On the Optimal Use of "Dirt Taxes" and Credit Subsidies in the Presence of Private Financing Constraints

Transcription

1 On the Optimal Use of "Dirt Taxes" an Creit Subsiies in the Presence of Private Financing Constraints Florian Ho mann Roman Inerst y Ulf Moslener z February 2011 Abstract We consier an economy where prouction generates externalities, which can be reuce by aitional expenitures. This requires to raise outsie nancing, which is "costly" ue to a stanar agency problem vis-á-vis outsie investors. Policy is constraine as agents are privately informe about their marginal cost of avoiing externalities. We rst erive the optimal linear "irt tax", which is strictly lower than the Pigou tax, as reucing externalities requires aitional, costly funing an as through reistributing resources in the economy a higher tax reuces average prouctive e ciency. We show how this can be improve through nonlinear "irt taxes", which are thus no longer equivalent to a system of traing "pollution rights". E ciency can be further improve through tying grants to creit, as is frequently observe in practice. University of Frankfurt, IMFS. fho mann@ nance.uni-frankfurt.e. y University of Frankfurt an Imperial College Lonon. Aress for corresponence: University of Frankfurt, IMFS, Grüneburgplatz 1, Frankfurt am Main, Germany. inerst@ nance.unifrankfurt.e. z Frankfurt School of Finance an Management, UNEP Center for Climate Finance.

2 1 Introuction Pigou (1920) showe that the optimal tax on a goo that generates externalities shoul be equal to the marginal external amage that arises from the consumption of that goo. For instance, to internalize potential environmental amages, rms aoption of i erent technologies shoul be steere through taxing the emission of greenhouse gasses or, more irectly, the choice of particular technologies. In practice, however, governments also employ policies that subsiize the aoption of particular, more environmentally frienly policies, either through grants or through cheaper creit. One aim of our research is to inform policymakers about the optimal mix of these instruments. We consier an economy where agents must invest to prouce. An they must invest more in orer to prouce with less negative externalities, which can be observe an thus taxe appropriately. (A version of our moel applies to the case where the aoption of more or less polluting technologies is itself contractible.) The key feature of our moel is the introuction of funing constraints. For this we enow agents with a limite amount of resources an grant them access to an external creit market ("outsie nance"). (In our basic moel, we restrict the government s role to that of transferring resources, instea of acting, in aition, as a nancial intermeiary with respect to "foreign investors".) The amount of external nancing that is raise interferes with prouctive e ciency ue a stanar moral-hazar problem, which is moele through the provision of unobservable e ort that a ects the likelihoo of realizing high instea of low nal output. In such a setting, we explore rst the optimal taxation of the externality, combine with a buget-balancing transfer, that maximizes the objective function of a utilitarian policymaker. This can be implemente also through the free istribution of a constraine amount of "pollution rights", which are then trae in a market. We then introuce nonlinear irt taxes as well as policies that are contingent on the amount of outsie nance ("creit") that is raise to (purportely) reuce pollution. Our framework may be particularly suitable to escribe policies that are aime towars rms an that inten to implement a large, costly change in their behavior. The costs of achieving the necessary aoption of i erent technologies shoul then have a non-negligible impact on rms funing requirements. We aress this in a moel where rms costs of outsie nancing are erive from rst principles. 1

3 When policy is restricte to a simple linear tax or, likewise, to the allocation of marketable "pollution rights", we n two reasons for why the tax is strictly lower than the optimal Pigou tax. First, the optimal tax - or, likewise, the aggregate amount of permissible pollution - takes into account the enogenous costs of raising nance so as to reuce negative externalities. Note that this e ect oes not hinge on the presence of heterogeneous agents with i erent marginal costs of avoiance: It is not the irt tax per se that generates a istortion, but more simply the aitional costs of higher avoiance an, consequently, the requirement to raise more outsie nance, as this increases agency costs. With a homogeneous agent, in equilibrium the agent is actually fully reimburse for his taxes, at least in expectation, an the incremental prouctive ine ciency arises only through the nee to raise aitional nance so as to cover the incremental costs of avoiance. With heterogeneous agents, there is a secon rationale for why the optimal linear tax is strictly lower than the " rst-best" Pigou tax. As tax procees are istribute back to agents in equilibrium, a higher tax shifts resources to rms that ue to lower marginal avoiance costs are alreay avantage an nee less outsie nancing. We show that this reistribution, which only arises in the presence of a policy aime at reucing externalities, reuces average prouctive e ciency. In our framework, a linear tax is, however, not the optimal mechanism. We rst explore the use of nonlinear taxes. We n that the optimal nonlinear tax rst ecreases with the externalities that an agent generates an then increases again; or, likewise, for rms the marginal bene t from avoiance is highest at relatively low levels (the " rst units") an at relatively high levels, which are only reache by rms with very low marginal costs of avoiance. As we explore, this non-linearity mirrors similar observations in the theory of optimal income taxation with private information an an utilitarian policymaker. In our framework, a key implication is that compare to such a nonlinear tax, a system of traable "pollution rights", which is akin to a linear tax, woul be ine cient. Furthermore, we show that e ciency can be further improve by linking transfers to the outsie nance that agents raise so as to (purportely) reuce externalities. In a nutshell, by combining a tax on externalities with grants that are linke to creit, the reistribution that is generate by the tax can be counteracte. Thereby, agents with lower avoiance costs n it less pro table to mimic those with higher avoiance costs, so 2

4 as to claim aitional grants. This follows simply as raising more-than-necessary outsie nance exacerbates the agency problem vis-á-vis outsie investors. As a consequence, coupling taxes on externalities with grants linke to loans - as is frequently observe in practice - may be an e cient instrument, as it allows to improve aggregate prouctive e ciency. As note above, our results with a linear tax eviate from the " rst-best" Pigou rule, which woul prescribe to set the tax so as to thereby fully internalizes the marginal social amage from pollution. It also contrasts with the notion of a "ouble ivien", which woul prescribe to set the tax still higher, as the revenues coul then be use to reuce istortions that woul arise from other taxes, e.g., on income (cf. Gaube 2005 for an exposition). However, as was recognize early on (cf. Bovenberg an e Mooij 1994; Parry 1996), the tax on externalities may itself a istortions in prouction or consumption. 1 However, recent contributions in the public nance literature have restore the " rstbest" Pigou rule, most notably through the use of nonlinear income taxes that are use to compensate for the aforementione istortions that a tax on externalities imposes on prouction an consumption in these moels (cf. on nonlinear taxes, in particular, Kaplov 2006; Jacobs an e Moij 2010). 2 Importantly, we solve in our moel for the optimal mechanism an allow the government to tax both the externality as well as observe output. Still, in the presence of outsie nancing requirements an agency problems the wege between the optimal marginal tax an the Pigou benchmark remains. Outsie the public nance literature, our paper is somewhat relate to the literature that analyzes the e ect of liability on environmental care. In some of this literature (cf. the survey in Boyer et al. 2007), compensation for amages is restricte by agents lim- 1 Goulner (1995) terme the positive e ect from lowering other istorting taxes the "revenue-recycling" e ect an the negative istortionary impact the "tax-interaction" e ect. Bovenberg an e Mooij (1994) argue that in the presence of preexisting istortionary taxes, an optimal pollution tax shoul lie below the Pigouvian tax. The intuition is that the collective goo of environmental quality competes with other collective goos, so that the marginal cost of increasing environmental quality through a public policy increases with the marginal cost of public funs. (In more etail, a pollution tax then further istorts the composition of consumption, even if tax revenues are recycle in the form of lower tax rates on labour income.) Cf. also Atkinson an Stern (1974) on the ning that the secon-best provision of public goos uner private information will be lower than the rst-best provision. 2 As pointe out in Kaplov (2006), the key is to recognize that the environmental tax will inuce, ceteris paribus, reistribution an that this a ects the relationship between the Pigouvian tax increment an social marginal amages. Cf. earlier in a similar vein Diamon an Mirrlees (1971), who state that istributional concerns o not justify violating prouction e ciency if the government can optimally ajust taxes (on consumption). 3

5 ite resources or the limite liability embee in the nancial structure that they use to nance prouction. Imposing an extene liability also on the proviers of outsie nance may then impact e ciency, in particular in the presence of nancial frictions an imperfect nancial markets (cf. Pitchfor 1995, Tirole 2010). Finally, the interaction between private nancial frictions an public policy has also been aresse in the literature on entrepreneurship that examines various rationales for policy intervention, in particular the possible spillover e ects create by start-ups. Boaway an Tremblay (2005) o er a broa overview, which mainly focuses on tax consierations. The rest of this paper is organize as follows. Section 2 introuces the economy. Section 3 erives some preliminary results. In Sections 4 an 5 we solve for the optimal linear an nonlinear tax on the externality. Section 6 shows that aing a tax on output oes not alter results. In Section 7 we allow the government to use, as an aitional instrument, a grant linke to the size of the loan. Section 8 summarizes our results. 2 The Economy Agents an Enowments. We consier an economy populate by a unit mass of agents inexe by i 2 I = [0; 1]. There are two points of time: t = 0 an t = 1. In our basic case, each agent is originally enowe with zero resources, but he has access to the same prouction technology that pays out in the nal perio t = 1. Abstracting rst from both the presence of a policymaker an the presence of externalities, starting prouction in t = 0 requires the investment of I 0 0 an generates in t = 1 either zero output or an output of x > 0. 3 The likelihoo of a positive outcome epens on the non-observable, real-value e ort e that the respective agent exerts. For our purpose it will be convenient to make agents utility separable in e ort cost (as well as in the consumption of externalities, in what follows). The respective e ort cost function is enote by c(e), while the likelihoo of positive output x is given by p(e). For what follows, we only nee that the functions are continuously i erentiable, so that a marginal change in contractual parameters has inee a marginal impact on e ciency through a ecting the choice of e ort. However, it is convenient to stipulate, in aition, that c 00 < 0, p 00 0, c 0 (0) = 0, an p 0 (0) > In our moel, the investment outlay an output are both measure in the same unit of "resources" or capital. 4 For instance, we coul stipulate that p(e) = e, c(e) = e 2 =(2), where is then taken to be su ciently large so as to ensure that p(e) < 1 hols in equilibrium. 4

6 Note that as presently all agents in the economy start with zero resources, they will have to raise capital from an external capital market. (To be speci c, we may think of this as raising capital abroa.) We stipulate that the agents utility is linear in the resources that they consume an that they o not iscount future consumption, which is why in our moel the only rationale for borrowing (i.e., raising outsie nance) is for prouction. In terms of contracting with outsie investors, we stipulate that the output realization is veri able an can thus be part of a nancial contract. Externalities. Prouction generates negative externalities, which can be reuce by aitional investment. Precisely, we stipulate that when the prouction of agent i creates y i 0 units of these externalities, then this a ects all other agents equally an, thereby, generates the social loss y i with > 0. Recall that we stipulate that utility is also aitively separable in externalities. That is, when in t = 2 agent i is left with w i resources for consumption an has exerte e ort e i, then his total utility is u i = w i c(e i ) y j j: (1) Note that this implies that an agent s private incentives to reuce his own externality y i are zero (given his "small size" compare to the uniform istribution of the externality over all agents). Externalities are veri able. An agent s cost of avoiing externalities epen on his type. Presently, this will be the only source of heterogeneity between agents. De ne the strictly positive real-value type by i, where we assume that i is, for all i 2 I, inepenently an ientically istribute accoring to the istribution function F (), permitting a ensity function f() > 0 for all 2 [; ]. Generating a level of externalities y, e.g., by using the respective technology mix or by running prouction with the respective safety provisions, is associate with a particular prouction cost. It is convenient, albeit this is without loss of generality, to stipulate that there is a given maximum level of externalities y (per business) an, consequently, to enote the respective avoie externalities by a = y y. Then, we capture avoiance costs by the twice continuously i erentiable function K(a; ) with K(0) = 0, K 0 (a = 0) = 0, an K 00 > 0. Further, we stipulate that j2i 2 K(a; ) a < 0: (2) 5

7 That is: Higher types have everywhere lower marginal costs of avoiance. For instance, we coul stipulate that K(a; ) = k(a)=. Note that the respective costs are incurre, together with the investment I 0, right when prouction starts in t = 0. Further, observe that the agent s economic success (x or zero) oes not irectly interact with the generation of externalities. However, as we show below, it interacts inirectly through the incurre avoiance costs an thus the nee to raise more external nance. That there is no irect interaction of prouce externalities an the likelihoo of high output allows to restrict contracts with external investors to repayments that are conitional only on realize output (as this is then a su cient statistic for e ort). Alternative Interpretation. Our chosen set-up, where the reuction in negative externalities is a function of investment, allows also for the following alternative interpretation. The following interpretation may be particularly suitable when introucing later policies that are contingent on raise outsie nance. We coul think of y or likewise a = y y as a (continuous) veri able technology choice, e.g., the "amount" of more fuel-e cient equipment that is installe or of energy-e cient builing materials that are use when setting up prouction. Though the purchase costs may be the same for all agents, agents costs of installation or, alternatively, their total opportunity costs may i er, given the builings, equipment, an technologies that they alreay own. For higher, the associate costs are lower accoring to conition (2). Note nally that a given choice of y an a coul then be associate with a i erent istribution over externalities (which, however, is no longer type-epenent). Without loss of generality, any policy coul then target irectly the aoption of a. When y an a capture the aoption of a particular, veri able technology, agents utility function (1) an thus also the policymaker s objective function coul be rewritten accoringly, namely as a function of expecte externalities, without changing results. Feasible Policies. In our main analysis we introuce a utilitarian policymaker, who maximizes the expecte utility of all agents: E R u i i (in contrast to, for instance, only implementing policies that woul lea to a Pareto improvement; cf. our iscussion below). For simplicity, we refer to the policymaker as the government. We consier various 6

8 policy instruments as we go along. In particular, we consier rst a simple, linear tax on externalities, couple with a transfer that is pai out of the receipts. This can also be implemente through a market for pollution rights. We further consier non-linear taxes an, subsequently, more general mechanisms that can also conition on rms nancial nees (i.e., the creit that they have to raise). Throughout our main analysis we also restrict the government to policies that, conitional on the generate externalities, reistribute resources only at the initial state. As we show below, we can inee ignore without loss of generality taxes that are levie on nal output. This hols even when the "rights" to these taxes coul be istribute initially an coul thus be sol to outsie investors, as to thereby reuce agents nee to raise outsie nance backe only by promises on own future output. Also, note that we o not allow the government to raise nance on behalf of agents. That is, the government is presently not allowe to itself borrow funs from outsie the economy. This coul be justi e on various grouns. For instance, in our two-stage worl outsie creitors may not n it possible to successfully ask repayment from a strategically efaulting sovereign. However, we show below that having the government acting as an intermeiary to raise nance oes by itself not a ect the outcome. It woul merely a an alternative way to implement a given outcome that woul, otherwise, be obtaine through having agents raise more funs themselves. When the government controls all ows of creit, however, then this gives rise to an aitional control variable, though we show that the same outcome is obtaine when the government oes not act as a nancial intermeiary but merely conitions taxes or grants on the amount of outsie nance that agents privately raise. 3 Preliminary Results 3.1 The Outsie Financing Problem Consier thus the problem of an agent who must raise capital L (a "loan") to start prouction. As he can only pay back in case output is positive, the contract with outsie investors can be restricte to a single variable: The repayment R in case the output equals x. Given some repayment R, the agent s uniquely optimal e ort level e is given by the 7

9 rst-orer conition p 0 (e )(x R) c 0 (e ) = 0: (3) (Recall that we assume that the externality a ects each agent only "in the aggregate".) This can then be substitute to obtain the investors break-even requirement p(e )R = L: (4) While (3) an (4) together may have multiple solutions, we pick in what follows the pair (R; e ) that has the lowest value R an, consequently, achieves the highest payo for the agent. Clearly, this is the unique equilibrium in a game where either outsie investors compete or the agent makes a take-it-or-leave-it o er. Further, it is immeiate that e is then strictly ecreasing in L while R is strictly increasing. Denote! = p(e )(x R) c(e ); which given the bining break-even constraint (4) is the total expecte surplus net of the funing expenitures. By the previous iscussion we can write this as a function of L:!(L). With L = 0 an thus R = 0, the agent woul choose a rst-best value solving p 0 (e F B )x c 0 (e F B ) = 0, thereby realizing a total surplus of!(0). Clearly, it hols that e < e F B whenever L > 0. When we thus compare the total net surplus at the benchmark with L = 0 an at any other choice L > 0, we have!(l) <!(0) L: This i erence beyon the change in funing requirements captures the crucial ine ciency that arises from the outsie nancing problem. It follows as the agent shirks when he no longer realizes the full bene ts from putting in higher e ort an, thereby, pushing up the likelihoo of success. Observe further that from the break-even constraint (4) the agent is the resiual claimant, so that it follows immeiately that!(l) is strictly ecreasing in L: As more of the output has to be promise away as repayment, the agent s incentives to exert e ort further ecline, resulting in a further reuction of e ciency. This captures the key ine ciency that arises from the agency problem ue to non-observable e ort an the nee to raise outsie nance. 8

10 So far, we restricte attention to eterministic contracts with outsie investors. Given risk neutrality of both agents an outsie investors, without loss of generality the most general, stochastic contract is escribe as follows. Note rst that (L;!(L)) escribes investors an the agent s expecte payo s once nancing is sunk (net of e ort costs, but gross of initial capital L). A contract with investors coul now prescribe the following, next to the initial provision of capital L: A istribution over values L n with E[L n ] = L so that when a particular value L n is rawn, the contract that is then implemente generates for the investor the expecte repayment of L j an, consequently, the expecte payo!(l n ) for the agent. Clearly, by optimality the chosen lottery woul maximize E[!(L n )] subject to E[L n ] = L. Denote the respective value by b!(l) = max E[!(L n )]. Importantly, while it is immeiate that also b!(l) is strictly ecreasing in L, it is also concave: b! 00 (L) 0 at points of i erentiability. (It is continuous an i erentiable almost everywhere.) The argument follows simply by contraiction: If this was not the case, then we coul n from Jensen s inequality three values L 1 < L < L 2 an two probabilities 1 an 2 so that L = L L 2 2 while b!(l) < b!(l 1 ) 1 + b!(l 2 ) 2, contraicting the construction of b!() at L. Lemma 1 If the agent nees initial nancing of L, then uner the optimal contract that lets outsie investors just break even he realizes the expecte payo (net of costs of e ort) b!(l), which is continuous, strictly ecreasing with b! 0 (L) < 1, an concave. In what follows, we n it more convenient to suppose that alreay the payo function!(l), for which we i not mae use of lotteries, is strictly concave. This clearly implies that lotteries are not optimal. Still, our subsequent results hol generally when we use, instea, b!(l). 3.2 A Secon-Best Benchmark without Reistribution The object function of a utilitarian government is to maximize the expecte utility of all agents: E[u i ] = [!(L i ) y i ] i: (This uses that!(l i ) alreay takes into account the investment costs as these are fune by outsie investors.) Suppose for a moment that the government coul control irectly 9

11 the activities of each agent, e.g., as i was observable. Abstracting from other transfers, i.e., transfers between agents, the government woul then choose y i so as to maximize s i =!(L i ) y i with L i = I 0 + K(y y; i ): Proposition 1 If the government coul observe each agent s type an interfere only by controlling irectly the actions of any agent i, then epening on = i the agent woul choose a unique level of externalities y SB () an thus a unique level of avoiance a SB () = y y SB (). This is strictly increasing in an etermine by K 1 (a SB (); )! 0 (I 0 + K(a SB (); )) = : (5) Proof. For agent i, the erivative of s i is given by s i y i =! 0 K 1, where K 1 > 0 an! 0 < 1. Note that then 2 s i (y i ) 2 =!0 K 11 + (K 1 ) 2! 00 < 0; where we use K 11 > 0 an! 00 < 0, so that there is a unique value y SB (). By implicit i erentiation we then obtain y SB () = K 12! 0 + K 1 K 2! 00! 0 K 11 + (K 1 ) 2! 00 < 0; (6) using that K 2 < 0, as for all y > 0 we have K 12 < 0 while K(0; ) = 0. Q.E.D. When! 0 = 1, then conition (5) woul simply state that for each the costs of avoiing externalities shoul equal the marginal social bene ts,. That is: K 1 (a SB (); ) =. Note that from K 12 < 0, the level of avoiance woul then still be strictly increasing in. With external nancing, however, we have that! 0 < 1 (cf. Lemma 1), so that at the secon-best value a SB () the marginal avoiance costs are still strictly below. In light of the subsequent analysis, for which the "avoiance scheule" a SB () will serve as a benchmark, the following case istinction will be of interest. Take again the case where! 0 = Note now that 1 an enote the " rst-best value" by a F B (), which satis es K 1 (a F B (); ) =, with a F B() = K 12(a F B (); ) K 11 (a F B (); ) > 0: K(a F B(); ) > 0 if K 2 + a F B() K 1 > 0 () K 2 K 11 > K 1 K 12 : (7) 10

12 That is, when conition (7) hols, then the marginal avoiance cost avantage of a higher type is su ciently large so that in the rst-best benchmark he woul invest strictly more to reuce externalities, implying that he woul have to raise strictly more outsie nance. As is easily checke, this is, for instance, the case when 5 K(a; ) = k(a) = 1 a 2 2 : Instea, when the converse of conition (7) hols everywhere, then the irect e ect (K 2 < 0) ominates an higher types woul invest less uner the rst-best choice of externalities. These observations exten also to the secon-best benchmark. Corollary 1 When conition (7) hols everywhere, then uner the secon-best benchmark (cf. Proposition 1) the cost of avoiance K(a SB (); ) an thus the total outsie nance are strictly increasing in. When the converse hols strictly everywhere, then uner the secon-best benchmark the cost of avoiance K(a SB (); ) are strictly ecreasing in. Proof. We have which is just conition (7). Q.E.D. K(a SB(); ) > 0 if K 2 + a SB() K 1 > 0 K 12! 0 + K 1 K 2! 00 () K 2! 0 K 11 + (K 1 ) 2! K 00 1 > 0 () K 2 K 11 > K 1 K 12 ; This case istinction is now crucial when consiering the optimality conition (5). When conition (7) hols, then together with the concavity of! from Lemma 1, we have that the term! 0 > 1 is strictly increasing in. Compare to the rst-best benchmark, the nee to raise outsie nance then ampens the avoiance of externalities more for high-type than for low-type agents. The opposite is the case when conition (7) oes not hol. 3.3 A Benchmark with Reistribution Suppose now that the government can also reistribute resources. Such a reistribution is accomplishe through transfers. Precisely, suppose now that the government coul control 5 More generally, this hols when k 0 =k > k 00 =k 0. 11

13 not only y i but that it coul also stipulate a transfer T i for each agent. Consequently, the respective agent woul then have to raise the funs L i = I 0 + K(a i ; i ) + T i. Again we stipulate that the government can observe the agent s type = i. The respective Lagrangian for the government s problem is then L =E[u i ] + T i i; (8) taking into account the constraint that the integral over all transfers must sum up to zero. Recall now that for the benchmark in Proposition 1 higher-type agents always have a higher level of avoiance, a SB (). Still, conition (7) or the converse of it (provie either hols globally) etermine whether total expenitures to reuce externalities an thus the nee to raise outsie nance are increasing or ecreasing in. Recall further, that when expenitures are increasing in, then this reuces prouctive e ciency for high-type agents, while the opposite hols when expenitures are ecreasing in. In either case, as! is strictly concave by Lemma 1, the utilitarian government woul want to reistribute resources an thus ajust the nee to raise nance across agents. Conition (7) etermines the irection of the optimal reistribution. Note that in our moel this scope for e ciency-enhancing reistribution hinges entirely on the agency problem vis-á-vis outsie investors, together with the i erence in agents costs of avoiing the negative externalities. (When a = 0 hols for all agents, then also! 0 is equalize across agents, as they all raise only outsie nance equal to I 0.) Proposition 2 Suppose the government coul observe each agent s type an stipulate, next to the level of avoiance, a transfer. Then the respective type-epenent transfer T RD () woul be chosen to equalize each agent s nee to raise nance, L RD () = L, while the respective level of avoiance y RD () woul satisfy K 1 (a RD (); ) = with > 1. (9) This is everywhere strictly lower than the rst-best level of avoiance, an it compares to the secon-best benchmark (cf. Proposition 1) as follows: i) When conition (7) hols, then there is reistribution to high-type agents. With reistribution high-type agents have a strictly higher level of avoiance compare to the benchmark without reistribution (a RD () > a SB ()), while low-type agents have a strictly lower level of avoiance. 12

14 ii) If the converse of conition (7) hols, then there is reistribution to low-type agents, who then have a strictly higher level of avoiance uner reistribution, while high-type agents then have a strictly lower level compare to the benchmark without reistribution. Proof. From the Lagrangian we have for type the rst-orer conitions w.r.t. y() an T () K 1 (a(); )! 0 (I 0 + K(a(); ) + T ()) = 0; from which we obtain (9) as well as > 1 as! 0 < that L RD () = L = (! 0 ) 1 ().! 0 (I 0 + K(a(); ) + T ()) + = 0; 1. We then have also the requirement Suppose that at some we have that a SB () = a RD () an thus y SB () = y RD (). Comparing slopes at this type, using (6) an implicitly i erentiating (9), we have for the two strictly positive slopes that: a SB () < a RD() () K 12 + K 1 K 2! 00 =! 0 K 11 + (K 1 ) 2! 00 =! 0 < K 12 K 11 () K 2 K 1 K 12 K 11 > 0; when a SB () = a RD () (10) as in conition (7). It is then straightforwar to show that a SB () a RD () or the converse can not hol everywhere. Q.E.D. Clearly, absent problems of private information the utilitarian government optimally uses reistribution until prouctive e ciency is equate at each agent, i.e., until all! 0 are equalize. This is the case when the respective nees to raise nance are equalize, so that ultimately!(l RD ()) =!(L) an thus also! 0 (L RD ()) =! 0 (L) no longer epen on the agent s ientity. 4 Linear Tax So far we have assume that the government can observe i an, thus, control irectly the activities of each agent. From now on, we suppose that i is the agent s private information. 13

15 In this Section, we restrict the government to the following simple policy. Depening on the volume of prouce externalities, y i, each agent is taxe accoring to the function 6 (y) = 0 + y: (11) Here, with a slight abuse of notation, is the per-unit tax on the externality, while the xe component 0 takes into account the overall istribution that is achieve by (optimally) making the government s buget just balance: 0 + y i i = 0: (12) Without loss of generality, we stipulate that the agent must "purchase" the respective pollution rights when starting prouction in t = 0. Consequently, without resources on his own, an agent must raise outsie nance equal to L(y; ) = I 0 + K(y y; ) + (y): (13) Optimal Avoiance. Given the tax scheme (y) = 0 + y, we consier rst the program of an iniviual agent. The agent chooses y i an, consequently, has to raise L(y i ; i ), as given by (13). Dropping the subscript i, an agent of type thus chooses y to maximize! (L(y; )) with L(y; ) given by (13). Lemma 2 Suppose the government imposes a (buget-breaking) tax-cum-transfer T (y) = 0 + y. Then, an agent of type chooses the optimal level of externalities y () an thus a unique level of avoiance a () = y y () so that K 1 (a (); ) = ; (14) from which a () is strictly increasing in both an. Still, higher-type agents invest less in avoiance an thus nee to raise less outsie nance: L() = K 2 (a (); ) < 0: (15) Proof. We have! y = ( K 1(y y; ))! 0 (L(y; )) ; 6 Strictly speaking, the tax scheule is an (a ne) two-part tari. 14

16 which is strictly quasiconcave an yiels the rst-orer conition (14). i erentiation an using (2) we have further that From implicit a () a () = = K 12 (a (); ) K 11 (a (); ) > 0; 1 K 11 (a (); ) > 0: Finally, expression (15) follows from substituting the rst-orer conition (14) into the total erivative of L(). Q.E.D. Hence, with a linear tax on externalities, each agent chooses a level of avoiance where the marginal nancial bene ts arising from a reuction in the incurre tax are equal to the marginal cost of avoiance. Importantly, prouctive e ciency, as expresse through the slope! 0, plays no role in this trae-o. Moreover, note that regarless of whether the previously iscusse conition (7) hols, uner the agent s optimal choice his nee to raise outsie nance is always strictly ecreasing in his type. In fact, as now the agent chooses his privately optimal level of avoiance, this follows immeiately from optimality, as otherwise higher-type agents coul not enjoy a higher expecte utility!(l). Optimal Tax. Given the agent s optimal ecision, using Lemma 2, the government s program is now to maximize E[u i ] = [!(L(y (); )) y ()] F () (16) subject to the buget-breaking constraint (cf. (12)) 0 + y ()F () = 0: (17) Take for a moment the benchmark without nancial constraints, so that everywhere! 0 () = 1. Then, from substitution of (17) into (16) while using the agent s rst-orer conition (cf. Lemma 2) we obtain the Pigou rule =. (This obviously then implements the rst-best outcome, espite agents private information about their marginal cost of avoiance.) The following result characterizes the optimal linear tax when agents must raise outsie nance. 15

17 Proposition 3 The optimal linear per-unit tax satis es R! 0 (L(y (); ))F () =!0 (L(y (); )) y () R which implies that is strictly smaller than. y () R y ( 0 )F ( 0 ) F () ; F () (18) Proof. We can substitute from (17) to obtain for each type the nancing requirement L(y (); ) = I 0 + K(y y (); ) + y () y ( 0 )F ( 0 ); so that E[u i] " y () equals R y ( 0 )F ( 0 ) + y () ( K 1 ()) R y ( 0 ) y () # F ( 0 )! 0 () F (): Substituting the rst-orer conition (14) for y (), = K 1, this gives rise to the rstorer conition (18). From R remains to prove that! 0 () y () y () F () < 0, given that y () is strictly ecreasing, it y ( 0 )F ( 0 ) F () < 0: (19) To see this, note rst that, next to y () < 0, we have from expression (15) an Lemma 1 that (!0 (L(y; ))) = L()!00 > 0: (20) De ne now the unique type b where y ( b ) = E[y ] = R y ( 0 )F ( 0 ), while y () > E[y ] hols for < b an y () < E[y ()] hols for > b. We can now rewrite the left-han sie of (19) as LS =! 0 () [y () E[y ]] F () + < b > b! 0 () [y () E[y )]] F (): (21) There, the terms in the rst integral are all strictly negative an the terms in the secon integral are all strictly positive. Given strict monotonicity of! 0 (), we can thus erive the upper boun LS <! 0 (L(y; b )) [y () E[y ]] F () +! 0 (L(y; b )) [y () E[y ]] F () < b < b =! 0 (L(y; b )) y () y ( 0 )F ( 0 ) F () = 0. 16

18 Q.E.D. From (18) an optimal tax is strictly lower than. for two reasons. The rst is capture by the multiplier on the left-han sie (cf. Lemma 1 for! 0 < capture by the term! 0 () y ()! 0 (L(y (); ))F () > 1 (22) 1); given R y () F () < 0, the secon is y ( 0 )F ( 0 ) F () > 0; (23) that is subtracte on the right-han sie. We iscuss both in turn. The term (22) is similar to the respective multiplier obtaine in the secon-best benchmark in expression (5). It captures the fact that to lower externalities, agents must raise outsie nance, which ue to the agency problem involves an aitional "shaow cost", namely in the form of lower e ciency as e ort becomes ine ciently low. (Formally,! 0 < 1.) Next, the term (23) captures the e ciency implications of the reistribution of resources that goes han-in-han with the applie taxation, namely from agents with higher marginal avoiance costs to agents with lower marginal avoiance costs. (Note that when = 0, then the type oes not a ect utilities as y i = y an, consequently, avoiance costs are zero.) The impact of reistribution on aggregate prouctive e ciency is negative. This follows from the following two observations: First, with a linear tax high-type agents incur, uner the optimal choice y (), strictly lower costs of avoiance; secon,! is strictly concave. As the tax on externalities shifts resources to high-type agents, this reuces the agency problem of high-type agents but increases the agency problem of low-type agents. Thus, it makes the alreay more prouctive high-type agents (enogenously) still more prouctive, while further reucing prouctivity of low-type agents. This leas to a reuction in aggregate e ciency of prouction in the economy. In what follows, we explore i erent possibilities of how the government coul improve on the linear tax. Our previous iscussion suggests, in particular, the following ways. First, to avoi the iscusse reistribution to agents with lower marginal costs of avoiance, a better policy coul try to link transfers not only to the generate externalities but also 17

19 to the nee to raise outsie nance. Secon, even when this option is not feasible so that transfers can still conition only on the generate externalities, there coul be scope for nonlinear taxes. Before exploring this option in the following Section??, we rst iscuss an alternative implementation of the optimal linear tax scheme an shortly compare the outcome uner such a scheme with the secon-best benchmark, where i is observable. Alternative Implementation. It is straightforwar to see that the government coul implement the outcome i erently as follows. It coul set a total maximum capacity for externalities Y an allocate this uniformly (an for free) across all agents. (Thus each agent receives the same capacity, which we may write as Y i = Y, as there is the measure one of agents in the economy.) These capacities or "pollution rights" are then trae in the market. When is the resulting price, we obviously have that K 1 (a (); ) =, as previously in (14), together with R a ()F () = y Y. This uniquely links to Y, an vice versa. The equivalence of the two policy instruments can then be seen immeiately from substituting into the funing retirement (13) L(y (); ) = I 0 + K(y y (); ) + (y () Y ) = I 0 + K(y y (); ) + (y () y ()F ()); which is just the same as uner the linear tax, after substituting the "break-even" constraint for taxes (17). Comparison with Secon-Best Benchmark. Before moving on, we compare the outcome with a linear tax to those obtaine in the previously erive benchmarks, where the government can control irectly agents actions, without being impee by private information. There is an interesting comparison with respect to the secon-best benchmark without reistribution. With a linear tax we have observe that high-type agents always en up raising less nance, as L() is strictly ecreasing. Moreover, the linear tax implies a strictly positive transfer to high-type agents, which further reuces their nee to raise external nance. In the secon-best benchmark without reistribution, recall that conition (7) etermines whether avoiance expenitures an thus the nee for outsie nance increase or ecrease in. In analogy to the comparative analysis in Proposition 2, we then have the following result. 18

20 Corollary 2 The outcome with a linear tax compares as follows to the secon-best benchmark without reistribution (cf. Proposition 1). i) When conition (7) hols, high-type agents have a strictly higher level of avoiance with the linear tax (a () > a SB ()), while low-type agents have a strictly higher level of avoiance in the secon-best benchmark ii) When the converse of conition (7) hols, high-type agents have a strictly higher level of avoiance in the secon-best benchmark an low-type agents with the linear tax. Proof. Using the agents rst-orer conition K 1 (a (); ) =, we have R R = K 1 (y y (); )! 0 F () +!0 y () y ( 0 )F ( 0 ) F () R : F () From this we have that a () = K 12(y y (); ) K 11 (y y (); ) : y () Hence, we have in analogy to the proof of Proposition 2 that at types where expenitures are the same (i.e., where the functions cross) a SB () < a () () K 12 + K 1 K 2! 00 =! 0 K 11 + (K 1 ) 2! 00 =! 0 < K 12 K 11 ; which is just conition (7). The claim then follows as a is not higher or lower everywhere uner either regime. Q.E.D. 5 Nonlinear Taxes Consier now a general tax function (y). As is stanar, we analyze rst a irect, incentive compatible mechanism. Then, for each type there is a tax T () as well as a prescribe level of externalities y() that must be incentive compatible. That is, for all types 2 it must hol that where! (L(y(); T (); )! L(y( b ); T ( b ); ) L(y(); T (); ) = I 0 + K(y y(); ) + T (); L(y( b ); T ( b ); ) = I 0 + K(y y( b ); ) + T ( b ): for all b 2, (24) 19

21 (In wors, type must not strictly prefer to preten to be any other type b.) It is convenient to express the following optimization problem purely in terms of (permitte) externalities y(), rather than optimal avoiance a(). For ease of exposition only we presently assume that the menu f(y(); T ())g is continuously i erentiable. In this case, the incentive constraint (24) woul hol locally if! L(y( b ); ) b = T 0 () y 0 ()K 1 (y y(); ) = 0: (25) b= Further, we presently assume that the " rst-orer approach" is vali such that (25) is suf- cient to ensure global incentive compatibility. As is immeiate from the single-crossing property (2), note that this requires also that the characterize function y() be nonincreasing. De ne now (with some abuse of notation) the payo function uner truthtelling 7 u() =! (L(y(); T (); )) : We know that by incentive compatibility u() is nonecreasing an continuous an thus a.e. continuously i erentiable with L(y( b ); T ( b ); ) =! 0 K 2 (y b= y(); ) > 0 when y() > 0. (26) Control Problem. To solve for the optimal menu we set up the government s optimal control problem. For this we make the following choices. With some abuse of notation e ne L() = I 0 + K(y y(); ) + T (); which we take as the state variable. As thus u() =!(L()), we have from (26) that L() = K 2 (y y(); ) < 0: (27) Further, from T () = L() [I 0 + K(y y(); )] 7 This still presumes that taxes an subsiies are fully "use" to increase or reuce the amount of funs that must be raise externally (instea of being immeiately consume or save for consumption in the nal perio). It can be shown that this restriction is without loss of generality. 20

22 we can substitute pointwise for T (), which leaves us with the single control variable y(). The objective is thus to maximize [!(L()) y()] F () subject to the "law of motion" (27) an the buget balance conition [L() K(y y(); ) I 0 ] F () = 0: With the Hamiltonian then given by H = [!(L()) y()] f() + [L() K(y y(); ) I 0 ] f() + ()K 2 (y y(); ); an optimal solution must satisfy the rst-orer conition for y() an for the costate variable f() [ + K 1 (y y(); )] ()K 12 (y y(); ) = = 0 (), f() [! 0 (L()) + ] + 0 () = 0: (29) There are no terminal conitions, an the transversality conitions are given by lim () = 0; (30)! () = 0: (31) lim! Using (30) an (31), we thus obtain from integrating (29) an () = (! 0 (L(#)) + ) f(#)# = = (! 0 (L(#)) + ) f(#)#! 0 (L(#))F (#) > 1: (32) Note that expresses the marginal bene ts when the economy s resource constraint was marginally relaxe (e.g., by some initial enowment that coul be allocate by the government). Now, from (27) an the concavity of!(l()), we have that! 0 (L()) is increasing in. Thus, making use of (32), 0 () = f() (! 0 (L(#))! 0 (L())) F (#) 21

23 is rst positive an then negative, i.e., () is rst increasing an then ecreasing. Clearly the transversality conitions then imply that () 0 hols everywhere. Rearranging now the rst-orer conition for the control y() in (28), we have 8 K 1 (y y(); ) = + () f() K 12(y y(); ); (33) which, using K 12 < 0, 0 an > 1, shows rst that K 1 () <, i.e., also with nonlinear taxes externalities are for all types higher than uner the ( rst-best) Pigou tax. Moreover, note that it hols only at the bounaries an (when they are nite) that () = 0 an thus K 1 (y y(); ) = an K 1 (y y(); ) = ; while K 1 (y y(); ) < for all 2 (; ): Recall that we substitute out the tax T (). We obtain the slope from T 0 () = L 0 () K 2 () + y 0 ()K 1 () = y 0 ()K 1 () 0: That is, as in the case of linear taxes, the nonlinear tax on the externality still involves a transfer from low-type agents to high-type agents, given that T 0 () 0 (an strictly so where y 0 () < 0). 9 8 Presently, we assume that the rst-orer approach applies an that y(), as characterize by (33) is nonincreasing. Di erentiating (33) wrt we obtain (K 12 (y y(); ) y 0 ()K 1 (y y(); )) = () K 12 (y y(); ) + () f() f() (K 122(y y(); ) y 0 ()K 112 (y y(); )) an thus y 0 () = () 1 f() K 112() () f() K 1 () K 12 () + () f() K 122() K 12 () : 9 In fact, incentive compatibility implies that in both cases, i.e., with linear an nonlinear taxes, the marginal tax w.r.t. the agent s type is given by y 0 ()K 1 (). (For the linear tax we can use that T 0 () = y 0 () an that K 1 () =.) 22

24 Implementation. (y) = T ((y)), then If we were to implement the prescribe menu with a nonlinear tax 0 (y) = T 0 () y = T 0 () y 0 () = K 1(y y; (y)) (34) = 1 + ((y)) f((y)) K 12(y y; (y)) where when y 0 < 0 hols strictly we use (y) = y 1 (y()). This yiels next 00 (y) = () f() K () y f() K () K 12 : (35) f() Recall now from the transversality conitions (30)-(31) that at the bounaries an (when they are nite) we have () = 0. Denote now the lowest an highest realize level of externalities by y l = y() < y h = y(): Further, recall that 0 > 0 for low an 0 < 1 for high, while y bunching). Using then we have () = 1 f() f 2 () [f()0 () + f 0 ()()] 00 (y l ) = y K 0 () 12 f() < 0; 00 (y h ) = y K 0 () 12 f() > 0: < 0 (when there is no In wors, uner the optimal nonlinear tax (y), at very high levels of y the marginal tax 0 (y) > 0 becomes strictly ecreasing, while at very low levels of y the marginal tax is strictly increasing. In terms of the achieve avoiance a this result reas as follows. For the rst "units" of avoiance the marginal bene ts are strictly ecreasing, while when avoiance is alreay very high, the marginal bene ts of further avoiance are strictly increasing. Proposition 4 Uner the optimal nonlinear tax (y) the marginal tax rate is highest at the two extremes, y l an y h. This implies that when the generate externality is still high, then the marginal tax is strictly increasing. Instea, when the generate externality is alreay low, then the marginal bene ts from further reucing pollution are strictly increasing. 23

25 Put i erently, by Proposition 4 the optimal nonlinear tax rewars a reuction of externalities in particular at very high an very low realizations, i.e., the " rst units" an the "last units". The intuition for this result is as follows. At the heart is the attempt to restrict the reistribution that is mae to high-type agents. Recall that reistribution is higher when the marginal tax is higher. Note, further, that () is the marginal increase in social welfare resulting from shifting one unit of require nancing from types below to types above, i.e., from ampening reistribution to higher types. Hence, the marginal tax rate shoul be lower the higher (), as can be seen from expression (34). Obviously, at the lower bounary, there is no nee to istort the implemente choice of externalities (away from the rule K 1 = ) as there is no one bene tting from reistribution below. At the other extreme, when =, there is clearly no longer a bene t from "istorting" the marginal tax since there is noboy contributing to reistribution above. 6 Taxes on Output As we explore in etail in the preceing two sections, the government s problem is that it has only a single instrument for its two goals of, rst, a ecting the generation of externalities an, secon, limiting the amount of resources that are thereby reistribute to high-type agents. This reistribution lowers prouctive e ciency. It coul now be aske whether the government coul not improve overall e ciency through taxing the nal output, which is zero or x for all agents. For instance, one coul imagine that the government coul then reistribute to agents these rights to future taxes, which agents coul then plege to outsie investors so as to, thereby, reuce their own nee to raise nance. It is, however, straightforwar to see that such a policy is of little use. For brevity s sake we restrict ourselves to a somewhat informal argument. Suppose thus that the mechanism woul also specify a tax on positive outcome z() < x. 10 Taking this as well as the resulting nancial nee L() into account, this gives rise to a level of e ort an, thereby, to some expecte probability of success p(). Denote by () = z()p() the resulting expecte levy on output. As these can be istribute among agents, who can 10 Importantly, this can not conition on the subsequently agree repayment R with the respective investors, which woul proxy for a making the mechanism contingent on the size of external nance (cf., however, the following section). 24

26 then plege it to outsie investors, we now have the aggregate resource constraint [T () + ()] F () = 0: The key observation is now the following. For a given agent, the respective total tax payments over the two perios can be lumpe into the rst perio without a ecting the outcome. Precisely, the agent then faces the transfer e T () = T ()+() up-front an thus has to increase his external nance by the amount (). This, however, leaves both the agent s utility an total prouctive e ciency completely unchange as it oes not a ect p(). 7 Loan-Base Grants So far, we have consiere a class of policy interventions with transfers that conition only on the generate externalities. A more general policy coul link transfers also to the amount of outsie nance raise, thus, trying to limit the reistribution to agents with lower marginal costs of avoiance. Hence, in this section, we allow the government to set, apart from the optimal (linear or nonlinear) tax on externalities, also a creit subsiy epening on the size of the loan raise by a particular rm. Such a policy can realistically only conition on whether a loan of a certain size is raise, however, not on whether this is really spent on avoiance, as the concrete usage of the funs within the rm is usually unobservable. Neither can the government restrict the rm to raise, apart from the subsiize loan, aitional outsie nancing elsewhere. Within this basic framework, we erive in the following two sections the optimal creit subsiy complementing, rst, a linear an, secon, a nonlinear tax on externalities. Presently, the focus is solely on showing how loan-base grants allow to improve e ciency through reucing the reistribution to high-type agents that is implie by a tax on externalities. Linear Tax with Subsiize Loans. We rst consier the case where the government is restricte to a linear tax on externalities or, likewise, a system of traable rights. In aition, the government can now specify a transfer t() that can be i erent from a xe, type-epenent istribution t() = 0. For what follows we take it as given that, 25