A Theory of Optimal Green Defaults

This paper develops an analytical framework for studying the Baumol–Oates efficiency of traditional single instrument abatement policies vis-à-vis green defaults in the face of price inertia and passive choice by subpopulations. In this special case of behavioural heterogeneity, command and control approaches can outperform price-based instruments while pure tax/subsidy schemes need to be adjusted in order to achieve politically desired levels of abatement. We also prove that choice-preserving nudges are superior to any single-instrument policy in this case. An average marginal abatement cost rule is developed to optimize the green defaults and traditional policies of standards and prices under different degrees of market rigidity.


Introduction
Behavioural economists have observed a great number of seemingly irrational human be haviours, including inertia to price incentives and a stickiness of 'defaults' caused by indifference or a naive belief in experts and authorities. These behavioural traits are often observed in subpopulations such as elderly people or youngsters, but are also ev ident in comparisons of small and large firms. This gives rise to the profound problem of behavioural heterogeneity in economic policy. The notion that individuals, firms and corporations respond differently to economic stimuli and default rules has obvious con sequences for the effectiveness and optimality of regulation. The long-standing debate about 'Quantities versus Prices' (Weitzman [25]) is just one of the candidates potentially affected by this; others are theories of enforcement (Malik [10]) and rules of liability and contract law (Jolls et al. [8]). A more recent approach within behavioural economics is nudging (Thaler and Sunstein [23]). A nudge, according to Thaler and Sunstein, is "any aspect of the choice architecture that alters people's behavior in a predictable way without forbidding any options or significantly changing their economic incentives." In their classic "cafeteria example" a 2 The model We consider emitting firms of K different technolgies. These technologies are ordered with respect to abatement costs C k (a), k = 1, 2, ..., K where a is abatement. The technolgies exhibit the following properties C 1 (a) < C 2 (a) < · · · < C K (a), ≥a > 0 (1) To keep the analysis simple we assume a quadratic abatement cost function C k (a) = b k a 2 , k = {1, 2, · · · , K}, b 1 < b 2 < · · · < b K (2) 2 which satisfies (1). Finally, we assume that there are N k , k = {1, 2, · · · , K} emitters within each technology class. Hence, the total sum of emitters amounts to Environmental policy enters into the model by utilizing the price-standard approach in troduced by Baumol and Oates [1]. The public authority introduces an emission reduction max max max max requirement. We start from unrestricted emissions {e , e , · · · , e } where e are 1 2 K k emissions from technology k unrestriced by policy measures. We could also imagine that these amounts refer to historic emissions from a base year. The authority fixes a minimum total reduction obligation for all emitting firms such that The efficient allocation of abatement across emitters can be achieved by minimizing total abatement costs, i.e.
subject to (4). From the first order conditions we know that where > is the Lagrangean. Since the abatement cost functions are the same within each technology class we have The optimal allocation of abatement effort a = {a 1 , a 2 , · · · , a } equates marginal abate-K ment costs to each other. From the literature it is well known that a single emission price achieves an efficient allocation of both emissions and abatement levels.
Each emitter of technology class k minimizes her abatmement costs while taking into account the costs of paying for her residual emissions.
where δ is a tax rate and 5 = {0, 1}. If 5 = 0 the environmental authority apply a tax-based approach. If 5 = 1 the policy maker adopts subsidies to encourage abatement efforts. We could also conceive δ as the equilibrium price on a certificate market where 5 = 0 refers to an auction approach and 5 = / could be described as grandfathering approach. Notice, that we have omitted index i because each emitting firm in class k choose the same abatement level a k . Since each emitter sets her marginal abatement costs equal to the tax rate an efficient allocation of emissions is achieved (see (6)). The environmental target is achieved by choosing the tax such that the reduction goal (4) is satisfied. Hence, δ * is chosen such that where a k solves (8). Alternatively, a command and control approach could be applied. In this case an overall uniform lower limit of abatement levels A is introduced such that the environmental objective is achieved. To do so, the limit per emitter is set such that /E A = ∑ K (10) k=1 N k Needless to say, this policy is inefficient if marginal abatement cost functions differ.

The performance of traditional policy instruments under be havioural heterogeneity
In the following we analyse how the policy instruments reviewed in the previous section affect the allocation of abatement levels when emitters do not always behave rationally across the population, i.e. when they choose their abatement levels so as to minimize their costs. Let us assume that emitting firms behave either rationally with probaibility γ or exhibit passive behaviour with probability (1 − γ) by negelecting any optimizing efforts (inertia). We assume that these probabilities are independent of technology type k. Passive behaviour implies that firms do not respond to price incentives such as taxes or subsidies. They only comply with measures imposed in the context of an command and control approach. If the policy maker relies only on a price oriented policy (taxes, subsidies, etc.), she will have to take this heterogeneous behaviour into account. The same applies to the determination of a single overall abatement level for firms. In addition, the policy has to take account the stochastic nature of abatement behaviour, which requires the caclulation of expected values.
To begin with, let us assume that the policy maker relies exclusively on a tax approach. In this case the environmental authority needs to calculate expected abatement levels. Take e.g. technology class k consisting of N k emitters. It is a standard tenet of probability theory 1 that the probability of one out of two events occuring i times (i : 0 < i < N K ) is distributed according to the binomial distribution function Here, the one event is that emitters behave rationally. The complementary event is that emitters behave passively. Accordingly, this probability applies to N k − i firms behaving passively. To calculate the expected aggregate abatement level under behavioural heterogeneity we first have to determine the expected abatement level of each technology class. If a firm behaves rationally it will choose abatement according to (8), thus leading to a level a k ' . Hence, expected aggregate emissions of class k will be 2 ( ) The same procedure is applied to passively behaving firms that melely simply comply with a abatement prescription A 0. The probability of this behaviour is 1 − γ. Hence, the expected abatement level of technology class k is The aggregated expected abatement level of all k technology classes follows directly by adding (12) and (13) across all classes: It is a straightforward matter to calculate the aggregate expected abatement costs in a similar way: In the following we analyse the performance of three traditional policy instruments under the behavioural heterogeneity introduced above.

Tax policy
Under a tax regime, rational emitters behave according to (8), leading to abatement levels a ' k . Notice, that 5 = 0. Since the policy maker relies solely on an emissions tax there are no constraints on quantity, hence A = 0. Thus the tax rate δ = δ has to be fixed such that the expected aggregate abatement level complies with the environmental target (see (4)).Thus we have

Abatement subsidies
Again, the policy maker introduces a price incentive to subsidize abatement. In this case, (8) is valid for 5 = 1. The subsidy rate has to be chosen such that (16) is met.

Direct abatement regulation
In this case, the regulatory authority does not rely on prices. Instead, it fixes an abatement standard A > 0 with which each emitting firm has to comply. Thus, all firms will abate according to this provision irrespective of their behavioural type. T he standard A = A must be set such that the environmental target is achieved, i.e.
From the traditional model we know that a price setting policy always guarantees that the environmental target will be achieved in an efficient way, provided the emission price level is set properly. In contrast to this, introducing a mandatory abatement level is inef ficient as it leads to higher aggregated abatement costs than in the case of a tax/subsidy approach. At this point it is interesting to analyse whether this assessment applies also in a world where emitters display heterogeneous behaviour.

Result 1
• In the case of behavioral heterogeneity the tax/subsidy rate must be higher than in the case of a pure price-standard approach, i.e.¯. δ > δ * • A price oriented policy is not always better than a quantity-setting regulation. The relative performance depends on technological properties, the distribution of technol ogy classes and the proportion of rational firms. Proof: The first part follows directly by comparing (9) with (16). The second part will be shown by example in section 2.3 below.
Obviously, if a certain proportion of firms is price resistant, a tax policy can only achieve the abatement target if the tax rate is higher than in the classical price standard approach case. This inertia calls for the use of an additional instrument, namely, the green defaults introduced in the following section.

Green defaults as optimal policy mix
We have assumed that a certain proportion of emitters do not actively choose their emis sion levels by minimizing their abatement costs. One remedy against this inertia could be a command and control-style quantity regulation which forces emitters to a achieve certain level of abatement irrespective of their abatement costs. The other more flexible option is to introduce green defaults. In our model a green default is a requirement of a fixed abatement level with which each emitter has to comply. However, in contrast to the traditional quantity constraint of a command and control policy, here emitters are allowed to deviate from the prescribed level if they wish to do so. The decision to deviate leads to additional costs (benefits) in terms of tax payments (subsidies). If the active � � emitter increases her abatement level beyond the default value she receives subsidies; if she reduces her abatement efforts she is required to pay tax. This mechanism resemble the 'flexibility regulations' of the early U.S. Clean Air Act ('bubble' and 'offsets') but can also be seen in any standard-based rule, for example a speed limit, which gives rise to monetary sanctions, fines or liability, if the choice of speeding-and-paying can be modeled as a deliberate economic decision of the agent 3 .
To capture this mechanism in the model we rewrite (8) as follows: As before, δ is a tax rate to be fixed and A is the default value. The rational emitter of technology class k minimizes total abatement costs leading to k In contrast to rational firms passive emitters will stick to the green default A. Having thus described the behaviour of all emitting firms we can now proceed to the task the policy maker has to solve. As in the traditional policy setting the task consists in choosing the policy that minimizes aggregate abatement costs. Since emission behaviour is random the relevant object func tion is expected aggregate abatement costs derived in (15). In contrast to the traditional setting, the policy maker does not choose between a price or a quantity approach but instead seeks to utilize both instruments. Thus, green nudging can be modelled as the application of a policy mix approach consisting of prices (taxes/subsidies) and quantities (green defaults). Minimizing aggregate expected abatement costs (15) subject to the environmental con straint leads to the following first order conditions 4 where > is the Lagrangean. Bearing in mind that ' > 0 and utilizing (19) we can transform (22) to In the more general terms of nudging it would be any behavioral standard which preserves the choice of disobedience (over-obedience) with the consequence of additional psychological or situational economic costs (benefits) such as the 'costs' of changing TV settings from energy-saving to time-saving stand-by modes or the 'benefits' of having healthy foods within arms reach. 4 These conditions are also suffcient to characterize the optimal policy due to the assumed properties of the abatement cost functions.
The green default should be set such that the average marginal abatement costs of all emitters is equated to the tax rate, which is in turn equal to the marginal abatement costs of rational firms. We call this optimality condition the average marginal abatement cost rule (AMAC). This rule is easily explained if one multiplies the nominator and the denominator by (1 − γ) exhibiting the ratio as the average marginal abatement costs of passive firms 5 . Since passive emitters do not exploit efficiency gains by equating their marginal abatement costs to the tax rate the policy maker should set the green default such that these firms' average marginal abatement costs are equal to the marginal abatement costs of each rational firm. Together with the environmental constraint (20) the AMAC can be used to determine the optimal policy mix {δ o , A o } whose properties are summarized in the following proposition.

Result 2
1. Optimality of policy mix: It is optimal to introduce a green default, i.e. a pure price policy approach is suboptimal. An exclusively quantity approach is likewise suboptimal. In formal terms: The tax rate of the policy mix approach is higher than the tax rate of the price standard approach. Formally: Hence, under a green default approach every rational emitter will invest more in abatement than in the price standard approach case.
Proof: See appendix.
The first assertion simply states that both instruments should be utilized. Price responsive firms react to the tax rate and passive emitters follow the green default. Since aggregated abatement costs are to be minimized both instruments are deployed such that the marginal abatement costs of both behavioural groups are equated, as stated by the average marginal abatement cost rule (36). Figure 1 illustrate this finding. The AMAC-rule is satisfied at the point {A o , δ o }. From figure 1 one also can easily see that δ > δ o In the pure tax approach the tax rate δ is chosen such that the abatement target /E is achieved (point pT). The tax rate must be set rather high to meet this target. Aggregated abatement costs can be reduced by lifting the abatement level of passive emitters and thereby making it possible to reduce the tax rate. The first part of the proposition and the tax relation ¯is easily to understand. This δ > δ o does not extend to the relation between δ o and δ * which follows from the AMAC-rule and the convexity of the abatement cost curve. The proof contained in the appendix shows that under the green default the sum of abatement levels of rational firms is always higher than the sum of regulated abatements of the passive emitters, i.e.
Inserting this relation into (20) immediately yields the result, that the tax rate δ o would lead to higher total abatement efforts than required in the textbook case, i.e.
It is interesting to observe that the presence of passive emitters (γ < 1) requires a higher tax rate on emissions than in the classical price standard approach despite the introduction of a green default that secures the abatement by passive emitters.

Corollary 1 • The strictness of environmental policy decreases (increases) with the fraction of rational (passive) emitters. In formal terms:
λδ o λA o < 0 and < 0 (28) λγ λγ • Aggregated abatement costs decrease as the proportion of price responsive firms in creases, i.e.

λAAC < 0 (29) λγ
This corollary re-emphasizes the findings of result 2 by following the optimal policy tra jectory with respect to the fraction of rational firms. Starting from the textbook case γ = 1, it becomes clear that environmental policy must be strengthened. Both, the tax rate and the green default must be increased as γ decreases in order to guarantee the environmental target /E. The larger the fraction of passive emitters the higher the green standard and, at the same time, the higher the tax rate. The fraction of passive emitters can be seen as a measure of the rigidity of markets. More rigid markets demand stronger regulatory interventions. In the case of pure taxes, regulators need to consider that some passive parties can not be moved by taxes. They will have to adopt higher taxes for rationally acting emitters to achieve the overall desired level of abatement. The impact on aggregate abatement costs (see (15)) is also rather obvious. The larger the fraction of rational firms minimizing their abatement costs the lower aggregate abatement costs are.

Example: two technology classes
The results obtained can be exemplified for the case of two technolgies b 1 < b 2 and N 1 = N 2 emitters within each technology class. Utilizing the abatement cost function (2) we can derive from (19) the optimal abatement efforts of rational emitters  Similarly we can draw iso-abatement-costs curves by utilizing (15). Iso-abatement cost curves can be obtained by inserting (30) into (15) for a predefined cost level CC. The red curve gives the iso-abatement cost curve for the cost level which relates to the optimal solution characterized by (23) and (20). Iso-abatement curves to the south west (north east) represent lower (higher) costs. The optimal policy mix {t o , A o } is located where the slopes of both curves are equated. Figure 1 also enables us to demonstrate the findings of Result 1. A pure tax policy approach (point pT 7 ) leads to a tax ratethat is higher than the optimal tax rate. Vice versa, the abatement provision given by the pure command and control approach (point CaC 8 ) is higher than the green default A o . Needless to say, points (pT) and (CaC) lead to higher aggregate abatement costs than under the optimal policy mix. Whether pT or CaC leads to higher aggregate abatement costs (see result 1) has yet to be evaluated. In the following Figure we have drawn total abatement costs for the pure tax case (hatched surface) and for the command-and-control approach (plain surface). The hatched surface is obtained by inserting the pure tax policy approach into the sum of both abatement cost functions. From (16) we obtain which can be solved for δ. Inserting δ into the quadratic abatement cost function yields the total 9 Notice, that the aggregated abatement costs depend on technologies (b i ) and on the pro portion of rational emitters. The surface is drawn as a function of b 1 and b 2 . Similarly, total abatement costs under a command and control approach can also be derived. Uti lizing the abatement cost functions and (17) we get Again, total costs are a function of b 1 and b 2 . The figure 10 clearly shows that the success of both pure policies depends on the structure of technologies. If technologies are very diverse, i.e. if the absolute amount of the difference of b 1 − b 2 is large, then the pure tax policy approach fares better than the command -and-control-approach, and vice versa. This result is easily understood. In the pure tax case the tax rate must be rather high due to the fact that only a fraction of all emitters will respond. Hence, to fulfill the standard /E the policy must be fairly strict, leading to relatively high aggregate abatement costs due to the convexity of cost curves (Note, that passive firms do not abate in the pure tax case). By contrast, an overall standard applied to all emitters can be relatively weak since all firms have to fulfill the provision A. However, this command and control approach is rather inefficient in the case of heterogeneous marginal abatement cost curves. Thus, there are two countervailing forces that affect total abatement costs under the two regulatory regimes. On the one hand, the pure tax approach is highly efficient but covers only the proportion of rational emitters.
On the other hand, we have the command and control approach covering all emitters but being inefficient. The more technologies differ the greater this inefficiency is. This can be seen by the area in the b 1 − b 2 plane where aggregate abatement costs under a pure tax policy are higher than under the command and control approach decreases if the fraction of rational firms increases. If we equate (32) and (33) and solve for, say, b 2 we get the two solutions Both solutions are the mathematical expresion of the cut lines of the hatched with the plain surface. If we derive (34) with respect to γ we can observe that the slope of the left (right) solution for b 2 decrases (increases). If the fraction of rational firms increases (decreases) the set of b 1 − b 2 -combinations where total abatement costs under a pure tax policy are higher than under a command and contral approach contracts (broadens). The implication is quite obvious. The more rational emitters there are the more appropriate is a price-oriented policy approach and vice versa: the more passive firms exist the more a command and control approach is appropriate.

Summary and outlook
This paper has demonstrated that the existence of subpopulations with price resistant behaviour and deliberate defaulting has important implications for the design of envi ronmental policy. Uniform command and control approaches can outperform price-based instruments, and pure tax/subsidy schemes need to be adjusted to achieve the politically desired level of total emissions. As a choice-preserving policy mix green defaults are su perior to any single-instrument policy in the case of behavioural heterogeneity. Much remains to be done. There are many examples of the effectiveness of nudges in envi ronmental policies, for example in policies to improve energy efficiency and the choice of green energy (Shogren et al. [17] and [18]). However, most of the literature is concerned with individual and household environmental behaviour. Few studies have looked at orga nizational nudges despite pervasive evidence of managerial inattention and organizational routines (Sims [20]; Wiederholt [26]; Henderson and Kaplan [7]). Both of these could help reveal important factors that contribute to price inertia and deliberate defaulting in firms. Additionally, the interplay of consumer inertia and dynamic pricing (Zhao et al. [9]) has not been sufficiently considered in the nudging literature. Time-variant pricing of energy usage, for example, would increase the difference in profits from behavioural inertia com pared to rational choice and consequently reduce the probability of deliberate defaulting within a population, as we demonstrate in this paper. The more general question arising from these analyses is how the fraction γ of behavioural inertia within a population could be reduced by means of tailored policy efforts. While environmental awareness raising and education could clearly contribute to a lower γ and thus to greater regulatory efficiency according to our findings, they would also involve additional costs of implementation etc. Consideration of these additional instrument costs and benefits as well as the remaining behavioural inertia may lead to even more policy mixing. Green defaults combined with any such instrument may prove superior to both traditional single-instrument policies (such as quotas and taxes) and stand-alone nudges. Finally we would expect important insights to be gained from applying our theory of optimal nudging to different special cases. For example, empirical findings show that the default inclusion of natural hazards in general homeowners insurance makes a considerable difference in the uptake of natural hazards insurance compared to a rule of separation of both lines (Schwarze and Wagner [17]). Based on our theory of optimal defaults we would expect that 'insurance nudging' is economically superior to mandatory insurance (with no choice-preserving mechanism) or direct regulation of private risk mitigation. Another promising area of study, following the recent findings of Parry et al. [15], would be to compare emissions standards (which allow a trade-off between activity level and technology choice) and technology fixes such as fuel efficiency standards for cars.

Proof of result 2
To prove the first statement assume, per absurdum, that A o = 0. From (23) (20). Next assume, per absurdum, that From the environmental constraint (20) it follows that τ o = 0 so as to reduce the abatement 0 levels to zero, i.e. a = 0. But from (23) it follows that τ o > 0 which is a contradiction.
k To prove the second statement we utilize the assumed quadratic form of the abatement cost function (see (2). Utilizing (19) AMAC can be rewritten as: Multiplying both sides by N j and summing up leads to From the assumed properties of the abatement costs function and from (8) we know, that ' ' ' a 1 > a · · · > a 2 > K 13 Taking these properties into account it follows from (39), that Define the difference Then it follows from (20) Since a is a positive monotone function with respect to τ a comparison of (9) with (43) shows k immediately that τ o > τ * .

Proof of corollary 1
Differentiating (23) and (20) with respect to π leads to From (44) it is easy to calculate and where the determinant of the system matrix is | | < 0. The second part of the corollary can be proved by recalling (15) Utilizing the envelope theorem the derivative of AAC with respect to π is The sign follows from the AMAC-rule (36) and the cost minimizing behaviour of rational firms (see (19)).