to be locally optimal. Finally, monotonicity makes the local …rst order condition su¢ cient for a global optimum in truth-telling. R Substituting t( ) = q( ) + q( )d into the objective one has !! Z Z V (q ( )) q( )d q( )+ f ( ; e) d (28) Integration by parts delivers the representation in terms of expected surplus net of the agent’s R ;e) expected virtual surplus (Myerson (1981)), V (q ( )) + Ff (( ;e) q ( ) f ( ; e) d : R Consider now the e¤ort constraint. After substitution of t( ) = q( ) + q( )d one has Z (t ( ) q ( )) dF ( ; e) = = Z Z Z q( )d dF ( ; e) F ( ; e) q ( ) d Di¤erentiating, and integrating by parts, using the property of nonmoving supports, one has Z Fe ( ; e) q ( ) d = Z Z Fe ( ; e) d q ( ) d where the inequality follows from the implementability condition q ( ) more since R Z Fee ( ; e) q ( ) d = Fee ( ; e) d = 0 and R Z Fee ( ; e) d Z Fee ( ; e) d q ( ) d 08 and q ( ) 0 0: Di¤erentiating once 0 08 : Hence, if (11) and (12) hold, then the agent faces a strictly concave problem in e¤ort and the …rst order condition in conjunction R with the implementability conditions t( ) = q( ) + q( )d and q ( ) 08 is necessary and su¢ cient for e 2 arg max e 8 <Z : (t ( ) q ( )) dF ( ; e) 9 = g (e) ; To see the necessity part, suppose that (11) does not hold. For concreteness, suppose that R Fe ( ; e) < 0 on ( ; 1 ) and Fe ( ; e) 0 else such that Fe ( ; e) d = 0: A contract that satis…es 29 the implementability condition is q~ ( ) = q~ > 0 for Z 2[ ; 1) and q~ ( ) = 0 else. But then Fe ( ; e) q~ ( ) d < 08e and the …rst order condition is neither necessary nor su¢ cient for the optimal choice of e: Likewise suppose that (11) does hold but that (12) does not hold and suppose that Fee ( ; e) > 0 on ( ; 1 ) R and Fee ( ; e) 0 else such that Fee ( ; e) d = 0: In this case under the implementable contract R q~ ( ) ; Fe ( ; e) q~ ( ) d > 08e but Z Fee ( ; e) q~ ( ) d > 08e Consequently, the …rst order condition is neither necessary (the optimal choice may be e = 0) nor su¢ cient (the value of e that solves the …rst order condition may correspond to a minimum.) Proof of Proposition 2. The proof is given in two parts. In the …rst part I establish the properties of the conditional expectation function that follow from the assumptions; in the second part I use these characteristics to establish the properties of the distribution of the conditional expectation. Part I: The conditional expectation is given by Z Z h ( j s; e) d = H ( j s; e) d Di¤erentiating with respect to e I have e (s; e) = Z He ( j s; e) d (s; e) = Z Hee ( j s; e) d and ee It follows immediately that 0 for s 2 (s; s~) then ee (s; e) that e e e (s; e) = 0 and ee (s; e) = 0 for s 2 fs; s~; sg : Moreover, if He ( j s; e) > (s; e) < 0 for s 2 (s; s~) ; likewise, if Hee ( j s; e) 0 for s 2 (s; s~) : Similarly, He ( j s; e) < 0 and Hee ( j s; e) (s; e) > 0 and Part II: The cdf of ee (s; e) 0 for s 2 (s; s~) then 0 for s 2 (~ s; s) implies 0 for s 2 (~ s; s) : 1 is given by F ( ; e) = ( ; e) for 2 ; : Moreover, if a function is increasing (decreasing) and concave (convex), then its inverse is decreasing (increasing) and convex (concave). Therefore, I have Fe ( ; e) > 0 and Fee ( ; e) 30 0 for 2 ; ~ ; and Fe ( ; e) < 0 and Fee ( ; e) 0 for n o 2 ; ~; : ~; 2 : Obviously it is also true that Fe ( ; e) = 0 and Fee ( ; e) = 0 for 1 Proof of Proposition 3. Recall that s = expectation = ( ; e) is the signal that generates the conditional (s; e) : So, the probability that the conditional expectation is smaller or equal to is 1 F ( ; e) = ( ; e) because the distribution of s is uniform. Hence, the density of 1 f ( ; e) = ( ; e) = s 1 ( is 1 ( ; e) ; e) where the second equality uses the inverse function theorem. By the assumption that Hs ( j s; e) < 0; we have s (s; e) > 0: Boundedness of Hs ( j s; e) implies that s (s; e) < 1; and so f ( ; e) > 0 for all : The inverse hazard rate becomes F ( ; e) = f ( ; e) Di¤erentiating with respect to @ F ( ; e) @ f ( ; e) 1 ( ; e) 1 s ( ; e) ; e I obtain = = 1 s 1 s( 1 1+ ( ; e) ; e + ( ; e) ; e) ( ; e) 1 1 ss 1 s( ( ; e) ss s( 1 1 ( ; e) ; e ( ; e) ; e) ( ; e) ; e ( ; e) ; e) Thus, @ F ( ; e) @ f ( ; e) 0,1+ s (s; e) (s; e) ss s 0 Di¤erentiating once more, I have @ 2 F ( ; e) @ 2 f ( ; e) 1 ss = ( = s ( s( 1 ( ; e) ; e 2 ( ; e) ; e)) 1 + 1 @ 1 ( ; e) ; e) @s s ( ; e) ss s( 1 1 1 sss ( ; e) ; e ( ; e) ; e) ( ; e) ; e ! 1 s ( s ( ( ; e) ; e 1 ss 1 3 ( ; e) ; e)) which establishes the desired result. Recall the structure of the example: Examples. (s; e) = B + 1 3 (a + bx (s) + ey (s)) : The following statements are true: i) ii) s (s; e) > 0 , bxs (s) + eys (s) > 0; s (s; e) + s ss (s; e) > 0 , b [xs (s) + sxss (s)] + e [ys (s) + syss (s)] > 0; 31 ( ; e) ; e 2 @ iii) @s h s ss (s;e) s (s;e) i 0,[ ss (s; e) + s sss (s; e)] (s; e) s s( 2 ss (s; e)) [b (xss (s) + sxsss (s)) + e (yss (s) + sysss (s))] [bxs (s) + eys (s)] 0, 2 s (bxss (s) + eyss (s)) 0 The idea in the following examples is that for e small enough, properties i through iii depend crucially on x (s) : Example I: x (s) = 9 5 xs (s) + sxss (s) = h i s ss (s;e) @ have @s s (s;e) 6 5 1 5 (s 4 5s 2 3) : Since xs (s) = 2 5 (3 s) > 0; for e small enough property i holds. > 0; so property ii holds for e small enough. Since xss (s) = 2 5 < 0; I 0 for e small enough. Example II: x (s) = small; since xss (s) = 81 19 2 19 + 1 19 2 (s + 9) : xs (s) = 2 19 (s + 9) > 0; so property one is satis…ed for e h i s ss (s;e) @ > 0 property ii is satis…ed for small e; To see that property @s s (s;e) 0 holds in this example, consider …rst the case where e = 0 (for this speci…c example). In that case h i s ss (s;e) @ > 0 if @s s (s;e) b (xss (s)) bxs (s) 2 s (bxss (s)) > 0 , b2 4 (s + 9) 361 Obviously this condition holds. By continuity, for e small enough @ @s b2 s h s 4 >0 361 i ss (s;e) s (s;e) 0. Proof of Proposition 4. e = 0 is optimal for the agent if and only if q( ) = q for all t q and 0 for all : The best such contract from the principal’s perspective solves Z max (V (q) t) dF ( ; e) q;t s:t: t q 0 The optimal contract in this class satis…es Vq (q)jq=^q = and t^ = q^: This contract is very costly to the principal, because he pays the agent always as if this one had the highest possible cost. Suppose instead the principal o¤ers the contract q BM ( ; e) = Vq 1 + F ( ; e) f ( ; e) (29) This contract corresponds to the case where the principal neglects his in‡uence on the agent’s e¤ort choice but o¤ers a contract which elicits information truthfully. For simplicity in this argument we assume that + F ( ;e) f ( ;e) is non-decreasing in ; however this is not essential. Even with some bunching, the principal manages to get some share from the surplus. And since the principal extracts some rents, this contract dominates the contract t^; q^ : I now prove that there exist e¤ort levels such that the principal’s contract o¤er is a best reply to the agent’s choice of e¤ort and the agent’s choice of e¤ort is consistent with the contract o¤ered; 32 that is, in addition to (29) ; it must also be true that Z Fe ( ; e^) q BM ( ; e) d =0 ge (^ e) (30) e^=e Consider the agent’s utility as a function of e^ and e : Z F ( ; e^) q BM ( ; e) d g (^ e) Under our assumptions, q BM ( ; e) is di¤erentiable in e: Hence, the agent’s utility is continuous in e and e^ and strictly concave in e^: By the theorem of the maximum, the maximizer correspondence of the agent’s utility function with respect to e^ is upper hemicontinuous. By strict concavity in e^; the maximizer correspondence is in fact a function. Since a single valued correspondence is upper hemicontinuous if and only if it is continuous as a function, it follows that the maximizer of the agent’s utility function is a continuous function of the principal’s conjectured e¤ort level. Formally, let e^ = r q BM ( ; e) denote the agent’s optimal choice of e¤ort when the principal o¤ers contract q BM ( ; e) : De…ne (e) Z Fe ; r q BM ( ; e) Vq 1 + F ( ; e) f ( ; e) d ge r q BM ( ; e) (31) An equilibrium e¤ort (that satis…es both (29) and (30)) is then de…ned as a solution to the equation (e) = 0; or, equivalently, as …xed point satisfying e = r q BM ( ; e) : Such a …xed point must exist, because I have r q BM ( ; e) e=0 > 0 and r q BM ( ; e) e=e < e: To see the …rst point, notice that the family of distributions has a monotone inverse hazard rate for all e: Therefore, q BM ( ; 0) is a strictly monotonic contract, and the agent has a strictly positive incentive to acquire information. To see the second point, notice that the marginal cost of e¤ort goes to in…nity as e approaches e: Since r q BM ( ; e) is a continuous function, it must have a …xed point by Brouwer’s …xed point theorem. Proof of Proposition 5. The proof is split into two parts. In the …rst part, I show that the multiplier is negative for e < e~ and that is positive for e > e~: In the second part, I give su¢ cient conditions for a small increase in the e¤ort level to be bene…cial (detrimental, respectively) to the principal around = 0: Part i) If e < e then < 0; if e > e then > 0: By the de…nition of the smallest …xed point, we know thatr q BM ( ; e) > e for e < e~: To make sure that the agent indeed chooses e; the principal must reduce the agent’s incentive to acquire 33 ~ and increasing production for information. This is achieved by reducing production for ~: From the condition of optimality, Vq (q ( )) = + we conclude that < 0 since Fe ( ; e) F ( ; e) f ( ; e) Fe ( ; e) f ( ; e) ~ and Fe ( ; e) 0 for > ~: The proof for e > e~ 0 for is analogous and therefore omitted. Part ii) The marginal e¤ect of a small increase in e around a point where Let W (e) max q( ) 8 R > < 9 > ;e) V (q ( )) + Ff (( ;e) q ( ) f ( ; e) d = R > ; Fe ( ; e) q ( ) d ge (e) + > : Invoking the envelope theorem I have around a point where We (e) = Z V (q ( )) q( )+ Integrating by parts, and noting that Fe ( ; e) = Fe We (e) = Substituting for q ( ) = @ @ I obtain Recall that Vqq (q) Vq (q) (q) = + f ( ;e) ) Vqq (q( )) ; Z We (e) = Z for F ( ;e) f ( ;e) Z = Vq (q ( )) Vq (q ( )) @ Vqq (q ( )) + F ( ;e) @ + f ( ;e) Vq (q) Vqq (q) = q( )d 1 (q) fe ( ; e) d ) q ( ) Fe ( ; e) d F ( ;e) f ( ;e) so that =0 !! ; e = 0; I can write (Vq (q ( )) F ( ;e) ( =0: and let ; and multiplying by F ( ; e) f ( ; e) (q) = Vq (q( )) F ( ;e) + f ( ;e) =1 Fe ( ; e) d 1 (q) : Then, recollecting terms, I can write Z We (e) = 0 @ (q ( )) F ( ;e) f ( ;e) ;e) + Ff (( ;e) @ @ after another integration by parts, using the fact that have We (e) = Notice that expression R @ @ Z 0 Z @ Fe ( ; e) d 2 @ 4 Fe ( ; e) d (q ( )) @ R 1 F ( ; e) A + Fe ( ; e) d f ( ; e) Fe ( ; e) d = 0 for F ( ;e) f ( ;e) ;e) + Ff (( ;e) @ @ = and for 31 F ( ; e) 5A + d f ( ; e) = ;I 0 by proposition 2. Thus, to prove the result, it su¢ ces to sign the [ ]. De…ne X( ) (q ( )) F ( ;e) f ( ;e) ;e) + Ff (( ;e) 34 @ @ + F ( ; e) f ( ; e) Performing the di¤erentiation, I have X ( ) = F ( ;e) f ( ;e) ;e) + Ff (( ;e) q (q ( )) q ( ) + (q ( )) @ F ( ;e) @ f ( ;e) + + (q ( )) @ @ F ( ;e) f ( ;e) 2 F ( ;e) f ( ;e) F ( ;e) f ( ;e) ;e) + Ff (( ;e) @2 @ 2 + @ @ + F ( ; e) f ( ; e) F ( ; e) f ( ; e) + F ( ; e) f ( ; e) To sign, these expressions, notice that convexity (concavity, respectively) of the inverse hazard rate is equivalent to F ( ;e) f ( ;e) ) @@ ( )( F ( ;e) f ( ;e) : @ F ( ; e) @ f ( ; e) Convexity of the inverse hazard rate implies that @ F ( ; e) @ f ( ; e) F ( ; e) f ( ; e) 0 Concavity of the inverse hazard rate implies that @ F ( ; e) @ f ( ; e) Note …nally that by de…nition q (q) = (q ( )) q (q) , ( (q))2 F ( ; e) f ( ; e) @ F ( ; e) @ f ( ; e) which implies that sign 0: Then it is now easy to see hazard rate implies that that X ( ) q (q) q (q) = sign q (q) and recall that 0, together with a convex inverse 0: Result ii) follows from observing that the …rst and last terms on the right-hand side of X ( ) change sign for smaller as q (q) 0 and a concave inverse hazard rate, and that the middle term becomes is decreased. Proof of Lemma 1. Suppose the cost function is changed to g^ (e) = g (e) + g (e) where is a parameter that takes values in the interval [ ; ], and where < 1: Notice that the function g^ (e) is an Inada cost function for any such , and an interior solution is guaranteed. The marginal cost to the agent of exerting e¤ort e is now g^e (e) = ge (e) + ge (e) : The multiplier is equal to the change in the principal’s utility due to a change in ge (e) : Since e is a constant, I can de…ne c( ) ge (e) : Let W (c) denote the welfare of the principal as a function of c Z F ( ; e) W (c) = max V (q ( )) + q ( ) f ( ; e) d f ( ; e) q( ) ! Z + Fe ( ; e) q ( ) d ge (e) c and let q ( ) denote the optimal quantity schedule for c = 0: Finally, let W (0) denote the value of welfare for c = 0 (that is, = 0): From the envelope theorem, I have Wc (c) = 35 I now provide bounds on the multiplier. I distinguish two cases, a) c ( ) ? 0 i¤ > 0 and b) < 0: Since ? 0 I directly state my results in terms of c: Case a): If c > 0; then the principal must do at least as well as under the following contract. Let q ( ) denote the optimal production schedule for c = 0 then the principal can o¤er the contract where q^ ( ) = q ( ) + " Fe ( ; e) f ( ; e) Notice that by construction the expected level of production under the schedules q^ ( ) and q ( ) Z Z Fe ( ;e) f ( ; e) d = are the same, since Fe ( ; e) d = 0: However; the schedule q^ ( ) has more f ( ;e) variance than the schedule q ( ) : " is de…ned by the agent’s …rst-order condition with respect to e Z Fe ( ; e) q ( ) + " Fe ( ; e) f ( ; e) d = ge (e) + c (32) Using Z Fe ( ; e) q ( ) d = ge (e) and Z Fe ( ; e) f ( ; e) d = 0 f ( ; e) I can solve (32) for ": c "= (33) Fe ( ;e) f ( ;e) V ar The welfare of the principal satis…es W (c) Z V Z V (q ( )) = W (0) q ( )+" " Z Fe ( ; e) f ( ; e) + + + F ( ; e) f ( ; e) F ( ; e) f ( ; e) F ( ; e) f ( ; e) q ( )+" q ( )+" Fe ( ; e) f ( ; e) Fe ( ; e) f ( ; e) f ( ; e) d f ( ; e) d Fe ( ; e) f ( ; e) d f ( ; e) where the …rst inequality is due to the de…nition of W (c) and the second inequality uses the fact ;e) that q ( ) + " Ffe(( ;e) is a mean-preserving spread of q ( ) and that V ( ) is concave. Since has mean zero, the integral in the last line is just equal to the covariance between Fe ( ;e) f ( ;e) : Thus, W (c) W (0) "Cov 36 + F ( ; e) Fe ( ; e) ; f ( ; e) f ( ; e) + Fe ( ;e) f ( ;e) F ( ;e) f ( ;e) and Substituting for the value of " from (33) ; I have Cov W (c) W (0) + c F ( ;e) Fe ( ;e) f ( ;e) ; f ( ;e) Fe ( ;e) f ( ;e) V ar Dividing and taking limits as c ! 0 I have lim W (c) Cov W (0) = c c!0 + V ar F ( ;e) Fe ( ;e) f ( ;e) ; f ( ;e) Fe ( ;e) f ( ;e) Thus, Cov + F ( ;e) Fe ( ;e) f ( ;e) ; f ( ;e) Fe ( ;e) f ( ;e) V ar From a standard result, Cov + r F ( ;e) Fe ( ;e) f ( ;e) ; f ( ;e) V ar + F ( ;e) f ( ;e) r ;e) V ar + Ff (( ;e) r ;e) V ar Ffe(( ;e) r Fe ( ;e) f ( ;e) V ar : Hence, (34) Case b) c < 0: In this case, the principal can do at least as well as by o¤ering the contract 8 > < q( ) for < ~ q^ ( ) = > ~ : q ( ) + " Fe ( ;e) for f ( ;e) " is again de…ned by the …rst-order condition for e¤ort " Z Fe ( ; e) Fe ( ; e) d = f ( ; e) c ~ Solving for "; I can write "= 1 F ~; e Notice that " < 0: I have W (c) Z V Z V (q ( )) = W (0) q( )+1 " Z + h c V ar ~" Fe ( ; e) f ( ; e) + F ( ; e) f ( ; e) F ( ; e) f ( ; e) Fe ( ;e) f ( ;e) + ~ +E F ( ; e) f ( ; e) q( )+1 ~" Fe ( ; e) f ( ; e) d f ( ; e) ~ 37 Fe ( ;e) f ( ;e) q( )+1 Fe ( ; e) f ( ; e) ~ ~" i Fe ( ; e) f ( ; e) f ( ; e) d f ( ; e) d ;e) where the …rst inequality uses the de…nition of W (c) ; the second uses the fact that " Ffe(( ;e) is non-negative for ~; so the principal’s utility is at least as high as when he does not consume the additional quantity at all. Substituting from the agent’s …rst-order condition for " I can write W (c) W (0) c Z Fe ( ;e) f ( ;e) f + F ( ;e) f ( ;e) Z ;e) Fe ( ; e) Ffe(( ;e) d ( ; e) d ~ ~ Dividing by c < 0 I have W (c) W (0) Z Fe ( ;e) f ( ;e) f + F ( ;e) f ( ;e) Z ;e) Fe ( ; e) Ffe(( ;e) d + F ( ;e) f ( ;e) Z ;e) Fe ( ; e) Ffe(( ;e) d ( ; e) d ~ c ~ Rewriting the left-hand side, I can state that W (0) W (c) Z Fe ( ;e) f ( ;e) f ( ; e) d ~ c ~ Taking limits as c goes to zero, I obtain the left-side di¤erential of W with respect to c: Thus, lim c!0 W (0) W (c) c Z + F ( ;e) f ( ;e) Fe ( ;e) f ( ;e) f Z ;e) Fe ( ; e) Ffe(( ;e) d ( ; e) d ~ = ~ and hence Z + F ( ;e) f ( ;e) Fe ( ;e) f ( ;e) f Z ;e) Fe ( ; e) Ffe(( ;e) d ( ; e) d ~ ~ The integral in the numerator can be written as 2 Cov + Ff (( 6 1 F ~; e 4 ;e) +E + Ff (( ;e) 38 ;e) Fe ( ;e) ;e) ; f ( ;e) ~ E Fe ( ;e) f ( ;e) ~ ~ 3 7 5 while the integral in the denominator can be written as 1 F ~; e Fe ( ; e) f ( ; e) V ar ~ +E Fe ( ; e) f ( ; e) where the expectations are taken with respect to the random variable ~ : The ratio of the two terms satis…es Cov + F ( ;e) Fe ( ;e) f ( ;e) ; f ( ;e) Fe ( ;e) f ( ;e) V ar r V ar r V ar V ar r V ar Fe ( ;e) f ( ;e) Fe ( ;e) f ( ;e) Fe ( ;e) f ( ;e) Fe ( ;e) f ( ;e) V ar +E r V ar + F ( ;e) f ( ;e) V ar r V ar Fe ( ;e) f ( ;e) +E r V ar Fe ( ;e) f ( ;e) + + Fe ( ;e) f ( ;e) E Fe ( ;e) f ( ;e) +E + F ( ;e) f ( ;e) +E +E + Fe ( ;e) f ( ;e) F ( ;e) f ( ;e) E Fe ( ;e) f ( ;e) 2 F ( ;e) f ( ;e) 2 F ( ;e) f ( ;e) Fe ( ;e) f ( ;e) where the …rst inequality uses again the fact that Cov (A; B) inequality uses the observation that E I have Fe ( ;e) f ( ;e) p p V ar (A) V ar (B); the second ~ < 0; and the third is trivial. It follows that r ;e) V ar + Ff (( ;e) r ;e) V ar Ffe(( ;e) Proof of Result 1. The agent’s ex ante expected utility can be written 8 9 > > Z(e) < Z = max 0; q ( ) d dF ( ; e) > > : ; (e) There are three cases to consider: i) < (e) and (e) < (corresponding to an actual e¤ort level that is lower than the one that the principal wishes to implement); ii) (e) < iii) (e) = and and < (e) ; and = (e) : Case iii) corresponds to the case that I have already analyzed in the main model; so I disregard this case here. 39 i) In this case, Z Z(e) (e) Since dF ( ;e) d 0 for all (e) q( )d = 0 for 8 > < Z max 0; q ( ) d > : 9 > = > ; dF ( ; e) = q ( ) d dF ( ; e) (35) ; it is true that q ( ) d dF ( ; e) + Z Z q ( ) d dF ( ; e) = 0 (e) But then, (35) is equivalent to 8 > Z(e) < Z max 0; q ( ) d > : (e) Finally, after an integration by parts, q ( ) d dF ( ; e) Z(e) Z (e) < (e) and (e) < Z(e) Z Z Z (e) and therefore I have = F = Z ;e Z 9 > = > ; dF ( ; e) = q( )d Z Z F ( ; e) q ( ) d dF ( ; e) Z q( )d + Z q ( ) F ( ; e) d q ( ) F ( ; e) d where the …nal equality uses the fact that F ( ; e) = 0 for all (e) ; and hence F ( ; e) = 0: Case ii) In this case, I can write 8 9 > > Z(e) Z Z Z < Z = max 0; q ( ) d dF ( ; e) = F ( ; e) q ( ) d + q ( ) d dF ( ; e) > > : ; (36) (e) The …rst term on the right-hand side of (36) is computed using the fact that 8 9 > > Z Z Z Z < Z = max 0; q ( ) d dF ( ; e) = q ( ) d dF ( ; e) = F ( ; e) q ( ) d > > : ; (e) (e) Since the lowest type the agent can announce is ; he will always do so when (e) : On the Z other hand, if < (e) ; the agent rejects the contract, since q ( ) d < 0 over this range. 40 Finally, again after an integration by parts, I can write F ( ; e) Z q( )d + Z Z q ( ) d dF ( ; e) = F ( ; e) Z F ( ; e) = e e Z q( )d + ;e Z Z q( )d q ( ) F ( ; e) d q ( ) F ( ; e) d Since F ( (e) ; e) = 0 for all e; I can di¤erentiate totally and Proof of Proposition 7. have f ( (e) ; e) Z q( )d + F (e) + Fe ( (e) ; e) = 0: At (e) = ; I have Fe ( ; e) = f ( ; e) e (e) > 0 since (e) < 0: Therefore, for a contract that implements a high e¤ort level ( > 0), production at the top is going to be unusually high. A similar argument can be used to show that production at the bottom is smaller than the Baron Myerson quantity for the case where > 0: Proof of Proposition 8. The proof is split into two parts. In part i I derive the properties of the conditional expectation function. In part ii I use these properties to derive those of the ex ante distribution of : Part i: Properties of the conditional expectation function From Milgrom (1981) it follows directly that for s 2 (~ s; s) : Likewise, @ hi ( js;i) @ h( js;i) @ hi ( js;i) @ h( js;i) > 0 for s 2 (~ s; s) implies Hi ( j s; i) < 0 < 0 for s 2 (s; s~) implies Hi ( j s; i) > 0 for s 2 (s; s~) : Since i (s; i) = Z Hi ( j s; i) d this proves that i Finally, I show that i (s; i) < 0 for s 2 (s; s~) and i (s; i) > 0 for s 2 (~ s; s) (s; i) = 0 for s 2 fs; s~; sg : To see this, note that one can write for s 2 fs; s~; sg @ hi ( j s; i) H ( j s; i) = 0 @ h ( j s; i) Integrating I have Z @ hi ( j s; i) H ( j s; i) d = 0 @ h ( j s; i) 41 Integrating by parts, I obtain he s; i h s; i Z Since h ( j s; i) is a density for all i; I have @ hi ( js;i) @ h( js;i) = 0; it follows that hi ( js;i) h( js;i) hi ( j s; i) h ( j s; i) d = 0 h ( j s; i) Z hi ( j s; i) d = 0: It follows that hi ( h( js;i) = 0: From js;i) = 0: Finally, from the fact that h ( j s; i) > 0 for all follows that hi ( j s; i) = 0 for all : Hence, for s 2 fs; s~; sg it (s; i) is independent of i: Part ii: Properties of F ( ; e) : has a nonmoving support F ( ; e) = Since l (i; e) has full support for all e; the distribution of 08e and F E ; e = 18e. Hence Fe ( ; e) = Fe ; e = 0. By the law of iterated expectations E = R R R for all e: Since dF ( ; e) = F ( ; e) d ; this is equivalent to Fe ( ; e) d = 08e: By an integration by parts F ( ; e) Z = 1 Fi ( ; i) dL (i; e) 0 Z 1 = Fi ( ; i) L (i; e)j0 since Fi ( ; i) is locally constant for 1 1 0 i ( ; i) L (i; e) di 2 = [ (s; i) ; (s; i)] : Taking derivatives with respect to e; since L (1; e) = 18e; I have Fe ( ; e) = Z 0 1 1 i ( ; i) Le (i; e) di From part i, I have i ( ; i) T 0 , S~ and hence Fe ( ; e) > 0 f or 2 ;~ Fe ( ; e) < 0 f or 2 ~; Since Lee (i; e) and Le (i; e) have opposing signs for all i; I have also Fee ( ; e) < 0 f or 2 ;~ Fee ( ; e) > 0 f or 2 ~; 42 10 References Armstrong, M., and Rochet, J.-C., (1999) ”Multi-dimensional Screening: A User’s Guide” European Economic Review 43, 959-979 Athey, S., and Levin, J. (2001) “The Value of Information in Monotone Decision Problems”working paper, Stanford University Baron, D. and Myerson, R. 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