Faculty of Business and Law School of Accounting, Economics and Finance ECONOMICS SERIES SWP 2006/23 The Value of Information in a Principal-Agent Model with Moral Hazard: The Ex Ante Contracting Case Randy Silvers The working papers are a series of manuscripts in their draft form. Please do not quote without obtaining the author s consent as these works are in their draft form. The views expressed in this paper are those of the author and not necessarily endorsed by the School or IBISWorld Pty Ltd.
The Value of Information in a Principal-Agent Model with Moral Hazard: The Ex Ante Contracting Case Randy Silvers School of Accounting, Economics and Finance Deakin University January 2013 Abstract We examine a principal-agent model with moral hazard in which the technology the vector of probability distributions from the agent s actions to the possible outcomes is initially unknown. The principal offers a menu of contracts. After this offer, information about the technology is obtained and the principal selects the operative contract from the menu. We consider that information is null, private to the principal, or public. We show that: (i) if with private information, the principal conditions the action on the signal, then she strictly prefers both public to private information and private to null information; (ii) if with public information, the principal does not condition the action implemented on the signal, then she prefers both null and private to public information; (iii) the value of better information can be non-monotonic both with private and with public information; and (iv) the value of better information can be greater either with private or with public information. KEYWORDS: Moral Hazard, Ex Ante Contracting, Informed Principal, Technology, Value of Information. JEL Classification: D82, D86. I would like to thank Hector Chade and Eddie Schlee, for their helpful comments and suggestions. The paper also benefitted from comments by seminar participants at Arizona State University, the University of New South Wales, the University of Melbourne, and Deakin University.
1 Introduction In a principal-agent model with moral hazard, the principal offers an incentive scheme to an agent in order to induce him to choose an unobservable action; the action she wants to implement depends upon several parameters, which are common knowledge in the standard model. Several studies have extended the standard model to allow the agent or the principal to have private information before the incentive scheme is offered. 1 Others have compared situations where one or both players obtain information after the incentive scheme has been offered. They consider how the principal, the agent, or both obtaining information and at what stage(s), affect the expected costs to provide incentives and thereby the actions implemented. However, they do not consider how these different expected costs induce the principal to alter the action implemented when doing so affects the expected utilities of the incentive schemes. 2 That is, there are situations in which, after the agent accepts the incentive scheme, the principal obtains information that may induce her to implement different actions, possibly yielding the agent different expected utilities. For instance, when a firm hires a manager to head operations in a new market, the firm may not have information about local demand and the returns to the manager s different actions. Also, in the Reserve Officer Training Corps, cadets enlist and are committed to the corps. Their assignment upon graduation depends upon the needs of the military four years after enlistment, needs that were uncertain when the cadet enlisted. 3 Additionally, government procurement contracts often involve the winning firm having to modify the project as more information is obtained. In order to understand the effects of different distributions of information in these situations, we consider the technology to be unknown at the time the principal offers the incentive scheme. The technology the vector of conditional probability distributions from the agent s unobservable actions to the observable outcomes determines both the principal s expected revenues and expected 1 For the former, see for example Holmström (1979), Myerson (1982), and Sobel (1993); for the latter, see for example Myerson (1983), Maskin and Tirole (1992), Inderst (2001), and Chade and Silvers (2002). 2 See for example Lewis and Sappington (1997), Lizzeri et al (2002), Nafziger (2009), and Ederer (2010). In each, the agent is risk neutral and the payments are set solely to provide incentives to choose a particular action consequently, there is no tradeoff between providing incentives versus utility. 3 During their training periods, they select preferences of job assignment (e.g., aviation or military intelligence) and undergo interim evaluations. Moreover, pay depends upon job and location; promotion and career prospects depend upon these and upon how hard the cadet works. 1
costs from implementing an action, and the agent s expected utility from choosing an action. If the agent accepts the incentive scheme, then the principal subsequently observes a noisy signal that is correlated with one of two possible technologies that is in effect. We call this timing ex ante contracting; with it, the incentive scheme is a menu of contracts, each of which specifies a payment from the principal to the agent contingent upon the outcome. This menu is similar to the menu in Maskin and Tirole (1992) and Laffont and Martimort (2002, Chapter 9.1) except that the contracts induce hidden action by a risk averse agent. We consider three possibilities regarding the noisy signal: the posterior belief after observing the signal equals the prior belief (null information); and if the posterior belief does not equal the prior, then either only the principal observes the signal, and then she makes an announcement of it to the agent (private information), or the agent also observes the signal (public information); additionally, we consider better information (greater correlation between this signal and the technology). We perform two sets of comparative statics about information: (i) null versus private versus public information; and (ii) better private and better public information. Importantly, if information is private, then in addition to the moral hazard incentive constraints for the agent, there are hidden-information ones for the principal to truthfully announce the signal she observed these assure the agent that he uses the right probability distributions to evaluate the expected utility of each action. We show that: (i) if with private information the principal conditions the action implemented on her announcement, then the principal prefers both public to private information, and private to null information; (ii) if with public information the principal does not condition the action implemented on the signal, then the principal is indifferent between null and private information, and prefers either to public information; (iii) both the value of better private and better public information can be non-monotonic; and (iv) when the principal conditions the action implemented on her announcement (or the signal), the value of better information may be greater either with private or with public information. The second point implies that the value of public information vis-á-vis null information can be negative, even if the principal conditions the action implemented on the signal. With ex ante contracting, individual rationality implies that only the expected utility of the menu, not of each contract, must be at least the agent s reservation utility. The principal is thus able to provide him different expected utilities from the different contracts i.e., she may want 2
to tradeoff utility provision as each signal affects her expected cost to provide incentives for the agent to choose an action. Alternatively, if she does not want to condition the action on the signal, then she prefers to implement the action with the same contract regardless of the signal i.e., if possible, she trades off incentive provision by committing to a more expensive contract given one signal but a less expensive contract given the other signal. The timing and symmetry of information together determine whether the principal can insure against the possibility of having a higher expected cost to implement an action and whether or not the agent can hold pessimistic beliefs about which technology the principal has. Consequently, they affect the tradeoff between providing incentives and utility to the agent, and thereby affect the expected cost to implement the profile of actions. Thus, our work adds to the growing literatures on the timing of information, the symmetry of information, and the value of better information. Regarding the timing of information, by obtaining information sooner, a principal can tailor the action that she induces the agent to take to the state of the world. However, this often makes it more costly to induce the agent to undertake any action, particularly if the information is public see, e.g., Lizzeri et al (2002), Nafziger (2009), and Ederer (2010). As in these papers, our results show that the principal prefers information to arrive later; however, this arises even when the agent observes the signal before he chooses his action. Moreover, our results arise for a different reason; rather than save on incentive provision, the principal is able to reduce cost by trading off utility provision. The timing in our model is similar to that in both Demski and Sappington (1987) and Laffont and Martimort (2002, Chapter 9.1) an uninformed principal offers an uninformed agent a contract, then a signal is observed and then the agent takes an action. However, Demski and Sappington (1987) consider the agent, not Nature, choosing the information structure, and the signal is private to the agent, not either private to the principal or public. Closer to our model is Laffont and Martimort (2002, Chapter 9.1), who also consider a model of ex ante contracting and contrast private with public information; however, in their model, the possible states of the world satisfy a single-crossing property and the agent chooses a contractible production level rather than an unobservable action. In their model, compared to public information, with private information, only the action (the production level) for the higher type is distorted and always in the same direction, the expected utilities provided are always adjusted in the same direction, and the principal strictly prefers public information. In our model, by contrast, there are 3
possible distortions in the action implemented in either direction for either type, the principal can adjust the expected utilities from each contract in either direction, and she may prefer private to public information. Regarding the symmetry of information, in contrast to the model we examine, suppose that the principal has private information when she offers the menu. Myerson (1983) showed that the principal does best by withholding the private information until after the agent either accepts or rejects the menu; nevertheless, as Maskin and Tirole (1992) showed, a privately informed principal may not attain her complete information payoff. These results arise because the principal offers the menu after she has obtained information; thus, the agent may hold pessimistic beliefs that preclude certain menus from being offered. As such, the principal generally prefers public information (see also Mezzetti and Tsoulouhas, 2000). In contrast with these, in our model the principal may prefer private to public information, even complete public information. Lastly, Gjesdal (1982, Proposition 3) showed that the value of worse information can be positive for the principal. This arises when imposing risk on the agent allows her to implement a higher action at the same expected cost. 4 Gjesdal considered information that is related to the action the agent will choose; i.e., better information yields greater control over the agent s action, which is endogenous. We obtain a similar result, but in a different context; when the information is related to the principal s technology, which is exogenous, worse information may decrease her expected cost to implement the same action. Interestingly, in both, she offers a random incentive scheme to a risk averse agent. In Gjesdal, the randomization occurs after the outcome has been realized, whereas in our model, it occurs before the agent chooses his action. The paper is structured as follows: In Section 2, the model is laid out. The results are presented in Section 3, first for private information and then for public information. In Section 4, we examine the robustness of the results. In Section 5, we discuss two features that are central to our framework: the timing of contracting and the importance of commitment. Section 6 concludes. 4 Although the agent is risk averse, if his coefficient of absolute risk aversion is decreasing as he chooses a higher action, then it is possible that, knowing that his payment will be randomized after the outcome is realized, he chooses a higher action, which yields the principal greater benefits, with which she can compensate the agent for his increased risk, and both are better off. 4
2 Model 2.1 Preferences, Actions, and Technologies The agent is a risk-averse expected-utility maximizer with additively separable von Neumann- Morgenstern utility function over payment and action, given by U(I,a) = V (I) a, with V (I) > 0,V (I) < 0; I such that lim I I V (I) =. He chooses an action a {a 1,...,a M } = A where 0 < a 1 < a 2 <... < a M < and M 2. The action specifies a probability distribution over the possible outcomes q n {q 1,...,q N } where 0 < q 1 < q 2 <... < q N < and N 2. The technology is the vector of probability distributions for each of the agent s actions. The principal is risk-neutral and offers the agent a menu of contracts. She is endowed with one of two technologies, Π 1 or Π 2. For k {1,2}, Π k = {π k (a 1 ),...,π k (a M )} where π k (a) is the conditional probability distribution; thus, π kn (a) is the conditional probability that q n is realized. Since we are interested only in the consequences of information about the technology, we make no assumption about the ordering of technologies. The results hold whether one technology is preferred to the other only for some actions or for all actions. 5 Let λ (0,1) be the prior probability that the principal has Π 1. 2.2 Information Nature sends a signal, z k {z 1,z 2 } correlated with the technology. For both technologies, there is a probability distribution described by the single parameter ζ [0.5, 1], which is the conditional probability that z k is sent if she has Π k. This pair of probability distributions is called an information structure. If ζ = 0.5, then the principal and agent have null information. If ζ > 0.5, then, if only the principal observes the signal, then she has private information, but if both she and the 5 Nevertheless, two rankings have economic interpretations that fit our model. First, if Π 1 is generated from Π 2 by a mean-preserving spread in the likelihood ratio distribution function, then the principal prefers Π 1 to Π 2 in that she can implement any action at a lower expected cost (Kim, 1995); and second, if Π 1 first-order stochastically dominates (FOSD) Π 2, then she prefers Π 1 to Π 2 in that each action yields her greater benefits. In an employer-employee context, regarding the former, Π 1 could correspond to a less volatile environment, so that, loosely speaking, the contractible outcome more accurately reflects the action that the agent chose; as to the latter, Π 1 could correspond to a more productive employee who is more likely to realize a greater output than is Π 2. It is not uncommon that a worker does not know when the menu is offered the volatility of the environment or his own productivity, but the employer can become more certain of either, by accumulating economic data, or by conducting interviews and interim evaluations. 5
agent observe it, then they have public information. Lastly, as ζ increases, information is better. Now, define the following three environments: Null Information: ζ = 0.5; Private Information: ζ (0.5,1] and only the principal observes z k ; and Public Information: ζ (0.5,1] and both the principal and agent observe z k. After she observes z k, the principal updates her beliefs, denoted by ρ k, about the technology. By Bayes rule, ρ 1 = λζ Pr(z 1 ) is the probability that the principal has Π 1 conditional upon observing z 1, where Pr(z 1 ) = λζ + (1 λ)(1 ζ) is the probability of observing z 1 ; similarly, ρ 2 = λ(1 ζ) Pr(z 2 ) is the probability that the principal has Π 1 conditional upon observing z 2, where Pr(z 2 ) = λ(1 ζ)+ (1 λ)ζ is the probability of observing z 2. The agent also updates his beliefs, denoted by α, about the technology. Let θ k denote the principal s type. When her beliefs are ρ k, denote by p(a;ρ k ) = ρ k π 1 (a)+(1 ρ k )π 2 (a) the expected conditional probability distribution if the agent chooses a. The expected conditional probability that q n is realized is p n (a;ρ k ). Similarly, let p(a;α) and p n (a;α) denote the agent s expected conditional probability distribution and probability that q n is realized. 2.3 Timings Our objectives are to compare and contrast the resulting Perfect Bayesian Equilibria (PBE), both between Null, Private, and Public Information, and within Private and Public Information as information is better. These comparisons are made at the offer date the stage, prior to the signal being observed, when the principal offers the agent a menu of contracts. Each contract in the menu specifies a payment from the principal to the agent contingent upon the realized outcome. The operative contract is determined either by the principal s announcement (Private Information) or by the publicly observed signal (Public Information). These environments are summarized in the following timelines. In each, if the agent rejects the menu, then the game ends and the agent receives Ū while the principal receives 0. For k,k {1,2}, where k may or may not equal k, and N denotes Nature, the timings are as follows: 6
P updates ρ k A updates α = α(k ) P offers menu A accepts or rejects menu N selects Π 1 or Π 2 N sends z k P announces z k Private Information A chooses a N selects q n Payoffs made Public Information P updates ρ k A updates α = α(k) = ρ k Note that in Public Information, α = ρ k. However, in Private Information, α = α(k ) depends upon the principal s possibly false announcement z k of the signal she observed, so that α(k ) = ρ k ; by the revelation principle, we can restrict attention to PBE in which the principal s announcement is truthful. The timing of Null Information can be given by either, as ρ k = λ so that α = λ for both z k. 2.4 Contracts and Constraints If the principal announces z k (Private Information) or z k is observed (Public Information), let I n (k) denote the outcome-contingent payment from the principal to the agent, with I n (k) [I, ) n {1,...,N}. Then I(k) = {I 1 (k),...,i N (k)} is the contract, and {I(1),I(2)} is the menu. Formally, a menu is a mapping from {z 1,z 2 } [I, ) n [I, ) n. We say that a contract I(k) implements a if it is incentive compatible for the agent to choose a i.e., a satisfies a argmax N n=1 p n(a;α)v (I n (k)) a or a A (1) ã N [p n (a;α) p n (ã;α)]v (I n (k)) a ã. n=1 The contract I(k) that implements a yields the agent an expected utility denoted by u k = EU(I(k)) = N n=1 p n(a;α)v (I n (k)) a. Note that the constant payment Ī(u k) = V 1 (u k + a 1 ) is the least-cost contract that implements a 1 and provides the agent expected utility u k. By making the acceptance/rejection decision prior to learning anything about the principal s type, only the menu, not each contract, must yield the agent expected utility at least Ū. If it does, then the menu is individually rational: 7
(2) Pr(z 1 )u 1 + Pr(z 2 )u 2 Ū. Denote by B k (a) the expected benefit (revenue) for θ k from implementing a: B k (a) = N n=1 p n(a;ρ k )q n. θ k s expected profit from implementing a 1 is B k (a 1 ) Ī(u k). Similarly, for any k,k {1,2}, denote by C k (I(k );a) the expected cost for θ k to implement a with I(k ): C k (I(k );a) = N n=1 p n(a;ρ k )I n (k ). In Private Information, the agent does not observe the signal; thus, the principal can announce a different signal than she observed. The menu is incentive compatible for the principal if, for each z k, the principal has the incentive to truthfully announce z k. θ k if announces truthfully if and only (PIC kk ) B k (a ) C k (I(k );a ) B k (a) C k (I(k);a), where I(k ) implements a and I(k) implements a. A menu is incentive compatible if it is both incentive compatible for the principal and both contracts are incentive compatible for the agent. Finally, a menu is feasible if it is both individually rational and incentive compatible. 2.5 Equilibrium Menus We call a pair of actions that the contracts implement, {a(z 1 ),a(z 2 )}, an action profile. If a(z 1 ) = a(z 2 ) then we say that the action profile is constant, else it is non-constant. Note that it is always feasible for the principal to implement a constant action profile. For a given action profile, the principal selects the feasible {I(1), I(2)} that solves the following in Public Information: (Public Info Prog) min I(1),I(2) N N Pr(z 1 ) p n (a;ρ 1 )I n (1) + Pr(z 2 ) p n (a;ρ 2 )I n (2) n=1 n=1 subject to: I(1) implements a(z 1 ),I(2) implements a(z 2 ), and (2), and in Private Information: 8
(Private Info Prog) min I(1),I(2) N N Pr(z 1 ) p n (a;ρ 1 )I n (1) + Pr(z 2 ) p n (a;ρ 2 )I n (2) n=1 n=1 subject to: I(1) implements a(z 1 ),I(2) implements a(z 2 ), For each program, the solution yields C 1 (I(1);a(z 1 )) and C 2 (I(2);a(z 2 )). PIC 12,PIC 21, and (2). Then the principal selects the action profile that yields the greatest expected profit; i.e., {a (z 1 ),a (z 2 )} argmax Pr(z 1 )[B 1 (a(z 1 )) C 1 (I(1);a(z 1 ))]+ {a(z 1 ),a(z 2 )} {A A} Pr(z 2 )[B 2 (a(z 2 )) C 2 (I(2);a(z 2 ))], where a (z k ) denotes the action that the principal implements in the profit-maximizing profile if she observes z k and the menu is feasible. In Public Information, let I (k) denote the corresponding least-cost contract in this menu; I (k) yields the agent expected utility u k solution to (Public Info Prog) is then a menu {I (1),I (2)}. = N n=1 p n(a (z k );α(k))v (I n (k)) a (z k ). The Note that with ex ante contracting, the principal can provide more utility than Ū from one contract and less from another; when she does, we say that the principal trades off utility provision. It is useful to denote the cost-minimizing contract for which u k = Ū, by I (k), and when also the agent s beliefs α = λ, by I (λ). In Private Information, we assume that, for the action profile under consideration, I (1) cannot be part of a separating menu since PIC 12 would be violated. 6 Clearly, if {I (1),I (2)} with I (1) I (2) is a feasible menu in Private Information, then private information has no consequences. Let Î(1) denote the corresponding least-cost contract in the profit-maximizing (and feasible) menu. Î(1) clearly cannot satisfy the same incentive compatibility constraints with strict equality that I (1) satisfies (note that ρ 1 > ρ 2 ). The separating solution to (Private Info Prog) is then {Î(1),I (2)}. Note that {I (λ),i (λ)} is a pooling solution to (Private Info Prog). Because the principal solves a minimization problem and, at the offer date, she has no private information, the agent does not update his beliefs at this stage; thus, any menu yields each player a 6 Because the technology does not satisfy the single-crossing property, it is also possible that I (2) cannot be part of a menu since PIC 21 could be violated. 9
particular expected payoff (rather than a non-degenerate set of expected payoffs). It is possible that the principal is indifferent between two menus, but then they would yield her the same expected profit, and, as Lemma 1 below shows, the agent is also indifferent. The characteristics of the possible equilibrium menus are relegated to Appendix 7.1. 3 Results 3.1 Preliminaries Throughout, we maintain the following assumption: 7 Assumption 1 The minimum payment from each contract is strictly greater than I. We first show that, with ex ante contracting, the agent is indifferent between the environments since if a menu ever yielded more than Ū, there exists a less expensive menu that implements the same action profile. Proofs of all lemmas and propositions are in Appendix 7.2. Lemma 1 Agent s Expected Utility For any environment, every equilibrium menu yields the agent expected utility exactly Ū. Assumption 1 is critical to Lemma 1. Suppose that one contract, I(k), had a minimum payment equal to I. Then this contract could provide more than Ū at least cost because in order to be incentive compatible for the agent, the contract can only become high-powered enough by increasing some payments without a compensating decrease in other payments. Nevertheless, the agent may still receive only Ū, if the other contract can be adjusted to provide less than Ū. Only if in this other contract, I(k ), the minimum payment also equals I or if PIC kk binds, could the agent receive more than Ū from the equilibrium menu. The primary impact of Lemma 1 is that the principal s preferences for null, private, or public information, or for better or worse information, depend upon how the different information affects the expected costs to implement action profiles and, thereby, the desirable action profile. Without Lemma 1, the profit-maximizing menu may cede the agent rents; moreover, those rents can differ 7 This assumption holds as long as, for each contract, the principal is able to make the contract more high-powered not only by increasing some payments, but also by decreasing the smallest payment i.e., no minimum payment or limited liability constraint binds. 10
across environments, or with better information, thereby obscuring the tradeoff between incentive and utility provision. Since we can focus on how different environments and better information affect the principal, intuitively, our results are consequences of the following four effects: (i) better public information increases the expected cost to implement an action; (ii) private rather than public information can either increase the expected cost by imposing a signaling cost on one type, or decrease the expected cost by allowing the principal to trade off incentive provision between her types; (iii) both with private and with public information, the principal may implement different actions due to these changes in the expected costs, thereby possibly decreasing her expected revenue; and (iv) the principal can reduce her expected cost by trading off utility provision between the two contracts. In order to see the first, note that before the offer date, if information is public, the principal faces a gamble over the possible expected costs to implement an action. 8 Even though the principal is risk neutral, this gamble has a negative expected return since the expected cost reduction upon observing z 1 and implementing a with a less high-powered contract is smaller than the expected cost increase upon observing z 2 and implementing a with a more high-powered contract. As Lemma 1 shows, the agent receives the same expected utility from any menu. Thus, any separating menu imposes a mean-utility preserving increase in risk on the risk averse agent; such a gamble has a higher expected cost (see Diamond and Stiglitz, 1974). In other words, the agent faces a gamble over both the contract and the outcomes; this gamble is a spread of the contract over only the outcomes, which is the pooling contract that he could receive if the agent s posterior beliefs equal his prior beliefs. Consequently, if the principal implements a constant action profile, then she does so at a lower expected cost by trading off incentive provision i.e., she insures herself against having a higher expected cost to implement that action by offering a pooling menu, if such a pooling menu is feasible, which it is in Private Information but not in Public Information. Lemma 2 Expected Cost: Null vs. Public Information If the principal implements a constant action profile, then the expected cost of the menu {I (1),I (2)} with probabilities Pr(z 1 ) and Pr(z 2 ) is greater than the expected cost of a pooling menu with I (λ). 8 If the technologies are related as in Kim (1995), then C 1(I (1); a) < C 2(I (2); a) holds for all a. All we need is that these expected costs are unequal for the action under consideration (see also Grossman and Hart, 1983, Proposition 13). 11
In our comparisons that follow, a constant action profile serves as a baseline. We do not seek to characterize conditions on primitives that induce the principal to implement a constant or a non-constant action profile. Rather, we seek to determine the consequences both of private and public information and of better information, for any action profile, and how these affect the action profile that the principal implements. 3.2 Private Information In a standard principal-agent contracting game with moral hazard, the contract that the principal offers determines the action the agent chooses; when the principal has private information before she offers the contract, the principal-agent contracting game becomes a signaling game in which the contract also signals the principal s private information to the agent (and thereby may affect his incentives). Our ex ante contracting game with private information differs from this signaling game in that, while the principal (the privately informed sender) selects the contract (the signal), the effective signal space (the menu) is endogenous rather than exogenous. That is, although θ k announces an exogenous signal z k or z k, her announcement determines the operative contract; the important distinction between ex post and ex ante contracting is that in the latter the principal proposes and the agent accepts or rejects the feasible set of contracts i.e., the effective signal space. The timing differs from that in Maskin and Tirole (1992) importantly in that in their paper, the principal is informed, rather than uninformed when she offers the menu. They showed that, as a consequence, the agent may hold pessimistic beliefs that prevent the principal from realizing her complete information payoff; 9 whereas, in our current setting, the agent cannot hold pessimistic beliefs when she offers the menu. 10 Then, in contrast with Maskin and Tirole (1992), that the agent s beliefs are the same as the principal s at the offer date allows the principal to do at least as well with private as with null 9 In their model, the principal and agent each have private information about different parameters; however, the agent makes his report simultaneously with the principal s selection of the contract, so that this selection cannot influence the agent s incentives. Nevertheless, there exist equilibria in which the principal offers a menu that is more expensive than that with complete information. The agent s out-of-equilibrium beliefs for menus that yield her her complete information payoff are such that the principal is believed to be the low type; this prevents the principal from offering this menu since the agent would reject it. 10 After the principal selects the contract by making her announcement, the agent updates his beliefs, which may induce him to choose an action different from what he would choose if information were public. 12
information. 11 Essentially, if the principal implements a constant action profile, then she pools if this is feasible, which it is if she has private information. Proposition 1 Null vs. Private Information If the principal implements a constant action profile in Null Information, then she weakly prefers Private Information to Null Information; if she implements a non-constant action profile in Private Information, then the preference is strict. For example, consider a firm that wants to transfer a manager, who is currently contracted with the firm, to head operations in a new market. It will pay to learn the value of higher actions by the manager, if it would want the manager to choose different actions (e.g., if the returns depend upon the sensitivity of demand), even if the manager would not know the value of his actions. In the next subsection, we show that the firm would gain even more if the manager would learn this. In Null Information, the principal implements a constant action profile with the menu {I (λ),i (λ)}. In Private Information, this same menu also implements the same constant action profile because the agent s beliefs after her announcement are still α( ) = λ. Recall that, before she observes the signal, the principal faces the risk of observing the signal correlated with the technology that has a greater expected marginal cost of utility provision; she avoids this risk by trading off both incentive and utility provisions between the contracts. If the principal chooses to implement a non-constant action profile, then she is necessarily better off. Intuitively, for each signal and action implemented, there is a cost-minimizing contract that yields the agent expected utility u k. The expected cost of this contract is increasing in u k. For a given action profile, the principal minimizes the expected cost to provide Ū, by comparing the expected marginal costs of utility provision for each contract; the lower is the expected marginal cost of utility provision to implement an action given a signal, the more expected utility that the principal provides in the corresponding contract. Consider now Private Information and the impact of better private information. Gjesdal (1982) defined two ways that better information can benefit the principal: (i) it can allow better risksharing and so have marginal insurance value; and (ii) it can induce the principal to implement a different action profile, and so have marginal incentive informativeness. 11 This is true so long as the principal does not implement a non-constant action profile with null information. 13
The marginal insurance value of better private information in our model depends upon two factors: the changes to the probabilities of observing each signal, and the expected marginal costs of utility provision from each signal. Fundamentally, better information about an exogenous state of the world is independent of the probability with which a signal is observed; rather, the value of better information is a function of how a signal affects the likelihood ratios of different states. Thus, we fix the probabilities of observing each signal. This leaves the changes in the expected marginal costs of utility provision for a given action profile. Since the expected cost to implement one action and provide the same expected utility increases with better information, the expected cost to implement an action profile may increase, even though the principal can further adjust the expected utilities that each contract provides. As to marginal incentive informativeness, if with worse information she implements a constant action profile, then either she still does with better information so that there is no marginal incentive informativeness, or she implements a non-constant action profile with a higher action upon observing one signal. 12 In Private Information, the principal implements either a constant or a non-constant action profile. First, consider that she implements a constant action profile with worse information. To be clear about the marginal insurance value of better private information, note that Lemma 2 implies that, when λ = 0.5: (3) 0.5C 1 (I (1);a) + 0.5C 2 (I (2);a) > C λ (I (λ);a). The left-hand side equals the principal s ex ante expected cost of implementing a since Pr(z 1 ) = Pr(z 2 ) = 0.5. Not only is the expected cost of utility provision for a given type convex in the amount of expected utility provided, but also (3) shows that the ability of the principal to trade off utility provision still does not reduce her expected cost below that of the pooling menu. 13 We are now ready to state our second main result, which does not require that λ = 0.5. 12 As in the proof of Proposition 1, simple algebra shows that if Pr(z 1 ζ) = Pr(z 2 ζ) = 0.5 and a(z 1) = a(z 2), then Pr(z 1 ζ)b 1 ζ (a(z 1)) + Pr(z 2 ζ)b 2 ζ (a(z 2)) is constant with respect to ζ; otherwise, the expected revenue from {a(z 1), a(z 2)} may increase or decrease with ζ. 13 The agent s incentive compatibility constraints become easier to satisfy for one type but more difficult to satisfy for the other type. These generate a less (respectively, more) high-powered incentive scheme, but the latter effect dominates the former effect. 14
Proposition 2 Value of Better Private Information Consider Private Information. For 1 ζ > ζ, if the principal implements a constant action profile with ζ, then her equilibrium expected profit with ζ is weakly greater than that with ζ. As in the analysis following Proposition 1, when the principal obtains private information, she cannot do worse than implementing a constant action profile, so that better information increases her expected profit if it induces her to implement a non-constant action profile. Now, consider that the principal implements a non-constant action profile with worse information. Note that she must offer a separating menu {Î(1),I (2)}, where Î(1) dissuades θ 2 from mimicking. 14 The value of better private information may be negative. This could follow if either her expected cost to implement the same action profile increases i.e., the marginal insurance value is negative, or she implements a lower action upon observing one of the signals i.e., the marginal incentive informativeness is negative. Fixing the probabilities of observing each signal, the expected cost to implement the non-constant action profile is 0.5C 1 (Î(1);a(z 1 )) + 0.5C 2 (I (2);a(z 2 )). To see how the marginal insurance value can be negative i.e., how the value of this expression increases with ζ, note that an increase in ζ causes a spread of ρ k. If the principal does not adjust the expected utilities that each contract provides, then C 2 (I (2);a(z 2 )) increases. Additionally, C 1 (Î(1);a(z 1 )) may also increase with better information if the principal incentive compatibility constraint becomes more difficult to satisfy. 15 That the principal can further adjust the expected utilities each contract provides, mitigates but need not overcome these effects. Thus, the expected cost to implement the same non-constant action profile may increase or decrease. Second, the marginal insurance value can be negative as well; any change in the expected cost to implement an action profile may induce her to implement a different action profile. However, since the expected cost to implement either action in the profile may increase, she may implement a lower action upon observing either signal, thereby lowering her expected revenue. 14 Since we assume no ordering of the technologies, this is without loss of generality; i.e., for some other action profile, it may be θ 2 who must dissuade θ 1 from mimicking. 15 This can happen because, although an increase in ρ 1 and a decrease in ρ 2 each lower the signaling cost, which is smaller as pn(a;1) p n(a;2) 1 increases, θ2 would have more to gain, and therefore would mimic some contracts that she would not with worse information. 15
4600 profit 4400 a 3, a 2 B Λ a 2 C Λ I Λ ; a 2 4000 3800 a 2, a 2 a 3, a 1 3600 Ζ 0.5 0.6 Ζ PrI 0.8 0.9 1 Figure 1: Principal s Expected Profit Functions In Private Information: Principal Pools for ζ [0.5, ˆζ PrivInf ) These effects are evidenced in Figure 1, which depicts the principal s expected profit in Example 1 as a function of ζ. The red, green, and blue expected profit functions correspond to the action profiles {a 2,a 2 }, {a 3,a 2 }, and {a 3,a 1 }; the solid portions indicate the upper envelope of these expected profit functions. As ζ increases, when the marginal incentive informativeness first exceeds the marginal insurance value (if this is negative) plus the signaling cost, the principal implements a non-constant action profile. Denote the smallest value of ζ at which the principal is indifferent between implementing a constant action profile and a non-constant action profile, by ˆζ PrivInf (0.673 in the example). 16 The value of better information is the slope of this function, except at ζ = 0.5 for ζ = 0.5, an increase in ζ not only yields better information, but also makes it asymmetric. Example 1 Non-Monotonic Value of Better Private Information The agent has log utility, disutility of action a {0.8,2,5}, and reservation utility Ū = 1. The.7.25.05 revenues are q = {250,4000,12000}. Let the technologies be given by Π 1 =.4.3.3 and.1.3.6 16 It is possible for the principal to implement a constant action profile for some ζ > ˆζ PrivInf, since the expected revenues change at a constant rate for any action profile, but the difference in expected costs between alternative menus may rise or fall with ζ. 16
.6.3.1 Π 2 =.5.3.2, with λ = 0.5. In Private Information, the principal maximizes expected profit.4.35.25 by implementing: {a 2,a 2 } for ζ [0.5, ˆζ PrivInf ] her expected profit is 4159.19; {a 3,a 2 } for ζ [ˆζ PrivInf,0.941] her expected profit monotonically increases to 4427.42 at ζ = 0.86 and then monotonically decreases, reaching 4338.98 at ζ = 0.941; and {a 3,a 1 } for ζ [0.941,1] her expected profit monotonically increases to 4614.81 at ζ = 1. 17 To complete the characterization of the principal s expected profit function, consider the possibility that the principal implements a non-constant action profile {a(z 1 ),a(z 2 )} at ζ = 0.5 with a(z 1 ) a(z 2 ). This could be profit-maximizing since she can implement {a(z 1 ),a(z 2 )} at a lower expected cost than the expected cost of the two constant action profiles {a(z 1 ),a(z 1 )} and {a(z 2 ),a(z 2 )}, but the expected revenues are the same. The possibly lower expected cost arises because the principal can trade off utility provision. If she gains by doing so, then there would then be a downward jump discontinuity in the expected profit function, since for any ζ > 0.5, one of the principal incentive compatibility constraints would bind, resulting in a discrete increase in expected cost. Lastly, note that three actions is minimal in order for the value of better information to be non-monotonic. If there were only two actions, then the principal implements a 1 with Ī(u k) which does not increase in cost with ζ. To summarize this subsection, if the principal implements a constant action profile with ζ 0.5, the marginal incentive informativeness is non-negative. Even though both the marginal insurance value is negative and possibly exacerbated by an increase to ζ, the principal would not have to pay more in expectation, unless the value due to marginal incentive informativeness exceeds the increase in expected cost. Alternatively, if she were to implement a non-constant action profile with ζ, then both the marginal insurance value and the marginal incentive informativeness of an increase to ζ can be positive or negative; moreover, the expected cost of separating can decrease or increase. 17 Solutions were computed for ζ [0.5, 1] incrementing by 0.01. The values of ζ at which the principal changes the profit-maximizing action profile are rounded to the nearest thousandth. 17
3.3 Public Information The previous subsection considered private information. Private information has two effects: first, if the principal does not want to condition the action implemented on the signal, then private information allows her to avoid imposing the mean-utility preserving increase in risk; but second, if she does want to condition the action implemented on the signal, then private information entails a signaling cost. Thus, the principal may prefer private to public information or vice versa. Unlike with private information, with public information the principal is precluded from offering the pooling menu it is no longer incentive compatible for the agent to choose a when he observes one signal, because his posterior beliefs would equal the principal s so that (1) is violated if z 2 is observed. That is, with public information, the principal cannot completely insure herself against having a higher expected cost to implement the action by trading off incentive provision. Thus, if she implements a constant action profile, public information has negative value for the principal; i.e., she would prefer that information had been private. However, if she implements a non-constant action profile, then, although the separating menu with private information implements the same action profile with public information, she does best by not having to pay a signaling cost and there is no loss from being unable to trade off incentive provision since this is impossible when she separates with private information. With public information, as ζ increases, the principal implements a non-constant action profile at a lower value of ζ than when she has private information because the expected cost of a separating menu is lower, but the expected revenue is the same as when she has private information. That is, ˆζ PubInf < ˆζ PrivInf, where ˆζ PubInf > 0.5 is the smallest value at which the principal is indifferent between implementing a constant versus a non-constant action profile with public information. 18 This lower expected cost of a separating menu also implies that, for any value of ζ for which the principal is better off by separating when she has private information, she certainly is better off by separating when information is public. We have then the following proposition: Proposition 3 Private vs. Public Information (a) If the principal implements a constant action profile with a > a 1 in Public Information with ζ, then she strictly prefers both Null Information and Private Information with ζ, to Public 18 Note also that the principal may implement a constant action profile for some ζ > ˆζ PubInf. 18
Information. (b) If the principal implements a non-constant action profile in Private Information with ζ, then she strictly prefers Public Information with ζ to Private Information with ζ. In other words, the principal: (i) would pay to keep her private information private if she would implement a constant action profile; (ii) would pay to acquire private information if and only if she would implement a non-constant instead of a constant action profile; and (iii) in that case, would pay even more to share this information with the agent. We are now ready to examine the value of better public information about the technology. We extend Example 1 to Public Information: Example 2 Non-Monotonic Value of Better Public Information Let all primitives be the same as in Example 1. In Public Information, the principal maximizes expected profit by implementing: {a 2,a 2 } for ζ [0.5, ˆζ PubInf ] her expected profit monotonically decreases from 4159.19 to 4158.74; {a 3,a 2 } for ζ [ˆζ PubInf,0.647] her expected profit monotonically increases to 4489.38; and {a 3,a 1 } for ζ [0.647,1] her expected profit monotonically increases to 5410.33 at ζ = 1. Figure 2 depicts the principal s expected profit functions from Example 2. As before, the red, green, and blue expected profit functions correspond to the action profiles {a 2,a 2 }, {a 3,a 2 }, and {a 3,a 1 }; the solid portions indicate the upper envelope of these expected profit functions. Note that: (i) ˆζ PrivInf > ˆζ PubInf = 0.539; (ii) although it is not optimal to implement {a 3,a 2 } for ζ > 0.647, the expected profit function for this action profile is non-monotonic, increasing to 4704.00 at ζ = 0.82 and then decreasing to 4253.19 at ζ = 1; (iii) the expected profit function for {a 2,a 2 } is everywhere monotonically decreasing, evincing the negative marginal insurance value of better public information; and (iv) when ζ rises above 0.647, the increase in the principal s expected cost of utility provision to implement a 2 upon observing z 2 induces her to implement a lower action, evincing the possibility of negative marginal incentive informativeness of better public information. Lastly, we compare and contrast the values of better private with better public information. In Figure 3, the expected profit functions for each of the three action profiles are compared for 19
profit 5400 5200 5000 a 3, a 1 4800 4600 a 3, a 2 4400 4200 a 2, a 2 4000 Ζ 0.5 Ζ PuI 0.6 0.7 0.8 0.9 1 Figure 2: Principal s Expected Profit Functions In Public Information: Principal Pools for ζ [0.5, ˆζ PubInf ) Private Information (brighter) and Public Information (darker). For ζ > ˆζ PrivInf, though the principal prefers Public to Private Information, the value of better information given by the slope of the principal s expected profit function may be larger or smaller if it is private than if it is public, since the signaling cost may increase or decrease. Moreover, comparing the expected profit functions for ζ [0.5, ˆζ PubInf ], the principal prefers Null to Public Information. That is, the value of better public information may be negative for ζ near 0.5. If the principal implements a non-constant action profile with public information, then although she strictly prefers public to private information, she may still have preferred null to public information. The negative value of better public information in panel (a) of Figure 3 illustrates that the principal obtaining private information is crucial to her gaining by trading off incentive provision, since with public information she is unable to do this. Just prior to Example 1, we noted two effects of a marginal increase in ζ when ζ = 0.5 and the signal is private to the principal. We were unable to determine the directions and magnitudes of these two effects: that due to better information for the principal versus that due to the induced asymmetry of information. We can now disentangle them. Having private information is beneficial if the principal implements a constant action profile because she is able to avoid the gamble on incentive provision; however, having private information is harmful if the principal implements a 20