Louis-Gaëtan Giraudet
email: giraudet@centre-cired.fr.

Sébastien Houde
email: shoude@umd.edu.

Double Moral Hazard and the Energy Efficiency Gap

Keywords: D86, Q41 Energy efficiency gap, moral hazard, energy-savings insurance, minimum quality standard

We investigate how moral hazard problems can cause sub-optimal investment in energy efficiency, a phenomenon known as the energy efficiency gap. We argue that such problems are likely to be important for home energy retrofits, where both the seller and the buyer can take hidden actions. The retrofit contractor may cut on the quality of installation to save costs, while the homeowner may rebound, that is, increase her use of energy services when provided with higher energy efficiency. We first formalize the double moral hazard problem described above and examine how the resulting energy efficiency gap can be reduced through minimum quality standards or energy-savings insurance. We then calibrate the model to the U.S. home insulation market and quantify the deadweight loss. We find that for a large range of market environments, the welfare gains from undoing moral hazard are substantially larger than the costs of quality audits. They are also about one order of magnitude larger than those from internalizing carbon dioxide externalities associated with the use of natural gas for space heating. Moral hazard problems are consistent with homeowners investing with implied discount rates in the 15-35% range. Finally, we find that minimum quality standards outperform energy-savings insurance.

Introduction

The rationale for government interventions promoting energy efficiency has been debated for more than three decades. At the heart of the debate are empirical studies finding that people apply abnormally high discount rates to energy efficiency investment decisions (see [START_REF] Train | Discount rates in consumers' energy-related decisions: A review of the literature[END_REF] for an early review). This suggests that some investment opportunities that are privately profitable -even if energy-use externalities are not internalized -are not undertaken. This empirical fact is known as the energy efficiency gap. [START_REF] Jaffe | The energy-efficiency gap: What does it mean?[END_REF] were the first to conceptualize this debate by emphasizing the difference between market failure (e.g., information asymmetries, technology spill-overs, energy price distortions) and "non-market failure" (e.g., consumer heterogeneity, hidden costs) explanations of the energy efficiency gap. They argued that only market failures could justify government intervention. More recently, this dichotomy has been enriched with the concept of behavioral anomalies to account for the fact that consumers may value energy savings in a way that is inconsistent with perfect rationality [START_REF] Attari | Public perceptions of energy consumption and savings[END_REF][START_REF] Gillingham | Bridging the Energy Efficiency Gap: Policy Insights from Economic Theory and Empirical Evidence[END_REF][START_REF] Allcott | Energy policy with externalities and internalities[END_REF].

In this paper, we aim to contribute to the debate by drawing attention to one market failure, which, to our knowledge, has been overlooked in the literature: moral hazard in the provision of quality in energy efficiency investments. For instance, the problem was not discussed in the most recent and exhaustive literature reviews on the energy efficiency gap [START_REF] Allcott | Is There an Energy Efficiency Gap?[END_REF][START_REF] Gillingham | Energy Efficiency Economics and Policy[END_REF]. We also shed light on little-discussed policy remedies such as energy-savings insurance and minimum quality standards for energy efficiency professionals.

Many energy efficiency technologies are considered to be credence goods, the performance of which is never perfectly known to the buyer [START_REF] Sorrell | Understanding barriers to energy efficiency[END_REF]). This characteristic is conducive to a variety of information problems, which have long been suspected to be the main source of market failures in energy efficiency markets [START_REF] Howarth | Market barriers to energy efficiency[END_REF][START_REF] Huntington | Editors' introduction[END_REF]. This is especially true in the building sector. Technological complexity may cause a general lack of understanding about energy-saving opportunities. However, evaluations of energy audits find that consumers respond less to information provision than to price signals, suggesting that the knowledge gap is small [START_REF] Palmer | Assessing the energyefficiency information gap: results from a survey of home energy auditors[END_REF][START_REF] Frondel | Heterogeneity in the Effect of Home Energy Audits: Theory and Evidence[END_REF]Murphy 2014b). Still, information may be comprehensible but asymmetrically distributed. Several studies have examined information asymmetries in rental housing, in which the landlord is supposedly more informed than the tenant about the energy efficiency performance of the dwelling. They find that a higher energy efficiency does not lead to a higher rent and that rented dwellings are less energy efficient than owner-occupied ones [START_REF] Levinson | Energy use by apartment tenants when landlords pay for utilities[END_REF][START_REF] Davis | Evaluating the Slow Adoption of Energy Efficient Investments: Are Renters Less Likely to Have Energy Efficient Appliances?[END_REF][START_REF] Gillingham | Split Incentives in Residential Energy Consumption[END_REF][START_REF] Burfurd | Stars and standards: Energy efficiency in rental markets[END_REF][START_REF] Myers | Asymmetric Information in Residential Rental Markets: Implications for the Energy Efficiency Gap[END_REF]. Fewer studies have examined information asymmetries in home sales, in which the seller is supposedly more informed than the buyer about the energy efficiency performance of the dwelling. Research conducted in the Netherlands suggests that information conveyed by energy performance certificates tends to be capitalized into sale prices, but only weakly [START_REF] Brounen | On the economics of energy labels in the housing market[END_REF]Murphy 2014a).

The information asymmetries we consider here are related, but different in nature. We examine energy efficiency projects in which a contractor may cut on the quality of installation 1 to save costs, while the buyer may rebound, i.e., increase her use of energy services when provided with higher energy efficiency. Both actions are unobservable to the other party. We refer to this problem as double moral hazard. Our focus is on the supply side of energy efficiency markets; we thus extend an analysis of information asymmetries so far confined to building sale and rental transactions. 2 Our contribution is threefold. We first formalize how moral hazard in the provision of quality leads to an energy efficiency gap. We then investigate policy tools that could be used to address this market failure. In the building sector, firms could offer energy-savings insurance [START_REF] Mills | Risk transfer via energy-savings insurance[END_REF]. However, we show that due to the unobservable homeowner's response, a complete insurance contract is not optimal. We also examine professional certification in the form of minimum quality standards. We suggest that these policies be part of the policy portfolio used to encourage energy efficiency, beside Pigouvian instruments [START_REF] Allcott | Energy policy with externalities and internalities[END_REF], energy efficiency subsidies [START_REF] Ito | Asymmetric Incentives in Subsidies: Evidence from a Large-Scale Electricity Rebate Program[END_REF][START_REF] Boomhower | A credible approach for measuring inframarginal participation in energy efficiency programs[END_REF], energy efficiency labels [START_REF] Houde | How Consumers Respond to Environmental Certification and the Value of Energy Information[END_REF], building codes [START_REF] Aroonruengsawat | The Impact of State Level Building Codes on Residential Electricity Consumption[END_REF][START_REF] Jacobsen | Are Building Codes Effective at Saving Energy? Evidence from Residential Billing Data in Florida[END_REF] and information provision [START_REF] Jessoe | Knowledge is (Less) Power: Experimental Evidence from Residential Energy Use[END_REF]. We finally quantify the size of the energy efficiency gap due to moral hazard 1 Such a quality shortfall may materialize as either inefficient labor or capital input. For instance, an insulation contractor may omit to fill wall cavities before installing insulation panels and/or install insulation panels of a low grade.

2

The hidden actions examined in our paper may propagate as hidden information in subsequent principalagent relationships. That is, the contractor's failure can be internalized as the homeowner's by prospective homebuyers or renters, leading to non-capitalization of energy efficiency performance in home sale prices or rental contracts. We left this fruitful question of the articulation between moral hazard and capitalization problems for future research.

in the U.S. insulation market, using data from the U.S. 2009 Residential Energy Consumption Survey (RECS). We perform sensitivity analysis and develop a close approximation of deadweight loss that only requires engineering data.

We find that over a large range of market environments, the deadweight losses of moral hazard are several times larger than the costs of quality audits. They are also approximately one order of magnitude larger than those from internalizing carbon dioxide externalities associated with natural gas use. Moral hazard problems lead to implied discount rates in the 15-35% range, instead of a 7% rate assumed in the absence of this market failure. Naively extrapolating our results to the U.S. population of homeowners using natural gas for space heating, undoing moral hazards associated with insulation could save at least 11 billion cubic feet of natural gas and 0.6 million tons of CO 2 annually, hence yielding $2.4 billion of present value benefits. Energy-savings insurance could close 75% of this gap and minimum quality standards could be even more welfare-improving.

The outline of the paper is as follows. Section 2 introduces the model. Section 3 examines quality standards and energy-savings insurance as policy solutions to the moral hazard problem. Section 4 provides numerical estimates of the energy efficiency gap due to moral hazard in the U.S. home insulation market and comparisons of policy instruments. Section 5 discusses sensitivity analysis and a sufficient statistic approach. Section 6 concludes.

Energy efficiency investments and double moral hazard

Our model builds upon the double moral hazard model of [START_REF] Cooper | Product Warranties and Double Moral Hazard[END_REF]. Investments in energy retrofits, which typically involve hidden actions from both the homeowner and the contractor, are considered as a canonical example. Other situations that give rise to one-sided moral hazard can be viewed as special cases of this general model; they are occasionally discussed in the text.

Setup

A homeowner uses energy for space heating. This energy service s, measured in indoor temperature, provides her with value V (s), multiplied by a taste parameter θ > 0 representing heterogeneity across consumers in the valuation of energy service. The homeowner expects to pay energy bill pE 0 (s), where E 0 (•) is the energy use and p the price of energy. The energy use is a random variable influenced by idiosyncratic factors, such as weather conditions and the architectural characteristics of the house. For simplicity, we use a deterministic framework; utility is quasi-linear and there is no risk aversion. The homeowner sets the intertemporal energy service vector s 0 θ so as to maximize expected utility U 0 (θ, s) over an investment lifetime of l years, discounted at some rate r:

(1)

U 0 (θ, s) ≡ l t=1 θV t (s t ) -p t E 0 t (s t ) (1 + r) -t
The homeowner can invest in retrofits to reduce her energy bill. In this setting, the homeowner is the principal and the contractor is the agent. Energy use after investment E is reported on homeowner's energy bill. Hence, it is common knowledge to both parties. Yet each one can take hidden actions s and q to influence it.

The homeowner chooses a stream of energy service s. This action is unobserved to the contractor and a higher energy service will induce a higher expected energy use. Likewise, the contractor provides a certain quality q in installing an energy-efficient equipment. We assume that the quality q is unidimensional and can be measured as the number of hours worked by the contractor. Unlike other amenities, the impact of this action on the energy efficiency performance cannot be fully assessed by the homeowner. The only thing that is known to both parties is that a higher quality of installation lowers expected energy use.

The homeowner considers future discounted benefits with expected utility U (θ, s, q), pays upfront cost for the retrofit (T > 0), and receives some fixed non-energy benefits net of the inconvenience costs generated by the investment (ǫ):

(2)

U (θ, s, q) ≡ l t=1 [θV t (s t ) -p t E t (s t , q)] (1 + r) -t -T + ǫ
In what follows, we assume time invariance of energy price, technology and consumer value function. We remove t subscripts and consider vector s as a scalar s constant over time. We further simplify the notations with a discount factor Γ such that:

(3) Γ ≡ Γ(r, l) ≡ l t=1 (1 + r) -t = 1 -(1 + r) -l r
Firms are homogenous in the industry. The profit of a representative contractor is the revenue from the sale minus the cost of the quality provided:

(4) Π(q) ≡ T -C(q)
The following assumptions hold (subscripts denote partial derivatives):

Assumption 1: Technology.

(i) At constant consumer behavior s, investment reduces energy use: E(s, q) < E 0 (s) ∀q ≥ q min , where q min is the minimum input.

(ii) Contracting parties' actions have opposite effects: E 0 s > 0, E s > 0 and E q < 0 (iii) Energy savings exhibit decreasing returns: -E 0 ss ≤ 0, -E ss ≤ 0 and -E qq ≤ 0 (iv) Contracting parties' actions are substitutes: E qs < 0 and E s < E 0 s (v) Non-energy benefits are not sufficient to motivate investment: ǫ ≤ C(q min ) Assumption 2: Behavior and preferences. Contracting parties are (i) value-maximizers, (ii) risk-neutral and (iii) have twice differentiable, concave value functions:

V ′ (•) > 0, V ′′ (•) ≤ 0 and -C ′ (•) < 0, -C ′′ (•) ≤ 0
Assumption 3: Market. The industry is competitive with free entry: Π(q) = 0.

Corollary: T ≡ C(q).

Assumptions 1(i)-(v) are mild: The energy service has a convex effect on expected energy use, and quality has diminishing returns on expected energy savings. Moreover, both factors impede each other: The marginal increase in expected energy savings due to increased quality is larger when the underlying energy service is high (e.g., a house heated in a cold climate) rather than low (e.g., a house heated in a warm climate). Reciprocally, the marginal increase in expected energy use due to increased energy service is lower when the quality installed is high rather than low.

Assumptions 2(i)-(iii) are meant to be as standard as possible, in order to isolate the moral hazard problem from possibly interacting with market failures and behavioral anomalies. Their generality is discussed in Section 5. Assumption 3 is not essential but simplifies the exposition. Whatever the structure of the market, home energy retrofits are very specific to a bundle of home and homeowner characteristics, and hence do not lend themselves to arbitrage. A monopolist could thus perfectly price discriminate. This would not change equilibrium quantities in the model, but only the surplus repartition.

Social versus private optimum

We will consider two equilibrium outcomes: a social (hereafter cooperative) optimum c and a private (hereafter non-cooperative) optimum nc. For any equilibrium situation j ∈ {c, nc}, the agreement between the homeowner and the contractor is a two-stage game that is solved backward. In the first stage, the homeowner of type θ invests if the net present value N P V j (θ) of investment is positive, given her beliefs about her future optimal energy service s j θ and the optimal quality q j θ offered to her by the contractor:

(5)

N P V j (θ) ≡ U (θ, s j θ , q j θ ) -U 0 (θ, s 0 θ ) ≥ 0
In the second stage, both agents determine their own action given their belief about the other party's action. We focus hereafter on this second stage, for a participating consumer of type θ.

Under perfect information, the contract between the two parties is set cooperatively so as to maximize joint expected surplus, subject to boundary conditions s ≥ s min and q ≥ q min . The optimal actions s c θ and q c θ that solve the first-order conditions for maximization 3 below will be such that their marginal benefit (in terms of value to the consumer and cost savings to the firm) equates their marginal effect on consumer's expected energy bill:

(6) ∀t θV ′ ≤ pE s with equality if s c θ > s min (7) C ′ ≥ -pE q Γ with equality if q c θ > q min
The cooperative optimum (s c θ , q c θ ) can be characterized as a reaction function equilibrium. Assuming interior solutions and applying the Implicit Function Theorem to the first-order conditions, we find that the reaction functions s * θ (q) and q * (s) are strictly increasing:

3 Throughout the paper, the objective functions are well-behaved and the first-order conditions discussed are necessary and sufficient for maximization.

(

) ∀t ds * θ dq = pE qs θV ′′ -pE ss > 0 (9) dq * ds = -pE sq C ′′ /Γ + pE qq > 0 8
Now if information is imperfect, the agreement is no longer cooperative. Both parties maximize their private expected value, given their beliefs about the other party's action and subject to boundary conditions s ≥ s min and q ≥ q min . While this yields the same reaction function as in the cooperative agreement s * θ (q) for the consumer, this does not hold for the contractor. He does not internalize the expected benefits that his action delivers to the homeowner and simply chooses the level of quality q nc that minimizes his cost: (10) ∀s q nc (s) = arg min

q≥q min C(q) = q min
Proposition 1. For a participating consumer of given type θ: (i) the private, non-cooperative equilibrium (s nc θ , q nc θ ) exists and is unique (ii) the social, cooperative equilibrium (s c θ , q c θ ) exists and is unique if and only if:

(11) dq ds * θ > dq * ds proof: (i)
The private equilibrium is uniquely defined as (s * θ (q min ), q min ). (ii) Likewise, if for at least one agent his or her optimal cooperative action is a corner solution, then the social equilibrium is uniquely defined. If optimal actions are interior for both agents, condition (11) implies that the composite function s * θ (q * (s)) defined for all s ≥ s min is a contraction mapping. Hence, by the Banach fixed-point theorem, it admits a unique fixed point.

The following proposition states that the two equilibria will involve unambiguous locations:

Recall from Assumption 1(ii) that q and s have an opposite effect on E(s, q). Hence, if both inputs increase simultaneously, as is the case when the parties move from the private optimum to the social optimum, the decrease in energy use due to the increase in quality is partly offset by the increase in energy service. This phenomenon is known as the rebound effect.

To the extreme, it can backfire, i.e., be such that energy use increases after energy efficiency investments. This case cannot be ruled out from our analysis, as E(s c θ , q c θ ), E(s nc θ , q nc θ ) and E(s 0 θ ) cannot be compared unambiguously.

We shall now make a distinction between two types of backfire rebound effect, which will prove useful later in the analysis.

Definition 1: Genuine backfire rebound effect. A genuine backfire rebound effect occurs if energy use after investment is larger than before investment: s > s 0 and E(s, q) > E 0 (s 0 ) Definition 2: Relative backfire rebound effect. A relative backfire rebound effect occurs between two investment options H and L if energy use after investment is larger in the more energy efficient option H: q H > q L , s H > s L and E(s

H , q H ) > E(s L , q L )

Consumer heterogeneity and aggregate welfare

We now turn to a continuum of consumers of mass 1. Consumers are assumed to all live in a similar dwelling and only differ with respect to their preference for energy service θ. The higher the value of θ, the higher the demand for energy service, hence the higher the quality offered by a cooperative firm; in contrast, the quality offered by a non-cooperative firm remains at minimum. This proposition is demonstrated in Appendix A.

For any equilibrium situation j ∈ {c, nc}, we have, by the Envelope Theorem:

(12) dN P V j dθ = V (s j θ ) -V (s 0 θ ) Γ
As V (•) is increasing and ∀θ s j θ > s 0 θ , Equation 12 means that the net present value of investment strictly increases with θ. Hence, if there exists a cutoff type θ j 0 such that N P V j (θ j 0 ) = 0, it is unique. In what follows, we are interested in this most relevant case; alternative cases are discussed in Appendix B. Assuming that F (•) is the cumulative distribution function of θ, participation to investment N j is given by: (13)

N j ≡ 1 -F (θ j 0 )
Finally, aggregate social welfare is the sum of utility before investment for those consumers who do not invest (θ ∈ [0, θ 0 )), plus the utility after investment for those who do invest (θ ≥ θ 0 ):

(14) W j ≡ θ j 0 0 U 0 (θ, s 0 θ )dF (θ) + +∞ θ j 0 U (θ, s j θ , q j θ )dF (θ)
Proposition 3. Assuming that condition ( 11) is satisfied for all consumers with θ > 0:

(i) the social optimum entails higher participation than the private optimum: N c ≥ N nc (ii) the social optimum entails higher aggregate welfare than the private optimum:

W c ≥ W nc proof: (i) Assume θ c 0 (respectively θ nc 0 )
is the cutoff value of θ in the social (respectively private) optimum. Proposition (2iii) imposes the following inequality:

N P V c (θ c 0 ) = 0 = N P V nc (θ nc 0 ) ≤ N P V c (θ nc 0 ). Since N P V j (•) is increasing, θ c 0 ≤ θ nc 0 . Hence, N c -N nc = θ nc 0 θ c 0 dF (θ) ≥ 0. (ii) W c -W nc = θ nc 0 θ c 0 N P V c (θ)dF (θ)+ +∞ θ nc 0 [U (θ, s c θ , q c θ )-U (θ, s nc θ , q nc θ )]dF (θ) ≥ 0.
This is a very general formalization of the energy efficiency gap: In the presence of moral hazard, investments in energy efficiency entail too low a quality of installation and too few homeowners participate. This result holds under very general assumptions of perfect rationality and risk-neutrality. Concretely, the homeowner does not have the technical skills to judge whether the retrofit has been properly completed, although she is aware that any defects will deter the energy performance of the investment. Anticipating that the contractor is aware of her limitations, she will expect him to save on installation costs and perform the job poorly. Any claim that he will provide the highest quality, enabling her to maximize energy savings, will be considered "cheap talk" by the homeowner. The contractor will not deviate from these expectations and indeed complete the lowest possible quality job. Quality will be non-contractible and thus underprovided.

Appendix C discusses some comparative statics with respect to a composite indicator of all market and behavioral features: ζ ≡ pΓ (r, l). This indicator is very similar to the investment inefficiency parameter proposed by [START_REF] Allcott | Is There an Energy Efficiency Gap?[END_REF]. Any value of p, r or l that does not reflect perfect competition, perfect rationality or perfect information translates into a biased ζ. Comparative statics of ζ thus provides insight into the interaction between moral hazard and other market failures or behavioral anomalies.

Policies

In this section, we examine some regulatory and incentive-based instruments that can be used to address moral hazard in energy efficiency markets.

Energy-savings insurance

Insurance is the most common way of addressing moral hazard problems. Energy-savings insurance or energy performance contracts typically have the contractor pay the consumer any shortfall in energy savings below a pre-agreed baseline. In our simple framework with no risk-aversion, insurance can be represented by a contract where the contractor bears a share k of the energy bill:

(15) U (θ, s, q) ≡ [θV (s) -(1 -k)pE(s, q)] Γ -I + ǫ (16) Π(q) ≡ I -C(q) -kpE(s, q)Γ
According to Assumption 3, the payment to the contractor is I = C(q) + k pE(s, q)Γ, where kpE(s, q)Γ is the actuarially fair insurance premium.

A new, opposite principal-agent relationship superposes to the previous one: Since the contractor now provides insurance, he is a principal and the homeowner is an agent. The implementation of this contract can be solved backward as a three-stage game played by the parties. In the third stage, each party determines non-cooperatively his or her own effort, given insurance coverage k and his or her belief about the other party's action. First-order conditions for maximization are:

(17) ∀t θV ′ ≤ (1 -k)pE s with equality if s i θ (k) > s min (18) C ′ ≥ -kpE q Γ with equality if q i θ (k) > q min
The optimal consumer's response is bounded above by a satiation value s max . 4 By the Implicit Function Theorem, the insurance reaction functions s * * θ (q, k) and q * * (s, k) are both increasing in k:

(19) ∀t ds * * θ dk = -pE s θV ′′ -(1 -k)pE ss > 0 (20) dq * * dk = -pE q C ′′ /Γ + kpE qq > 0
The implementation of such a contract partly solves the moral hazard, as it induces the contractor to offer some quality (Equation 18). At the same time, however, it gives rise to a second moral hazard: By lowering the homeowner's marginal value of energy service, it induces her to consume more energy. The energy service in Equation 17 is consumed to the socially optimal level defined by Equation 6 when the consumer is not insured (k = 0), whereas the quality in Equation 10 is offered to the socially optimal level defined by Equation 7when the firm offers full insurance (k = 1). Since k cannot be simultaneously equal to 0 and 1, insurance cannot achieve the social optimum. At best, both parties will agree 4 Satiation is needed in the model to handle full insurance (k = 1), which brings the marginal value of energy service in Equation 17 to zero. It could be introduced as the argument of the maximum of a parabolic utility function. Alternatively, in our model, satiation is introduced as an upper bound on the value of s. This specification allows for more flexibility in the numerical section, without loss of generality.

on an incomplete insurance contract k ∈ (0, 1). We recover here the result established by [START_REF] Cooper | Product Warranties and Double Moral Hazard[END_REF]. For any insurance k, the agreement (s i θ (k), q i θ (k)) will be a Nash equilibrium determined by the intersection of each party's reaction function s * * θ (q, k) and q * * (s, k). These inputs will be higher than in the private optimum; however, their location relative to the social optimum is ambiguous.

Note that if consumer's types are imperfectly observable to the contractor, a screening issue arises. Consumers with the highest use of energy service may self-select into the insurance contract that offers the highest energy savings coverage. Assuming this away, the optimal value kθ that sustains the Nash equilibrium to each type is determined cooperatively in the second stage of the game, so as to maximize joint expected surplus:

(21) ∀θ kθ = arg max k∈[0,1] [U (θ, s i θ (k), q i θ (k)) + Π(q i θ (k))]
The first-order conditions for maximization in the second stage can be found in Appendix D. Lastly, in the first stage, the homeowner chooses whether or not to invest, depending on her net present value for the investment and given her beliefs about the contractor's action and the optimal insurance coverage.

Note that if the consumer were not optimizing her energy service and consuming a constant level of it (e.g., a tenant who does not pay for her energy bill, or an employee in a commercial building), then the second moral hazard would not occur. The optimal insurance contract would feature full coverage and bring the parties to the social optimum.

Minimum quality standard

In our framework, a minimum quality standard translates into a perfectly enforced minimum labour requirement q.5 Yet such an instrument may cause two classic types of deadweight loss. First, compliance with the standard still needs to be monitored, which generates costs M (q). These costs do not occur with an energy-savings insurance. Second, minimum quality standards abstract from consumer heterogeneity. A minimum standard q can only be the optimal level of quality to one homeowner type, but it is suboptimal to all others (since, according to Proposition 1, the optimal quality is unique to each type θ). As a result, a uniform standard is strictly suboptimal over the population.

The optimal minimum standard will be set at a value q that maximizes the collective surplus, subject to the participation constraint:

(22) Maximize q θ 0 0 U 0 (θ, s 0 θ )dF (θ) + +∞ θ 0 [U (θ, s * θ (q), q) -M (q)] dF (θ)
subject to N P V (θ 0 , s * θ 0 (q), q) -M (q) ≥ 0

As developed in Appendix E, the first-order condition for maximization will be:

(23

) +∞ θ 0 ∂U (θ, s * θ (q), q) ∂ q -M ′ dF (θ) = 0
In words, the optimal standard will equalize the sum of marginal disutilities (net of marginal monitoring costs) of participants for whom the standard is too tight with the sum of marginal utilities (net of marginal monitoring costs) of participants who would have been willing to invest beyond the standard.

Intervention rules with interacting energy market failures

As we have just seen, addressing moral hazard problems through energy-savings insurance or quality standards can improve social welfare. Both instruments are, however, secondbest. Uniform quality standards cannot eliminate the gap, because of the heterogeneity in consumers' valuation of energy services. Energy-savings insurance is incomplete because moral hazard is bilateral.

Yet public intervention to address moral hazard problems may not be systematically justified if they interact with energy market failures. Assume that every unit of energy used generates a linear external cost p x , discounted over the relevant time period with a discount factor Γ x . For instance, p x is positive for environmental or energy security externalities, and negative for average-cost energy pricing. Expected consumer utility before and after investment is now:

(24)      U 0 x (θ, s) ≡ U 0 (θ, s) -p x E 0 (s)Γ x U x (θ, s, q) ≡ U (θ, s, q) -p x E(s, q)Γ x
These new utility functions allow one to define new net present value N P V x and aggregate welfare W x functions as in Equations 5 and 14, respectively. The optimal actions that internalize external costs are denoted by superscript x.

Proposition 4. In a world subject to both energy market failures and energy efficiency moral hazard: (i) When energy market failures are corrected, it is desirable to also undo moral hazard problems:

W c,x x ≥ W nc,x
x (ii) If no consumer is prone to a genuine backfire rebound effect, then it is desirable to correct energy market failures. This holds whether or not moral hazard problems are addressed:

∀θ E(s c θ , q c θ ) ≤ E 0 (s 0 θ ) ⇒ W c,x x ≥ W c x and E(s nc θ , q nc θ ) ≤ E 0 (s 0 θ ) ⇒ W nc,x x ≥ W nc x (iii) If
E(s c θ , q c θ ) ≤ E(s nc θ , q nc θ ) ≤ E 0 (s 0 θ ) ⇒ W c x ≥ W nc x proof: See Appendix F.
As long as energy efficiency does not backfire, correcting energy market failures is desirable, regardless of whether or not the contracting parties overcome the moral hazard. Indeed, social welfare cannot be maximized if the parties do not account for the broader distortions associated with their actions. However, the reciprocal needs not be true: If energy market failures are not (or cannot be) corrected, then it might be desirable to maintain, rather than undo, the moral hazard. This can actually occur if energy efficiency backfires. As a result, energy market failures would be larger.

A numerical illustration: Home weatherization

The building sector is believed to have the largest and most cost-effective potential for energy savings and carbon dioxide emissions reduction (Levine, Ürge-Vorsatz, Blok, Geng, Harvey, Lang, Levermore, Mongameli Mehlwana, Mirasgedis, Novikova, Rilling, and Yoshino 2007). Weatherization measures account for the bulk of this potential. According to a widely publicized although controversial6 study by [START_REF] Mckinsey | Strategic plan to reduce the energy impact of air conditioners[END_REF], improvements of building shells and heating, ventilation and air conditioning (HVAC) systems could save 3 quadrillion end-use BTUs in the U.S. by 2020. Two-third of this amount would be achieved in existing homes. Yet this technical potential could remain partly untapped if moral hazard problems were to remain unaddressed.

The sources of moral hazard in home weatherization

Home weatherization technologies involve a significant installation input. If completed poorly by professionals, installation can be the source of many defects. This includes, for instance, an improper connection of ducts in HVAC systems, an imperfect filling of wall cavities before insulation installation or infiltrations around windowpanes. Detecting such defects is technically possible through a blower door test or thermographic screening. Yet these tests come at a substantial cost to the consumer. 7 Overall, there is little data available about the prevalence of these defects. Some analyses suggest it is sizeable on the extensive margin. As of 2008, only 15% of central air conditioning installations in existing dwellings met satisfactory quality specifications in California (Messenger 2008). The intensive margin is particularly ill-documented. [START_REF] Metcalf | Measuring the Energy Savings from Home Improvement Investments: Evidence from Monthly Billing Data[END_REF] find that actual returns to attic insulation are around 10%, which is far from promises made by engineers and product manufacturers of 50%. The authors do not specifically investigate installation defects as an explanation for this gap, but their result provides suggestive evidence that poor quality is an issue in home weatherization.

The home energy retrofit industry is very fragmented. For instance, the HVAC industry in California is characterized by small firms offering low wages, with a very large number of quality problems reported [START_REF] Zabin | Chapter Four: Case studies of the HVAC, residential retrofit, and commercial lighting control sectors[END_REF]. This can be interpreted as a low quality market equilibrium, similar to what we have described through the model as the private optimum. Moreover, [START_REF] Zabin | Chapter Four: Case studies of the HVAC, residential retrofit, and commercial lighting control sectors[END_REF] also find that barriers to entry are low and that annual turnover is as high as 25%, which suggests that the competitive assumption made in our model is reasonable.

Various types of voluntary quality certifications exist in the marketplace, most notably those provided by the Building Performance Institute (BPI) and the Residential Energy Services Network (RESNET) in the U.S. These programs typically ensure that professional workers and contracting companies are trained to the best practices and that their performance is regularly tested. As of today, less than one percent of the professionals are certified, which suggests that the industry self-regulation has not been successful in addressing quality problems. In France, starting in 2014, public subsidies for home energy retrofits will be given 7 Moreover, they are meant to be conducted before a retrofit, to help determine what measures should be undertaken. They are almost never conducted after the job is completed to check the quality of installation.

only if the job is completed by a certified contractor. This "eco-conditionnality" rule is an interesting way of addressing both moral hazard and other problems. 8

Energy-savings insurance or energy performance contracts have been offered by energy service companies for about twenty years in the commercial sector [START_REF] Mills | Risk transfer via energy-savings insurance[END_REF]. In contrast, these contracts are almost absent from the residential sector. 9 As we have seen through the model, such contracts may be welfare-improving but cannot achieve full efficiency due to the existence of a rebound effect, which typically ranges from 10 to 30% in space heating use [START_REF] Sorrell | Empirical estimates of the direct rebound effect: A review[END_REF]. In contrast, in the commercial sector, building occupants are expected to adopt a constant behavior, as they do not pay for the investment nor the operating costs. The absence of moral hazard on the consumer side may explain why contractors are more likely to provide guarantee payments in the commercial sector.

This overview of the home retrofit industry shows that the model provides qualitative insight into some real-world facts. We now use it to conduct a quantitative assessment of the welfare implications of moral hazard problems. We focus on natural gas use for space heating and investments in wall insulation in U.S. homes.

Functional forms used in the simulations

The homeowner sets temperature s, measured in Fahrenheit ( • F), above a minimum comfort level s min . The value V (•) that she derives from this energy service is bounded above by V max , which corresponds to a maximum budget dedicated to space heating. The function is increasing and concave, and takes the following form:

(25) V (s) ≡ V max 1 -e -α(s-s min ) with s min ≤ s ≤ s max ,
where α > 0 is a calibrated parameter.

The use of natural gas E 0 , measured in thousand cubic feet of natural gas (MCF), increases with indoor temperature (at an increasing rate) with a constant calibrated elasticity γ > 1: 8 The "other problems" justifying subsidies for energy efficiency can be either technology spillovers or energy-use externalities. In the latter case, though, subsidies are only a second-best solution [START_REF] Giraudet | Efficiency and distributional impacts of tradable white certificates compared to taxes, subsidies and regulations[END_REF]). 9 GreenHomes America, Inc., NJ-PA Energy Group, LLC. and EcoWatt Energy, LLC. are the few examples we have found of companies offering energy-savings insurance in the U.S. residential sector.

(26) E 0 (s) ≡ β(s -s min ) γ with s min ≤ s ≤ s max ,
where parameter β > 0 is calibrated so as to convert Fahrenheit degrees into thousand cubic feet.

Investment in wall insulation of efficiency G(q) lowers energy use as follows:

(27) E(s, q) ≡ (1 -G(q)) E 0 (s).
Efficiency is increasing in the quality q offered by the contractor (at a decreasing rate), within two limits 0 < G min < G max < 1:

(28) G(q) ≡ G min + (G max -G min ) 1 -e -ω(q-q min ) with q min ≤ q ≤ q max ,
where ω > 0 is a calibrated parameter.

The contractor bears a fixed cost K, which corresponds to a minimum labor input q min . As the contractor provides the homeowner with a higher quality, he needs to have installers work longer and mobilize higher skills, which results in higher wages. As a result, cost increases quadratically in the number of worker.hours q:

(29) C(q) ≡ K + ρ(q -q min ) + φ 2 (q -q min ) 2 with q min ≤ q ≤ q max , where ρ > 0 and φ > 0 are calibrated parameters.

Data and calibration

Homeowners' characteristics are drawn from the U.S. Department of Energy's Residential Energy Consumption Survey (RECS) of 2009. We use information on indoor temperature, energy use, energy expenditure and income contained in the online database. We first extract a preliminary sample of 4,306 U.S. households who own and occupy their house and pay for natural gas for space heating. We then remove households who declare a winter daytime temperature below 60 • F or above 80 • F and thereby obtain a working sample of 4,266 households. This sample covers 35% of the complete dataset. Summary statistics are provided in Table 1.

In the reference scenario, we ignore potential behavioral anomalies and assume that homeowners discount future energy expenditures at a normal rate of 7%. They do it over the complete lifetime of an insulation project (35 years), thus assuming full capitalization of energy savings. To keep consistency with data, we use the price of natural gas derived from the RECS sample ($11.14/MCF), assuming away potential distortions in this market [START_REF] Davis | Do Americans consume too little natural gas? An empirical test of marginal cost pricing[END_REF]. All of these assumptions are subsequently relaxed in the sensitivity analysis. We consider environmental damages caused by carbon dioxide emissions associated with natural gas use and value them at $33/tCO 2 .

Fewer data are available to parameterize the supply side of the insulation market. We therefore use best guesses based on the engineering literature and drawn from discussions with practitioners. Our assumptions are detailed in Table 2.

In the RECS sample, the annual fraction of homeowners investing in insulation is 3.4%. 10 Our model is calibrated so that this rate is replicated in the private optimum and can be doubled at best. That is, participation among potential investors is set to 50% in the private optimum. A participation of 100% in the model can thus be interpreted as an annual insulation rate of 6.8% in the total population.

The calibration procedure leads to net non-energy benefits of $2,035. This means that attributes such as aesthetics or acoustic comfort yield benefits that exceed the hassle factor associated with insulation. Moreover, net non-energy benefits are necessary to induce the median homeowner to invest in insulation. All calibration targets are outlined in Table 3 and the calibration procedure is detailed in Appendix G.

The temperature distribution found in the RECS sample is fitted with a log-normal distribution of homeowners' types with parameters µ = 0 and σ = 1. Yet we do not intend to reproduce such a large heterogeneity, which may be partly driven by variables omitted in our model. Therefore, in the reference scenario, we assume a narrower distribution of homeowners' types with log-normal parameters µ = 0 and σ = 0.25, from the 0.5th percentile (θ = 0.53) to the 99.5th percentile (θ = 1.90). The model fit is illustrated in Figure 1. With these structural and numerical assumptions, the homeowner of type θ = 1 is both the median of the distribution and the marginal participant in the private optimum. The model satisfies the necessary and sufficient conditions for existence and uniqueness of private and social equilibria (Proposition 1).

Quantification of the energy efficiency gap

Simulation results are illustrated in the figures and detailed in Table 4 for the median homeowner and Table 5 for the population average. In Figure 3, various equilibria are mapped 10 6.8% of the population declare having insulation installed in the last two years (variable AGEINS=1).

Note that the 3.4% rate is close to 2.9%, which would be the annual rate if investment occurred once every 35 years.

in the framework proposed by Jaffe and Stavins 11 (1994), so as to visualize the trade-offs between economic efficiency and energy efficiency. Without internalization of energy-use externalities, the private optimum generates modest improvements in either welfare or energy efficiency, compared to the equilibrium before investment. In contrast, when the contractor cooperates to undo the moral hazard, both welfare and energy efficiency improvements become substantial. Average quality moves from 24 to 47 worker.hours (roughly a one workday gap), thus moving average energy efficiency from 3% to 27%. As a consequence, the average cost of quality increases from $2,400 to $2,830. 12 Lifetime discounted welfare net of environmental damages increases by $1,723 (or $1,249 if environmental damages are not accounted for). This number is several times larger than the cost of a home energy audit, estimated to be $347 on average [START_REF] Palmer | Assessing the energyefficiency information gap: results from a survey of home energy auditors[END_REF]. Therefore, government intervention intended to undo the moral hazard could be welfare improving.

Further improvements along both the energy efficiency and welfare dimensions can occur if environmental damages are internalized through a carbon price. In Jaffe and Stavins' words, the social optimum is then moved from the "Narrow economists' optimum" to the "True social optimum". The average welfare gains from undoing the moral hazard ($1,723) are one order of magnitude larger than those from internalizing energy-use externalities ($162). This proportion reflects the difference between the marginal inefficiency due to moral hazard, namely the unit of energy that could have been cut by optimal investment (valued at energy price p = $11.14/MCF), and the social cost of environmental damages (valued at p CO 2 = $33/tCO 2 = $1.69/MCF). Recall that here we assume away potential distortions in the price of natural gas, an assumption we then relax in sensitivity analysis.

The higher welfare level in the true social optimum is not general to the model, but due to the absence of backfire rebound effects in our calibration (see Proposition 4(iii)). Indeed, as reported in Table 4, the "genuine" rebound effect is 31% in the private optimum and 34% in the social optimum. In addition, we find a "relative" rebound effect of 33%, meaning that 33% of energy efficiency gains are taken back when the economy moves from the private to the social optimum. These numbers are in the upper range of the estimates reported by [START_REF] Sorrell | Empirical estimates of the direct rebound effect: A review[END_REF] for space heating.

11 The authors have refined their conceptual diagram over the years. The version we specifically refer to first appeared in [START_REF] Jaffe | Economics of energy efficiency[END_REF].

12 If the supply side could perfectly price discriminate, the cost of quality would be equal to the sum of the zero-profit price and the homeowner's net present value. For the median homeowner, the cost of quality would be $4,102, instead of $2,830 under perfect competition.

The moral hazard market failure can be restated as an average implied discount rate of 20%. This value is computed by solving and averaging the discount rate corresponding to each θ that matches the quality found in the social optimum with the net present value found in the private optimum, discounted at 7% by assumption.

Figure 2 illustrates, with the median homeowner, how equilibria are formed through reaction function intersections. The reaction functions are mildly upward sloping. The consumer's energy service varies by no more than 2 Degrees Fahrenheit with the quality of installation. Such a sensitivity is consistent with the values found in the literature [START_REF] Hirst | Indoor temperature changes in retrofit homes[END_REF]. While the quality offered by the contractor is always at the minimum if he behaves non-cooperatively, it mildly increases with consumer's energy service if he behaves cooperatively. As predicted by Proposition 2, the social optimum implements both a higher quality and a higher energy service than the private optimum. The figure also pictures some comparative statics of the energy price. Pricing energy-use externalities shifts the consumer's reaction function inward and the contractor's upward (c.f., Equations 35 and 36). As discussed in Appendix C, the final location of the equilibrium -at a higher quality and a lower energy service -is not general but specific to model parameters.

Efficiency of policy instruments

To isolate the ability of minimum quality standards and energy-savings insurance to specifically address moral hazard problems, we focus here on their welfare effects gross of environmental damages. Richer results can be found in Tables 4 and5.

If homeowner types were perfectly observable, the government would implement standards corresponding to each homeowner's optimal quality. Moreover, insuring contractors could design optimal contracts for every homeowner (Figure 6). On average, such contracts stipulate a coverage of 33% and close the energy efficiency gap by 77% along the welfare dimension. As illustrated in Figure 4, insurance shifts the reaction functions toward higher parties' actions (c.f. Equations 19 and 20). For the median homeowner, the resulting equilibrium entails a quality level that is intermediate between the social and private one and an energy service that is higher than in the social optimum. Again, this positioning is contingent upon our calibration, not general to the model. Under full insurance, the contractor offers the socially optimal quality, but the marginal energy costs are zero. The homeowner thus consumes her maximum amount of energy service (s max ).

In practice, homeowners' types are unobservable. The insuring contractor cannot offer each homeowner her optimal contract, just like the government cannot implement as many standards as there are homeowner types. Rather, they both implement uniform instruments.

The optimal minimum quality standard is given by Equation 22. Let us assume first that monitoring a standard is costless. Increasing its stringency from 24 to 47 worker.hours increases both economic efficiency and energy efficiency (Figure 7). Further tightening increases energy efficiency but not economic efficiency: The standard becomes too stringent for most of the people. The optimal standard is very close to the quality that is optimal to the median homeowner (47.1 worker.hours). As shown in Figure 5, such a standard is too tight to the 5th percentile homeowner (type θ = 0.66), who would have been better-off with a standard of 45.6 worker.hours. It is too loose to the 95th percentile homeowner (type θ = 1.51), who would have been better-off with a standard of 48.4 hours. Therefore, there is a narrow quality range in which the standard can be set so that average welfare is very close to the social optimum.

A similar pattern is observed with uniform insurance. Increasing insurance coverage up to 30% increases both energy efficiency and economic efficiency (Figure 7). Between 30% and 40%, the optimal uniform insurance contract is very close to the situation where consumers are all offered their optimal contract. Again, welfare losses due to heterogeneity are negligible. Energy efficiency then increases at the expense of economic efficiency up to a coverage of 60%, a situation discussed by Jaffe and Stavins as a "Technologist's optimum". Beyond that point, energy efficiency starts decreasing too, until the curve hits the horizontal axis, for a coverage of k = 100%, at the welfare level enjoyed before investment. That is, under full insurance, the total cost paid to the contractor is so high that no homeowner can be left with a positive net present value, hence none invests. Otherwise, indoor temperature would be set at corner s max = 80 • F, leading to an annual natural gas use of 79 MCF. This would imply a prohibitive lifetime discounted energy bill of $11,442, fully borne by the contractor and passed on to the homeowner as an insurance premium.

Overall, we find that the average welfare gain under the optimal standard is $290 greater than the gain obtained with optimal insurance (and $720 higher if environmental damages are accounted for). Unlike insurance, a standard entails monitoring costs, estimated for each realization at $347 from [START_REF] Palmer | Assessing the energyefficiency information gap: results from a survey of home energy auditors[END_REF]. In practice, only a fraction of realizations can be randomly monitored, so this estimate is an upper bound of the true monitoring cost. When accounting for monitoring costs, the welfare difference between the standard and insurance becomes ambiguous. However, our assumption of insurance contracts running over 35 years certainly overestimates the welfare gains from insurance. Therefore, the superiority of minimum quality standards over energy-savings insurance seems to be a robust result.

Discussion

Sensitivity analysis

We first examine the sensitivity of the model to market barrier parameters: a larger heterogeneity in homeowners' valuations, a lower initial insulation rate, higher insulation costs and a higher absolute valuation of energy service (Table 6, Figure 8). We find that compared to the reference scenario, absolute welfare levels vary by a large magnitude, but the deadweight loss from moral hazard is relatively stable ($1,085-1,260). Moving from the private optimum to the social optimum increases welfare by 3-5%. Implied discount rates amount to 17-20%.

We then introduce market failures and behavioral anomalies, the comparative statics of which is formalized in Appendix C (Table 7, Figure 8). We model non-capitalization of energy savings by setting investment lifetime to 10 years, the typical period of residency in a dwelling. We also model undistorted natural gas price by removing the 49.7% markup above marginal cost estimated by [START_REF] Davis | Do Americans consume too little natural gas? An empirical test of marginal cost pricing[END_REF]; the price is then $5.80/MCF. Lastly, we model undervaluation of energy savings with a 20% discount rate. Compared to the reference scenario, introducing non-capitalization and undervaluation of energy savings decreases the absolute welfare level and increases the implied discount rate. Removing energy price distortions increases absolute welfare and lowers the implied discount rate. In addition, unlike market barriers, these alternative assumptions significantly reduce the deadweight loss from moral hazard, which amounts to $289-517 in absolute terms and 1-3% in relative terms.

If homeowners undervalue energy savings (r = 20%) and ignore environmental damages, then they might not perceive optimal quality as beneficial: the welfare gains ($289) are below the upper bound of the monitoring cost ($347). This does not mean, however, that public intervention is not warranted. The situation with undistorted price of natural gas (p = $5.80) is the closest to a market where no inefficiency other than moral hazard and carbon dioxide emission externalities occurs. In this most relevant context, the deadweight loss from moral hazard ($486, or $912 if environmental damages are accounted for) is still above the monitoring cost, though by a smaller margin than in our reference scenario. This strengthens the case for government intervention aimed at undoing the moral hazard.

This approximation can be used as a sufficient statistics to guide future empirical work on the issue. Tables 6 and7 (fourth row) show that ∆ q W underestimates the exact deadweight loss by no more than 9% in absolute value across scenarios.

We now extrapolate our reference scenario to the U.S. population. Recall that the homeowners using natural gas for space heating covered 35% of the RECS dataset. Moreover, we confined our attention to 3.4% of that subpopulation (hence 1.2% of the complete dataset), the fraction of households investing in insulation annually. Applying these shares to a number of U.S. households of 115 million (U.S. Census Bureau value for 2008-2012), our analysis covered approximately 1.4 million households. On average, the moral hazard associated with insulation represents for each household a quality shortfall of one workday, a natural gas overuse of 8 MCF, 0.4 tons of CO 2 emissions and $1,723 of present value deadweight loss (net of environmental damages). At the population scale, undoing moral hazard could thus save 11 billion cubic feet of natural gas and 0.6 million tons of CO 2 annually, equivalent to $2.4 billion present value benefits.

Extensions

Considering risk-aversion would be a natural extension of the model. As a matter of fact, the performance of energy efficiency technologies depends on volatile factors, such as weather conditions or energy prices, to which consumers are likely to be adverse. Moreover, the home retrofit industry is made up of small industries that have little room to diversify risks (Lutzenhiser, 1994). In our model, risk-averse homeowners would expect higher energy expenditures than a certainty equivalent, hence demand less energy service. Risk-averse firms would respond with a lower quality in the social optimum. Overall, the introduction of risk-aversion on both sides of the market would reduce the size of the energy efficiency gap compared to a riskless situation.

Instead of assuming homogeneous firms that all fail to offer quality in equilibrium, we could assume heterogeneous firms. Within our informational structure, a fringe of firms may adopt reputation or signaling strategies and supply a better, or even optimal quality. Such private forces would reduce the size of the energy efficiency gap.

Conclusion

We examined how moral hazard problems can generate an energy efficiency gap, and how this information asymmetry may interact with energy market failures. Taking home energy retrofits as an example, we show that if the quality of installation offered by a retrofit contractor is unobserved by the homeowner, then the contractor will cut quality in equilibrium. This leads to a suboptimal level of energy efficiency along both the intensive and extensive margins: The quality offered to consumers is too low and there are too few consumers investing.

Numerical simulations calibrated to the U.S. home insulation market suggest that the potential welfare gains from undoing moral hazard are larger than the cost of quality audits. This holds for a large range of market environments, most notably when moral hazard is the only market failure in place. Therefore, undoing moral hazard is welfare-improving. The moral hazard problem is consistent with implied discount rates in the 15-35% range. It also induces an energy efficiency gap that is substantially larger than the one induced by energyuse externalities. Our sensitivity analysis illustrates that the deadweight loss associated with moral hazard can be closely approximated using only engineering data. Extrapolating our results to the U.S. population of homeowners using natural gas for space heating, we find that increasing the quality of each insulation installation by one workday could save 11 billion cubic feet and 0.6 million tons of CO 2 each year, equivalent to $2.4 billion present value benefits. This insight is relatively new. While most investigations of the energy efficiency gap have focused on the role of possible undervaluation of energy savings by consumers and the discrepancy between ex ante engineering estimates and ex post estimates [START_REF] Gillingham | Energy Efficiency Economics and Policy[END_REF][START_REF] Allcott | Is There an Energy Efficiency Gap?[END_REF], ours underlines the importance of considering the behavior of the firms supplying energy efficiency. The presence of moral hazard offers a simple explanation for the systematic overestimation of energy savings by engineering models.

Our analysis provides motivation for energy efficiency policies that would go beyond the internalization of energy-use externalities. This recommendation holds as long as consumers are not prone to a backfire rebound effect -a reasonable hypothesis [START_REF] Sorrell | Empirical estimates of the direct rebound effect: A review[END_REF][START_REF] Gillingham | Energy policy: The rebound effect is overplayed[END_REF][START_REF] Borenstein | A Microeconomic Framework for Evaluating Energy Efficiency Rebound And Some Implications[END_REF]. When addressing the moral hazard, the first-best outcome can only be attained to the extent that energy performance and consumer preferences can be made perfectly observable. Since no technology can meet that goal at an affordable cost yet, government intervention will only generate second-best outcomes: Minimum quality standards do not address consumer heterogeneity and energysavings insurance raises a second moral hazard. However, our numerical results suggest that the former can bring social welfare very close to its optimal level. Similarly, even with modest coverage, insurance contracts can deliver welfare gains that are economically important.

(34)

N P V j (θ) ≡ θ V (s j θ ) -V (s 0 θ ) -p E(s j θ , q j θ ) -E 0 (s 0 θ ) ≥0 Γ -T + ǫ ≤0
The right inequality is given by Assumption 1(v). The left inequality comes from the following inequalities: θV (s 0 θ ) -pE 0 (s 0 θ ) ≤ θV (s 0 θ ) -pE(s 0 θ , q j θ ) ≤ θV (s j θ ) -pE(s j θ , q j θ ). The former is due to technological assumptions about E and E 0 and the latter is due to s j θ maximizing U .

Thanks to Proposition 5, equilibrium actions s j θ and q j θ decrease with θ. As they are bounded below by s min and q min , the limit of N P V (θ) when θ tends toward zero is finite.

if lim θ→0 N P V (θ) ≥ 0 then all consumers participate. Participation is given by N j ≡ +∞ 0 dF (θ) = 1.

If lim θ→0 N P V (θ) < 0 and lim θ→+∞ N P V (θ) > 0 then by Equation 12, there exists a unique cutoff type θ 0 , as discussed in the text.

If lim θ→0 N P V (θ) < 0 and lim θ→+∞ N P V (θ) ≤ 0 then participation is nil. In this case, the gross utility gains accruing to the homeowner never offset the increase in the payment to the contractor.

Appendix C. Comparative statics with respect to market and behavioral features

Recall that ζ ≡ pΓ(r, l). A higher ζ is equivalent to a higher energy price p or a higher Γ, that is, a lower discount rate r or a longer lifetime l.

Applying the Implicit Function Theorem to Equations 6 and 7, we see that an increase in ζ shifts reaction functions s * θ (•) downward and q * (•) upward:

(35) ∀t ds * θ dζ = E s θV ′′ /ζ -E ss < 0 (36) dq * dζ = -E q C ′′ /ζ + E qq > 0
By the same reasoning as in Proposition 5, a higher ζ entails a higher energy service in private equilibrium. But optimal actions cannot be compared unambiguously in the private and social equilibria.

The influence of ζ on N P V * , established by the Envelope Theorem, depends on the consumer's reaction to higher energy efficiency: (37) dN

P V * dζ = -E(s * θ , q * θ ) -E 0 (s 0 θ )
As long as energy efficiency investments decrease energy use for all consumers, the net present value is increasing in ζ. By the same type of reasoning as in Proposition 5, this leads to a higher participation and a higher average welfare. This conclusion is reversed if all consumers are subject to a genuine backfire rebound effect, i.e., ∀θ E(s * θ , q * θ ) > E 0 (s 0 θ ). In this case, a higher ζ decreases participation and average welfare.

(40) dθ 0 dq U 0 (θ 0 (q), s 0 θ ) -U (θ 0 (q), s * θ (q), q) + M (q) + +∞ θ 0 (q)

∂U (θ, s * θ (q), q) ∂ q -M ′ dF (θ) = 0

Recognizing that U 0 (θ 0 (q), s 0 θ ) -U (θ 0 (q), s * θ (q), q) = -N P V (θ 0 (q), s * θ 0 (q), q) and using the binding constraint leads to the result (Equation 23).

Note that if participation to investment is nil without the standard, no standard will be welfare-improving. In contrast, if participation is full without the standard, the constraint will not be binding and the optimal standard will be defined by the following first-order condition:

(41

) +∞ 0 ∂U (θ, s * θ (q), q) ∂ q -M ′ dF (θ) = 0 0 [U x (θ, s x θ , q x θ ) -U 0 x (θ, s 0 )]dF (θ) + +∞ θ 0 ∆U x dF (θ).
The first and third integrands of the right-hand side are positive (see proof (i) just above). The second integrand is also positive, since ∀θ ≥ θ x 0 U x (θ, s x θ , q x θ ) ≥ U 0 x (θ, s 0x ) ≥ U 0 x (θ, s 0 ). Therefore, aggregate welfare is larger when externalities are internalized:

W c,x x ≥ W c
x . The exact same reasoning leads Notes: Adding "Zero-profit price of insulation" and "Homeowner's net present value" gives the insulation price that would be charged by a perfectly discriminating monopolist. Notes: "Energy efficiency" is averaged over the whole population. The average over participants is obtained by dividing "Energy Efficiency" by "Participation rate". Adding "Zero profit price of insulation" and "Homeowner's net present value" gives the insulation price that would be charged by a perfectly discriminating monopolist. 1). The model output is the probability distribution function of s 0 θ , calculated with the triangle method and assuming a log-normal distribution of θ with parameters µ = 0 and σ = 0.25. 6 and7 .

Tables and Figures

Figure 1 .

 1 Figure 1. Model fit to RECS sample data. The Gaussian estimator is the normal distribution of temperature with parameters µ = 69.3 and σ = 3.4, the mean and standard deviation of the RECS sample (Table1). The model output is the probability distribution function of s 0 θ , calculated with the triangle method and assuming a log-normal distribution of θ with parameters µ = 0 and σ = 0.25.

Figure 2 .

 2 Figure 2. Reaction functions, with a homeowner of median type θ = 1. (c) refers to the social optimum, (nc) to the private optimum and (c, x) and (nc, x) to the same optima in the presence of a carbon price of $33/tCO 2 .

Figure 3 .

 3 Figure 3. The energy efficiency gap. The horizontal axis represents average lifetime discounted welfare, net of environmental damages valued at $33/tCO 2 . The vertical axis represents average energy efficiency (with the value 0% attributed to non-participating homeowners)..

Figure 4 .

 4 Figure 4. Reaction functions under energy-savings insurance, with a homeowner of median type θ = 1. (c) refers to the social optimum, (nc) to the private optimum, (i( k)) to the equilibrium induced by insurance with optimal coverage k and (i(1)) to the full insurance equilibrium.

Figure 5 .

 5 Figure5. Reaction functions with a minimum quality standard. The standard is optimal to the median homeowner (θ = 1), but suboptimal to all others. For instance, it is too tight to the 5th percentile of the homeowners' distribution (θ = 0.66) and too loose to the 95th percentile (θ = 1.51).

Figure 6 .

 6 Figure 6. Net present value and participation with respect to homeowner's type θ. The net present value (gross of environmental damages) of investment in insulation reads on the right vertical axis. The intersection of each curve with the zero horizontal axis determines the cutoff type θ 0 of the marginal participant in investment. For each cutoff type on the horizontal axis (from the 0.5th to 95.5th percentile of the θ distribution), participation across the population is determined by the value of the complementary cumulative distribution (CCDF) of θ, which reads on the left vertical axis.

Figure 7 .

 7 Figure7. Economic and energy efficiency of policy instruments. The welfare displayed on the horizontal axis is the average lifetime discounted welfare gross of environmental damages. Each mark of the uniform standard parametric curve represents an additional worker.hour of labour requirement, from q min to q max . The stringency of the standard increases counter-clockwise. Each mark of the uniform insurance curve represents an incremental 10% of insurance coverage, from 0 to 100%. Insurance coverage increases counterclockwise.

Figure 8 .

 8 Figure 8. Sensitivity analysis. The welfare displayed on the vertical axis is the average lifetime discounted welfare gross of environmental damages. Scenario assumptions are detailed in Tables6 and 7.

Table 1 .

 1 Summary Statistics of the RECS Sample(n = 4, 266)

				RECS
			Unit	Entry	Median Mean	SD	Min	Max
	Temperature	when	• F	TEMPHOME	69.0	69.3	3.4	60.0	80.0
	someone is home dur-
	ing the day (winter)
	Natural Gas cost for	$	DOLNGSPH	562	624	386	33	3,591
	space heating, 2009
	Natural Gas usage for	MCF	BTUNGSPH 1.023 * 10 -3	50.2	55.0	34.0	0.6	337.2
	space heating, 2009
	Price paid for natural	$/MCF DOLNGSPH * 1.023 BTUNGSPH * 10 3	11.14	11.95	5.69	3.75 190.73
	gas for space heating
	Gross household in-	$	MONEYPY	65,000 78,928 50,727 2,500 170,000
	come, 2009
	Income share dedi-		DOLNGSPH MONEYPY	0.82% 1.54% 3.20% 0.02% 65.92%
	cated to natural gas
	for space heating
	Notes: "Natural gas cost for space heating" is measured in thousand BTU in RECS, here converted in MCF.
	"Gross household income" is measured with 24 income ranges; we identify each income range with its upper
	value and assume an average income of $170,000 for the top category, which is consistent with U.S. Census
	Bureau 2009 data for owner-occupiers.

Table 2 .

 2 Model parameters

	Parameter	Symbol Value Unit	Source
	Minimum indoor tem-	s min	60 • F	0.8th percentile of the RECS preliminary sample
	perature
	Maximum indoor tem-	s max	80 • F	99.9th percentile of the RECS preliminary sample
	perature
	Minimum labour in-	q min	24 worker.hours One workday = 24 worker.hours, e.g., three installers
	put			working 8 hours a day (Best guess)
	Maximum labour in-	q max	72 worker.hours Three workdays (Best guess)
	put
	Maximum valuation	V max	2,816 $	The 95th percentile of the income share dedicated to
	of energy service			space heating in the RECS sample is 4.3%. Apply-
				ing this fraction to the median income of the sample
				($65,000) leads to a maximum budget for space heating
				of $2,816.
	Minimum energy effi-	G min	5%	(Best guess)
	ciency of insulation
	Maximum energy effi-	G max	30%	(Best guess)
	ciency of insulation
	Fixed cost of wall in-	K	2,400 $	(Best guess)
	sulation
	Physical lifetime of in-	l	35 years	(Best guess)
	sulation investment
	Discount rate	r	7%	Value recommended to assess private investment (U.S.
				OMB).
	Price of natural gas	p	11.14 $/MCF	Median price of the RECS sample
	Carbon price	p CO 2	1.69 $/MCF	Equivalent to a social cost of carbon of $33/tCO 2 in
				2010, which is the value recommended for impact anal-
				ysis in the U.S. (White House, 2013)

Table 3 .

 3 Calibration targets

	Calibration target	Expression Target value Source
	Optimal temperature before in-	s 0 θ=1	69 • F	Median temperature of the RECS
	vestment to the median home-			sample
	owner
	Optimal annual energy use before	E 0 (s 0 θ=1 )	50 MCF	Median annual natural gas use for
	investment to the median home-			space heating of the RECS sam-
	owner			ple
	Price-elasticity of energy demand	See App.G	-0.4	Middle value of the [-0.03;-
	before investment to the median			0.76] range found in the lit-
	homeowner			erature by Gillingham, Newell,
				and Palmer (2009) for short-term
				price-elasticities of natural gas
				use
	Participation to insulation invest-	See App.G	50%	See text (Section 4.3).
	ment in the private optimum
	Expected energy savings at max-	G(q max )	99%G max
	imum quality level
	Wage for insulation workers at	C ′ (q min )	$15/hour	According to the U.S. Bureau of
	minimum quality level			Labour and Statistics, the median
				pay for insulation workers was
				$16.88/hour in 2010. According
				to Zabin, Lester, and Halpern-
				Finnerty (2011), the lower range
				of insulation wages in California
				is $10-15/hour
	Wage for insulation workers at	C ′ (q max )	$30/hour	Upper range of the values re-
	maximum quality level			ported by Zabin, Lester, and
				Halpern-Finnerty (2011) for Cal-
				ifornia
	Notes: Parameters α, β, γ, ω, ρ, φ and ǫ (Section 4.2) are calibrated so as to allow the model to replicate these
	targets. The procedure is detailed in Appendix G.

Table 4 .

 4 Simulation results, median homeowner (θ = 1)

	Uninternalized environmental damages	Internalized environmental damages

Table 5 .

 5 Simulation results, averaged over the population of total mass 1

	Uninternalized environmental damages	Internalized environmental damages

Table 6 .

 6 Sensitivity to market barrier parameters

	Reference	Large	Low initial	High cost	High valua-
		homeowner	insulation	of insula-	tion of en-
		heterogene-	rate	tion	ergy service
		ity
		σ=1	initial	F C=$3000,	Vmax=$4000
			rate=3.5%	C ′ (q min )=$20,
				C ′ (qmax)=$50

Table 7 .

 7 Sensitivity to market failure and behavioral anomaly parameters

	Reference	Non-	Undistorted	Undervaluation
		capitalization	price	of	of energy
		of energy	natural gas	savings
		savings
		l=10 years	p=$5.80/MCFr=20%

In practice, minimum quality standards could target materials used in retrofits and specific tasks.

See Allcott and Greenstone (2012) for a constructive critique of the study.

Acknowledgements

The authors started this research at the Precourt Energy Efficiency Center (PEEC), Stanford University. We thank PEEC for funding and James Sweeney for encouragements and comments. We are grateful to Tiger Adolf, Rick Diamond, Matt Golden, Evan Mills and Carol Zabin for discussions that motivated this work. We thank Frédéric Branger, Lawrence Goulder, Franck Lecocq, Marie-Laure Nauleau, Philippe Quirion, and Adrien Vogt-Schilb for comments on earlier drafts. We also thank seminar participants at CIRED, Stanford MS&E, EconomiX, Paris School of Economics, as well as participants to the WEAI Conference in San Francisco, the AEAA Conference in Seattle, the USAEE Conference in Austin, the AERE Conference in Banff, the AFSE Meeting in Aix-en-Provence, the EAERE Conference in Toulouse, the URV Workshop in Reus, the ZEW-Harvard-Duke Workshop in Mannheim and the FAERE Meeting in Le Havre.

Overall, the results of the sensitivity analysis suggest that the magnitude of the deadweight loss varies considerably with the investment lifetime, the energy price and the discount rate, but varies less with the structural parameters reflecting market barriers.

Approximating the deadweight loss from moral hazard

We seek to approximate the exact deadweight loss associated with the quality shortfall caused by the moral hazard: ∆ q W ≡ W c -W nc . A first step is to examine the marginal welfare change induced by a marginal change in quality, which is equivalent to solving Equation 22with M (q) = 0. Envelope conditions allow us to neglect changes in participation (see Appendix E) and, for each participant, the benefits from increased heating comfort (see Appendix H). For a participating homeowner, the marginal benefits from a higher quality are:

Integrating between q nc and q c (with q nc ≤ q c according to Proposition 2i) gives the following proxy for ∆ q W : (31)

∆ q W = -p∆ q E(s nc , q)Γ -∆ q C(q)

The error associated with integrating infinitesimal changes is positive and equal to the private benefits from increased heating comfort and the social benefits from increased participation. Therefore, ∆ q W provides a lower bound of the exact average deadweight loss:

The formula is quite intuitive. It weighs the cost of quality against its benefits in terms of gross energy savings. This corresponds to the net present value calculation typically performed by engineers, in the sense that it only takes into account technological information. It does not require knowledge of the utility function for energy service V (•) nor its specific effect on energy use ∂E/∂s. Therefore, the direct rebound effect can be ignored. Still, the formula contains the key parameters of the market and behavioral environment p, l and r which were found to be the most important in the sensitivity analysis.

Appendix A. Comparative statics with respect to consumer type θ

Applying the Implicit Function Theorem to Equation 6:

Therefore, for any given quality q offered by the contractor, a higher valuation of energy service shifts the consumer's reaction function upward:

Appendix B. Participation to investment

For any equilibrium situation j, participation will depend on the limits of the net present value function, the sign of which is indeterminate:

Appendix D. Optimal insurance coverage

The first-order condition for finding the optimal insurance contract from Equation 21is:

Plugging in Equations 17 and 18 and further rearranging gives the equation that solves the optimal coverage k:

Appendix E. Optimal minimum quality standard

Assuming that the cutoff type exists and is unique, the constraint in Equation 22 is binding.

The optimization program can be solved by simply maximizing the objective function and assuming that θ 0 is an implicit function θ 0 (q) defined by the constraint. Applying the Leibniz integral rule and the Envelope Theorem leads to the following first-order condition for maximization:

Appendix G. Model calibration

Parameters α, β and γ (Section 4.2) are computed so as to allow the model to replicate calibration targets 1, 2 and 3 (Table 3). For θ = 1 and s 0 = 69 • F, this leads to:

(42)

Note that if we assume linear technology (γ = 1), then the value of V max that solves the system is $2,719. This provides a lower bound for V max , as V max is increasing in γ:

Parameter ǫ, representing the non-energy benefits of insulation, is computed so as to allow the model to replicate calibration target 4. As the marginal investing homeowner has type θ = 1, ǫ is such that N P V (θ = 1, s nc θ=1 , q nc θ=1 ) = 0, which leads to ǫ = $2, 035.

Parameter ω is computed so as to allow the model to replicate calibration target 5:

Parameters ρ and φ are computed so as to allow the model to replicate calibration targets 6 and 7:

Appendix H. Derivation of the sufficient statistic

For a participating homeowner, the exact deadweight loss ∆ q W is:

(46) W c -W nc = l t=1 [V (s c ) -V (s nc ) -p (E(s c , q c ) -E(s nc , q nc ))] δ t -[C(q c ) -C(q nc )]

We recognize that:

(47) ∆ q W = ∆ q W + [V (s c ) -V (s nc ) -p (E(s c , q c ) -E(s nc , q c ))]

The term in brackets is positive because s c maximizes the function V (•) -E(•, q c ). Therefore, ∆ q W ≤ ∆ q W .