Faculty of Business and Law SCHOOL OF ACCOUNTING, ECONOMICS AND FINANCE School Working Paper - Economic Series 2006 SWP 2006/24 The Value of Symmetric Information in an Agency Model with Moral Hazard: The Ex Post 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 Please do not quote without obtaining the author s consent as these works are in their draft form.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.
The Value of Symmetric Information in an Agency Model with Moral Hazard: The Ex Post Contracting Case Randy Silvers School of Accounting, Economics and Finance Deakin University October 2006 Abstract In a principal-agent environment with moral hazard, when contracting occurs after the principal receives information about her technology, the principal cannot insure against the possibility that the technology is less informative. From an ex ante perspective, we show that: (i) the principal is worse off by acquiring private information if the agent will know that she is informed; (ii) the value of public information is negative if the principal implements the same action profile; and (iii) although the agent prefers that the principal has private information, there exists a transfer and a contract that make both players better off with complete information. KEYWORDS: Moral Hazard, Principal-Agent, Informed Principal, Information, Technology, Ex Post Contracting. JEL Classification: D82, D86. I would like to thank Ed Schlee, Alejandro Manelli, and Hector Chade for their helpful comments and suggestions. The paper also benefited from comments by seminar participants at Deakin University, LaTrobe University, and the Australian Economic Society Meetings 2004.
1 Introduction In contract theory, a common situation is a principal-agent model in which a risk neutral principal hires a risk averse agent, who chooses an unobservable effort level. The principal designs an incentive scheme that pays the agent a wage which depends on an observable outcome, which is a stochastic function of the agent s effort. The incentive scheme must both induce the agent to accept the contract and exert a certain amount of effort (Holmström, 1979; Grossman and Hart, 1983). Several authors have extended this model to allow the agent to also have private information (Holmström, 1979; Myerson, 1982; Sobel, 1993). Others have extended it to allow the principal to have private information (Myerson, 1983; Maskin and Tirole, 1992; Inderst, 2001; Chade and Silvers 2002). Grossman and Hart (1983) showed that when the principal and agent have symmetric information about the principal s technology, the principal prefers to have a more informative technology. The technology is a Markov matrix that relates the agent s action to observable outcomes. We define it more formally in the next section but, loosely speaking, the technology, in part, captures the agent s returns to effort, and a more informative technology is akin to greater returns to effort. Having a more informative technology thus implies that the principal is better able to control the action that the agent chooses and thereby implement any action at a lower cost than if she had a less informative technology. In some agency relationships, the principal has much more experience or knowledge than the agent and so has private information about the returns to the agent s effort; examples include the relationships between a landlord and a sharecropper, an insurer and an insuree, and a contractor and a subcontractor. Information has three potential effects on a player s expected utility in agency models with moral hazard: It allows the player to make a decision that is more closely aligned with the state of nature, changes an opponent s strategies, and affects risk-sharing possibilities. The model that we develop here can be viewed as a step toward determining the effect of each. To motivate the ideas in this paper, consider a few examples. First, insurance companies and governments often have more experience than individual insurees or contracted firms about the technology. An insurance company often knows the probability of an accident or of theft of a specific type of vehicle in a specific area better than the individual does. These probabilities not only depend upon the neighborhood s characteristics, which a driver may not know, but also upon the driver s care and attentiveness. Likewise, a government often knows the probability of a product failing better than an individual firm does. This probability not only depends upon the intended uses, which the contracted firm may not know, but also upon the firm s quality control procedures. Moreover, in situations such as these, the principal and agent will not meet until after the principal knows the technology. A question that arises is, would the principal benefit from rules that would compel this information to become public? As a second example, consider an automobile manufacturer that introduces a new or improved 1
safety device reducing the expected accident costs, or a health insurer that wants to implement a new testing procedure that will reduce expected illness costs. The insurer may not know by how much the expected accident or illness costs will decline, and so would not be able to offer even the second-best contract. What would the effects of a third-best contract be? Still, in other situations, it may be possible for the agent to learn the technology, or for the principal to inform the agent. For example, an executive to whom a firm has offered a position may find it beneficial to research the firm s prospects and profitability conditional upon his effort prior to accepting or rejecting the contract. In this situation, would the agent prefer to remain ignorant? Moreover, could the firm and the executive agree to some ex ante transfer and a contract that, together, yield a Pareto improvement? In this paper, we attempt to answer these questions by examining a principal-agent model with moral hazard in which the principal will obtain information about her technology before offering the contract. We ask whether, and under what conditions, the principal would prefer to obtain private information rather than having null information, and when the principal would prefer public information. The consequences on the agent s utility and the actions implemented are also explored. It is shown that, if the principal were to implement the same action regardless of which technology she has, then the principal prefers null to private perfect information, and the agency prefers symmetric null to symmetric imperfect or perfect information. In addition, we show that, even though the agent prefers the principal to have private information, there exists a transfer and a contract such that the agency would be better off if the principal s private information became public. In the first example above, our results indicate that the principal would want to enact rules that would compel this information to become public after she learns it, even though once she knows the technology, she might not want it to become public. Regarding the second example, the third-best contract may result in drivers exerting too much or too little care, or doctors over- or under-prescribing the new procedure, relative to second-best, and may yield the insurer negative profit. Finally, in the third example, the results obtained herein indicate that the agent would prefer to remain ignorant, but the firm could effect a transfer that the executive would accept and make both players better off. The model analyzed in this paper is related to the literature on the value of a more informative technology (Gjesdal, 1982; Grossman and Hart, 1983; Kim, 1995; Jewitt, 1997; Chade and Silvers, 2002). Our results imply that acquiring a more informative technology may not be beneficial for the principal unless she can make this knowledge public. Maskin and Tirole (1992) examined a model in which the principal has private information that affects the agent s expected utility. In such a model with common values, the agent receives his reservation utility but the principal may not attain her complete information payoff. Chade and Silvers (2002) extended this model to examine the situation in which the agent takes an unobservable action and the principal s technology is her private information. They showed that there exist equilibria in which the principal with the more informative technology earns less profit than the one with the less informative technology, and there 2
exist equilibria in which the agent receives more than his reservation utility. Our work is also related to the literature that examines the value of symmetric information in agency models (Sobel, 1993; Inderst, 2001). In their papers, they showed that the principal gains, but the agent does not, when the agent has more, as opposed to null, information; 1 we extend this result to a slightly different situation. We show that, although symmetric information makes the agent worse off, there exists a transfer that could make the agent willing to become better informed. In Sobel s (1993) model, the agent acquires information about his disutility of effort. By acquiring information, the agent implements a higher action but risk-sharing possibilities are reduced. It is worth highlighting that, in Sobel s model, the agent becomes privately informed. In contrast, in our model, it is the principal who acquires private information; moreover, the information itself is about the technology and not the disutility of effort. We show that the principal, and agency, would gain in expectation from making information public, even though one type of principal would lose. Inderst (2001) examined a model with a privately informed principal and a risk neutral agent. Importantly, the type of private information the profitability of the firm satisfies the singlecrossing property. He showed that the presence of private information can distort actions downward, making the principal better off, but that the agent still receives his reservation utility in the leastcost separating equilibrium. Thus, the principal and agency would be better off from having symmetric information. Our analysis tests the robustness of this result in a different context, where the private information does not possess the single-crossing property. Not only can actions be distorted in either direction, but also, since one type of principal and the agent can each be better off when the principal has private information, it is not always the case that the principal and agency prefer symmetric to asymmetric information. It is worth noting that there is empirical evidence that, in some situations, the principal has private information. In the insurer-insuree relationship, Chiappori and Salanié (2000) showed that there is no evidence of adverse selection in the French auto insurance market for drivers with no or little experience. They suggest, 2 as does Schlesinger (1997), 3 that in the insurance industry, the insurer may have private information. Cohen (2002) observed the same result about inexperienced drivers; however, she also determined that drivers with more than three years of experience do have private information when they switch insurers. In addition, the longer is the tenure of a driver with his insurance carrier, the weaker the correlation is between coverage and accidents. 4 In comparing the menus that in- 1 Sobel shows, through an example, that this result may fail when more than two outcomes are possible. 2 On page 73, they write:... the information at the company s disposal is extremely rich and that, in most cases, the asymmetry, if any, is in favor of the company. 3 In his Presidential Address Presented at the 1997 ARIA Meeting, he stated that, compared to individual drivers, insurance company actuaries will have a much better probability prediction for the probability that a driver will experience an automobile accident within the next 10,000 miles of driving, and further that the overall evidence shows a very uninformed population when it comes to insurance. 4 She notes that the coverage-accidents correlation is also consistent with moral hazard, so that her results imply the presence of adverse selection or moral hazard or both. However, moral hazard cannot fully explain the decline in the coverage-accidents correlation. 3
surers offer drivers and the selected contracts, she finds that insurers make greater profit on repeat customers than new customers. Our model suggests one reason for this is the asymmetry of information that prevents an insurer from offering the complete information contract so that she offers a more expensive one. Fluet (1999) showed that the contracts that individuals sign with insurers and those that fleets sign may differ because the number and size of claims that a fleet files in a period provides a more informative signal about the level of care that the fleet has taken. Our model provides an additional reason why such contracts may differ: The fleet company has symmetric information; whereas, the individual driver has incomplete information. Like our current paper, a companion piece (Silvers 2006) examines the value of symmetric information; however, that paper examines ex ante contracting and shows that the principal gains by acquiring information, even if it is private, and may actually be worse off if it is public. The paper is structured as follows: The model is laid out in the next section and the results are presented in section 3. Section 4 concludes. The appendix contains a characterization of the equilibria and the equilibrium contracts, a description of two refinements, and the proofs of lemmas. 2 Model In a principal-agent model with moral hazard, the agent chooses an action a m from a set {a 1,...,a M } where 0 < a 1 < a 2 <... < a M < and M 2. The action that the agent chose determines, through a stochastic process, the outcome q n from a set {q 1,...,q N } where 0 < q 1 < q 2 <... < q N < and N 2. The agent has additively separable von Neumann-Morgenstern utility function over income and effort, given by U(I,a) = V (I) a, with V (I) > 0,V (I) < 0. He is a risk averse, expected utility maximizer. Further, we assume that I such that lim I I V (I) =. Define h V 1 ( ). The conditional probability of outcome q n given that the agent chose a m is denoted by π n (a m ). A technology is a Markov matrix whose elements are π n (a m ). The principal is risk neutral and maximizes expected profit. There are two possible technologies, Π 1 and Π 0, with which she can be endowed. 5 These technologies are related by a stochastic matrix, R, which transforms Π 1 to Π 0 by Π 0 = Π 1 R T. This implies that Π 1 is more informative than Π 0. Let λ [0,1] be the prior probability that the principal has Π 1 so that π λn (a m ), is the conditional probability that outcome q n is realized given that the agent chose a m. Then, Π λ = λπ 1 +(1 λ)π 0. We also write Π λ for the principal who believes she has Π 1 with probability λ. One commonly assumed property of technologies is the monotone likelihood ratio property (MLRP). This states that the relative likelihood of a higher outcome to a lower outcome is increasing in the action; formally, m m, n n, π λn(a m ) π λn (a m ) π λn(a m) π λn (a m).6 We assume that both technologies 5 Throughout, we use male pronouns to refer to the agent and female pronouns to refer to the principal. 6 Equivalently, this states that the likelihood of an outcome resulting from a higher versus a lower action, is increasing in the outcome; i.e., π λn (a m) π λn (a m ) π λn(a m) π λn (a m ). 4
have MLRP, and further that Π λ also satisfies MLRP (this last assumption is required since MLRP does not necessarily carry forward to convex combinations of technologies). We make the additional assumption that both Π 1 and Π 0 satisfy the convexity of the distribution function condition (CDFC). Denote the c.d.f. of Π λ generated when the agent selects a m by F(ñ,a m ) = ñ n=1 π λn(a m ). If for i < j < k, and for ι (0,1) such that a j = ιa i + (1 ι)a k, F(ñ,a j ) ιf(ñ,a i ) + (1 ι)f(ñ,a k ), then the technology satisfies CDFC. Intuitively, CDFC implies that the returns to the action are stochastically decreasing. Finally, note that Π 1 and Π 0 satisfying CDFC is sufficient for Π λ to possess this. With MLRP and CDFC, the symmetric information contract is monotone in the outcome and incentive compatibility needs only to be checked for lower actions i.e., the principal does not have to worry about the agent selecting a higher action than she wants to implement (Salanié 1997). For any q n, let I λn R denote the payment. The set of such payments for all possible outcomes is a contract and is given by I λ = {I λ1,...,i λn }. Because we will often compare contracts that implement different actions, when necessary, we write I λ (a m ) and I λn (a m ) for the contract and wage, respectively. Given a contract, the principal s expected cost will also depend upon her type. For Π λ1 to implement a m with I λ2, this cost is C λ1 (I λ2 (a m )) = N n=1 π λ 1 n(a m )I λ2 n. Note that C λ1 (I λ2 (a m )) = λ 1 C 1 (I λ2 (a m )) + (1 λ 1 )C 0 (I λ2 (a m )). Similarly, the principal receives revenue (a benefit) that is realized when an outcome occurs. The expected benefit from implementing a m is B λ (a m ) = N n=1 π λn(a m )q n. We assume revenue equivalence so that B 1 (a m ) = B λ (a m ) = B 0 (a m ). If the agent believes that the principal is Π λ and he chooses a m, then a contract is individually rational if it yields expected utility at least equal to his reservation utility, Ū; i.e., N π λn (a m )V (I λn ) a m Ū (1) n=1 If the agent believes the principal is Π λ, then a contract is incentive compatible if it induces the agent to choose the action that the principal wants to implement; i.e., a argmax N n=1 π λn(a) a {a 1,...,a M } V (I n ) a or m m N [π λn (a m ) π λn (a m )]V (I λn (a m )) a m a m (2) n=1 Denote the individual rationality and incentive compatibility constraints corresponding to implementing a m by IR(λ,a m ) and IC(λ,a m,a m ), respectively, when the agent believes that the principal is Π λ. In order to implement a 1, the principal offers a flat wage, Ī = h(ū+a 1) and so receives B(a 1 ) Ī. Any contract that implements a higher action must satisfy both (1) and (2). There is a stochastic relationship between the technology that the principal has and a signal that she receives. We denote by z k the signal that she receives and by Z = {z 1,z 2 } the signal 5
space. If her technology is Π l,l {0,1}, then denote by ζ lk the probability that she receives z k. The principal and agent have a common prior probability, λ, that the principal will have Π 1. Therefore, the probability that the principal has Π 1 conditional upon observing z k is, by Bayes λζ rule, λ(z k ) = 1k λζ 1k +(1 λ)ζ 0k. An information structure is a Markov matrix [ ] ζ 01 ζ 02 ζ = ζ 11 ζ 12 where ζ lk 0, k ζ 0k = k ζ 1k = 1. The probability of observing z k is then prob(z k ) = λζ 1k + (1 λ)ζ 0k. The information structure determines how accurately the principal knows her technology. If ζ 01 = ζ 12 = 0 then the principal will know her technology with certainty. We then say that she has perfect information. The opposite, null information, arises when the signal that the principal receives conveys no information about her technology; therefore, her interim beliefs equal the prior beliefs. Also, if a player has null information, we will call him ignorant. If the technology and signal are imprecisely correlated, then we say that the principal has imperfect information. The agent, too, can have perfect, null, or imperfect information. The agent observes an event, s S, that contains the signal that the principal received. We can think of S as a coarsening of Z. An information function, t : Z S where #Z #S, determines the level of symmetry between the principal and the agent. If the agent s information function is constant, t(z 1 ) = t(z 2 ), then the agent will have null information, and if the information function is non-constant, t(z 1 ) t(z 2 ), then the agent and principal will have symmetric information. If the agent s information function is constant and the principal has imperfect or perfect information, then the principal has private information. If the agent s information function is one-to-one and the principal has perfect information, then we call the situation one of complete information. We restrict attention to those situations in which either the principal and agent have symmetric information, or the principal has private information. An information structure and an information function together constitute an environment. We consider the following five environments: Complete Information [ ] 0 1 Where ζ = and the agent s information function is one-to-one, the principal and the 1 0 agent have symmetric and perfect information. Grossman and Hart (1983) examined this environment. Symmetric Null Information [ ] 1 0 Where ζ =, both the principal and the agent are ignorant. The signal that the 1 0 principal receives, and the event that the agent observes, regardless of whether the information 6
function is constant or one-to-one, are entirely uninformative so that their posterior beliefs of the principal s technology equal λ. Symmetric Imperfect Information [ ] ζ 01 ζ 02 Where ζ = and the agent s information function is one-to-one, the principal and ζ 11 ζ 12 agent have the same posterior belief of the principal s technology, that she has Π λ(z1 ) or Π λ(z2 ). Asymmetric Perfect Information [ ] 0 1 Where ζ = and the agent s information function is constant, the principal has private, 1 0 perfect information of her technology. The agent is ignorant. Chade and Silvers (2002) examined this environment, except that we generalize here to more than two actions and more than two outcomes. Asymmetric Imperfect Information [ ] ζ 01 ζ 02 Where ζ = and the agent s information function is constant, the principal has ζ 11 ζ 12 imperfect, but private, information about her technology. The agent is ignorant. Because ζ 11 > ζ 01, λ(z 1 ) > λ > λ(z 2 ). If ζ 01 = ζ 12 = 0, then this environment reduces to Asymmetric Perfect Information; whereas, if ζ 01 = ζ 11 = 1, then this environment reduces to Symmetric Null Information. In each of these five environments, we compare the possible profits, utilities, and actions implemented that can arise in the possible Perfect Bayesian Equilibria (PBE). The timing in this ex post contracting game is as follows: 1. Nature chooses an information structure and an information function; 2. The principal and the agent are informed of these choices; 3. Nature chooses a technology; 4. Nature sends a signal to the principal according to the choices in 1; 5. the principal updates her prior to her posterior beliefs about the technology she has, having received z k ; the agent updates his prior to his interim beliefs, having observed the event according to his information function; and then the principal offers a contract to the agent; 6. the agent updates his interim to his posterior beliefs, having received the contract offer; he then chooses whether to reject or accept the contract; if the agent rejects it, then the game ends and he receives Ū while the principal receives 0; else 7. the agent then chooses an action; and 7
8. Nature chooses an outcome according to the technology from 3 and the action choice from 7; payoffs are made. For each a m, Π λ solves the following program: N Min π λn (a m )I λn s.t. (1) and (2) (3) n=1 The solution is a contract, Iλ. Π λ maximizes expected profit by implementing the action a that satisfies a argmax B(a) C λ (I λ (a)) (4) a {a 1,...,a M } If the principal has private information, then it is possible for one type of principal to mimic another. An equilibrium contract must guarantee that one type does not mimic the other. Π λ will not mimic Π λ only if B(a m ) C λ (I λ (a m )) B(a m ) C λ (I λ (a m )) (5) If Π λ would mimic Π λ, then Π λ solves the amended program: N Min π λn (a m )I λn n=1 s.t. (1) and (2) and (5) (6) We assume that there is no natural separation. Denote the solution to (6) by Î λ. It is least-cost for Π λ among the set of contracts that satisfy IR(λ,a m ) and that Π λ would not mimic where λ < λ and Π λ is another possible type. Î λ clearly cannot satisfy the same incentive compatibility constraints with strict equality that Iλ satisfies. Let, M = { mi : Îλ satisfies IC(λ,a m,a mi ) with equality}. A summary and characterization of the separating and pooling equilibria in each of the environments is relegated to the appendix. For an environment with asymmetric information, a specific {λ,π 1,R}, can admit multiple equilibria, as is the case in many signaling games. Thus, we have payoff correspondences for the principal and the agent. Denote by Y f the equilibrium payoff set for a player corresponding to one specification {λ;π 1,R}. Let y f be a particular equilibrium payoff in this set. Y f is a real-valued set that need not be convex. Following Shannon (1995), we have: Definition 1 Ranking of Sets Let Y f and Y g be two real-valued sets. Y f is strong set order greater than Y g if y f Y f and y g Y g, both max(y f,y g ) Y f and min(y f,y g ) Y g. 8
Y g is completely lower than Y f if y f Y f and y g Y g, y g y f. Finally, Y g is weakly lower than Y f if y f Y f and y g Y g, either max(y f,y g ) Y f or min(y f,y g ) Y g. 3 Results Chade and Silvers (2002) showed that the agent earns strictly more than his reservation utility from the least-cost separating contract. The following lemma extends this result from the two-actiontwo-outcome case to the M-action-N-outcome case. Lemma 1 Consider the Asymmetric Perfect Information environment. Assume λ > λ. Let Π λ implement a m and Π λ implement a m, where a m a m and suppose a m > a 1. The least-cost separating contract for Π λ yields the agent strictly more than his reservation utility. Proof: See Appendix 5.3. Lemma 2 Grossman and Hart (1983, Proposition 2). Consider the Complete Information environment. The agent receives expected utility exactly Ū; i.e., I λ exactly. satisfies individual rationality Lemma 3 Grossman and Hart (1983, Proposition 13). Consider the Complete Information environment, and that Π λ is a Blackwell transformation of Π λ. Π λ s cost of her perfect information contract is greater than Π λ s cost of her perfect information contract; i.e., C λ (I λ (a m )) > C λ (I λ (a m)). In Chade and Silvers (2002), if the principal has private information about her technology, the agent would be worse off if information became symmetric and the principal would know that he had learned her technology. The next proposition extends this result to the more general case, as well as to different information structures including symmetric, imperfect, or null information. Proposition 1 Positive Value of the Principal Having Private Information for the Agent The agent s equilibrium payoff sets in each of the Complete, Symmetric Null, and Symmetric Imperfect Information environments, is completely lower than his equilibrium payoff set in Asymmetric Perfect Information; i.e., the agent prefers that the principal has private information to having symmetric information. For the equilibrium payoff from the Asymmetric Perfect Information equilibrium payoff set that pertains to the least-cost separating contract for Π 1, the weak inequality in the definition of the completely lower than ranking can be strengthened to a strict inequality. In addition, this result is robust to the Intuitive Criterion and may be Undefeated. Proof: In Complete Information, Lemma 2 shows that the agent receives exactly Ū from any contract that is part of a PBE. 9
We now show that there are equilibria in the Asymmetric Perfect Information environment in which the agent receives strictly more than Ū. We then show that this result holds for the least-cost separating equilibrium, which survives the Intuitive Criterion and is the only separating equilibrium that can be an Undefeated Equilibrium. Consider the pooling contract Iλ of Asymmetric Perfect Information. Increase the largest wage that occurs with positive probability in this contract. This altered contract is still incentive compatible and now yields the agent strictly more than Ū. Let the agent believe that the principal is Π λ if this contract is offered but Π 0 if Iλ is offered.7 Neither Π 1 nor Π 0 can offer the contract that yields exactly Ū. So long as neither Π 1 nor Π 0 were indifferent between offering Iλ and deviating, there exist pooling equilibria that yield the agent more than Ū. In any separating PBE, the agent must receive at least Ū from both Π 0 and Π 1 ; therefore, he cannot do worse than in Complete Information or Symmetric Null Information or Symmetric Imperfect Information. Lemma 1 shows that the agent receives strictly more than Ū in the least-cost separating equilibrium from the contract that Π 1 offers. Increasing a wage in Î1 that does not violate incentive compatibility yields a contract that provides even more utility, that Π 0 would not mimic, and that Π 1 would still offer provided that she was not indifferent between offering Î and deviating. Thus, there exist many separating equilibria in which the agent receives more than Ū from the contract that Π 1 offers. In appendix 5.3, we prove that the least-cost separating equilibrium survives the Intuitive Criterion and is the only possible separating equilibrium that is Undefeated. Thus, if the principal had private information and the agent could choose to become symmetrically informed, he would prefer to remain ignorant if the principal were also to know that he became informed. Next, we determine when the principal is worse off from obtaining private information. Before showing that, if she implements a constant action profile with private information, then the principal prefers symmetric to having private information, we establish a lemma that the principal s cost of symmetric information contracts is convex in the technology. Lemma 4 Convexity in Technology of the Expected Cost of Symmetric Information Contracts Let λ be the prior probability of the principal having Π 1 and, without loss of generality, let (1 λ ) < λ. The expected cost of symmetric information contracts for Π λ and Π (1 λ ) is greater than the cost of a pooling contract, Iλ. Proof: See Appendix 5.3. It is easier to understand the sense in which the technology is convex by appealing to the notion of a mean-utility preserving increase in risk. A contract I implements an action and specifies wages; therefore, it generates a distribution over the possible wages conditional on q n. Let F(q n ;I) denote 7 This is permissible as a PBE requires, but puts no restrictions on, out-of-equilibrium beliefs. 10
the c.d.f. over the possible wages generated by I. Following Diamond and Stiglitz (1974), we have: Definition 2 Mean-Utility Preserving Increase in Risk Let F(q n,a,π λ ) denote the c.d.f. of q n induced by the technology Π λ and the action a. The distribution together with the contract induce a distribution of utilities. Let F(v n,a,π λ ;I) denote the distribution of V (I) induced by F(q n,a,π λ ) when I is offered and v n = V (I n ). Let f(v n,a,π λ ;I) denote the p.d.f. of F(v n,a,π λ ;I). Let v denote the utility of the maximum payoff over all wages in I and I. A distribution F(v n,a,π λ ;I ) is a mean-utility preserving increase in risk of the distribution F(v n,a,π λ ;I ) if it satisfies the following two conditions: 1. F(vn,a,Π λ ;I ) is a spread of F(v n,a,π λ ;I ) in the sense that: ṽ ( F(vn,a,Π λ ;I ) F(v n,a,π λ ;I ) ) dv 0 ṽ < ; and v(i) 2. v v(i) ( F(vn,a,Π λ ;I ) F(v n,a,π λ ;I ) ) dv = 0. By the concavity of the agent s utility function, the expected cost of I λ and I (1 λ ) 8 must have a higher expected cost than the contract that pays I λn with probability p λ π λ n(a m ) and I λn with probability (1 p λ )π (1 λ )n(a m ) i.e., the contract that pays I λn with probability π λn(a m ). The lottery that pays I λ n with probability p λ π λ n(a m ) and I (1 λ )n with probability (1 p λ )π (1 λ )n(a m ) is a mean-utility preserving spread of the lottery that pays I λn with probability π λn(a m ). To see this, consider the distribution formed by moving the weight π λ1 (a m ) on I λ1 to I (1 λ )1 and I λ 1 with probabilities (1 p λ ) and p λ, and for n 2, I λn occurs with probability π λn(a m ). This is a spread of the distribution induced by I λ (a m ) and π λ (a m ). By repeating this process for each n, the (1 p λ ) : p λ gamble on I(1 λ ) and I λ is a spread of I λ. Thus, the first condition of being a mean-utility preserving spread is satisfied. For the second condition, the (1 p λ ) : p λ gamble on I(1 λ ) and I λ yields Ū as does I λ. Thus, the concavity of the agent s utility function implies that the (1 p λ ) : p λ gamble on I(1 λ ) : I λ is more costly. That is, p λ C λ (Iλ (a m)) + (1 p λ )C (1 λ )(I(1 λ ) (a m)) > C λ (Iλ (a m)) (7) In particular, when the principal has perfect information, λ(z 1 ) = 1 so that λ in the lemma can be taken to be 1 and (7) becomes λc 1 (I1 (a m)) + (1 λ)c 0 (I0 (a m)) > C λ (Iλ (a m)), where the left-hand side equals the principal s ex ante expected cost of implementing a m if the agency is to become informed. The lemma implies that the cost of the complete information contracts is convex in λ. That is, if λ > λ > λ, then 8 This expectation is over the possible signals z k that the principal can receive with probabilities p λ and its complement; whereas, the cost of each of these contracts is an expectation, over the outcomes q n with probabilities given by the density as determined by the principal s actual technology and the action that the agent chooses. 11
p λ C λ (I λ (a m)) + (1 p λ )C (1 λ )(I (1 λ ) (a m)) > p λ C λ (I λ (a m)) + (1 p λ )C (1 λ )(I (1 λ ) (a m)) (8) Proposition 2 Negative Value of Private Information for the Principal Unless She Implements the Low Action When She Has Null Information (a) Assume that the principal implements a m given z 1 or z 2 in Asymmetric Perfect Information. The principal s equilibrium payoff set in Asymmetric Perfect Information is completely lower than her equilibrium payoff set in Symmetric Null Information. (b) Assume that the principal implements a 1 in Symmetric Null Information. The equilibrium payoff set from Symmetric Null Information is completely lower than both the equilibrium payoff sets from Asymmetric Imperfect Information and that from Asymmetric Perfect Information. (c) Assume that the principal implements a 1 given either z 1 or z 2 in Asymmetric Imperfect Information, but she implements a m > a 1 given z 1 in Asymmetric Perfect Information. The equilibrium payoff set from Asymmetric Imperfect Information is completely lower than that from Asymmetric Perfect Information. Proof: For part (a), because the principal implements the same action regardless of the environment, the differences in the equilibrium payoff sets are due entirely to differences in costs of implementing the action. When the principal is ignorant, the equilibrium contract that implements a m is I λ. When the principal has perfect information about her technology, there exist both separating and pooling equilibria. In the pooling equilibria, Iλ is a possible contract, but there is a multiplicity of pooling equilibrium contracts, the remainder of which cost the principal more than this does. Thus, in pooling equilibria, she cannot do better, does just as well with only one contract, and otherwise does worse than if she had null information. Without loss of generality, the separating equilibria that can exist have Π 0 offer I0 and Π 1 offer a contract that dissuades mimicking. This means that she cannot offer I1. The least-cost separating contract is Î1, which necessarily costs her more than I1 costs. Therefore, λc 1 (Î1) + (1 λ)c 0 (I 0 ) > λc 1(I 1 ) + (1 λ)c 0(I 0 ) > C λ (I λ ) = λc 1(I λ ) + (1 λ)c 0(I λ ) (9) The first inequality follows because Π 1 cannot offer I1 ; the second inequality follows from Lemma 4 and (7), and the equality follows from simple algebra. All other pooling contracts cost 12
the principal more than I λ costs, and all other separating contracts for Π 1 cost her more than Î1 costs. To see that (b) is true, consider the separating equilibria in Asymmetric Imperfect Information. When she receives z 2, the principal also implements a 1 since she does not mimic in the separating equilibrium and with symmetric null information, she implements a 1. As λ(z 2 ) < λ, by Lemma 3, for all a m > a 1, C λ(z2 )(I λ(z 2 ) (a m)) > C λ (I λ (a m)) which implies that B(a 1 ) h(ū) > B(a m) C λ(z2 )(I λ(z 2 ) ). As λ(z 1) > λ, if she implements a m > a 1 when she receives z 1, then it must be that B(a m ) C λ(z1 )(Îλ(z 1 )(a m )) > B(a 1 ) h(ū). Thus, ( ) ) λ B(a m ) C λ(z1 )(Îλ(z 1 )(a m )) + (1 λ) (B(a 1 ) h(ū) > λ(b(a 1 ) h(ū)) + (1 λ)(b(a 1) h(ū)) = B(a 1) h(ū) To complete the proof of part (b), we need to show that the principal cannot do any worse in Asymmetric Imperfect Information than in Symmetric Null Information. As implementing the low action is the least possible profit that a principal can attain, she cannot do worse. Moreover, because in the least-cost pooling equilibrium the principal implements a m > a 1 with I λ (a m), she would receive the same profit in any pooling equilibrium of Asymmetric Imperfect Information. As Asymmetric Perfect Information is a special case of Asymmetric Imperfect Information with λ(z 1 ) = 1 and λ(z 2 ) = 0, the second claim follows. Finally, (c) follows analogously; if the principal implements a 1 in Asymmetric Imperfect Information whether she receives z 1 or z 2, then her profit equals B(a 1 ) h(ū), which is less than her profit in any separating equilibrium of Asymmetric Perfect Information. We have shown that neither would the agent choose to acquire symmetric information, nor would the principal choose to acquire private, perfect information about her technology, if the acquisition of information were to become common knowledge. 9 Moreover, the agency, too, may choose voluntarily to remain ignorant rather than become publicly informed of the technology. Gjesdal (1982) defined two ways in which a more informative information structure can have value for the agency. We can adapt Gjesdal s definitions to our context. 10 A more informative information structure has marginal insurance value if, given that the principal implements the same action, she does so at a lower cost through better risk-sharing, and a more informative information 9 This result arises with any uncertainty. It may be interesting, as a technical aside, to note that neither the principal s nor the agent s correspondence is lower hemi-continuous at λ = 1 or λ = 0. Additionally, if the agent holds sufficiently pessimistic beliefs, then neither correspondence is upper hemi-continuous at λ = 1 or λ = 0. A proof is available from the author upon request. 10 In his model, he considered the marginal value of an additional information structure (which he called an information system) rather than a Blackwell improvement of the existing information structure. As Holmström (1979) does, Gjesdal appeals to the statistical concept of sufficiency, but adapts it to a control rather than an inference context. 13
structure has marginal incentive informativeness if it induces the principal to implement a higher action than she does with the less informative information structure. In our model, the marginal insurance value of better public information of the technology is negative; therefore, if the principal implements the same action, so that there is no marginal incentive informativeness, the agency is made worse off by becoming symmetrically informed. We claim without proof that if both Π 1 and Π 0 implement a m in Complete Information, then so too would Π λ. Proposition 3 Negative Value of Public Perfect Information for the Agency (a) Assume that Π 1 and Π 0 implement a m > a 1 in Complete Information. The agency is better off in Symmetric Null Information than in Complete Information. 11 (b) Assume that Π λ implements a 1 in Symmetric Null Information. The agency is better off in Complete Information than in Symmetric Null Information. Proof: Consider (a). By Lemma 4, the principal s expected cost in Symmetric Null Information is less than her expected cost in Complete Information. Because the agent receives Ū in each equilibrium by Lemma 2, the agency is better off in Symmetric Null Information than in Complete Information when the principal implements the same action in the latter environment. To see (b), by Lemma 3, for all a m > a 1, C 0 (I 0 (a m)) > C λ (I λ (a m)) which implies that B(a 1 ) h(ū) > B(a m) C 0 (I 0 ) since the principal implements a 1 in Symmetric Null Information. Consider Complete Information. When she receives z 2, the principal would also implement a 1. If she implements a m > a 1 when she receives z 1, then it must be that B(a m ) C 1 (I 1 (a m)) > B(a 1 ) h(ū). Thus, ( ) ) λ B(a m ) C 1 (I1(a m )) + (1 λ) (B(a 1 ) h(ū) > λ(b(a 1 ) h(ū)) + (1 λ)(b(a 1) h(ū)) = B(a 1) h(ū) As implementing the low action is the least possible profit that a principal can attain, she cannot do worse. This completes the proof of the second claim. In order to understand this result, it is helpful to see how symmetric information contracts change with λ. Definition 3 Continuity of Contracts A contract I λ is continuous in λ if for any ɛ > 0 such that λ λ < ɛ, there exists δ > 0 with I λ n I λn < δ n. That is, I λ converges uniformly for each wage in the contract: I λ I λ as λ λ. 11 If both types implement a 1, then they, and Π λ, offer Ī. 14
Utility Π Λf a 2 Π Λf a 2 ΛΠ 1f a 2 ΛΠ 1s a 2 Π Λs a 2 1 Λ Π Λ0s a 2 Π Λs a 2 1 Λ Π 0 f a 2 I 0 f I f CE I Λ1 f I 1 f I 1 s I Λ1 s I s CE I 0 s Wage Figure 1: Mean-Utility Preserving Increase in Risk with Two Actions and Two Outcomes Suppose that in Complete Information, the principal implements a non-constant action profile. The principal may still prefer to have null to complete information, since it may allow her to implement the higher action when she has Π 0 (the action that she would implement given z 1 with complete information). This seems to imply that when she has Π 0, she is implementing a higher action than is optimal; however, this is not true since she implements the higher action with I λ, which costs Π λ less than I 0 costs Π 0. For larger values of λ, when she is endowed with Π 0, the principal gains by implementing the higher action with a less expensive contract, Iλ ; moreover, this occurs often enough to offset the relative loss of implementing the higher action with a more expensive contract when she is Π 1. As λ decreases, though she is experiencing this relative gain more frequently, it is rapidly becoming smaller as I λ I 0, and the cost for Π λ is convex. The intuition is more easily understood when there are two possible actions, a 2 > a 1, and two possible outcomes, q s > q f. Figure 1 shows the agent s utility function and depicts the six possible wages from the contracts I1, I 0, and I λ and two certainty equivalents ICE f and Is CE. Dashed lines of the same style are part of the same contract. Each corresponding ex ante probability of receiving a wage is indicated. For example, with complete information, the agent faces a compound lottery, first receiving either the contract I1 or I 0, and then receiving the wage specified by the contract and the outcome. Thus, he will receive I0f if both the principal is Π 0 with probability (1 λ), and q f is realized with probability π 0f (a 2 ) assuming that the agent chooses a 2. Thus, the gamble on I1f and I 0f with probabilities λπ 1f (a 2 ) λπ 1f(a2 )+(1 λ)π 0f (a 2 ) and (1 λ)π 0f (a 2 ) λπ 1f(a2 )+(1 λ)π 0f (a 2 ) has the certainty equivalent If CE, and the analogous gamble on I1s and I 0s has the certainty 15
equivalent Is CE. Each gamble costs more than its certainty equivalent. And, since the gamble on If CE and Is CE has the same probabilities as Iλ does, yields exactly Ū, and is a mean-utility preserving increase in risk, it costs more than the π λf (a 2 ) : π λs (a 2 ) gamble on Iλf : I λs does. The action profile that the principal chooses to implement plays a key role in whether the principal or agency prefers a more informative information structure because the principal cannot insure the agent against the possibility of being Π 0. Chade and Silvers (2002) showed that when information is private and perfect, actions can be distorted relative to symmetric information in either direction. As in their paper, in our model, it is possible for actions to be distorted in either direction when comparing symmetric against private information. Let a m > a m. It may be that the principal would implement a m if she had perfect or imperfect information and information were symmetric, but chooses to implement a m when she has null or private information. When the principal has private information, she is precluded from offering certain contracts, so that the higher type must now offer a contract that is more expensive. This increase in costs may more than offset the additional revenue from implementing a m versus a m so that the agent s ignorance distorts the implemented action downwards. The same may be true in a pooling equilibrium. Alternatively, it is possible that the principal implements a m when informed, but a m when she is ignorant, so long as the agent is ignorant. As was shown above, the principal s ex ante expected cost when she remains ignorant is less than that when she becomes informed. In addition, the two technologies are related by a stochastic matrix, which makes the incentive compatibility constraints corresponding to Π 0 harder to satisfy than those corresponding to Π 1 ; however, the individual rationality constraint for the former may be easier to satisfy than that for the latter. 12 Ignorance thereby allows the principal to implement a m with a more relaxed incentive compatibility constraint compared to when it is common knowledge that she has Π 0, and a more relaxed individual rationality constraint compared to when it is common knowledge that she has Π 1. Thus, the agent s ignorance may distort the implemented action upwards. A natural extension of Proposition 3 is that, when information is symmetric, the agency prefers a less informative information structure to a more informative one, except either when the principal implements a 1 with the less informative information structure, or when the principal implements a different action profile with the more informative information structure. The imperfect information may induce the principal to implement a higher action when she receives z 1 because the cost of I λ(z 1 ) has declined enough. Or, she may implement a lower action when she receives z 2 because, due to symmetric information, the cost of Iλ(z 2 ) has risen too much. In other words, a public, Blackwell improvement in the information structure can only have positive value for the agency if the more informative information structure induces the principal to implement a different action profile than the one she implements with the less informative technology; however, even this is not sufficient for the agency to prefer the more informative 12 Incentive compatibility is related to the difference in probabilities of realizing outcomes from one action versus another action, while individual rationality is related to the level of the probabilities of the implemented action. 16
information structure to the less informative one. Corollary 1 Negative Value of Public Imperfect Information for the Agency Let ζ be more informative in the sense of Blackwell than ζ. (a) Assume that Π λ(z1 ζ) and Π λ(z2 ζ) implement a m > a 1 in Symmetric Imperfect Information. The value of public information for the agency is negative. 13 (b) Assume that Π λ(z1 ζ ) and Π λ(z2 ζ ) implement a 1 in Symmetric Imperfect Information. The value of public information for the agency is nonnegative. Proof: Some algebra shows that the Blackwell improvement implies that λ(z 1 ζ) > λ(z 1 ζ ) > λ > λ(z 2 ζ ) > λ(z 2 ζ). The result follows as in the proof of Proposition 3 except that by Lemma 4, (8) holds where λ = λ(z 1 ζ) and, without loss of generality, λ = λ(z 1 ζ ) after the Blackwell improvement changes ζ to ζ. Thus, for the second part, C λ(z2 )(Iλ(z 2 ) ) > C λ(iλ ). If the principal implements a 1 given z 2 but a m > a 1 given z 1, she may be better or worse off with the Blackwell improvement. Though C λ(z1 ζ)(iλ(z 1 ζ) (a m)) < C λ(z1 ζ )(Iλ(z 1 ζ ) (a m)), thereby increasing her profit when she receives z 1, she may receive z 1 less often, thereby decreasing her expected profit. In Silvers (2006), it is shown that when the principal offers the contract before she receives the signal z k, this result is reversed; i.e., with ex ante contracting, the principal and agency prefer a more informative information structure. In some contracting situations, the principal may be able to pay the agent a fee to induce a different distribution of information or timing of contracting. If there is some transfer or any other type of contract such that the principal and agent can both be better off with one information distribution or contract timing than with another, then we say that the former is superior to the latter using the Potential Pareto Criterion. Proposition 4 Positive Value of Symmetric Information for the Agency (a) Assume Π 1 and Π 0 implement a m > a 1 given z 1 or z 2 in Complete Information. Complete Information is superior to any separating equilibrium in Asymmetric Perfect Information by the Potential Pareto Criterion. Symmetric Null Information is superior to Asymmetric Perfect Information by the Potential Pareto Criterion. (b) Assume that Π 1 and Π 0 implement a m1 and a m0 a m1, respectively, in Asymmetric Perfect Information. The agency is better off in Complete Information than in Asymmetric Perfect Information. 13 If both types implement a 1, then they, and Π λ, offer Ī. 17