When is Knowledge Acquisition Socially Beneficial in the Laffont–Tirole Regulatory Framework?

1.1 Motivation and Overview

In a series of influential articles Laffont and Tirole^[1] have developed a theoretical framework for the regulation of a firm under the assumption that “Regulators face a double asymmetry of information, called adverse selection and moral hazard respectively” (Tirole 2015, p. 1670). Whereas the Laffont–Tirole regulator can observe the firm’s accountancy costs she can neither observe the firm’s true cost-type (adverse selection) nor its chosen effort level (moral hazard). Incentive compatible contracts have to ensure that cost-efficient firms do not mimic the accounting costs of less cost-efficient firms through the choice of inefficiently low effort levels. Within the Laffont–Tirole regulatory framework any benefits from better information strictly decrease in the marginal deadweight losses from taxes, denoted λ ≥ 0, whereby these benefits become zero if there are no deadweight losses from taxes, i.e. if λ = 0. The reason for this peculiar feature of the Laffont–Tirole regulatory framework is the assumption of an utilitarian regulator who maximizes the overall social welfare, which includes the firm’s utility. Whenever the regulator has to transfer money to the firm in order to incentivize her to reveal her true type in a second best contract, such transfers would only decrease the overall aggregate welfare if they are financed through taxes that come with deadweight losses. Without such deadweight losses from taxation publicly funded transfers required for incentive compatibility would simply redistribute aggregate welfare towards the firm. In other words, the primary issue of welfare losses from asymmetric information in the Laffont–Tirole regulatory framework is not this asymmetric information per se but rather the inefficiencies of a distortive tax system which makes incentive transfers from the regulator to the firm socially costly.

Because any information acquisition by the regulator must, eventually, also be paid for with taxes, the question about the optimal trade-off between benefits and costs of better information takes on the following specific form within the Laffont–Tirole regulatory framework:

We address this research question through the construction of a stylized model of ‘knowledge acquisition’ as a special case of ‘information acquisition’. General models of information acquisition would consider ‘signals’ that are used to update a prior distribution about the firm’s possible types towards a more precise posterior distribution. Our notion of knowledge acquisition corresponds to the formal special case of information acquisition according to which Bayesian updating can rule out some cost-type(s) as impossible. In our opinion, better knowledge – rather than just more precise posteriors in general – describes the bulk of real-life expert advice. For instance, when people ask a medical doctor for advice about their symptoms they expect the medical expert to narrow down the possible causes for these symptoms. Arguably, ‘experts’ who cannot reduce the support of the prior distribution but rather declare that ‘anything remains possible but with updated posteriors’ would be soon out of business as they do not inspire much trust in non-experts. The reason is that any statistical ex post evaluation makes it so much harder to judge the quality of an expert than the simple observation whether the expert’s opinion is verified or falsified.

More specifically, we assume that the Laffont–Tirole regulator of our model could employ an expert who possesses a knowledge-generating technology which is characterized by the following two features:^[2]

Assumptions on knowledge-generating expert information.

1. The technology strictly reduces the uncertainty about the firm’s possible cost-types by excluding some cost-type(s) as impossible;

2. The technology is imperfect in the sense that it cannot reduce all uncertainty about the firm’s true cost-type.

To capture simultaneously both features of the expert’s technology in the most parsimonious way, we distinguish between three possible cost-types β ₁ < β ₂ < β ₃ of the firm. Without knowledge acquisition the regulator is completely uncertain about the firm’s true cost-type whereby she resolves this uncertainty through the prior belief μ such that μ(β _i ) > 0 for all i. If the regulator acquires better knowledge through the expert, she becomes able to rule out exactly one cost-type as impossible whereby the prior μ is accordingly updated. For the sake of analytical rigor – as well as possible future extensions – we explicitly construct a state space such that better knowledge formally corresponds to a refined information partition of the state space. This information partition reveals in any given state of the world exactly two cost-types β _i   and β _j with i < j as possible whereas the cost-type β _k , k ≠ i, j, is ruled out as impossible.

Based on our parsimonious model of knowledge acquisition, we derive a closed-form expression for the welfare gains from knowledge acquisition within the Laffont–Tirole regulatory framework. Denote by E _μ (W ^Unc) the expected welfare from Laffont–Tirole regulation for the situation in which the regulator decides to remain completely uncertain about the firm’s possible cost-types. This expected welfare is pinned down by the second-best regulatory contract with the firm for the three cost-types {β ₁, β ₂, β ₃}, which we explicitly derive in this paper. Next, denote by E _μ (W ^Kno) the expected welfare that the Laffont–Tirole regulator achieves when she decides to hire the expert in order to acquire better knowledge about the firm’s cost-type. This expected welfare is pinned down as the expectation over the (expected) welfare from the three second-best regulatory contracts for all combinations of two possible costs types, i.e. for {β ₁, β ₂}, {β ₁, β ₃}, and {β ₂, β ₃}, respectively. Such optimal contracts for two cost-types are standard in the literature. The expected welfare gain from better knowledge is then formally defined as the difference between the expected welfare from these two scenarios

To keep our analysis as simple as possible, we impose two technically convenient restrictions on the parameter space. Firstly, we assume that all states of the world are equally likely, implying μ ( β i ) = 1 3 for all i. Secondly, we assume that the three possible cost-types come with a fixed difference

Under these simplifying assumptions, we derive the following closed-form expression for the expected welfare gain E _μ (ΔW) from better knowledge as a function in the ‘deadweight losses from taxes’ parameter λ:

Within the Laffont–Tirole regulatory framework, the acquisition of better knowledge must always result into some non-negative expected welfare gain because better knowledge helps to save on expected deadweight losses from taxes associated with incentive transfers to the firm. These savings are the greater the greater the deadweight losses from taxes. Beyond the general insight that expected welfare gains from knowledge acquisition increase in the deadweight losses from taxes, the expression (2) provides a specific functional form for this relationship. By (2), expected welfare gains are a strictly increasing, unbounded, and strictly concave function in the marginal deadweight losses from taxes.

We are now in the position to answer our research question about the values of the ‘marginal deadweight losses from taxes’ parameter for which the benefits from knowledge acquisition outweigh the corresponding costs and vice versa. The benefits of knowledge acquisition are given – under our modeling assumptions – as the closed-form expression (2) for expected welfare gains. To capture the costs of knowledge acquisitions, we assume that the regulator can hire the expert at the fixed fee L ≥ 0. Because this fee has to be financed through taxes, the overall social costs from knowledge acquisition – including deadweight losses from taxes – are given as (1 + λ)L. For a fixed value of the ‘marginal deadweight losses from taxes’ parameter λ, the overall expected welfare gain from the acquisition of knowledge is thus given as the difference

In other words, the social benefits from knowledge acquisition strictly outweigh the corresponding social costs if and only if

Analyzing inequality (3) gives us – for our specific modeling assumptions – the following answer to our research question:

Answer to our Question:

1. If the expert fee for knowledge acquisition is too costly in the specific sense that

then knowledge acquisition is never socially beneficial irrespective of the value of the‘marginal deadweight losses from taxes’ parameter λ.

1. If we have instead that

then knowledge acquisition is strictly socially beneficial for all values of the‘marginal deadweight losses from taxes’ parameter λ ∈ (λ ^*, ∞) such that the critical threshold is given as

1.2 Contribution and Relationship to the Literature

The main benefits of the Laffont–Tirole regulatory framework are its theoretical insights into the basic trade-offs for regulation under asymmetric information. According to Tirole (2015) this framework highlights “two broad principles”:

The first is obvious. Authorities should attempt at reducing the asymmetry of information: by collecting data of course […]. The second principle is that one size does not fit all: one should let the regulated firm make use of its information. (p. 1671)

This paper takes a critical look at the first principle whereby we argue that the overall benefits of better information are less ‘obvious’ whenever information acquisition is costly. In the Laffont–Tirole regulatory framework both the benefits and the costs from knowledge acquisitions strictly increase in the value of the ‘marginal deadweight losses from taxes’ parameter λ. In general, both curves – of benefits and costs, respectively – might therefore never intersect or intersect arbitrarily many times. Under our modeling assumptions, we show the following:

1. If the knowledge acquisition cost- and the cost-type-difference parameters – L and Δβ, respectively – satisfy inequality (4), the regulator should abstain from knowledge acquisition and rather regulate under complete uncertainty;

2. Else, there exists exactly one critical threshold value λ ^* > 0 – as pinned down by Equation (6) – such that regulation under better knowledge is (strictly) superior to regulation under complete uncertainty if and only if the ‘marginal deadweight losses from taxes’ parameter satisfies λ > λ ^* (That is, if inequality (5) holds, the cost- and benefit curves intersect exactly once, namely at λ ^*).

To bring our theoretical analysis closer to regulatory practice, one would need empirical estimates about the relevant parameters – L, Δβ, and λ – in question. On the one hand, there exists a large literature in public finance which works with plausible values or/and estimates for deadweight losses from taxes, typically within discussions about possible benefits (or the lack thereof) of selected tax reforms.^[3] To quote from Jacobs (2018):

Many applied cost–benefit analyses multiply the cost of public projects with a measure for the marginal cost of public funds that is larger than one. As a result, public projects are less likely to pass a cost–benefit test. For example, Heckman et al. (2010) evaluate the Perry Preschool Program and add 50 cents per dollar spent to account for the deadweight costs of taxation. Many other examples can be given, but the message is clear: The marginal cost of public funds has a tremendous impact on how governments should evaluate the desirability of public policies. (p.884)

On the other hand, we are not aware about any plausible values or/and estimates for relevant parameters of knowledge acquisitions costs L and cost-type differences Δβ. The problem from an empirical perspective is that the relevance of the Laffont–Tirole regulatory framework for regulatory practice appears to be rather limited. Compare, e.g. Rogerson (2003) who writes about the Laffont–Tirole regulatory framework:

Two related problems with applying this theory in practice have been that the economic logic and the underlying mathematics involved in calculating the optimal menu are quite complex, and the principal must be able to specify the agent’s entire disutility of effort function in order to calculate the optimal menu. As a result, the model has not been widely used in practice to either calculate actual incentive contracts or even to develop useful qualitative guidance about the nature of the optimal solution and how it is affected by various economic factors. (p. 919)

Whereas our analysis provides a better theoretical understanding of the welfare trade-offs implicit to the Laffont–Tirole regulatory framework, our model is, admittedly, rather far away from any real-life applications in regulatory practice. Nevertheless, we hope that our main insight – according to which information acquisition through a regulator is only socially beneficial for economies with sufficiently high deadweight losses from taxation – might serve as guidance for future research which is more applied in nature. For example, Laffont (2003, 2005) argues that optimal regulation for developing countries should be different from regulation in developed countries (also see Estache and Wren-Lewis 2009). As one important driver for such difference in regulation, Laffont identifies the inefficient fiscal system in developing countries. Translated into our model, greater fiscal inefficiency of a developing country naturally corresponds to a larger value of the ‘marginal deadweight losses from taxes’ parameter. According to our main insight, it would then be beneficial for regulator in developing rather than in developed countries to acquire better knowledge.

Our theoretical approach is also related to two different strands in the theoretical literature that look into the role of better information within a regulatory context.^[4] The first strand includes the articles by Cremer and Khalil (1992), Crémer, Khalil, and Rochet (1998a, 1998b), and Szalay (2009). These authors take up the Baron and Myerson (1982) model about the regulation of a monopoly with an unknown cost-structure and ask when is it optimal for the regulator to provide the firm with incentives to acquire better information about its own cost-structure. The differences to this literature are that (i) in our model the regulator would acquire better information – through the expert – directly and (ii) that the regulator is utilitarian in that she also cares about the firm’s welfare, which brings in the deadweight losses from taxation as social cost drivers.

The second strand – represented by Lewis (1996) and Magesan and Turner (2010) – introduces asymmetric information into economic policy models of regulation by Stigler (1971) and Becker (1985). For example, Magesan and Turner (2010) consider a situation where a regulator faces firms with different negative externality (i.e. pollution) and profitability characteristics that are unknown to the regulator. In their model the (constrained) regulator has to decide whether she rather acquires information about the social costs of pollution or of the profitability of the firm. This decision matters for the regulatory success because firms with higher profitability might be able to block any regulatory measures through legal processes. In contrast to our approach – where better information is only valuable through better ‘contracts’ with the regulated firm – information is valuable in Magesan and Turner (2010) because it increases the chances that the regulation of a firm gets actually implemented through the political process.

Although the drivers for the costs and benefits from better information in these two strands of models are quite different from the Laffont–Tirole regulatory framework, this literature shares with our paper the motivation to investigate the benefits of better information in connection with the associated costs of information acquisition.

The remainder of this paper is structured as follows. Section 2 recalls the Laffont–Tirole regulatory framework. Section 3 derives the optimal allocation under complete uncertainty. Section 4 provides the general (Blackwell) argument why expert information must always result in (weakly) superior allocations. Section 5 solves for the optimal allocation under the assumption that expert information generates better knowledge. In Section 6 we compare the expected welfare gain from knowledge acquisition with the corresponding acquisition costs. Section 7 concludes. Formal proofs are relegated to the Appendix.

3.1 Discussion: Parameter Conditions

The literature derives the optimal allocation of the Laffont–Tirole regulatory framework either for a continuous interval of cost-types or for two different cost-types only.^[6] The challenge in deriving Proposition 1 for three different cost-types is to identify the parameter conditions (9) and (10) that ensure the existence of an internal optimum.

The optimal allocation of Proposition 1 is derived under the assumption that the local downward incentive constraints (=LDICs) – according to which cost-type β _i has no strict incentive to mimic the contracted costs of type β _i+1 – are binding. By a standard argument, the Spence–Mirrlees single crossing condition ensures that binding LDICs also imply the remaining downward incentives constraints (=DICs) to hold, which means in our case that cost-type β ₁ has no strict incentive to mimic cost-type β ₃. However, as the Spence–Mirrlees single crossing condition is not applicable to our model, we have to ensure directly that this DIC also holds for the allocation of Proposition 1. This is done by the likelihood ratio condition (9) which is equivalent to e 2 * ≥ e 3 * so that the optimal effort levels become monotonic in the firm’s cost-type.

Turn now to the size of Δβ which needs to be sufficiently small for the optimal allocation to be characterized by Proposition 1. Condition (10) is mathematically equivalent to

which implies, together with e 2 * ≥ e 3 * , that

Consequently, inequality (10) ensures the existence of some e i ′ ∈ [ 0 , 1 ] such that ( β i − e i ′ ) = C i + 1 * for i ∈ {1, 2} so that the more efficient cost-type β _i is physically able to mimic the contracted costs C i + i * of the less efficient type β i + 1 . Without condition (10) the LDICs would become superfluous to the effect that these LDICs are no longer binding in an optimum as assumed by the optimal allocation (t ^*, e ^*) of Proposition 1.

We conclude this section with an illustrating example.