Regulation under Imperfect Observability: The Dynamics of Incentive Contracts Revisited 1

Transcription

1 Regulation under Imperfect Observability: The Dynamics of Incentive Contracts Revisited 1 Thomas D. Jeitschko Department of Economics 110 Marshall-Adams Hall Michigan State University East Lansing, MI Phone: 001 (517) Fax: 001 (517) jeitschk@msu.edu preliminary draft November I wish to thank Len Mirman, J.-J. Laffont, Ed Schlee, and participants of the Conference in Honor of J.-J. Laffont, Toulouse June/July 2005 for helpful advice and encouragement.

2 Abstract We consider a two-period regulatory environment in which contracts generally last only for one period, or in which longer-term contracts are open to renegotiation after the first period. The setting chosen is that studied by Laffont and Tirole in analyzing the ratchet effect, however, it is assumed here that cost observations are imperfect. As a result of imperfect cost observability, contracts can either be based on cost observations alone, or additional messages from the firm to the regulator. It is shown that if learning is the only concern, then additional messages are used to avoid the noise introduced by imperfect observability. On the other hand, if the ratchet effect is a dominant concern, contracts are based solely on the imperfect cost observation. In this case the first period contract accounts for experimentation and signal dampening effects. This contract is separating, and more likely to be feasible than under perfect cost observations. Moreover, such a contract leads to universal deviations from first-best levels, which results in a upward ratcheting of effort levels of the efficient type firm. The results are discussed in the context of a continuum of types and under renegotiation. Moreover, general noise distributions and larger contract spaces are also considered. Keywords: Dynamic Contracts, Dynamic Agency, Ratchet Effect, Experimentation, Signal Dampening, Regulation, Cheap Talk JEL classifications: C73 (Stochastic and Dynamic Games), D8 (Information and Uncertainty), L1 (Market Structure, Firm Strategy, and Market Performance)

3 1 Introduction In a string of seminal contributions Laffont and Tirole (1986, 1987, 1988, 1990, 1993) laid the foundations for the way that modern economic theory looks at regulation and procurement in the presence of asymmetric information. Starting with a static model they show how agency schemes are used to regulate firms on the basis of the data that becomes available in the course of the operation of the firm (i.e., costs). They demonstrate that for these schemes to work, data need not be observed perfectly, that is, even if costs are affected by unobservable noise (e.g., accounting errors), the optimal incentive scheme yields the same results as under perfect observation. In their latter works, regulation is analyzed in a dynamic setting in which the regulator cannot commit to long-term contracts. These papers lead to an exploration of the celebrated ratchet effect. The critical insight from this work is that it is hard to elicit private information from the firm. That is, type-revealing separating contracts are generally not feasible. Indeed, separation is only possible if future payoffs are discounted sufficiently strongly. In these instances, efficient firms operate at first best levels, and other types of firms costs are subject to distortions above first best levels. In contrast to the static model, Laffont and Tirole conduct the dynamic analysis only in a deterministic setting with perfect cost observations. In this paper we expand on recent results in dynamic stochastic agency (Jeitschko and Mirman, 2002, and Jeitschko, Mirman and Salgueiro, 2002), to revisit Laffont and Tirole s classic work with imperfect observability, and consider a dynamic version of their static analysis. Adding noise to cost observations may be more appropriate than considering deterministic models in many settings, and the resulting analysis differs substantially, both qualitatively and quantitatively, from deterministic models on the ratchet effect. It is shown that contracts based solely on cost observations deviate from their deterministic counter-parts, due to imperfect inferences that result from stochastic costs. In order to better appreciate the resulting 1

4 deviations, the importance of learning and the ratchet effect are analyzed separately. To this end, consider first the event that the ratchet effect can be circumvented. This scenario arises if, for instance, the intrinsic costs of the firm are tied to technology and the management of the regulated firm is replaced between periods. Then current management is not concerned with dynamic incentives associated with information revelation. However, the regulator would nonetheless still like to obtain information about the intrinsic efficiency of the firm. Thus, the first period contract is affected by the regulator s desire to improve the inference he draws from the first period imperfect cost observation and, consequently, the regulator alters the contract in order to obtain better information about the cost structure. That is, the regulator uses contract design in order to experiment. Experimentation leads to greater distortions away from first best cost levels for inefficient firms, and more importantly experimentation results in excessive cost reductions, below first-best levels, for the most efficient type of firm. The added distortions caused by experimentation are due to the (possibly restrictive) assumption that regulatory contracts are based solely on imperfect cost observations. Indeed, such costly distortions can be avoided if the regulator uses additional messages between the firm and the regulatory agency as the basis for contracts. That is, cheap talk a la Crawford and Sobel (1982) can be used to improve upon the outcome. Specifically, the firm can be asked to give a deterministic message about its type to the regulator, and payoffs of the firm can be conditioned on this message in addition to the cost observations. This results in a contract whose economic implications are analogous to a deterministic setting thus, obviating the need for costly distortions due to experimentation. While imperfect observability has no impact on real variables in expectation in multi-period settings provided that the ratchet effect can be circumvented and contracts are based on more than just the cost observation imperfect cost observations do affect equilibrium actions when the ratchet effect becomes a concern. Again, consider first contracts based solely on imperfectly observed 2

5 costs. When the ratchet effect is present, imperfect cost observations have two informational effects between periods. First, the aforementioned experimentation effect that is used to reduce deviations from first-best actions in the second period. Second, the ratchet effect that necessitates additional high upfront rents to the most efficient type of firm, that are designed to compensate for reductions in future information rents due to learning. In deterministic settings this latter effect leads to the potential for the take-the-money-and-run strategy, which makes separation of firm types difficult. However, under imperfect observability, learning is impeded. This preserves future information rents, on the one hand, and makes current deception less effective, on the other hand. As a consequence, the additional upfront rents to the most efficient firm are reduced so that separation is more likely to occur under imperfect observability than in deterministic settings. Thus, in contrast to deterministic settings with much pooling, separation of firm s actions is readily supported in stochastic settings. Since the upfront rents to the efficient firm are, thus, functions of the imperfect observability of costs, the regulator can manipulate contract terms in order to further reduce upfront rents to the most efficient firm. This is termed signal dampening, as a reduction in the informativeness of the cost observation (i.e., the signal) preserves future rents and reduces payoffs from deception. While signal dampening (reducing the flow of information) runs counter to experimentation (increasing the amount of learning), the former is a dominant effect. Consequently, costs of inefficient firms are closer to first-best levels, and more importantly the most efficient firm is induced to produce at costs above the first-best level. The distortions resulting from experimentation and signal dampening are, in a sense, costly. And, to some degree, these distortions can be avoided when contracts are not only based on imperfect cost observations, but also on additional messages. However, unlike the case of pure experimentation, when the ratchet effect is of concern, additional messages are not used, even when available. That is, the contract based solely on cost observations is optimal, despite the distortions implied 3

6 by experimentation and signal dampening. The intuition is that learning (i.e., obtaining information) is costly to the regulator this is crux of the incentive implications of the ratchet effect so an environment with imperfect cost observations is to the benefit of the regulator, who can manipulate contract terms in such an environment in order to obtain the optimal degree of information dissemination between periods. The paper concludes with a discussion of long-term commitment and renegotiation in stochastic settings. It is shown that short-term contracts and long-term renegotiation-proof contracts are more similar to one-another in stochastic settings when compared to deterministic settings. That is, some of the insights from the analysis of short-term contracts in stochastic settings carry over to renegotiation-proof long-term contracts. For similar reasons, much pooling can be avoided in the case of a continuum of types. Indeed, if contracts are based solely on the imperfect cost observation, then partial pooling is not even feasible. 2 The Model The model used is a variation of the model of regulation introduced by Laffont and Tirole (1993). The addition of imperfect observability follows along the lines of Jeitschko et al. (2002), which analysis a dynamic stochastic setting of procurement and managerial compensation. Suppose that an indivisible public project is to be executed by a firm in each of two periods, τ = 1, 2. The per-period (gross) value of the project to consumers is denoted by S, and the firm s per-period costs of executing the project are given by C τ (e, β) = β f e τ /β m + ɛ τ, τ = 1, 2. (1) The β are time-invariant intrinsic efficiency parameters. There are two possible values of the intrinsic cost parameters referred to as types ) ) denoted by β := (β f,β m and β := (β f, β m with 1 β f,β m β f, β m and β f := β f β f < 1. 1 The intrinsic cost is private information, so that 1 Throughout, the base model and notation are those of Laffont and Tirole with only 4

7 the manager of the firm knows the value of β, but outsiders including regulators do not. Regulators beliefs in period τ concerning the project s intrinsic costs are denoted by ν τ,sothatpr{β = β} =ν τ,τ= 1, 2. ) In each period τ = 1, 2 the manager applies effort e τ [0,β f in order to decrease costs below their intrinsic level of β f. The manager experiences disutility from effort that can be expressed in monetary terms by the function ψ(e). 2 The manager s disutility increases with effort, ψ > 0fore>0, at an increasing rate, ψ > 0, and satisfies ψ(0) = 0. For the sake of concreteness and in order to obtain closed-form solutions, let ψ(e) =.5e 2. Finally, ɛ τ captures unobservable shocks to costs. For instance, one can think of ɛ τ as an independent observation or accounting error on costs. In any event, ɛ is unobserved both ex ante and ex post 3 and has distribution G(ɛ) with continuous density g. The distribution G is homoscedastic (i.e., independent of e and τ) with zero mean on the connected support Φ. Again, for the sake of concreteness and in order to obtain closed-form solutions, suppose that in each period ɛ τ is distributed independently and uniformly on an interval of length φ for some φ>0, large. Even though neither the intrinsic costs β, nor the manager s effort e, nor the random shocks to costs ɛ are observable, overall costs C are observed by the regulator and it is assumed that the regulator reimburses these costs to the firm. In addition to this reimbursement, firm management is compensated for its cost-reducing efforts by being given a perperiod (net) monetary transfer of t τ. Letting the manager s per-period utility be denoted by U τ, we thus have, U τ = t τ.5e 2 τ. (2) minor alterations. Here, for instance, while Laffont and Tirole consider only differences in fixed costs, β f, we also allow for differences in the marginal cost-reducing impact of effort yielding additional and richer results. Moreover, we bound costs away from 1 in order to assure positive effort levels when we introduce a specific functional form for the disutility from effort. 2 The period-subscript τ is suppressed wherever this does not create ambiguities. 3 Of course, this does not preclude that inferences are drawn about the realization of ɛ τ on the basis of the observed cost C τ. 5

8 If the firm does not produce, a reservation payoff of 0 is obtained. The shadow cost of funds to the regulator are given by λ>0. That is, since reimbursements to the firm can only be financed through distortionary taxation, each $1 paid to the firm, costs the taxpayer $(1 + λ). The regulator s objective is to maximize total societal welfare, which is comprised of the value of the public project net of costs to taxpayers plus the manager s utility. Thus, per-period social welfare is given by, W τ = S (1 + λ)(c τ + t τ ) + U τ, (3) where C τ is given by (1) and U τ is given by (2). We assume that S is sufficiently large that production takes place in equilibrium regardless of the firm s cost type i.e., the service provided is considered essential. 4 The regulator has a discount factor of δ r and the manager one of δ m. We distinguish two types of (short-term) contracts. Those in which the payment transfer to the firm, t, is based solely on the observed cost C, i.e., t τ = t(c τ ). And those in which additional messages m { m, m } from the manager to the regulator are used in the contract, i.e., t τ = t(c τ,m). Let e and e denote the manager s type-dependent effort level, given a contract t( ), i.e., e(β, t( )). Then we introduce the following two definitions; Definition 1 (Separation in Distributions, SID) Anycontractthatis either based solely on the cost accounting report, or that is based on the accounting report in addition to a non-informative message (i.e., with m = m), is called separating in distributions (SID), if [ ( )] [ ( )] C := E C e,β E C e, β =: C, where E denotes the expectations operator with respect to ɛ. Conversely, 4 Otherwise see Jain, 2004 (EER), or JJM (N1-N3). 6

9 Definition 2 (Separation in Messages, SIM) Any contract in which the manager sends type-dependent distinct messages, i.e., m m, is called separating in messages (SIM). In a deterministic setting a SID-contract yields a standard separating equilibrium. However, unlike a deterministic separating contract, with imperfect cost observations, a SID-contract does not necessarily lead to complete learning and full information. Instead it may result in incomplete learning and gradual information dissemination. In contrast to the SID-contract and a separating contract in deterministic settings, for a given cost observation, a SIM-contract may give distinct (type-dependent) payments to the firm. However, similar to a separating contract in a deterministic setting, the SIM-contract is fully type-revealing. Finally, we use the following notation for equilibrium expected payoffs: U(ν) denotes the static expected equilibrium payoff of a low cost firm when the regulator has beliefs given by ν. U(ν) is defined analogously (the standard result that U(ν) = 0, ν holds). W (ν) denotes the static average first-best (expected) social welfare, given beliefs of ν. W SB (ν) denotes the static second-best (expected) social welfare, given beliefs of ν. 3 Static Contracts under Perfect Cost Observations... [Insert standard static analysis here]... And we obtain our first critical result: 7

10 Lemma 1 Social welfare is increasing in the accuracy of information, so that information is valuable. Thus, W (ν) νw (1) + (1 ν)w (0) >W SB (ν), ν (0, 1). (4) Proof The Lemma can easily be proved analytically, but it follows more generally from the convexity of the value function, which is shown by { } use of the Envelope Theorem. 5 Specifically, letting C(ν) := C,t; C,t ν denote the optimal contract, given beliefs of ν, note that for arbitrary λ, x 1,x 0 (0, 1): W SB (λx 1 + (1 λ)x 0 ) = E [W (λx 1 + (1 λ)x 0 C (λx 1 + (1 λ)x 0 ))] λe [W (x 1 C (λx 1 + (1 λ)x 0 ))] + (1 λ)e [W (x 0 C (λx 1 + (1 λ)x 0 ))] λe [W (x 1 C (x 1 ))] + (1 λ)e [W (x 0 C (x 0 ))] = λw (x 1 ) + (1 λ)w (x 0 ) W (λx 1 + (1 λ)x 0 ). Now let λ = ν,x 1 = 1andx 0 = 0 and the result obtains, with the second inequality being strict for λ = ν (0, 1). Having established that information is valuable, the remainder of the paper is concerned with the cost of obtaining information and ascertaining when despite its value information is best not obtained because doing so is too costly. 4 Dynamics without the Ratchet Effect Suppose for the moment that the ratchet effect is of no concern. This is the case if, for instance, the services of another firm that has the same (unknown) intrinsic cost can be used in the second period, or if the management of the firm can be replaced between periods. In general, we capture such cases by setting δ m = 0. 5 See Lemma 1, Jeitschko and Mirman (2002). 8

11 4.1 Benchmark: Perfect Cost Observations Under perfect cost observations, the first period equilibrium contract specification without the ratchet effect are straight forward. Since the firm s first period constraints are unaffected by the second period, they are identical to those in a static model. Consequently, a standard static second-best forcing contract is offered to management in the first period. Letting ν generically denote the regulator s beliefs, the second-best entails the (type-dependent) efforts and expected transfers 6 of, e = 1 and t = [ ( ) 2 ( λν 1 2 (1 + λ)(1 ν) β f λ ν ) 2 ] (1 + λ)(1 ν) β f, (5) and λν e = 1 (1 + λ)(1 ν) β f and t = 1 ( ) 2 λν 1 2 (1 + λ)(1 ν) β f. (6) Resulting in (type-dependent) expected costs of λν C = β f 1, and C = β f 1 + (1 + λ)(1 ν) β f. Since this contract is separating, the regulator learns the intrinsic cost associated with producing the public service. Consequently, in the second period, a first-best contract is offered to the new management of the firm, which entails efforts and transfers of so that (type-dependent) first-best costs of are obtained. e = 1 and t = 1/2, (7) C = β f 1 and C = β f 1 (8) 4.2 SID-Contracts without the Ratchet Effect The key that allows the regulator to obtain the first-best in the second period under perfect cost observations is the complete information dissemination between periods. However, under imperfect cost observations this is no longer the case under contracts that rely solely on cost 6 Actual transfers are both a function of the contract terms and observed/realized costs neither of which, however, affect payoffs. 9

12 observations, even if these contracts induce the two types of firm to target distinct levels of costs. That is, with a first period SID-contract, asymmetric information may prevail in the second period. To see this, note that with a SID-contract in the first period, the regulator s second period updated (i.e., posterior) beliefs are captured in the following lemma. Here, we let C 1 denote the first period observed costs, whereas C 1 and C 1 denote the equilibrium expected costs of the two types of firm. Thus, in equilibrium, if β = β then C 1 = C 1 + ɛ 1 and if β = β then C 1 = C 1 + ɛ 1. Lemma 2 Given a first period SID-contract, the regulator s posterior belief is given by ν 2 = ν 2 ( C 1 C 1, C 1 ) = ( ) 1, if C 1, C 1.5φ, [ ] ν 1, if C 1 C 1.5φ, C 1 +.5φ, 0, if C 1 ( C 1 +.5φ, ). Note that the support of the distribution of costs is extended to include the entire real line, with posterior beliefs chosen to be monotone on the entire support. Thus, given the uniform distribution of noise, the regulator offers the standard second best contract in the second period with probability ( { }) ( { }) max 0, 1 C 1 C 1 φ ; and with complementary probability of min 1, C 1 C 1 φ the first best outcome is obtained. Given that the additive noise term in the cost observation integrates to 0, the second period expected social welfare is, thus, given by E[W 2 ] = W (ν 1 ) Pr{ν 2 ν 1 }+W SB (ν 1 ) Pr{ν 2 = ν 1 }. (9) In other words, with a SID-contract in the first period, second period welfare is either the expected first best welfare (given in the first line of Equation 9), or it is the second best welfare depending on whether the firm s type is revealed by the first period cost observation. However, the likelihood of first period costs revealing the firm s type is a function of the first period equilibrium cost targets, C 1 and C 1. Hence, 10

13 Proposition 1 (Experimentation) In a SID-contract in the first period, the firm s equilibrium effort levels are distorted in order to generate more information. Specifically, the efficient firm is induced to make supraoptimal efforts at cost reductions and the inefficient type is induced to reduce efforts below standard second-best levels. Proof The regulator s objective is to maximize total (discounted) social welfare. The first period problem is analogous to choosing contract terms that maximize ) W 1 + δ r (W (ν 1 ) W SB (ν 1 ) Pr{ν 2 ν 1 }, (10) subject to the standard static incentive constraints. Since the expected first best social welfare is greater than second best social welfare (Lemma 1), total social welfare is increasing in the likelihood of obtaining full information. { } Recall that Pr{ν 2 ν 1 } = min 1, C 1 C 1 φ. Thus, the likelihood of obtaining full information is decreasing in C 1 and increasing in C 1 ;and the regulator experiments in order to maximize total social welfare by decreasing C 1 below the first best level and increasing C 1 above the second best level. This is achieved by inducing the efficient firm to make supra-optimal efforts and having the inefficient firm reduce its efforts even below regular second best levels. 4.3 SIM-Contracts without the Ratchet Effect The added distortions in the first period SID-contract are brought about by the regulator s desire to obtain information about the firm s type. As these distortions are costly, the regulator would like to be able to obtain information about the firm s costs in another way if this is possible. In particular, the revelation principle can be made use of, provided that contracts can be based on signals other than only (imperfect) cost observations. 11

14 Corollary 1 (Full Revelation) A contract in which the firm announces its type and is punished for cost observations inconsistent with the type announcement is fully type-revealing (i.e., a SIM-contract) and is superior to a SID-contract. Thus, under a SIM-contract the possibility of a second best contract in the second period is averted and the regulator is assured the expected first best outcome in the second period. Consequently, the SIM-contract is informationally equivalent to the deterministic benchmark, and both settings yield the same expected payoffs to the firm and society overall. 5 Dynamics without the Desire to Learn Suppose that the current regulatory agency has no interest in using information that it can glean in this period about the firm. That is, its discount factor is δ r = 0. However, assume that the firm is concerned about future payoffs so that δ m > 0. This setting essentially presumes that the current regulator is not concerned with future social welfare which to some degree calls into question why one would expect the regulator to be concerned with current social welfare. However, the setting may be more plausible if the service to be provided by the firm is provided only once for any given regulatory jurisdiction, but the firm contracts in the second period with a distinct regulatory agency to provide the service in another jurisdiction. As a matter of convenience, we will thus refer to the first regulator and a second regulator who requires the provision of the good from the firm after all interaction with the first regulator is complete. The setting is also consistent with that of a lame duck government agency that feels obligations towards current constituents, but not their future welfare. If whatever information can be gleaned about the firm s costs in its initial interaction with the first regulator is accessible to the second regulator, the firm is faced with the ratchet effect: If the firm has low costs, it obtains an information rent when there is asymmetric information about 12

15 its type. However, if its type is revealed by the interaction with the first regulator, the second regulator takes advantage of this information dissemination and reduces the firm s information rent to zero. While the interaction between the first regulator and the firm is static from the perspective of the first regulator, the firm s concern about information dissemination feeds back into the relationship with the first regulator. Not because the first regulator has any interest in learning anything about the firm by revealing its type, but because the firm has an incentive to conceal its type to preserve indeed even increase its information rents when it contracts with the second regulator. 5.1 Benchmark: Perfectly Observable Costs Suppose, for purposes of obtaining a baseline, that costs are perfectly observable. If there are no dynamic concerns in the interaction between the first regulator and the firm, then a fully separating contract is obtained and the firm s type becomes common knowledge. A fully separating contract is indeed the preferred type of contract for the first regulator. However, in order to obtain such an outcome, the first regulator must now account for the low cost firm s dynamic incentives. In particular, the low cost firm will be reluctant to reveal its type for it will no longer be able to command an information rent from the second regulator once its type becomes public. Moreover, if the first contract is type-revealing, then a low cost firm that mimics the high cost firm in the first period, will be able to deceive the second regulator about the firm s true cost. The second regulator then (mistakenly) offers a first-best contract for a high cost firm. This contract yields an information rent to a low cost firm that is even greater than that obtained under asymmetric information. Thus, the low cost firm not only has a disincentive to reveal its type in the first period, it has a strong incentive to manipulate the second regulator s beliefs by mimicking the high cost firm. In order to account for these incentives, the high cost firm s incentive compatibility constraint in the first period must be modified. Specifically, in order to obtain separation of types in the first period, the low cost firm 13

16 must be given up-front its discounted gain from deception. The low cost firm s incentive compatibility constraint under a first period separating contract is, thus, t ( ) 2 β f C 1 β 2 m + δ mu(1) = t 1 1 ( ) 2 β 2 f C 1 β 2 m t 1 1 ( ) 2 β 2 f C 1 β 2 m + δ mu(0). (11) Necessitating a higher transfer payment to the low cost firm than in the truly static setting in order to obtain separation. It is this additional upfront transfer to a low cost firm that makes it difficult to obtain a separating contract in the first period when the ratchet effect is of concern. As Laffont and Tirole (1993) note, unless the firm s discount rate is sufficiently small a separating contract is not feasible, since the high cost firm will exert a high enough effort to be mistaken for the low cost firm in the first period in order to obtain the additional transfer, and then refuse the contract offered by the second regulator, who mistakenly believes the firm to have low cost: the so-called take-the-money-and-run strategy of a high cost firm SID-Contracts with two consecutive Regulators Consider now the case when costs are only imperfectly observed. If a SID-type contract is in effect between the first regulator and the firm, then the dynamic incentives of the firm must be modified when costs are only imperfectly observed and, thus, not necessarily fully type-revealing. Note from Lemma 2 that given a first period SID contract, information revelation may be incomplete so that posterior beliefs coincide with prior beliefs. This affects the low cost firm s dynamic incentive compatibility constraint in two ways. First, when choosing the equilibrium action, its type may not be revealed so that future information rents are retained. Second, when mimicking the high cost firm by attaining the 7 Kartasheva (2005) considers this issue for the case of distinct principals in detail, and examines what happens when incentive constraints become binding for both types of agent. 14

17 high expected costs in the first period, full deception need not occur, so the highly profitable first-best contract for a high cost firm may not become available to the low cost deceptive firm in the second period. As a consequence, the additional transfer to the low cost firm can be reduced by the probability that the first period imperfect cost observation will fully reveal the firm s type. Thus, the low cost firm s first period incentive compatibility constraint (11) becomes 8 t ( ) 2 β f C 1 β 2 m + δ [ m U(1) Pr{ν2 ν 1 }+U(v 1 ) Pr{ν 2 = ν 1 } ] t t ( β f C 1 ) 2 β 2 m + δ m ( β f C 1 ) 2 β 2 m t [ U(ν1 ) Pr{ν 2 = ν 1 }+U(0) Pr{ν 2 ν 1 } ] ( β f C 1 ) 2 β 2 m + δ mu(0) Pr{ν 2 ν 1 }. (12) Since Pr{ν 2 ν 1 } 1, it is less likely that the take-the-money-and-run strategy is a binding concern when costs are only imperfectly observed. However, there is another important informational feature present. Notice that the likelihood of full information revelation, which determines the size of first period transfers, is itself a function of the first period { } targeted cost levels, since Pr{ν 2 ν 1 }=min 1, C 1 C 1 φ. Indeed, in choosing the first period targeted costs, the first regulator also determines the degree of information transmission and, hence, the size of the additional upfront transfer to the low cost firm. Hence, Proposition 2 (Signal Dampening) In a SID-contract with the first regulator, the firm s effort level is distorted in order to impede the flow of information, so as to preserve future information rents and reduce expected payoffs from deception. Specifically, the low cost firm exerts less-than 8 Since costs are only imperfectly observed, incentive compatibility must be assured for all effort levels that yield costs that are consistent with equilibrium observations. Thus, the low cost firm s incentive compatibility constraint is given by, ( ) e arg max t 1 C 1 C 1, C 1 1 )) 2 e2 + δ m U (ν 2 (C 1 C 1, C 1 df (C 1 e). However, using results from Caillaud, Guesnerie and Rey (1992), Jeitschko and Mirman (2002) establish that it is without loss of generality that only the two equilibrium cost targets are considered when establishing incentive compatibility. 15

18 first-best effort and the high cost firm is induced to increase efforts above standard second-best levels. Proof The first regulator s objective is to maximize the social welfare of the first period, subject to the standard participation constraints and the low cost firm s dynamic incentive compatibility constraint given by (12). The first period problem is, thus, analogous to choosing contract terms that maximize W 1 ν 1 λδ m U(0) Pr{ν 2 ν 1 }, (13) subject to the standard static constraints. Recall that Pr{ν 2 ν 1 } = { } min 1, C 1 C 1 φ. Thus, the likelihood of obtaining full information is decreasing in C 1 and increasing in C 1 and the regulator reduces the additional upfront payment by increasing C 1 above the first best level and decreasing C 1 below the second best level. This is achieved by inducing the efficient firm to make sub-optimal efforts and having the inefficient firm increase its efforts above regular second best levels. Since the key to reducing the upfront payment made to a low cost firm is to thwart information dissemination, the regulator s manipulation of information is termed signal dampening. 5.3 SIM-Contracts with two consecutive Regulators Just as is the case in Subsection 4.3, adding messages as a distinct dimension to the contract with the first regulator, can allow complete typeinference, despite imperfect cost observability. However, note that this goes against the first regulator s interests on two grounds. First, the first period cost levels cannot be manipulated to reduce upfront transfers. Second, complete revelation necessitates even further increased costly upfront transfers to the low cost firm. Since the first regulator does not benefit from any information thus obtained, the first regulator deliberately does not ask for any meaningful messages beyond the imperfect cost observability, even when this is otherwise feasible. Thus, 16

19 Corollary 2 (Non-optimality of SIM and the ratchet effect) If the ratchet effect is the sole dynamic concern in the first period, a SIM-type contract is eschewed in favor of a SID-type contract with signal dampening effects, even when a SIM contract is feasible in the first period. The insight is straightforward, given that the first regulator has no interest in obtaining information that arrives when it is too late to act upon. Indeed, with two consecutive regulators, it is only the second regulator that has an interest in using the information obtained through the first period contract. This raises the question, whether the second regulator might pay off the first regulator in order to affect contract terms that affect the flow of information. The most natural way to study this question, is to simply fully integrate the two regulators and, thus, consider the framework originally proposed by Laffont and Tirole. 6 Full Dynamics: The Laffont-Tirole Model with Noise Consider now the analysis of Laffont and Tirole (1993) with additive uniform noise. For sake of simplicity, let δ m = δ r = δ>0, so that both the regulator and the firm discount the future at the same positive rate. Since the analysis follows that of the deterministic setting closely when a SID-contract is not possible, assume for now that SID can be obtained in the first period. Then, from the preceding analysis in particular Equations (10) and (13) it is clear that the regulator s first period problem is analogous to the following maximization problem, ( ) max W {t 1,C 1 ;C 1 + δ W (ν 1 ) W SB (ν 1 ) ν 1 λu(0) Pr{ν 2 ν 1 }, s.t. 1,t 1} t 1 = 1 2 t 1 = 1 2 ( ) 2 β f C 1 β 2 m where t 1 = t 1 + δu(0) Pr{ν 2 ν 1 }. ( β f C 1 ) 2 β 2 m, (β f C 1 ) 2 β 2 m 1 2 ( β f C 1 ) 2 β 2 m, Proposition 3 (Optimality of SID) Given the choice between a contract structure that bases payments only on (imperfectly) observed costs (SID) and one that 17

20 takes into account additional messages in a meaningful way (SIM), the optimal contract is SID whenever either the likelihood of the firm being efficient (ν 1 )is not too small. Proof The concerns about the ratchet effect dominate those of learning, so that the logic of Proposition 2 carries over and dominates that of Proposition 1, whenever the consolidated dynamic effect in the objective of the regulator, namely ( W (ν 1 ) W SB (ν 1 ) ν 1 λu(0) ), is negative. Notice that A(ν 1 ) := ( W (ν 1 ) W SB (ν 1 ) ) is concave with A(0) = A(1) = 0. Since λu(0) isaconstant bounded away from 0, it follows that SID is optimal for ν 1 close to 1, and whenever A (0) <λu(0) SIDis optimal for all values of ν 1. This condition is given by 1 + λ 1 2 (1 β) > 1, so it holds for large λ and small β. CHECK THIS: The intuition for the condition in Proposition 3 is that when prior beliefs are fairly strong, i.e., close to 0 or 1, the value of learning becomes relatively small, whereas the added transfers made to the low cost firm become very large as this type of firm becomes less likely. Hence, when it is likely that the firm has high cost, the regulator is better off, on the margin, in reducing upfront transfers to the low cost firm, rather than obtaining better information. CHECK THIS: As a corollary to the proposition, it should be noted that as λ goes to 0, the magnitude of the informational aspect of the first period problem goes away, as the regulator becomes less concerned about leaving a rent for the firm. Given the results from the previous sections, it follows from Proposition 3 that Proposition 4 (SID and the Ratchet Effect) Whenever a SID contract is optimal, one observes a ratcheting of the low cost firm s cost target. Proof { In line with Propositions 1 and 2, since Pr{ν 2 ν 1 } = min 1, C 1 C 1 φ it follows that given a negative impact of the informational component on the regulator s first period objective, efforts of the low cost firm are suboptimal compared to the first best. Consequently, the cost target of the low cost firm, which is always set to the first best level in the second period, is decreased over time. }, 18

21 The result harks back to the early literature on the ratchet effect as a problem in long-term relationships governed by short-term contracts. A certain universal planning practice [...] operates like a ratchet in the planning mechanism, so that once a new high level of performance has been achieved, the next plan target [...] must usually be raised above it. [...] The ratchet principal applies not only to production targets, but to the planning of profit and cost targets as well. 9 7 Generalizations 7.1 Variations on Imperfect Observability Consider now extensions that deal specifically with the imperfect observability of costs. First, we consider larger contract spaces, to assess how mixed message affect effort levels and learning. We then briefly consider general distributions of noise and show how some of the results are affected by this The Ratchet Effect and Mixed Messages It has been shown that whenever the ratchet effect is of greater concern than learning by the regulator, the first period contract entails signal dampening so that the low cost firm employs less than first-best effort in reducing costs. This contract is optimal when compared to an alternative contract the uses additional messages in order to mimic the case of perfect cost observations. However, it is possible that the regulator may chose to use additional message, but not use these to obtain fully separating messages. Such a contract would have the feature that it is separating in distribution in the sense that first period transfers would only be based on (imperfect) cost observations, but updating could also be affected by messages. 9 Berliner (1957), pg

22 7.1.2 General Noise Distributions The uniform distribution used in the exposition has the great benefit that is permits only three possible posterior beliefs. This is different with a model with noise with full support, as is studied in Jeitschko and Mirman (2002). Indeed, suppose that ɛ is distributed on the entire real line with the density g satisfying the monotone likelihood ratio property (MLRP). Then under a SIDtype contract the posterior is given by ( ) ν 1 g ν 2 C 1 C 1, C 1 = ν 1 g + (1 ν 1 )g, (14) where g = g ( ) ( C 1 C 1 and g = g C 1 C 1 ). Now noise has full support and ν 2 (0, 1) so that the second period is characterized by asymmetric information as well. Nevertheless leaning and signal dampening play a role. In particular, the analogue to the regulator s future expected payoff, Equation 9 becomes ( ) E[W 2 ] = W SB (ν 2 ) ν 1 g + (1 ν 1 )g dc 1. (15) R Similarly, the ratchet concern as captured by the second term in Equation 13 becomes ν 1 λδ m R ( ) U(ν 2 ) g g dc 1. (16) Thus, if a SID-contract can be established in the first period, then the second period will resemble a static asymmetric information environment. However, the analysis must necessarily now include the case where the inefficient type of firm is shut down in the second period. Therefore, there is a nondifferentiability in the second period problem. Even absent considerations of shut-down in the second period, the existence of a SID-contract in the first period is sensitive to the nature of the noise in the distribution. In particular, note that the firm s first period indifference curves in the effort-transfer space are function of the second period expected payoffs, which depend on the first period effort. For a straightforward SID analysis to apply in the first period, first period indifference curves must be well-behaved. The following result obtains: 20

23 Proposition 5 For large enough variance, i.e., sufficiently unreliable cost observations, a SID contract is feasible and the Spence-Mirrlees single-crossing condition holds for first period preferences under a SID contract, i.e., ( ) ( ) Et e β Et e β U(ν 2 ) ( g g ) dc 1, e, (17) R ) where g = g (C 1 β f + e/β m. Proof The proof relies on the fact that as the variance of the distribution of the noise term increases, g g decreases so that the informational impact on the first period transfers becomes small. Note that this result is, in fact, a formalization of when the take-the-moneyand-run strategy can become a concern. The result is fairly intuitive in that the key is to reduce the informational effects by reducing the informativeness of the first period cost observation. However, it is nonetheless non-trivial in the sense that first period indifference curves are endogenous to the model and thus to the first period contract terms. In other words, first period indifference curves and thus the Spence-Mirrlees condition cannot be established independently of the first period equilibrium contract. 7.2 Variations on the Laffont and Tirole setting In this penultimate section we consider some of the issues that have been studied using the Laffont and Tirole framework and discuss how these are affected by imperfect observability of costs. We consider first how insights of the model version with a continuum of types may be affected and then turn to renegotiation-proof long-term contracts Continuum of Types The case of a continuum of types is much less tractable in the Laffont and Tirole framework, although it generally has nicer properties since the type distribution can be restricted to follow the monotone hazard rate property (MHRP). They demonstrate that with a continuum of types, much pooling occurs, that 21

24 is, there is a bunching of equilibrium effort levels for adjacent types, as complete separation of types is no longer possible (Proposition 1, Laffont and Tirole, 1988). While we leave a complete treatment of this case for another paper, it is worth pointing out that noise has some effects that destroy the underlying reasons that make separation hard to accomplish and lead to much pooling in the deterministic setting. Proposition 6 (Full Separation) Given imperfectly observed cost, full separation of agent s actions is feasible, even with a continuum of agent types. That is, SID-contracts are obtainable, whereas SIM-contracts are not. The proof that SIM-contracts are not attainable (i.e., that in deterministic settings there cannot be full separation in the continuous type case) relies on the fact that full separation yields full information. An implication of full information is that there are no information rents to be had in the second period. However, this means that, on the margin, information manipulation (i.e., slight deviations from first period actions), yields first-order rents in the second period yet come only at a second-order cost in the first period, since otherwise separation cannot have occurred a contradiction that precludes full separation in the continuous type case. In contrast, when signals are only imperfectly observed, there is no complete information dissemination. Consequently, all but the lowest types retain some information rents in the second period. Thus, small deviations in first period actions have only second-order effects in the second period. These second-order effects can be balanced with the first period second-order effects, so that separation is feasible. Indeed, an even stronger result follows: Corollary 3 (Infeasibility of Pooling) Proper partial pooling is not feasible under imperfect cost observations with a SID-type first period contract, when there is a continuum of firm types and noise has full support. In order to illustrate the underlying intuition for the Corollary, let the prior distribution of types be denoted by F(β) with continuous density f,thennote ( ) that the principal s posterior beliefs, denoted by ˆF ˆβ, given a continuum of 22

25 agent types and imperfect observability are: ˆβ ( ) β ˆF g (C 1 C(β)) df(β) ˆβ C1,C(β) =. (18) β β g (C 1 C(β)) df(β) The same argument made to show that separating is possible works to establish that pooling is not feasible Long-Term Renegotiation-Proof Contracts The preceding analysis has some immediate implications for long-term contracts that are renegotiation-proof. That is, two-period contracts that are agreed to at the outset, but whose terms can be renegotiated after conclusion of the first period if both the firm and the regulator agree to do so. The rationale for renegotiation stems from the fact that in a long-term contract, the regulator commits to preserving information rents of the efficient firm in the future, by preserving second-best arrangements with the inefficient type. However, under perfect cost observations, once the firm is revealed to be inefficient, gains to both the regulator and the firm can be obtained by moving to first best effort levels. Of course, in anticipation of this, the efficient firm, again has the incentive to mimic the inefficient firm making separation costly upfront. With imperfect cost observations the probability that renegotiations take place is diminished. In particular, the probability that under a SID-type contract renegotiation takes place is { min 1 ν 1, (1 ν 1 ) C } 1 C 1. (19) φ Consequently, a SID-type contract under imperfect cost observations is more easily sustainable compared to the perfect observation case. 8 Conclusion The model demonstrates that many of the difficulties that arise in longer-term relationships governed by short term contracts are functions of the amount of information dissemination in the course of the interaction. When observations 23

26 are imperfect, learning and information dissemination is impeded. This, generally, has two effects. First, contract design can be adjusted to eliminate added inefficiencies that are induced by noise (experimentation effects), however, as these are quite powerful, they may undermine ratchet effect considerations that stem from the fact that imperfect observation can serve as a commitment device to preserve uncertainty, when the principal cannot commit to long-term contracts. As a consequence, when the ratchet effect is the dominant concern, contracts will rely solely on imperfect observations. Contracts account for experimentation and signal dampening effects that are designed to steer the flow of information between periods. This results in universal deviations from first best effort levels, and, consequently, to a manipulation of all effort levels across time. These types of separating contracts are more likely to be feasible when compared to deterministic settings with perfect cost observations a feature that carries over to long-term renegotiation-proof contracts as well. However, under general assumptions concerning the nature of cost observationally, these contracts may break down when the precision of error terms, and thus reliability of observations, increase leading to a violation of single-crossing properties in the first period. 9 References Baron, D. P. and Besanko, D. (1984) Regulation and Information in a Continuing Relationship. Information Economics and Policy 1: Berliner, J.S.: Factory and manager in the USSR. Cambridge: Harvard University Press 1957 Caillaud, B., Guesnerie, R., and Rey, P. (1992) Noisy Observation in Adverse Selection Models. Review of Economic Studies 59: Crawford, V. and Sobel, J. (1982) Strategic Information Transmission. Econometrica 50:

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