Delegating Pollution Permits

We discuss the decision to delegate the regulation of pollution through sales of permits to a biased expert in a situation where the polluting firm has private information about its technology. We consider, in particular, constrained delegation where the government puts restrictions on the amount of pollution that the expert can sell permits for. We find that there is, in general, more delegation the higher is the probability of the firm being low-cost. This is not in line with the so-called uncertainty principle, which states that there is more delegation the more uncertainty the government faces.


I. Introduction
One way to regulate a polluting industry is to require a firm to purchase pollution permits sold by the government in order to be allowed to pollute. With such permits, the government can keep the pollution down, at the same time as the proceeds from the sale of these permits provide the government with needed revenue. The task of regulating polluting firms through permits gets complicated in situations where firms have private information about their production technology. A standard solution to such an asymmetry of information is for the government to offer a menu of combinations of transfers and pollution permits, such that the firm self-selects in its choice of combination according to its technology, creating a distortion in the pollution level of a high-cost firm and an information rent for a low-cost firm.
Regulation is in several countries left to independent regulatory agencies, to an extent that has been increasing over the last few decades (Gilardi, 2009). Such independent agencies open up for the hiring of biased experts to carry out dealing with the regulatee. In this paper, we discuss whether leaving the sale of pollution permits to biased experts is an improvement over the government itself doing it. The advantage of hiring an expert is that she has knowledge of the technology of the firm; in the present analysis, we discuss the case where she has perfect such knowledge and therefore does not have to spend resources at all on distorting the pollution level for a high-cost firm or offering an information rent to a low-cost firm that the government would without the expert. The disadvantage of hiring an expert is that she typically has preferences that are not perfectly aligned with those of the government; in the present analysis, we consider the case when the hired expert puts even more weight on having a low pollution level than the government does. 1 In order to get the most benefits out of delegation, the government will often use the statutes that delegate authority to an independent regulator to also put limits on that regulator's actions.
This way, the government creates what Gailmard (2009) calls "windows of discretion", and he 1 A further amendment, which we do not discuss here, would be to make the pollution permits tradable. There are mainly two reasons for sticking to the case of non-tradable permits. First, our concern is mainly cases of localized pollution, such as pollution of lake, river or coastline, where a permit to pollute cannot be transferred to other locations, so that tradability is not so crucial. Secondly, as argued by, e.g., Lewis (1996), there may be political restrictions making it difficult to introduce tradable permits. mentions examples in the US from the Interstate Commerce Commission, the Federal Trade Commission, and − of particular interest for us − the Environmental Protection Agency.
We model here this window of discretion as a requirement by the government that the independent bureaucrat, if hired, choose the firm's number of pollution permits from an interval.
There are two types of the firm's technology, low cost and high cost, which is known to the firm, and to the bureaucrat if she is hired, but which is not known to the government. In this setting, we find that there are two different ways constrained delegation works. One is what we call weak delegation, which implies capping the bias of the expert so that she is constrained from selling fewer pollution permits to the firm than the first-best quantity for the low-cost type.
By imposing such a restriction, the government ensures that, by hiring the expert, the firm will have the first-best level of pollution if it has low costs. But if the probability of the firm being high-cost is high and/or the distortion from the first-best of the expert's offer to a high-cost firm is large, then weak delegation becomes too costly. We show in our analysis that the government then may choose to resort to strict delegation, where in practice the government pinpoints the pollution level of the firm based on an ex-ante assessment of its cost. In the language of Gailmard (2009), the window of discretion can be wide, when weak delegation is the optimum for the government; it can be narrow when strict delegation is the optimum; and it can be closed, when no delegation is the optimum thing for the government to do.
The need to resort to strict delegation comes out of the government's informational disadvantage and becomes a solution in between weak delegation, with few restrictions in the hired expert, and no delegation to an expert at all. Strict delegation has two implications that make our results on delegation of pollution permits stand out relative to related analyses. First, with strict delegation there is distortion at the top, in the sense that the firm not even when it is low-cost is offered its first-best amount of permits. This aspect of our regulatory regime makes our model contrast with most other models of regulation under asymmetric information, where the most effective type's quantity is not distorted. 2 And note that this distortion at the top occurs because delegation, even when strict, turns out to be better for the government than no delegation at all, when there would be no distortion at the top.
Secondly, strict delegation occurs mainly when the probability that the firm is high-cost is high. It follows that the government does not necessarily respond to reduced uncertainty by allowing more delegation. Rather, if the government becomes more certain the firm is highcost, it is more likely to allow less delegation, that is, either strict or no delegation rather than weak delegation. This contrasts with the so-called uncertainty principle in studies of delegation in government, stating that the government is more prone to delegate tasks when uncertainty is large.
The model we develop to discuss these issues includes a firm with a polluting production that has private information about its production technology; in particular, the firm has either high or low production costs. The firm needs permits to pollute, which are sold by the government.
The government, in turn, lacks information about the firm's technology and can offer a menu of contracts, i.e., combinations of number of pollution permits and price for the permits, in order to solve its regulation problem as effectively as possible. Alternatively, it can leave the permits to an expert who has perfect information about the firm's technology but also is less interested in allowing pollution than the government is. In particular, we analyze the following game: First, the government decides whether or not to hire an expert to carry out the sale of pollution permits and, if hiring an expert, which constraints to put on the expert's actions. If an expert is hired, then the expert sells pollution permits to the firm, knowing its technology and therefore not having to resort to offering a contract menu. If an expert is not hired, then the government presents the firm with its menu of contracts. Finally, the firm buys pollution permits from either the expert or the government, depending on whether or not the government has delegated the permit sales to the expert, after which it uses its permits to produce its goods and sells them.
Our work relates to work on regulating a polluting firm under private information; see Lewis (1996) for a review of the early literature. We add to this by discussing how regulation will be changed by the hiring of a biased expert to do the regulation. One important finding in our analysis is that there may be "distortions at the top". This happens if, in optimum, the expert is subjected to strict delegation, when different types have the same pollution level.
Our work is also related to work, particularly in political science, discussing how and when a government delegates tasks to independent bureaucrats; see, e.g., Epstein and O'Halloran (1999), Huber and Shipan (2006), and Gailmard and Patty (2012). Two factors emphasized in this literature that are particularly pertinent to our paper are the ally principle and the uncertainty principle. According to the ally principle, the government is more interested in delegating tasks the less biased the bureaucrat is. This shows up also in our framework. The uncertainty principle says that the government is more interested in delegating tasks the more uncertain it is about the effects of its decisions. An important finding in our analysis is that this principle does not show up in the present framework of delegating the task of selling pollution permits. The uncertainty here is with respect to the firm's technology. This uncertainty would be small if the government were almost sure that the firm is either low cost or high cost. But we find that delegation is more likely the higher is the probability that the firm is low-cost.
Our work connects to studies in organizational economics on how to put constraints on a hired biased expert, i.e., an agent who is charged with carrying out tasks on a principal's behalf and who is both better informed than the principal and has interests that are misaligned with that of the principal; see, e.g., Gibbons et al. (2013) for a survey, Holmström (1984) for the seminal work, and Alonso and Matouschek (2008) and Amador and Bagwell (2013) for some recent work. Our situation differs from those previously studied in this literature in that the task to be delegated is the design of contracts. Our delegation problem has two distinct features. First, the task to be delegated is two-dimensional, since the contracts we study are two-dimensional. Secondly, the bias of the hired expert is type dependent: the bureaucrat's interest in avoiding pollution depends on the firm's technology, which is private information. In previous analyses of the delegation of multi-dimensional tasks, such as Frankel (2016), the expert's bias is assumed to be independent of the private information, an assumption that makes it possible to turn the multi-dimensional problem into a one-dimensional one. In our case of delegation of contract design with a type-dependent bias, we are able to obtain this conversion to a one-dimensional problem because the regulated firm's participation constraint is binding for the high-cost type in the optimal contract menu. We note, finally, that the distinction between weak and strict delegation that grows out of our analysis resembles the distinction made by Melumad and Shibano (1991) between communication dependent and communication independent decision rules in organizations.
The introduction of experts into models of environmental regulation is also done by Porteiro (2008) and Voss and Lingens (2018). Porteiro (2008) focuses on experts' role in the acquisition of information. In his analysis, the polluting firm may not know its technology, and he discusses how the presence of an information-collecting expert may affect the chance of such information being acquired. In particular, he compares an unbiased expert, whose costs of information collection has to be covered by the government, and a biased environmentalist who covers such costs herself. In contrast, the expert is already informed in our analysis, and we focus on how to use constrained delegation to balance the knowledge and bias of the expert in order to get out the most efficient regulation. Voss and Lingens (2018) discuss the incentive contract between the government and a biased regulator. In contrast, our focus is on the regulatory contract between the firm and the government, and we discuss how the government hiring a biased expert through constrained delegation can improve on that contract when incentive contracts for the expert are not feasible.
Finally, Kundu and Nilssen (2020) discuss the delegation of regulation to an independent bureaucrat in a setting that is related to, but still distinct from, the present one. There are especially two ways in which the two analyses differ. First, in Kundu and Nilssen (2020), the regulatory task to be delegated is one of procurement, where the government buys products from the firm, so that transfers are from the government to the producing firm. This means that the firm in that setting likes transfers while the government does not like them. In the situation discussed presently, the firm's production is sold to the market and the firm buys pollution permits from the government. In this case, transfers are from the firm to the government, and we have the opposite relationship: now, the government likes transfers while the firm does not like them. 3 Secondly, in Kundu and Nilssen (2020), the regulatory contract is about the quality of the firm's product, rather than the firm's pollution. And that quality is affected by both the firm's technology and the firm's efforts, in line with Laffont and Tirole (1993), so 3 The distinction between the frameworks of permits and procurement closely resembles the one made in Caillaud et al. (1988) between marketed and non-marketed goods. The production cost is covered with private funds in the permits framework and with public funds in the procurement framework. If the costs of raising public funds and private funds differ, the two frameworks do not yield the same social welfare. In particular, in the presence of positive (and higher) costs of raising public funds, the value of social welfare will be lower in the procurement framework than in the permits framework. This observation is also noted by Caillaud et al. (1988, Section 4). that contracts are three-dimensional, whereas there in the present model are no efforts. The bureaucrat, when hired, is biased in favor of high quality in Kundu and Nilssen (2020) and in favor of low pollution in the present work. In our view, it is at the outset not clear how a change of the direction of transfers would affect the analysis. Despite these differences, our findings share some crucial common features: the uncertainty principle does not hold, but at the same time constrained delegation shows up as an effective way of dealing with biased experts in the bureaucracy. One difference between the two analyses concerns the prevalence of no delegation when the probability of the firm having low costs is very high: this prevalence is higher in the present permits setting than in the procurement setting of Kundu and Nilssen (2020); we discuss this issue in more detail towards the end of Section IV.
The paper is organized as follows. In Section II, the model is presented. In Section III, we analyze the equilibrium of the model for various benchmark cases, such as when the government has complete information, and when the government can delegate but only without constraints.
In Section IV, we introduce constrained delegation and the concepts of weak and strong delegation, and we solve for the equilibrium outcome of mode of delegation and regulatory contract.
Section V concludes, while all proofs are relegated to an Appendix.

II. The model
We consider the problem of regulating a polluting firm under asymmetric information by requiring that the firm buy permits in order to be allowed to pollute. Following Laffont (1994) and Boyer and Laffont (1999), we assume that the firm's level of pollution is observable and verifiable. The firm provides a good that gives a positive surplus S. The cost of production is given by is the observable and verifiable pollution level chosen by the firm, and θ is a cost characteristic that is the firm's private information. With ∂C/∂θ > 0, a high θ implies a high cost and therefore low cost efficiency. For simplicity, we make the assumption that θ can take only two values, θ, θ , with 0 < θ < θ < K. 4 Let ν ∈ (0, 1) be the probability that the firm is low-cost with type θ = θ, so that the expected value of θ is The firm produces the good and sells it at a price p (θ). Since the price is chosen by a firm with private information, it is in principle a function of the firm's type θ. We assume, however, that the firm, whichever type it is, is able to extract the full surplus value when selling it on the market. This implies that p (θ) = S, for θ ∈ θ, θ . The government regulates production by issuing pollution permits, which the firm buys in order to be able to pollute. The transaction between the government and the firm on permits can be viewed as the government offering a contract α = (t, d) ∈ A := R + × [0, K]: by providing a transfer t to the government, the firm can keep its level of pollution at d. With this contract, the firm's payoff is (1) Society is adversely affected by the firm's pollution, with a disutility given by d 2 /2. 5 Moreover, along the lines of Laffont and Tirole (1993), we assume that the government faces a marginal cost of public funds that exceeds one. The government receiving an amount from the firm means that distortionary taxes can be reduced elsewhere in the economy by that same amount. Specifically, for every unit of transfer, the government benefits 1 + λ from it, where λ > 0. 6 Thus, the government's payoff from a contract α is the social value of the project: We assume that S ≥ K 2 /2, ensuring that the social value of the project is non-negative even at 4 Having only two types is a simplification that facilitates the analysis. A continuous type space would allow a richer analysis. We conjecture, for example, that, in a setting with a continuous type space, forms of delegation in between weak delegation and strict delegation will occur. An exploration of this is left for future research. 5 The specific functional forms we use for the firm's cost of production and society's cost of pollution are helpful in making the model analytically tractable and the discussion of the roles of weak and strict delegation instructive. We do not believe our results are very dependent on the functional forms used. 6 The assumption that λ > 0 is needed to ensure that the equilibrium we study here is unique. For a discussion of the notion that the marginal cost of public funds, here modeled as 1 + λ, is greater than 1, see, e.g., Ballard and Fullerton (1992). maximum pollution and zero transfer.
The government can delegate the regulatory decision-making to an independent regulator, or a bureaucrat. We assume that the bureaucrat is informed about the firm's cost. She can therefore implement a type-contingent regulatory policy. If the government delegates, then the bureaucrat has the authority to choose a regulatory policy according to her own preferences.
We assume that the bureaucrat is intrinsically motivated to keep pollution low. In particular, we assume that her payoff is given by where β ≥ 0 measures the bureaucratic drift. The higher is β, the more the bureaucrat is concerned about pollution.
The game proceeds as follows.
• Stage 1. The government decides whether or not to delegate the decision-making authority to an independent bureaucrat. If it does not delegate, then the authority remains with the government.
• Stage 2. The firm learns its type θ, which can be either θ with probability ν or θ with probability 1 − ν. The bureaucrat also learns the firm's type at zero cost.
• Stage 3. The player with decision making authority determines the permit contracts.
• Stage 4. The firm chooses whether or not to accept an offered contract. If it accepts, then it produces and sells its production. Payoffs are realized. The game ends.
We study the Perfect Bayesian equilibrium of the game. We solve the game by backward induction.

III. Analysis
The analysis at stage 4 is trivial. The firm only accepts contracts that satisfy the participation constraint U P (θ, α) ≥ 0 and extracts the full surplus.
At stage 3, the regulatory contract is determined. We first describe the contract α GI (θ) = (t GI (θ) , d GI (θ)) that the government chooses if it has complete information about θ. The contract for type θ solves the following problem: Note that, at the solution, the firm's participation constraint is binding, which gives U P (θ, α) = 0. After replacing U P in (4), we have that the first-order condition with respect to d is This equation states that, in optimum, marginal social damages from pollution, d, equals marginal social abatement costs, (1 + λ) θ. Since λ > 0, this implies that, in optimum, marginal social damages from pollution is larger than marginal private abatement costs, θ. We see from equation (5) that the optimal pollution level is increasing in λ; this is because the government's interest in receiving transfer from permits increases with λ.
If λ ≥ (K/θ) − 1, then the pollution is at the maximum level K, and there is also a maximum transfer: α GI (θ) = (S, K). In this case, essentially, government lets the producer go unregulated, with full pollution and no profit. For the analysis below, we restrict our attention to cases where government does not offer such a no-regulation contract to any type of firms under complete information. Formally, we impose the following restriction: With this assumption, the government sets the pollution level From the binding participation constraint, we get It follows from (6) that a high-cost firm is allowed to pollute more in equilibrium.
The government's expected payoff under full information is given by

No delegation
With no delegation at stage 1, the uninformed government offers an incentive-compatible pair of contracts (α, α) = (t, d) , t, d to the firm at stage 3. The contract pair solves the following problem: subject to where (ICH) and (ICL) are the two firm types' incentive-compatibility constraints and (IRH) and (IRL) are their individual-rationality constraints. We denote the solution with subscript GN . Define θ := θ − θ. The following Lemma describes the contract pair.
Lemma 1. Consider the case of no delegation. The optimal incentive-compatible contract pair to the firm is given by From (11), we have that which is decreasing in λ. With d = K, we have t = S and C (·, K) = 0, so that both revenue and cost, and hence profit, equal zero. Thus, while Assumption 1 ensures that this does not happen under complete information, that assumption still allows for it under asymmetric information.
In order to understand the outcome in Lemma 1 better, consider Figure (6) and (7), are depicted as the points A and B in Figure 1. Each of them is the optimal contract for the government that satisfies individual rationality on the part of the firm; this is illustrated by the dashed curves in Figure 1, showing indifference curves for the government for contracts to the two firm types. We recognize the feature of the model that a high-cost firm pollutes more than a low-cost firm.
The contracts that prevail under asymmetric information, given in Lemma 1, are depicted as points C and D in Figure 1. Note that there is no distortion at the top, in that information asymmetry does not affect the pollution level of the low-cost type, as a comparison of equations (6) and (10) shows; in Figure 1, this means that point C is vertically below point A. There is, however, an information rent accruing to the low-cost type when ν < ν * . 7 From equations (7) and (12), we have that this information rent equals This equation shows a trade-off for the government: it can lower the information rent to the 7 When ν ≥ ν * , we have d GN θ = K and t GN θ = S, and there is no need to incentivize the low-cost type to keep away from the high-cost contract. low-cost type, which is costly, only by increasing the pollution level of the high-cost type, which also is costly. In equilibrium, the government will do a bit of this trade-off, which is why the pollution level of the high-cost type is higher under asymmetric information than under complete information: contract D is to the right of contract B on the red zero-profit curve of the high-cost type in Figure 1.
The government's expected payoff under no delegation is given by

Full delegation
If the government delegates at stage 1, then an informed bureaucrat, who has complete information about θ, chooses the contract α BI (θ) = (t BI (θ) , d BI (θ)) at stage 3. In case of full delegation, the bureaucrat can choose any contract in A. The contract for type θ solves the following problem: subject to U P (θ, α) ≥ 0.
describes the optimal contract.
Lemma 2. Assume that the government delegates the decision-making authority to a bureaucrat. The contract α BI (θ) = (t BI (θ) , d BI (θ)) that the bureaucrat offers to a producer of type θ ∈ θ, θ is given by The bureaucrat's choice of pollution level is always below the government's choice because of her vested interest in reducing pollution. By setting pollution at a lower level, the bureaucrat increases the production cost, and thereby the compensatory transfer. In Figure 1, the bureaucrat's contracts to the low-cost and high-cost firm type are depicted as the points E and F, respectively. Because of the bureaucrat's knowledge of the firm's technology, the two points are on the firm's participation constraints. Because the bureaucrat is biased and dislikes pollution more than the government, points E and F are to the left of points A and B in Figure 1.
The government's ex-ante expected payoff under full delegation is given by

Comparison between full delegation and no delegation
The condition under which government prefers no delegation to full delegation at stage 1 is The following Proposition characterizes the government's preference over no delegation and full delegation.   space. The figure illustrates how the result in Proposition 1 is consistent with both the ally principle and the uncertainty principle. Note, first, that the effect of β is consistent with the ally principle, which suggests the government prefers to give more discretion to better aligned bureaucrats. This effect arises as the government's payoff under full delegation decreases with β whereas β has no impact on that payoff under no delegation. The intuition is that a bureaucrat with a low bias, as measured by β, is more easily trusted by the government and therefore more often will be delegated the regulation of the firm than a bureaucrat with a high bias.
Secondly, the effect of ν is consistent with the uncertainty principle, which suggests that the government prefers more bureaucratic discretion in situations with high uncertainty. To see this, observe that the government's no-delegation contract coincides with the full information contract at ν = 0 and ν = 1, and gives her a higher payoff than what she gets under full delegation. Further, her expected payoff from full delegation changes linearly with respect to ν, and in Lemma A.1 in the Appendix, we show that U N D G in (15) is convex in ν. These observations together imply that D is convex in ν and is positive as ν approaches 0 or 1. Therefore, D in (20) can take negative values only at an intermediate range of ν, i.e., when the uncertainty about the firm's type is high. It follows that the government's benefit from the bureaucrat's informational advantage is high in situations with high uncertainty. The intuition is that the value of the bureaucrat's knowledge about the firm's technology is greatest when the government's uncertainty about it is the greatest. When ν is close to either 0 or 1, the government is almost certain of the firm's tecnology and the benefit of the bureaucrat's knowledge is smaller.

IV. Constrained delegation
The government can improve its payoff from delegation by restricting the bureaucrat's choice set. As the bureaucrat has an interest in reducing pollution, her preferred pollution level is always below that of the consumer. The government can therefore improve its payoff by imposing a lower bound on the bureaucrat's choice of this level. But being uninformed, it cannot impose type-dependent bounds. In order to study the government's interest in setting bounds on pollution levels, we will be considering a bureaucrat choosing regulatory contracts α (θ) = (t (θ) , d (θ)) ∈ A, θ ∈ θ, θ under the constraint that d (θ) ∈ [d 1 , d 2 ] ⊆ [0, K]; it is this constraint that we call constrained delegation. To study it, we modify stage 1 of the game as follows.
• Stage 1 (modified). The government decides whether or not to delegate the decisionmaking authority to an independent bureaucrat. If it delegates, then the government chooses 0 ≤ d 1 ≤ d 2 ≤ K such that the bureaucrat can offer contract α (θ) = (t (θ) , d (θ)) ∈ A with the constraint that d (θ) ∈ [d 1 , d 2 ] ⊆ [0, K], θ ∈ θ, θ . If it does not delegate, then the authority remains with the government.
Below we first look at how constrained delegation affects the bureaucrat's choice of regulation contracts. Her optimal contract for type θ solves the following problem: subject to U P (θ, α) ≥ 0 and d ∈ [d 1 , d 2 ] .
We denote the solution with a superscript C and a subscript BI. The following Lemma describes the bureaucrat's optimal choice of contracts under constrained delegation.
Lemma 3. Assume that the government delegates the decision making authority with the con- The bureaucrat's preferred regulation contract for type θ ∈ θ, θ is given by α The following lemma describes the choice for the upper bound.
Lemma 5 implies that, if the government delegates with constraints, then two possibilities may arise. In the first case, the government chooses d 1 = d GI (θ). In response, the bureaucrat The government implements the full-information regulation contract if the firm is low-cost. There is distortion at the contract offered to a high-cost firm, as d BI θ < d GI θ , while the full-information contract is implemented if the firm is low-cost. We refer to this case as weakly constrained delegation, or simply weak delegation (WD), since the government puts rather weak constraints on the bureaucrat's contract choices. The government's expected payoff under weak delegation is given by In the second case, the government chooses , resulting in a uniform pollution level for both types of firm. The government's choice of d 1 is the optimal uniform pollution level. We refer to this case as strictly constrained delegation, or simply strict delegation (SD), since the constraints are stricter than under weak delegation. The government's expected payoff under strict delegation is given by In Figure 3, which is comparable to Figure 1 the probability of the firm being low-cost is sufficiently high. The costs of weak delegation lie in the distortion at the contract offered to a high-cost firm. This distortion increases with β and so does the government's preference for strict delegation over weak delegation. The following lemma documents how the two key parameters of our model, β and ν, affect the government's preference over the weak and the strict forms of delegation.
Lemma 6. There exists a threshold ν SD (β), which is increasing in β, such that U SD G > U W D G if and only if ν < ν SD (β).
Lemma 6 implies that the government implements either strict delegation or no delegation for ν < ν SD (β) and implements either weak delegation or no delegation for ν ∈ ν SD (β) , 1 .
First, consider the case ν < ν SD (β). Since the government's payoff under strict delegation and no delegation is invariant to β, her choice over these regimes would only depend on ν.
When there is no uncertainty about the firm's type, i.e., when ν = 0 or ν = 1, the government implements the relevant full-information contract and receives the same payoff under both no delegation and strict delegation. In Lemmas A.1 and A.2 in the Appendix, we show that both U N D G and U SD G are increasing and convex in ν. Moreover, since both payoffs are independent of β, we have that, if U SD G ≥ U N D G for some β, then this relationship holds for all β.
Next, consider the case ν ≥ ν SD (β). In this case, the government chooses between weak delegation and no delegation. We find that the difference in the government's expected payoff between the two regimes of no delegation and weak delegation,  Bureaucratic discretion reduces with bureaucratic drift β. For a given ν, weak delegation is dominated for a sufficiently large β and the government chooses to give the bureaucrat less discretion, through either strict or no delegation. This observation is in line with the ally principle.
Uncertainty, on the other hand, affects bureaucratic discretion non-trivially. The effect depends on how uncertainty changes − whether it happens because the firm is more likely to be low-cost or because it is more likely to be high-cost. It follows from Lemmas 6 and 7 that max ν SD (β) , ν N D (β) is increasing in β. If the firm is more likely to be low-cost, so that ν ≥ max ν SD (β) , ν N D (β) , then the government chooses weak delegation: weak delegation, without an information rent, implements the full-information contract for the low-cost firm, and this is the likely firm type when ν is high. If the firm instead is more likely to be highcost, so that ν < max ν SD (β) , ν N D (β) , then the government opts for giving the bureaucrat less discretion, through either strict or no delegation. The uncertainty principle, therefore, does not carry over to a setting where constrained delegation is possible: in our setting, weak delegation is more prevalent the more likely it is that the firm is low-cost. The reason is that strict delegation, which is an alternative to weak delegation, would be based on the government's exante assessment of the firm's technology, as illustrated in Figure 3. When ν is low, the distortion created in the low-cost type's pollution level is large, with the low-cost strict-delegation contract at G in Figure 3 being far from the low-cost weak-delegation contract at A; on top of that, this distortion is ex-ante likely, exactly because ν is low. The prevalence of weak delegation at high values of ν is in contrast to the case when only full delegation is available, in which the outcome is consistent with the uncertainty principle, as shown in Section III.
For ν < max ν SD (β) , ν N D (β) , Proposition 2 does not provide a full characterization, except that there will be either no or strict delegation, and that which of the two that dominates does not depend on β so that, as illustrated in Figure 4, there will be vertical lines splitting nodelegation regions and strict-delegation regions. The difficulty in assessing this case stems from the difference U N D G − U SD G being non-monotonic when ν < max ν SD (β) , ν N D (β) . In the Appendix, we provide a further discussion of the comparison between no delegation and strict delegation, including an example where the government, for ν < max ν SD (β) , ν N D (β) , will choose no delegation for low and high ν and strict delegation for ν in a middle range.
See, in particular, Figure A1 in the Appendix, illustrating an instance of non-monotonicity of Kundu and Nilssen (2020), discussing delegation in a procurement setting, do not find any occurrence of no delegation at large values of ν. In fact, it is shown there that, if no delegation at large values of ν does not occur for some specific value λ of λ, then it cannot occur for any λ > λ either. 8 The example we present in Figure A2 in the Appendix is a direct counterexample to that result for the present analysis of delegation in a permits setting: the numerical examples pictured in Figures 4 and A2 are identical except that λ is higher in Figure A2, where no delegation occurs at large values of ν, than in Figure 4, where it does not. This observation indicates that the prevalence of no delegation being chosen for both low and high values of ν is larger in the present permits setting than in the procurement setting of Kundu and Nilssen (2020).

V. Conclusion
In this paper, we have discussed how constrained delegation can improve the sales of pollution permits in situations where the polluting firm has private information about its technology. As documented by Gilardi (2009), there has been an increase in the independence of regulatory agencies over the last few decades. We would like to argue that it is important that this delegation is well understood. Our analysis shows that a carefully designed delegation, putting limits on what the hired biased experts can do, is efficient. In particular, we show how strict delegation, even though giving away very little regulatory decision power to the expert, is better than no delegation at all.
Definition. Define , as the government's full-information payoff when obtaining pollution level d from a firm of type θ.
It follows that Applying the envelope theorem, we get Proof of Lemma 1. We use the method of substitution to solve for the optimal contracts. This is the standard method used in the screening literature; see, for example, Bolton and Dewatripont (2005, Ch. 2). The government's expected payoff is increasing in transfers, and the firm's payoff is decreasing in transfers. The government would therefore prefer to increase transfer as much as possible subject to the firm's participation constraint, which can be achieved by having (ICL) and (IRH) binding without affecting the other constraints. The low-cost firm can always pretend to be the high-cost firm and get a payoff of t − θ K − d . In order to make it choose (t, d), the government therefore shares an information rent of IR d := θ K − d . Thus, Replacing t and t in (9) and using the fact that (IRH) and (ICL) together imply (IRL), we can rewrite the optimization problem as: From the first-order conditions of the unconstrained problem, we see that the pollution levels are given by Note that d GN θ ≥ d GN (θ), which ensures that (ICH) is satisfied at the unconstrained solution. The transfers given in the Lemma then follow.
The following lemma documents the convexity property of U N D G , which is used in the proof of Proposition 1.
Lemma A.1. The government's expected payoff in the no-delegation regime, U N D G , is increasing and convex in ν ∈ (0, 1).
Proof of Lemma A.1. By (A.7), the government's expected payoff in the no-delegation regime is where d GN (θ) and d GN θ are given in (10) and (11), respectively. Applying the envelope theorem, we get that which is strictly positive for all ν ∈ (0, 1) and therefore U N D G is strictly increasing.
Next, consider ν ∈ (ν * , 1). Then, d GN θ = K. From (A.9), which is independent of ν for ν ∈ (ν * , 1). Further, note that which implies that dU N D G /dν is continuous at ν = ν * . Therefore, dU N D G /dν is strictly positive and increasing in ν for ν < ν * and continuous at ν = ν * , and it remains constant at a strictly positive value at ν > ν * . From this, we find that dU N D G /dν is strictly positive and weakly increasing in ν for ν ∈ (0, 1), which proves convexity of U N D G .
Proof of Proposition 1. We prove the first part of the Proposition by showing that ∆D < 0 for To see that d∆D/dβ > 0, first note that U N D G is independent of β. By (18), government's expected payoff in the full-delegation regime is where f (θ, d) is given in (A.1) and d BI (θ) is given in (17). Therefore, We have df (θ, d BI (θ)) /dd > 0 for θ ∈ θ, θ , by (A.2) and the fact that d BI (θ) < d GI (θ) for β > 0. Further, dd BI (θ)/dβ = −(1 + λ) θ/(1 + β) 2 < 0. Together, the two inequalities We prove the second part of the proposition by showing that ∆D > 0 for ν = 0, 1, and D Proof of Lemma 3. Follows from replacing t by S − θ (K − d) in U B (θ, α) and using the firstorder condition of (21). The following lemma documents the convexity property of U SD G , which is used in the proof of Lemma 6.
Proof of Lemma 7. Since U SD G and U N D G are both convex and coincide at ν = 0 and ν = 1, the existence and monotonicity properties of ν N D (β) follow from arguments similar to the ones we used in characterizing ν SD (β) in the proof of Lemma 6.
Comparison between no delegation and strict delegation Therefore, Condition (A.17) holds if K is small and θ is large; the effect of λ is, however, non-monotone.
Since both U N D G and U SD G are increasing in ν and have the same value as ν approaches 0, we have that, if (A.17) holds, then no delegation is preferred to strict delegation as ν approaches 0.
(A.18) Condition (A.18) holds if K is large, θ is small, and λ is small. Since both U N D G and U SD G are increasing in ν and have the same value as ν approaches 1, we have that, if (A.18) holds, then strict delegation is preferred to no delegation as ν approaches 1. In Figure 4, we consider the following parameter specification: S = 50, K = 10, θ = 4, θ = 2, and λ = 0.15, which give d GI (θ) = 2.3, d GI θ = 4.6, K − d GI θ = 5.4, ((1 + λ) 2 θ)/2λ = 8.82, and 2 (1 + λ) θ = 4.6. Therefore, both (A.17) and (A.18) hold true, implying that no delegation is preferred to strict delegation for ν close to 0, and strict delegation is preferred to no delegation for ν close to 1. Let us continue our numerical example in the text, but now consider λ = 0.75, which also satisfies Assumption 1, and which gives d GI (θ) = 3.5, d GI θ = 7, K − d GI θ = 3, ((1 + λ) 2 θ)/2λ = 4.08, and 2 (1 + λ) θ = 7, thus satisfying (A.19). In Figure A1, the dotted curve plots U N D G − U SD G against the probability ν of the firm being low-cost, and it shows that only for an intermediate range of ν is strict delegation preferred to no delegation. Figure A2 illustrates the equilibrium regimes. Observe that, when ND is preferred to SD for large values of ν, the equilibrium regime will be a choice between no delegation and weak delegation. For ν > ν * , both U N D G and U W D G are linear in ν, and they coincide as ν approaches is linear in ν for ν > ν * and approaches 0 as ν approaches 1, which can be observed from the dashed curve that plots U N D G − U W D G against ν in Figure A1. 9 The choice between weak delegation and no delegation is completely determined by the slope of the linear part of U N D G − U W D G , which is decreasing in β. Therefore, weak (no) delegation is preferred for small (large) β.