Marion Podesta
email: marion.podesta@univ-savoie.fr

Jean-Christophe Poudou

Optimal Nonlinear Princing, Bundling Commodities and Contingent Services

Keywords: D42, L12, Q4 Bundling, Nonlinear pricing, Energy market JEL classi…cation: D42, L12, Q4 Tari…cation non-linéaire, Marchés énergétiques

This paper analyzes optimal nonlinear pricing for a monopoly supplying a bundle of a commodity and a related service, where the consumers' private information is captured by scalar variable. With constant marginal costs, we …nd the standard result where the good and the service in the bundle are lower than separately. However, under the increasing cost assumption, when the degree of the complementarity becomes su¢ciently high, the marginal price of separate good is lower than the good price in the bundle. Contrary to Martimort (1992), when good and service are perfectly complementary, we can not conclude that it is costly for consumers to sign two contracts from di¤erent shops than to buy the bundle. Because of asymmetric properties in the utility function, pro…tability result of bundling strategy depends, on the one hand, on the degree of complementarity between commodity and related service and on the other hand, on the degree of the optional service.

Résumé

Cet article analyse la tari…cation non linéaire optimale pour un monopole offrant un package comprenant un bien et un service attaché au bien, lorsque ce dernier augmente l'utilité marginale que les consommateurs accordent au bien. Nous examinons les mécanismes d'incitation de révélation des préférences dans le cas où l'information privée est représentée par un paramètre unidimensionnel. Sous l'hypothèse de coûts constants nous retrouvons le résultat standard où il est profitable pour les consommateurs d'acheter le bien et le service attaché sous forme de package plutôt que séparément. Cependant, avec des coûts croissants, lorsque le degré de complémentarité entre le bien et le service est su¢samment élevé, le prix marginal du bien séparé est plus faible que comme élément du package. Contrairement à Martimort (1992), lorsque le bien et le service sont parfaitement complémentaires, nous ne pouvons pas conclure qu'il est plus coûteux pour les consommateurs de signer deux contrats aux di¤érents magasins que d'acquérir le package. Du fait des

Introduction

Recently, a trend towards deregulation of utilities industries (as telecommunications or energy sectors) has been observed worldwide and has an impact on market structure. This new environment incites …rms to diversify their o¤ers, and they can compete with their new rivals with the help of bundles. Simultaneously, a convergence phenonenom appears and creates a strategic link between di¤erent markets. In telecommunications sectors, the process of convergence across multimedia and telecom markets involves triple-play or quadruple-play o¤ers. These packages include broadband, TV and …xed telephony at bundle discounts. In energy markets incumbents supply bi-energy bundles or o¤ers including energy and services. [START_REF] Adams | Commodity Bundling and the Burden of Monopoly[END_REF] Subsidiaries in telecommunications and energy sectors allow to incumbents to propose bundles with several goods or services, therefore incumbents keep a dominant position in these markets.

This paper deals with a private monopoly, mainly to depict a situation where market power is high. We study pricing strategy for a …rm supplying a good and a complementary optional service, when the service increases the good's marginal utility and where the consumers' private information is captured by scalar variable. [START_REF] Bateson | Managing Services Marketing[END_REF] In our model, we use the analysis of Martimort (1992, 1996) in a principal-agent context. Firm can sell good and service separately (independent pricing) or as a bundle (pure bundling [START_REF] Burstein | The Economics of Tie-In Sales[END_REF] ). Contrary to the Martimort's assumptions, there is no symmetry between the good and the related service in the consumers' utility function. In order to study the speci…city of the service, we use the nonlinear pricing. More precisely, the paper focuses on bundling to satisfy a fundamental need (space heating needs, for instance), where the good can be tied with a related service. [START_REF] Martimort | Multi-principaux avec Anti-selection[END_REF] We show that the pro…tability of bundling depends on the degree of complementarity between the good and the service, as well as on the optional character of the service. Martimort (1992, 1996), in a principal-agent model, introduces the possibility for a common agent to contract with multiple principals. He compares the cooperative situation, this is the case where …rms can o¤er a bundle, and the situation with noncooperation under the assumption of nonlinear pricing. The results depend on the complementarity or the substituability between activities controlled by each principal. If the goods are complements, bundling is an optimal strategy for the principals and consumers. However, if the goods are substitutes, the consumers' utility is better when there is noncooperation between the principals, but pro…ts are lower.

The bundling literature in the monopoly case shows that the optimal strategy is the mixed bundling one for it obtains a maximum surplus from the consumers if the correlation of reservation values is negative (Adams and Yellen, 1976, Schmalensee, 1984 and Mc Afee et al, 1989). With the mixed bundling strategy, the goods are available both separately and in a bundle. Since bundling allows the monopolist to extract more consumer rent, it makes higher pro…ts (since the correlation of consumers' reservation values is negative). However when competition increases, results are reversed. Bundling can reduce the …rms pro…t and increase the consumers' rent. [START_REF] Martimort | Exclusive Dealing, Common Agency, and Multiprincipals Incentive Theory[END_REF] Literature on bundling has not been really explored bundling strategy for a complementary service. In fact, models rarely integrate the speci…city of the service into the pricing strategy. The acknowledged fact is that when the service is personalized, its ex-ante valuation is di¢cult (Bateson, 1995). In the general case, the consumers' commodity value is better than the service value. Thus, a consumer has more di¢cult to give a benchmark price for a service, relatively to a good. When monopoly uses a bundling strategy, it has an additional tool to practice price discrimination. Moreover, nonlinear pricing is widely used in several markets (energy or telecommunication markets, for instance) to supply one or several goods. [START_REF] Maskin | Monopoly with Incomplete Information[END_REF] Firm proposes a menu of tari¤ to incite consumers to reveal their preferences [START_REF] Mcafee | Multiproduct Monopoly, Commodity Bundling, and Correlation of Values[END_REF] and can extract more consumers' surplus.

This paper considers the case where the monopoly practices nonlinear pricing for both a good and an attached service. The principal (monopoly) has two shops which can o¤er the commodity and the optional service. We compare marginal prices for the good and the service under an independent pricing strategy and when bundling is considered. In the bundling case, both shops coordinate their pricing strategy to provide a bundle. In the paper, two cases are considered. Firstly, monopoly costs are constants, the two shops are supposed to be only retailers. Secondly, we consider that the di¤erent shops produce the good and the service under increasing costs. However, incomplete information on consumers' type implies to implement an incentive compatible nonlinear pricing schedule in both cases. Our de…nition of bundling is slightly di¤erent from the standard industrial organization literature since we aim to study informational aspects of this kind of strategy by a monopolist. Here, we consider that two productive units (or shops) can separately produce the core good and the attached service but the commercial unit could sold either separately or together. It is thus in this business view that we study the informational gains and losses and the e¤ects of these practices on prices.

Under constant costs assumption, we show that it is less costly for consumers to buy the bundle and …rm has lower consumers' rent compared with the independent pricing strategy. Indeed when monopoly follows an independent pricing strategy, it would be worse o¤ to propose separating contracts. The monopoly proposes a contract for the commodity and another one for the service, it can capture more consumers' rent from the signature of each contract. However, when the monopoly provides a bundle it cannot duplicate the independent contracts and captures only once the consumers' rent.

With increasing costs and when the good and the service are perfectly complementary, the optionality of the service has an impact on the marginal price of separate good. The marginal price of the good is lower separately than as a component of the bundle. Because of asymmetries in the utility function, the pro…tability of bundling depends on the trade-o¤ between the degree of complementarity and the degree of the optional service.

The following section sets out the model. Firstly, we consider that the monopoly has constant production costs. The section 3 considers the case where the shops coordinate their o¤ers and propose a bundle, and in section 4, we remove this assumption to consider that the two shops provide the commodity and the service separately. The section 5 compares the di¤erent types of strategies. Secondly, in order not to limit our analysis at constant costs and to focus on the energy retailers concerns, we discuss to increasing costs in section 6. Finally, section 7 proposes few concluding remarks.

The model

The model focuses on energy demand to satisfy a fundamental need, for instance space heating, lighting or air conditioning needs for residential consumers. We consider a space heating need for instance, which can be provided by a couple of energy (x), gas or electricity and also a level of service (s). The …rm o¤ers a good (energy) and a related (complementary) service, it can be technical maintenance or energy consultancy. The gross utility is given by: u(H;

)
where H is the level of heating achieved and depends on the consumers' preferences denoted . Heating is obtained by a given technology which combined both the quantity of energy consumed x 0, and the related service level purchased s 0. Hence, the utility can be directly written as an increasing asymmetric function of x and s, let u(H; ) = u(x; s; ) which satis…es the following requirements for all y 0: 8y; u(y; 0; ) u(0; y; ) (H1)

Moreover the usual Spence-Mirless property is veri…ed and given by:

u x (x; s; ) > u s (x; s; ) 0 (H2)
where subscripts represent partial second derivatives. The (energetic) good is intrinsically preferred to the service since the component cannot provide heating alone. As a result, the technical service can be viewed as optional from the consumer's point of view as its marginal utility is always smaller than the one for the good for all types considered. Whenever u x (x; s; ) > u s (x; s; ), the service is not totally an option but neither essential. However if it is the case that u s (x; s; ) = 0, the service is viewed as a pure option from the consumer's point of view. We also assume strict concavity [START_REF] Mussa | Monopoly and Product Quality[END_REF] and complementarity between energy and service such as:

u xs (x; s; ) 0 (H3)
The preferences of consumers are given by a single-dimensional parameter 2 [ ; ] R + with a distribution function F ( ). [START_REF] Nalebu¤ | Bundling as an Entry Barrier[END_REF] In incomplete information, the monopoly knows the distribution of consumers' preference but it doesn't know exactly each consumer's reservation price. We also de…ne '( ) = 1 F ( ) f ( ) , the informational cost or virtual cost, which represents a cost for the …rm, and it is decreasing10 in : ' 0 ( ) 0. The information cost is generated by the informational advantage of the agent over the principal.

To simplify, the costs are supposed to be identical and increasing convex for both service and good units produced:

C(x; s) = w(x + s) + k 2 (x 2 + s 2 )
In order not to limit our study, we analyze two normalized cases. Firstly, we suppose that the shops sell their products at the market prices, in this case we suppose that marginal costs are constant. The linear cost function (with k = 0), when the monopoly is a simply retailer of the energy and the service, is given by:

C(x; s) = w(x + s) (H4)
with c(x) = C(x; 0) and c(s) = C(0; s) are respectively the cost function of the good and the service. Secondly, we remove this assumption to focus on the case where the shops produce energy and service sold to consumers, i.e when costs express increasing marginal slope so we normalize w = 0. The following increasing convex function satis…es the previous assumptions:

C(x; s) = k 2 (x 2 + s 2 ) (H5)
In order to simplify the analysis and without loss of generality, we set k = 1. Suppose that the monopolist sells its products using a nonlinear tari¤ =, the consumers' net utility is given by:

U = u(x; s; ) = 0 if x; s > 0 if x; s = 0
where = is the total expenditure paid if consumers buy either x units of good and s units of service in the bundling case, either x, or s in the independent pricing case. With the independent pricing strategy, = can be divided in two respective parts, t for the energy and for the related service: = = t + . With bundling strategy, the nonlinear tari¤ is denoted = = T . In each case, we consider nonlinear tari¤s with a …xed fee and a variable part.

The timing of the game is the following. First, the agent discovers his type. Then, the principals o¤er simultaneously the contracts, or equivalently the direct revelation mechanisms: T ( ^ ) or t( ^ ) and ( ^ ). Second, the agents accept or refuse the proposal they respectively receive. If the agents accept, they make their reports to the principal. Finally, the tari¤ T ( ^ ) or t( ^ ) and ( ^ ) as well as the quantities of good and attached service fx( ^ ); s( ^ )g are implemented.

In our setting, the …rm only cares about its expected pro…ts, and seeks to maximize E( ). Moreover, the consumer must have adequate incentives to reveal his type truthfully-incentive compatibility (IC)-and to participate in the mechanism voluntaryindividual rationality (IR). The program facing the monopoly is:

max E( ) = Z f=( ) C(x( ); s( ))gf ( )d (EP) U ( ) u(x( ); s( ); ) =( ) u(x( ^ ); s( ^ ); ) =( ^ ) (IC) U ( ) 0 (IR)
The(IC) constraint means that the …rm lets a consumers' surplus higher when the consumer reveals his preferences and chooses an optimal price schedule than when he lies about his preferences and would look like another type. With this constraint the monopoly encourages the consumers to reveal their preferences. With the (IR) constraint the …rm must incite the consumers to purchase by leaving them a positive net surplus.

To go further, we can specify a quasi-concave function for which assumptions above are ful…lled:

u(x; s) = (x + s) + 2 xs 2 x 2 2 s 2 (H6)
where 2 [0; 1] ; 2 [0; 1 2 ] and without loss of generality, we set = 1. The parameter represents the optional character of the service. If = 0, it is viewed as totally optional and respectively when = 1, it becomes essential to satisfy the fundamental need. The parameter is the degree of complementarity between the good and the attached service. If = 0, they are independent and contrary if = 1 2 , the good and the service are perfectly complementary.

The information cost '( ), previously de…ned, can be speci…ed with a uniform law as:

F ( ) = thus '( ) = (H7)
We will use this assumption in the sequel in particular to determine an implicit solution in the independent pricing case.

In the following, we will focus on interior allocation for good and service (x; s > 0), we will explicit the parametric restrictions involved when necessary. In sections 3 to 5, we …rst consider that (H4) is true, that is costs are constants and we relax this assumption in section 6, assuming that (H5) holds..

Nonlinear pricing schedule and bundling

In this section, the monopoly can o¤er a bundle composed of a good (energy) and a related complementary service (for instance technical consultancy). The monopoly has two separated shops which coordinate to propose a nonlinear tari¤ = = T , for the bundle.

Ex-post, the nonlinear tari¤ T is implemented as a three-part schedule and has the following form: T = Z + px + rs , Z = T px rs where Z is the …xed fee of the tari¤, p and r are respectively the energy and the service rates, and x and s are the relative quantities. Solving the restated problem (see Appendix B for details), the quantities of good and service are given by:

x B ( ) = ( '( ))(1 + 2 ) w(1 + 2 ) (1 4 2 ) and s B ( ) = ( '( ))( + 2 ) w(1 + 2 ) (1 4 2 )
The quantities [START_REF] Podesta | Optimal nonlinear pricing, bundling commodities and contingent services[END_REF] increase with consumers' willingness to pay. [START_REF] Schmalensee | Commodity Bundling by Single-Product Monopolies[END_REF] The degree of optionality has a more important impact for the service in the bundle than for the good and this as much as the degree of complementarity is higher. [START_REF] Schmalensee | Gaussian Demand and Commodity Bundling[END_REF] Therefore, T is the overall tari¤ that consumers pay to purchase the bundle. The …xed fee of the three-part tari¤ can be directly restated as:

Z( ) = T ( ) p B ( )x B ( ) r B ( )s B ( ):
Proposition 1 The marginal price of the service depends on the degree of optionality but it is not the case for the commodity price, these prices are given by:

p B ( ) = p F ( ) + '( ) r B ( ) = r F ( ) + '( )
Here p F ( ) and r F ( ) are respectively prices of good and related service in …rst-best situation. [START_REF] Wilson | Nonlinear Pricing[END_REF] The marginal price of the service is lower than the marginal price of the good because the service is optional. The monopoly less distorts the service price in relation to the good price and this e¤ect is due to the asymmetric information. As marginal price of the service is low, as well as its quantity, the information cost is captured by the …xed fee of the three part schedule. 15 Concerning good price in the bundle, it not depends on the optional character of the service.

In a more general setting, at the equilibrium, prices of commodity and service would verify (omitting arguments):

p B ( ) = u x (x B ; s B ; ) = c 0 (x B ) + '( )u x (x B ; s B ; ) r B ( ) = u s (x B ; s B ; ) = c 0 (s B ) + '( )u s (x B ; s B ; )
There are two e¤ects for marginal prices in bundle. First, marginal prices are higher than in complete information due to the information cost which is positive. Second, quantities are lower than in complete information situation. Indeed in this general setting, it is di¢cult to measure which e¤ect dominates. Proposition 1 clari…es these tradeo¤ in our speci…c case.

Separate sales and nonlinear tari¤

In this section, we consider the situation where two principals (the shop 1 is the energy retailer and the shop 2 is the service retailer) supply their contracts to a same type of agent under a nonlinear pricing de…ned by: = = t + . However, our analysis slightly di¤ers from Martimort (1992) as u(x; s; ) cannot be a symmetric function since energy and service don't ful…lled exactly the same intrinsic needs.

The agent maximizes his utility in relation to the rates of the good and the service bought and in relation to his report type ( ^ ) according his own type ( ). At equilibrium the agent report is truthfully, so the utility function for a consumer is given by:

U ( ) = max ^ x; ^ s u(x( ^ x ); s( ^ s ); ) t( ^ x ) ( ^ s ) when ( ^ x ; ^ s ) = ( ; )
11 For a positive level of attached service, we assume a condition on w i.e. w < w where w = ( +2 ) ( +2 ) 1+2

. [START_REF] Schmalensee | Commodity Bundling by Single-Product Monopolies[END_REF] See appendix B.1. [START_REF] Schmalensee | Gaussian Demand and Commodity Bundling[END_REF] Indeed,

dx B d = 2 ( '( )) 1 4 2 < ds B d = ( '( ))
1 4 2 : 14 These levels are detailed in the Appendix A. 15 Here dZ d ( ) = s F ( ) > 0. The …xed fee of the three part schedule Z is increased in for .

Here t is the price for the energy bought from the shop 1 and is the price for the level of service purchased from the shop 2. Ex-post, the reports for energy and service are truthfully: ( ^ x = ) and ( ^ s = ). The …rst order of incentive compatibility constraints are given by:

( ^ = ) u x (x( ); s( ); ) _ x( ) _ t( ) = 0 u s (x( ); s( ); ) _ s( ) _ ( ) = 0 (1)
The necessary optimal condition can allow us to write (with the envelope theorem):

_ U ( ) = u(x( ); s( ); )
This rent is increasing and must keep a positive value for all values of (under the IR constraint). Consequently, we minimize the consumer rent with low bound constraint in : U ( ) = 0:

The principals, the energy retailer (shop 1) and the service retailer (shop 2), must implemented a mechanism design which incite the consumer to tell the true and to reveal their preferences. Ex-post, the agent would be well advised to report his true type than he lies about their preferences ( ^ x ; ^ s ) = ( ; ). The second order incentive conditions are given by the sign of the hessian of U and are given in appendix C.1.

Sales of good

In this case we consider only the program of the shop 1. Ex-post, the two-part tari¤ for the energy supply is:

t = A + px , A = t px
where A is the …xed fee of the tari¤ and t is the overall price that consumers pay from energy purchase. The cost, with an independent pricing, is proportionate to the quantity of good bought and with constant costs (H4) is given by: C(x; 0) = c(x). The shop 1 maximizes its expected pro…ts (EP) under the …rst and the second order incentive constraints (1) with t( ) = t(x( )). In appendix C.1 using the analysis of Martimort (1992) we solve the shop 1 pro…t maximization problem. In general, if we note '( ) = 1 F ( ) f ( ) , the equilibrium price for the good is given by:

p IP ( ) = c 0 (x IP ( )) + '( ) u x (x IP ( ); s IP ( ); ) + I IP ( ) (2)
Comparing this price 16 with the energy price in the bundling case, the additional term I IP ( ) is positive thus it increases the marginal price of the separate good. Nevertheless, this term I IP ( ) is not in ‡uenced by the optionality of the service. In the separate sales, we can conclude that the independent good price is always higher than as component of the bundle.

Sales of service

For the sales of service, we consider only the program of the shop 2. As for the shop 1 in the previous section, we give the ex-post two-part tari¤ for the service supplied:

= B + rs , B = rs 16 With x( ) = x F ( ) and I IP ( ) = u s (x( ); s( ); )u xs (x( ); s( ); ) _ s( ) u xs (x( ); s( ); ) _ x( ) + u s (x( ); s( ); ) :
where B is the …xed fee of the tari¤ and is the overall price that consumers pay for the service bought. With (H4), costs are proportionate to the level of service chosen and are given by: C(0; s) = c(s).

In appendix C.2, we solve the shop 2 pro…t maximization problem. the equilibrium price of service is given by:

r IP ( ) = c 0 (s IP ( )) + '( ) u s (x IP ( ); s IP ( ); ) + J IP ( ) (3)
Comparing with the marginal price of service 17 as part of bundle, the term J IP ( ) increases the separate marginal price since it is positive. However, as for the good sales, this additional term J IP ( ) is not in ‡uenced by the optionality of the service. In the following section, we solve the system.

Independent pricing schedule

The contract is optimal if there is a couple fx ( ); s ( )g which satisfy the equation system ( 2) and ( 3), so called Hamilton-Jacobi system; -omitting arguments:

c 0 (x) + u x (x; s; ) '( ) u x (x; s; ) + I IP ( ) = 0 c 0 (s) + u s (x; s; ) '( ) u s (x; s; ) + J IP ( ) = 0 (4)
with x( ) = x F ( ) and s( ) = s F ( ). In general, one can conclude that the separate marginal prices of the good and the service are always higher than as a part of bundle.

As Martimort (1992), one …nd that it is costly for consumers to buy the good and the service separately due to the implicit competition between the two shops. In the general case, bundling is an optimal strategy for both principals and consumers.

In our speci…c setting (with constant costs), the Hamilton-Jacobi system (4) can be restated, as -omitting arguments-:

( (2 ) w x + 2 s ( ) 2 _ s 2 _ x+ = 0 (2 ) w s + 2 x ( ) 2 _ x 2 _ s+1 = 0
Using the method of Olsen-Osmundsen (2001) and under speci…cation (H7), there is an optimal linear solution f x ; s g which allow us to rewrite the program as:

x IP ( ) = x F ( ) + ( ) x (5) s IP ( ) = s F ( ) + ( ) s
Where x F ( ) and s F ( ) are respectively quantities of good and service. 18 At the equilibrium, values of x and s given in the appendix C.3 are negative, one can directly conclude that these quantities are sub-optimal than in the …rst-best situation. The marginal prices are given by:

p IP ( ) = p F ( ) + ( )(1 + Y ) r IP ( ) = r F ( ) + ( )( + Y )
17 With s( ) = s F ( ) and J IP ( ) = u x (x( ); s( ); )u xs (x( ); s( ); ) _ x( ) u xs (x( ); s( ); ) _ s( ) + u x (x( ); s( ); )) . 18 As in the bundling situation, we give the condition on w for which the level of attached service is positive: w =

(2 (4 )+ p 1+32 2 (1+2 ))+4 ( +2 ) 4 (1+2 )
> w.

with Y = y X and y = 16 2 + p 1 + 32 2 1 (here Y 1). These prices are both increasing with the optional character of the service. 19 Nevertheless, the parameter a¤ects more the information cost for the good than for the service. 20

Bundling strategy versus independent pricing

In this section, we compare marginal prices when shops coordinate on a bundling strategy and the case where there is no cooperation with an independent pricing strategy. At the equilibrium, the quantities 21 , under our speci…c assumptions, can be rewritten as:

x F ( ) > x B ( ) > x IP ( ) s F ( ) > s B ( ) > s IP ( )
In the bundling case, the optimal quantities of good and service are higher than in the separate sales. In the explicit case, we can rewrite marginal prices of the energy and the related service as:

p IP ( ) = p B ( ) + ( ) Y (6) r IP ( ) = r B ( ) + ( )Y
It implies that the marginal prices are higher when shops provide two di¤erent contracts than when they provide a bundle (since Y > 0). In the section 3, we have shown that the marginal price of the service in the bundle depends on the degree of its optionality. With an independent pricing strategy, the degree of the optional service has also an impact on the marginal price of good. The parameter has an in ‡uence on the information cost for the separate good.

When service is purely optional ( = 0), the marginal price of good is identical in an independent pricing strategy or in a bundling one, that is for all 2 , but it is not necessarily the case for the service rate. Indeed, the marginal price of service is always higher independently than in a bundle. As the service is not essential to satisfy the fundamental need, the monopoly …xes a lower price for the service in a bundle than independently and raises the price of good as a component of the bundle.

Proposition 2 One can conclude that:

p F ( ) < p B ( ) p IP ( ) r F ( ) r B ( ) < r IP ( )
The price gap between the energy price in a bundle and sold separately depends on the degree of the optional service.

The previous proposition states that the optional characteristic has an impact on the marginal prices of the independent good and the service bound. The discount when consumers purchase the good in the bundle rather than separately is informational (see 19 Indeed,

d(1+ y X ) d = y X and d( + y X ) d = 1: 20 Indeed, dp IP d = y'( ) dr IP d = '( ).
21 See appendix D.1. equation 6). Whereas, this informational rebate appears for the service when it is close to the …rst-best situation.

These results come from the asymmetric utility function (assumptions H1 and H2), since energy and service don't ful…lled exactly the same intrinsic needs. The independent pricing strategy implies ine¢ciencies due to the implicit competition between the two shops, the principals should coordinate to provide a bundle.

The global analysis is done in Martimort (1992, 1996) in the case where u( ) has strong symmetric properties. If it was the case here then any optimal independent nonlinear pricing scheme leads a continuum of allocations (x IP ( ); s IP ( )) such that22 :

s 1 ( ) = x 1 ( ) x IP ( ) = s IP ( ) x B ( ) = s B ( ) (7)
In this symmetric setting, (which would correspond to our limit case where = 1), Martimort (1992) has shown that independent pricing introduces ine¢ciencies due to the implicit competition between shop managers (principals) and of course bundling is dominant ex ante, that is E( B ) E( I ), since by de…nition bundling maximizes the total expected pro…t. In our model (where 1), one cannot directly conclude that the inequality (7) holds because of asymmetric properties (H1)-(H3) we consider. The optimal quantities of the good and the service when shops follow an independent pricing strategy are smaller than the cooperative situation that is:

s 1 ( ) = x 1 ( ) s IP ( ) x IP ( ) < x B ( ) = s B ( )
Concerning marginal prices, in the speci…c case, one can conclude that:

p F ( ) = r F ( ) < r B ( ) p B ( ) 7 r IP ( ) p IP ( )
In the benchmark, the marginal prices of good and service are equal, as well in the bundle when the service is considered as mandatory ( = 1). Moreover, prices are both lower in the benchmark than as components of the bundle. The information cost tends to increase rates in incomplete information case.

However, it exists a threshold of complementarity degree between good and service:

T ( ) = 1 2 1 2 2
1 , where the marginal price of the good in bundle can be higher than the level of separate service ( < T ( )). Finally, marginal price of the separate service is always lower than price of the separate good unless they are perfectly complementarity (if = 1 2 ). In order not to limit our analysis at constant costs and to focus the energy retailers concerns, the following section introduces the increasing costs assumption.

Increasing marginal costs

Until now, we have considered the case where the di¤erent shops are only retailers. In this section, we remove this assumption to consider the situation where the two shops produce good (energy) and service sold to consumers. In this way, contrary to the constant costs with market prices, principals have increasing costs, given by (H5) in section 2, introducing decreasing scale returns.

Bundling strategy

In the bundling strategy, the shops 1 and 2 can provide a bundle. At the equilibrium, quantities of good and service are given by:

x B ( ) = x F ( ) ( ) 1 + 2(1 2 ) and s B ( ) = s F ( ) ( ) + 2(1 2 ) (8)
where x F ( ) and s F ( ) are still quantities of good and attached service in the …rst-best situation (given in Appendix A). These quantities are increasing 23 with the type of consumers. Moreover, the impact of optionality degree is more important for the service bought than for the good, 24 and this as much as the degree of complementarity is high.

When the two shops follow a pure bundling strategy, they propose a unique nonlinear tari¤ for the bundle. Marginal prices of the good and the service, are given by:

p B ( ) = p F ( ) + ( ) (1 2 2 ) 2(1 2 ) r B ( ) = r F ( ) + ( ) ( 2 2
) 2( 12 )

As 2 [0; 1 2 ] and 2 [0; 1], the energy price is always higher in the bundle than in the complete information situation due to the information cost which is positive, even if the service is purely optional.

As well as in the bundling situation with constant costs, the e¤ect of the optional service increases marginal prices but this impact is more pronounced for the level of service and when the degree of complementarity is high. 25 The information cost is low for the marginal price of service on account of the optional character. 26 Therefore, the service is less costly than the good in the bundle, 27 this is due to the fact that consumers enjoy a higher utility by consuming energy relatively to the service.

Independent pricing strategy

Now the system (4) can be rewritten as:

( 2 2x + 2 s ( ) 2 _ s 2 _ x+ = 0 (2 ) 2s + 2 x ( ) 2 _ x 2 _ s+1 = 0
The method used is the one of Olsen-Osmundsen (2001), as in section 4. With an uniform law, it exists a linear optimal solution f x ; s g such that:

x IP ( ) = x F ( ) + ( ) x s IP ( ) = s F ( ) + ( ) s 23 See appendix B.2. 24 Indeed: dx B d = ( '( )) 2(1 2 ) < ds B d = ( '( ))
2( 12 ) . 25 Indeed:

dp B d = ( '( )) 2(1 2 ) < dr B d = '( )(1 2 2 )
2( 12 )

. 26 Here (1

2 2 ) ( 2 2 ) = (2 + 1)( 1)( 1) > 0 27 Indeed: (p B r B ) = ( +1)( 2 '( ) '( ) ) 2( +1) > 0:
At the equilibrium, the couple f x ; s g is given in appendix C. 4

and values of parameters

x and s are both negative. The marginal prices are given by:

p IP ( ) = p F ( ) + 1 D ( )( m + n) r IP ( ) = r F ( ) + 1 D ( )(m + n)
we show that 28 m > 0 n. These prices depend on the degree of complementarity as well as the degree of the optional service. Since m > n, the parameter of optionality a¤ects the information cost and more on marginal price of service than on the marginal price of good. The introduction of increasing costs has an impact on the good price contrary to section 4. Here the monopoly less distorts the good price than the service price.

Bundling versus independent pricing strategies

The …gure 1 compares bundling and independent pricing strategies with increasing costs.

Results di¤er from those of section 5 under the constant costs assumption.

Figure 1: Comparison between prices of good and service independently and in a bundle

There are two areas in the …gure 3. In the …rst area, below to the full line curve (where < ^ x ( )), the separate prices for the good and for the service are both higher than those as components of the bundle. This situation is the standard case where a consumer yields a discount for the bundle rather than the separates sales. This analysis holds when the good and the service are independent, where = 0. Usually the conventional wisdom (as 28 with m = 8 [START_REF] Martimort | Multi-principaux avec Anti-selection[END_REF] well as economic literature), tells us that it is less costly to produce a good and a service in a bundle rather than independently (saving costs for the packaging, for instance), so we expect that prices follow this cost saving. Here this is the case, but in the second case below, this argument doesn't hold anymore.

In the second area, above to the full line curve (where > ^ x ( )), the price of the separate good is lower than as component of the bundle and at the same time, the service price is less costly in the bundle compared to its separate price. Indeed:

p B ? p IP if 7 ^ x ( ) r B < r IP , 8 2 [0; 1] and 2 [0; 1 2 ]
and here ^ x ( ) = 1 4 ( p 8 2 + 1 1). It is interesting to see how, with increasing costs, the price of the independent good can be lower than the tied good. The introduction of increasing costs has a strong impact on the good price since results of section 5 are reversed. Nevertheless, there is no impact on the service price which is always lower in the bundle than sold separately.

When the good and the related service are perfectly complementary ( = 1 2 ), we are in the second area. The intuition suggests that the bundling strategy is more pro…table when the components of the bundle are complementary. It is straightforward for the monopoly to charge a high price for the good in the bundle. As the bundling strategy allows monopoly to practice price discrimination, consumers who have an intensive use of the good accept this high price. Consumers have more utility by consuming the good in relation to the service. The monopoly can charge a low price for the service as component of the bundle compared to the separates sales. Even if the price of the bundle is higher than the sum of the separates sales, the bundling strategy can be an e¤ective strategy. Some consumers consider that it is better to spend a large additional sum instead of making themselves the bundle. When the service is purely optional, = 0 (8 < 1

2 ), results are the same. 29

Comparative statics

Bundling strategy allows producers to restrict their unit production costs, with costs saving on packaging or storage for instance, or more generally, on the "selling costs". Simultaneously, when products are perfect complements, bundling literature advocates a high price for the good and the service sold in the bundle. In our analysis, the optionality of the service plays an important role and has an impact on the marginal price of the independent good. In this case, the good sold in the bundle is higher than sold independently.

With complementary goods, the analysis of our model is not as explicit as those of Martimort (1992) where utility function is symmetric. Martimort shows that the independent pricing strategy is ine¢cient for consumers, since it is costly to sign di¤erent contracts to di¤erent shops. Here, it is not explicitly the case as the price of the independent good can be lower than the price of the tied good. The optionality of the service as well as the increasing costs emphasize that independent pricing strategy can be e¤ective from consumers' point of view. The overall analysis is given by the …gure 2.

In this graphic s ( ) = f 2 [0; 1 2 ] j r IP = r F g. In order to understand the …gure 2 and table 2.1., we comment each areas.

p IP ( ) > p F ( ) r IP ( ) > r F ( ) p B ( ) > p F ( ) r B ( ) > r F ( ) p IP ( ) > p B ( ) r IP ( ) > r B ( ) Area 2 p IP ( ) > p F ( ) r IP ( ) > r F ( ) p B ( ) > p F ( ) r B ( ) < r F ( ) p IP ( ) < p B ( ) r IP ( ) > r B ( ) Area 3 p IP ( ) > p F ( ) r IP ( ) < r F ( ) p B ( ) > p F ( ) r B ( ) < r F ( ) p IP ( ) < p B ( ) r IP ( ) > r B ( )
In area 1, the standard case occurs: the information cost is taken into account for the monopoly, so that marginal prices are ranked with respect to the information cost.

In area 2, the information cost falls signi…cantly the marginal price of the service and it becomes smaller than its marginal cost. This e¤ect can be viewed as an "informational dumping e¤ect". 30 There is a trade-o¤ between the degree of complementarity and the degree of the optional service. The monopoly lowers its bundle price ("price e¤ect") but it makes up for quantities sold: the quantity e¤ect dominates the price e¤ect. However, the dumping e¤ect appears even when the degree of service is not totally optional to satisfy the essential need. Moreover as evoked in the previous section, bundling strategy allows monopoly to practice price discrimination, therefore consumers who have an intensive use of the good accept a higher price. This dumping e¤ect would possible because of using two-part tari¤, the monopoly can consequently rise the …xed part.

Analytically, the area 3 (where > s ( )) is the most interesting one since the dumping e¤ect appears for the service in both independent pricing and bundling strategies. As the service is optional, its price is lower than its marginal cost. This dumping e¤ect was underlined in the bundling case, nevertheless when the service is purely optional, the degree of complementarity must be high enough to enjoy this discount. 31 Moreover, this analysis holds when the components are perfect complements ( = [START_REF] Adams | Commodity Bundling and the Burden of Monopoly[END_REF] 2 ). The marginal price of service is lower than its marginal cost, moreover the quantities are sub-optimal 32 in relation to the complete information case. As the variable part of the tari¤ is minimized, the monopoly can compensate this loss through the …xed-fee of the tari¤: this e¤ect can be described as a "catching e¤ect".

Moreover, the marginal price of independent good is lower than its price in the bundle. The introduction of increasing costs has an impact on the good marginal price as this case not occurs with constant costs. This price discount for the independent good appears when the good and the service have a high degree of complementarity. With complement goods, the bundle is attractive for many consumers and the monopoly …xes a high marginal price for the good relatively to the independent pricing strategy.

One can not directly conclude that the overall analysis given by Martimort (equation ( 7) to section 5) holds because asymmetric properties (H1)-(H3) are considered. In our analysis framework, one can conclude that:

x F ( ) > x B ( ) ? x IP ( ) s F ( ) > s B ( ) > s IP ( )
and prices can be rewritten as follow:

p F ( ) < p IP ( ) ? p B ( ) r F ( ) ? r IP ( ) > r B ( )
As we compare bundling and independent pricing, considering marginal prices only, one can conclude that it is not always less costly for consumers to purchase each unit of good and service at two di¤erent shops rather than together in a bundle. To have a comparison of the entire cost, we should compare the consumer's rent in each con…guration (that is including the …x part of the tari¤).

Conclusion

This paper analyzes optimal nonlinear pricing when a …rm o¤ers a commodity and an attached service. We compare the bundling strategy and the independent pricing one. The pro…tability of bundling strategy depends on the degree of the optional service and 31 = p the degree of complementary between the good and the associated service. This paper deals with two cases concerning the cost function.

In a …rst time, we suppose that the shops sell their products at the market prices, they have constant marginal costs. In the general case, one can conclude that the separate marginal prices of the good and the service are always higher than as a part of bundle. As Martimort (1992), one …nd that it is costly for consumers to buy the good and the service separately due to the implicit competition between the two shops. In the general case, bundling is an optimal strategy for both principals and consumers.

With the speci…ed utility function, we can conclude that with bundling strategy, the optimal quantities are higher than those with an independent pricing strategy. If the service is purely optional i.e when = 0, marginal prices of the good sold independently and in a bundle are the same. However if > 0, this is the standard case where the good in the bundle is lower than separately. For the attached service, whatever its degree of optionality, the marginal price in bundle is always lower than sold independently. Comparing marginal prices, consumers prefer to buy the good and the attached service in a bundle as long as the degree of optionality is positive otherwise consumers are indi¤erent.

In a second time, we use increasing costs to describe the situation where both shops produce the good and the related service. We show that there is a trade-o¤ between the degree of complementarity and the optionality of the service. The overall analysis (see Figure 2) shows that there are three cases through di¤erent areas. Limiting the analysis to the polar cases ( = f0; 1 2 g; 8 ), the results are the following. With the speci…ed utility function, when the good and the service are independent ( = 0), if the monopoly commits to a bundling strategy then marginal prices are both lower then the separates sales. The monopoly prefers to leave a signi…cant unit margin and compensates to the quantities sold. Comparing marginal prices, consumers prefer to buy the good and the attached service in a bundle.

However, when the degree of the complementarity becomes su¢ciently high (when > f ^ x ( ); s ( )g), the marginal price of the separate good is lower than the good price in the bundle. The intuition suggests that the pro…tability of a bundling strategy depends on the degree of complementarity between the good and the attached service. It is straightforward for the monopoly to charge a higher price for good in bundle.

In the polar case when good and service are perfectly complementary, contrary to Martimort (1992) we can not conclude that it is costly for consumers to sign two contracts from di¤erent shops than to buy the bundle. With the asymmetric function, the marginal price of good can be lower independently than in the bundle whereas the service is more attractive when there is a cooperation between principals. Because of asymmetric properties in the utility function, pro…tability results of bundling strategy depends, on one hand, on the degree of complementarity between commodity and related service and on the other hand, on the degree of the optional service.

B.1 Constant costs

The …rst and the second order incentive compatibility constraints implemented with T ( ) = T (x( ); s( )) are given by: u x (x; s; ) _

x( ) + u s (x; s; ) _ s( ) _ T ( ) = 0 u x (x; s; ) _

x( ) + u s (x; s; ) _ s( ) 0

The variation of x B and s B in relation to with speci…ed utility function is given by:

_ x B = 1 + 2 (1 4 2 ) 0 and _ s B = 2 + (1 4 2 ) 0 B.

Increasing costs

The variation of x B and s B in relation to with the speci…ed utility function is given by: 12 ) 0 and _ s B = + 2( 12 ) 0

_ x B = 1 + 2(
C Nonlinear tari¤ with the independent pricing strategy

C.1 Supply of good

The shop 1 maximizes its expected pro…ts under the …rst and the second order incentive constraints with t( ) = t(x( )).The expected pro…ts of shop 1 can be rewritten as:

max U;x; ^ s E( ) = Z h u(x( ); s( ^ s ( )); ) c(x( )) U ( ) ( ^ s ( ))) i f ( )d _ U ( ) = u (x( ); s( ^ s ( )); ) ( ) U ( ) = 0 u s (x( ); s( ^ s ( )); ) _ s( ^ s ( )) _ ( ^ s ( )) = 0 ( ) (IC2)
where f ( ) is the density function of consumers preference. At the equilibrium, we have ^ s ( ) = .

From the analysis of Martimort (1992), to solve the program, we write the Hamiltonian under ( ) and ( ) constraints and :

H(U; x; ^ s ) = f ( )[ c(x) U ( ^ s ) + u(x; s( ^ s ); )] + ( )u (x; s( ^ s ); ) + ( )[u s (x; s( ^ s ); ) _ s( ^ s ) ( ^ s )]
The dynamic system of Hamilton-Jacobi (SHJ) in relation to the state variable has a form:

@H @U = _ ( ) , _ ( ) = f ( ) (1)
Therefore, according to the edge condition _ ( ) = f ( ) and according to the transversal condition we can restate: U ( ) = 0 and ( ) = 0 (10) After have rewritten the edge and transversal conditions we restate the SHJ with respect to the control variables:

@H @x = f ( )[ c 0 (x( )) + u x (x( ); s( ^ s ( )); )] + ( ) h u x (x( ); s( ^ s ( )); ) i (11)
+ ( )

h u xs (x( ); s( ^ s ( )); ) _ s( ^ s ( )) i = 0
If we suppose that the principals' contracts supply are truthfully ex-post we assume ^ s = , we can restate:

@H @ ^ s j ^ s= = f ( )[( _ ( ) + u s (x( ); s( ); ) _ s( )) (12)
+ ( )(u s (x( ); s( ); ) _ s( )) + ( )(u ss (x( ); s( ); )( _ s( )) 2 + u s (x( ); s( ); )• s( ) • ( ))] = 0
As the principals o¤er truthtelling tari¤ for agents to reveal their true type, the preference for the service is restated ^ s = , the derivative second order condition (IC2) with respect to is given by:

• s( )u s (x( ); s( ); ) + u ss (x( ); s( ))( _ s( )) 2 • ( ) (13)
+u s (x( ); s( ); ) _ s( ) + u xs (x( ); s( ); ) _ x( ) _ s( ) = 0
If the solution is separating then _ x( ) 6 = 0; _ s( ) 6 = 0, the second order incentive compatibility conditions are satis…ed and the relation (1) is given by ( ) = F ( ) + k where k is a constant. As F ( ) = 1; thus:

( ) = F ( ) + k = 0 ) k = 1 ( ) = F ( ) 1 = (1 F ( )) (14)
From the equations ( 13) and ( 14) into (12), we can restate the SHJ constraints:

(1 F ( ))u s (x( ); s( ); ) _ s( ) + ( ) [ u s (x( ); s( ); ) _ s( ) u xs (x( ); s( ); ) _ x( ) _ s( )] = 0 ( ) = (1 F ( )) u s (x( ); s( ); ) u xs (x( ); s( ); ) _ x( ) + u s (x( ); s( ); ) (15)
Under the analysis of Martimort (1992), by replacing the equation (15) into the equation (11), the SHJ can be restated as following:

f ( )[(u x (x( ); s( ); ) c 0 (x( ))) + ( (1 F ( ))u x (x( ); s( ); )) (16)
+( (1 F ( )) u s (x( ); s( ); )u xs (x( ); s( ); ) _ s( ) u xs (x( ); s( ); ) _

x( ) + u s (x( ); s( ); ) )] = 0

It is possible to assume '( ) = 1 F ( ) f ( ) the equation ( 16) can be rewritten as: With some simpli…cations: c 0 (x( )) + u x (x( ); s( ); ) '( )u x (x( ); s( ); )

'( ) u s (x( ); s( ); )u xs (x( ); s( ); ) _ s( ) u xs (x( ); s( ); ) _

x( ) + u s (x( ); s( ); ) = 0 with x( ) = x F ( ): With the speci…ed utility (H6), the SHJ can be rewritten as -omitting arguments-:

2 2x + 2 s ( ) 2 _ s 2 _ x + = 0 (18)
C.2 Supply of service

The shop 2 maximizes its expected pro…ts with ( ) = (s( )). Here, the tari¤ depends on the quantity of service chosen, thus shop 2 maximizes its expected pro…ts under the …rst and the second order incentive constraints:

max U;s; ^ x E( ) = Z h u(x( ^ x ( )); s( ); ) c(s( )) U ( ) t( ^ x ( )) i f ( )d _ U ( ) = u(x( ^ x ( )); s( ); ) ( ) U ( ) = 0 u x (x( ^ x ( )); s( ); ) _ x( ^ x ( )) _ t( ^ x ( )) = 0 ( ) (IC2)
where f ( ) is the density function of consumers preferences. At the equilibrium, we have ^ x ( ) = . To solve the program, we write the Hamiltonian under ( ) and ( ) constraints and under the analysis of Martimort (1992):

H(U; s; ^ x ) = f ( )[ c(s) U t( ^ x ) + u(x( ^ x ); s; )] + ( )u (x( ^ x ); s; ) + ( )[u x (x( ^ x ); s; ) _ x( ^ x ) t( ^ x )]
Here relations (1), (10) still holds and the optimal path of service is such that:

@H @s = f ( )[ c 0 (s( )) + u s (x( ^ x ( )); s( ); )] + ( ) h u s (x( ^ x ( )); s( ); ) i (19) + ( ) h u xs (x( ^ x ( )); s( ); ) _ x( ^ x ( )) i = 0
If we suppose that the principals' contracts supply are truthfully ex-post we assume ^ x = ; we can restate:

@H @ ^ x j ^ x= = f ( )[( _ t( ) + u x (x( ); s( ); ) _ x( )) (20)
+ ( )(u x (x( ); s( ); ) _ x( )) + ( )(u xx (x( ); s( ); )( _ x( )) 2 + u x (x( ); s( ); )• x( ) • t( ))] = 0
As the principals o¤er truthtelling tari¤ for agents to reveal their true type, the preference for the service is restated ^ x = ; the derivative second order condition (IC2) with respect to is given by:

• x( )u x (x( ); s( ); ) + u xx (x( ); s( ); )( _ x( )) 2 • t( ) (21) +u x (x( ); s( ); ) _ x( ) + u xs (x( ); s( ); ) _ x( ) _ s( ) = 0
If the solution is separating then _ s( ) 6 = 0; _ x( ) 6 = 0, the second order incentive compatibility conditions are satis…ed and ( 14) holds again. From equations ( 14) and ( 21) into (20), we can restate the SHJ constraints:

(1 F ( ))u x (x( ); s( ); ) _ x( ) + ( ) [ u x (x( ); s( ); ) _ x( ) u xs (x( ); s( ); ) _ x( ) _ s( )] = 0 ( ) = (1 F ( )) u x (x( ); s( ); ) u xs (x( ); s( ); ) _ s( ) + u x (x( ); s( ); ) (22)
By replacing equation ( 22) into (19), the SHJ can be restated as following: We do not attempt to investigate the global analysis of (25), but we …rst try to de…ne the solution when the service is purely optional that is if (u s 0). In this situation, (25) becomes:

f ( )[(u s (x( ); s( ); ) c 0 (s( ))) + ( (1 F ( ))u s (x( ); s( ); )) (23)
8 < : c 0 (x) + u x (x; s; ) '( )u x (x; s; ) = 0 c 0 (s) + u s (x; s; ) '( ) u x (x; s; ) u xs (x; s; ) _ x u xs (x; s; ) _ s + u x (x; s; ) we see from the …rst equation in (4) that x = x B (s) so we know the "reaction" function x B (s) is increasing 33 that is x B0 (s) > 0 so (4) writes knows: 8 < :

x = x B (s) c 0 (s) + u s (x; s; ) '( ) u x (x; s; ) u xs (x; s; ) x b0 (s) _ s u xs (x; s; ) _ s + u x (x; s; ) = 0

If second order incentive compatibility conditions are satis…ed, such that _ x ( ) ; _ s ( ) > 0, it must be true that u s (x; s; ) = c 0 (s) + '( ) u x (x;s; ) uxs(x;s; ) x b0 (s) _ s uxs(x;s; ) _ s+u x (x;s; ) > c 0 (s) hence 34 s IP ( ) s B ( ) and x IP ( ) = x b (s IP ( )) s B ( ) = x b (s B ( )) since x b0 (s) > 0).

The quantities of good and service are given by the equation ( 5) in section 4.3. At the equilibrium, the couple f x ; s g is given by: x = q 1 + 32 The comparison between marginal prices with an independent pricing strategy and in the …st-best situation, under a uniform law and with the speci…c utility funtion, is given by: , indeed x b ( ) = x b (s B ( )). Notice that if the service is purely optional (u s = u xs 0) then x b0 (s) > 0 since u xx (x; s; ) > 0 and cost are convex. 34 We have not prove that this allocation is not unique (as suspected from the analysis of Martimort, 1992) and incentive compatible. However, we admit this is the case.

p IP ( ) = p F ( ) + 1 X ( ) " 8

D Comparison between IP and bundling strategies D.1 Constant costs

The quantities comparison between the situation where the monopoly produces a bundle and when he/she buys its goods independently is given by:

x IP ( ) = x B ( ) (1 4 2 )X ( + 2 ) q 1 + 32 2 1 + 16 2 s IP ( ) = s B ( ) (1 4 2 )X (1 + 2 )

q 1 + 32 2 1
One can directly conclude that x IP ( ) < x B ( ) < x F ( ) and s IP ( ) < s B ( ) < s F ( ). With a bundling strategy, optimal quantities of good and service are higher than with an independent pricing strategy. However, these both quantities are suboptimal in relation to the …rst-best situation.

D.2 Increasing costs

When the service is purely optional, = 0 thus 8 < 1 2 , the marginal price of good is always smaller in bundle than separatly. With a uniform law on [1; 2] we can rewrite p IP = p B + 1 D ( 4 2 p 1 + 8 2 + 1). The marginal price of service depends on its optional character in order to satisfy the primary need but also to the degree of complementarity between the commodity and the contingent service. However when the service is purely optional, its marginal price is always higher than under the package form. With a uniform law on [1; 2], we can rewrite as follows r IP = r B + 1 D h

(2 2 1)(4 2 + p 1 + 8 2 1) i :

Figure 2 :

 2 Figure 2: Comparison between prices with independent pricing, bundling strategies and the …rst-best situation

7 p 17 4 0

 74 :424. 32 See appendix C.3.

+( ( 1 FC. 3

 13 ( )) u x (x( ); s( ); )(u xs (x( ); s( ); ) _x( ) u xs (x( ); s( ); ) _ s( ) + u x (x( ); s( ); ) )] = 0 and equation (23) rewrites:c 0 (s( )) + u s (x( ); s( ); ) '( )u s (x( ); s( ); )(24)'( )u x (x( ); s( ); )u xs (x( ); s( ); ) _ x( ) u xs (x( ); s( ); ) _ s( ) + u x (x( ); s( ); )) = 0with s( ) = s F ( ). Explicitely (24) writes: The problem solving systemWe can rewritte the equation (4) as the di¤erential equations system following:8 > > < > > : _ x = u s (x;s; ) uxs(x;s; ) [u s (x; s; ) '( )u s (x; s; ) c 0 (s)] [u x (x; s; ) c 0 (x)] (x; s; ) _ s = u x (x;s; ) uxs(x;s; ) [u x (x; s; ) '( )u x (x; s; ) c 0 (x)] [u s (x; s; ) c 0 (s)] (x; s; )(25)where(x; s; ) = [u s (x; s; ) c 0 (s)] [u x (x; s; ) c 0 (x) '( )u x (x; s; )][u x (x; s; ) c 0 (x)] '( )u s (x; s; ) with (x F ( ); s F ( ); ) = 0.

Table 2 .

 2 1: possible outcomes in relation to di¤erent areas

  2 (1 + 4 2 ) + 4 2 ( p 1 + 32 2 + 7) + 4 (8 2 + 1)At the equilibrium, the couple f x ; s g is given by: + 1) . For the values of parameters, x and s are negative.

	s =	1 4 2 X ) + 2 (4 + ) 1 4 1 4 2 1 + 32 2 (1 + 2 p	(26)
	where X = 2 3 +	p 1 + 32 2 .
	C.4 Increasing costs
	x = s =	2( ( p p 8 2 + 1 1) 8 2 + 1 1) + ( 2 2 + p 8 2 + 1 7) 2 + (1 + D p 8 2 + 1 + 1 4 1 2	p 8 2 + 1)	4 3
	where D = 2 1	2 (3	p 8 2

Pure bundling refers to the practice of selling two or more goods and/or services together in a bundle at a unique price.

The service for example may be technical maintenance or energy consultancy which enhances the gross utility for the good.

SeeReisinger (2006) and Economides (1993) in the duopoly case.

For example, for the gas supply, the industrial gas retailler uses a two-part tari¤. It comprises a uniform price for each unit of gas purchased plus a …xed fee payable if any positive amount is purchased.

Since consumers' willingness to pay is a private information, a problem of asymmetric information appears between …rm and consumers. These relations are described in principal-agent models.

More precisely this implies that u ss < 0; u xx < 0 and u ss u xx (u xx ) 2 > 0. Moreover we assume u xx (x; s; ) > 0.

The density function f ( ) is supposed to be everywhere non-negative.

This is an acceptable assumption as Bagnoli and Bergstrom (2005) have shown. As the type increases, the relative weight of types above decreases. The …rm is more concerned about the costs left below .

Solution s 1 ( ) = x 1 ( ) are such that: c 0 (x) + u x (x; x; ) 2'( )u x (x; x; ) = 0

See appendix D.2.

Dumping is often de…ned as the ability of a monopoly to …x a lower price in order to force the potential entrants to revise downwards their estimations of pro…tability on this market. Ex-ante, this allows the monopoly to deter potential competitors to enter the market.

Appendix A First Best

Without any further calculs, …rst-best levels for prices and quantities are given. More details could be found in Podesta and Poudou (2008).

A.1 Constant costs

Explicitely, with (H4), quantities of the good and the attached service are given by:

For a positive level of attached service, we assume a condition on w, i.e. w < w 0 where

. The derivative of the utility in relation to x and s gives the energy and relative service prices:

The equilibrium prices in the …rst-best situation are equal to marginal cost.

A.2 Increasing costs

Under increasing costs assumption (H5), quantities of the good and the attached service are given by x F ( ) = 1+ 2(1 2 ) and s F ( ) = + 2(1 2 ) . Energy and service marginal prices are given by p F ( ) = x F ( ) and r F ( ) = s F ( ).

B Nonlinear tari¤ and bundling

With bubdling, the …rm's objective (EP) can be implemented by an integration by parts:

which can allow us to write that _ U ( ) = u(x( ); s( ); ), and at the equilibrium the couple fx ( ); s ( )g satis…es the system:

Since '( )u x 0 and '( )u s 0, one can conclude that the quantities are suboptimal compared to the complete information situation (given above) for which '( ) would equated to zero for all .


T Giraud

A Paljic

L Leroy

"It" + "I": Virtual Embodiments as Hybrid Experiences

Keywords: Descriptors (according to ACM CCS), I3.7 [Computer Graphics], Three-Dimensional Graphics and Realism -Virtual reality / Animation

A dichotomy exists in the way virtual embodiments are currently studied: embodied entities are considered by conversational approaches as other selves whereas avatar approaches study them as users' hosts. Virtual reality applications such as in our case study often propose a different, in between embodiment experience. In the context of a virtual house for sale visit, this paper aims at examining the user's self-reported embodiment perception resulting from such a hybrid experience. To induce variability in this embodiment experience, we manipulated avatar representations (high versus low anthropomorphism) and frame of reference (egocentric versus exocentric). Results show the importance of the entity humanness to foster both experiences. When controlled by humanness, having a conversational experience appears uncorrelated to an avatar experience. This highlights the need to study these hybrid experiences as a combination of both approaches.

Introduction

To embody is the act of giving a body to an agent, a person or a system. Use of embodiments in interactive applications has a long history in Virtual Reality. But as one tries to trace it, two distinct approaches stand out despite early unified frameworks [START_REF] Thalmann | A New Generation of Synthetic Actors: the Interactive Perceptive Actors[END_REF]. Works on autonomous virtual agents represent a first line of research [START_REF] Catrambone R | ECA as User Interface Paradigm[END_REF]. In this perspective, embodiment aims to provide a natural interface with agent functionalities by using human-human interaction routines [START_REF] Catrambone R | ECA as User Interface Paradigm[END_REF]. The other line of research deals with user embodiment, for example in collaborative interfaces [BBF*95]. These incarnations of user presence in virtual environments are called Avatars. One line of research questions how users can communicate through non verbal behaviours [GCP*99]. This academic dichotomy seems pertinent as it appears transposable from everyday life: we are used to own our body, and interact with others' bodies. But with embodied entities, combinations of those experiences are possible. As an example, many computer games propose gameplaies combining autonomous and controlled aspects. The focus of this paper is on exploring the user's embodiment experience resulting from the interaction with a hybrid embodied entity. This bodily entity is a parallel combination of user controlled aspects and autonomous behaviours. An experiment is presented to evaluate the hybrid construction of the embodiment perception. To induce variability in this embodiment perception, we manipulated avatar representations (high versus low anthropomorphism) and frame of reference (egocentric versus exocentric).

Background on the two paradigms

Conversational approach

Embodied entities used as autonomous partners are called Embodied Conversational Agents (ECAs). Today, numerous approaches exist to design and evaluate ECAs [START_REF] Ruttkay | From Brows to Trust: Evaluating Embodied Conversational Agents (Human-Computer Interaction Series)[END_REF]. In this paper, we focus on the user perception of the agent embodiment. For conversational agent, this experience is firstly characterized by its naturalness. As they are designed to benefit from human to human interaction routines, naturalness is often linked to anthropomorphism. For example, users' social responses increase with humanlikeness computer representations [START_REF] Gong | How social is social responses to computers? The function of the degree of anthropomorphism in computer representations[END_REF]. Biological movements are also crucial for anthropomorphism [START_REF] Dalibard | Anthropomorphism of artificial agents: a comparative survey of expressive design and motion of virtual Characters and Social Robots[END_REF]. To complete the ECA believability, it should respond in a social coherent manner in the task context. These are the form and behavioural aspects of ECA realism [START_REF] Bailenson | The Effect of Behavioral Realism Form Realism of Real-Time Avatar Faces on Verbal Disclosure, Nonverbal Disclosure, Emotion Recognition, and Copresence in Dyadic Interaction[END_REF]. A social presence feeling results from this overall conversational naturalness. Biocca and colleagues defined it as "the feeling of being with another" [START_REF] Biocca | Toward a More Robust Theory and Measure of Social Presence: Review and Suggested Criteria. Presence: Teleoper[END_REF]. It induces the automatic generation of models of the intentionality of others. In a larger perspective, Vugt and colleagues propose a model which describes users' engagement as a result of aesthetics, ethics, realism, similarity, relevance and valence [VKH*07].

Avatar approach

Avatars are digital body representations controlled by the user. They personify presence, location, identity, activity, etc [BBF*95]. This experience of controlling an embodied entity has been studied in different ways. For example the Proteus effect is described as a user behaviour adaptation to conform to the avatar persona [START_REF] Bailenson | The Difference Between Being and Seeing: The Relative Contribution of Self-Perception and Priming to Behavioral Changes via Digital Self-Representation[END_REF]. Users will express more dominance if acting through a tall avatar. Such influence is mediated by immersion variables such as presence [START_REF] Fox | Virtual experiences, physical behaviors: The effect of presence on imitation of an eating avatar[END_REF]. Also, Mohler and colleagues observe that distance judgments are more accurate when acting through an avatar body [START_REF] Mohler | The Effect of Viewing a Self-Avatar on Distance Judgments in an HMD-Based Virtual Environment[END_REF]. Displacements synchrony, body movements' synchrony and body representation increased accuracy independently. Extreme embodiment experience can lead to body ownership experience. Initially done with a rubber hand, the experience of virtual body ownership has been replicated on whole body ownership [START_REF] Petkova | If I Were You: Perceptual Illusion of Body Swapping[END_REF]. This was done by altering the normal association between touch and its visual correlate. A recent work by Kalckert and Ehrsson make clear the distinction between ownership and agency [START_REF] Kalckert | Moving a Rubber Hand that Feels Like Your Own: A Dissociation of Ownership and Agency[END_REF]. Overall, body representation, sensory and action synchrony as well as person perspective contribute to different components of the embodiment experience.

Hybrid works

These two paradigms are well defined and their research field very active. Works on embodiment take one or the other perspective, but rarely cross them. Few studies deal with this question of hybridization. Gerhard and colleagues propose a hybrid avatar/agent model to enable a continuous embodiment presence [START_REF] Moore | Embodiment and copresence in collaborative interfaces[END_REF]. This model is sequential: autonomous agents are used when users are away. As a sequential approach, the hybrid embodiment experience is not questioned. To our knowledge, no work on the study of this possible parallel combination of experiences exists.

Experiment

This study aims at exploring the embodiment experience from both paradigms in a use case which includes some hybrid embodiment aspects. Those aspects of conversational and avatar experiences are present at the same time inducing a possible mixed experience.

Application and materials

The experiment takes place in the context of a specific use case proposed by the industrial Saint Gobain Recherche: the exploration of thermal comfort in a virtual house. A character is used to represent the comfort by its autonomous reactions to the environment. The user controls the character displacements. This combination of controlled and autonomous aspects makes this use case relevant for our research. The application is composed of several components:

• A virtual house with a living room meshed by a set of temperature data. Temperatures result from a simulation to ensure their realism (e.g. colder near a window).

• A 3D virtual character which reacts to the environment.

He has five waiting/idle behaviours triggered depending on the location: freezing idle if on cold temperatures to sweating idle if on hot temperatures.

• An immersive setup: users stand in front of a wall (2m*3m) with active stereoscopy and head tracking (Figure 1 (a)).

• A directional control of character displacements: users point with a Wiimote at a 2D mini-map of the sitting room to direct the character to a position (Figure 1 (d)).

Procedure and evaluation

To vary the embodiment experience, we designed four conditions (2*2 between subjects): Human (high anthropomorphic) / Sphere (low anthropomorphic) (Figure 1. (b)) * Egocentric reference frame / Exocentric reference frame (Figure 1. (c)) (Hu-Sp / Eg-Ex). In the egocentric reference frame condition, the user viewpoint is not collocated with the avatar body (1 meter behind) but is tethered to its displacements. Each participant is explained the use case scenario of the task: he visits a house for sale and has to explore the living room thanks to the 3D character in order to make an idea of its thermal comfort. Before starting the task, participants are invited to test the immersive system on a neutral scene (outside the virtual house) without the character. Then the user begins the task, exploring the virtual environment by moving the character. This exploration session lasts 3 minutes. Once finished, participants are invited to fill in the questionnaire with all affirmations in a randomized order. This part takes approximately 10 minutes. This paper focuses on the self-report analysis measures.

Perceived embodiments

Perceived embodiments refer to the two ways an embodied entity can be traditionally perceived: as an avatar or an ECA. Two scales are proposed to measure each type of experience. One consequence of a conversational experience is the simulations of "other minds" and the attribution of intentionality. To evaluate this intentionality attribution three items are proposed (Table 1). The resulting average is the Intentionality variable (Int). An avatar experience results in a better projection in place of the body in the virtual world: better distance estimation and better immersion [LSK*08]. Three items are proposed to evaluate this projection feeling (Table 1). The resulting average is the Projection variable (Pro). Each item of these two dimensions is a Likert type affirmation. Answers are ranging from 1 (I fully disagree) to 5 (I fully agree). How these two scales interact is of particular interest.

Perceived humanness

As shown by Konijn and colleagues [VKH*07], the overall embodiment perception is a result of concurrent intermediate variables. In this paper, we focus on variables related to the humanness of the embodied entity: Realism (Rea), Anthropomorphism (Ant) and Similarity (Sim). Indeed these variables are known to foster the experience from both approaches [VKH*07]. Each scale is composed of four items, and each item is a Likert type affirmation (Table 1). They are ranging from 1 (I fully disagree) to 5 (I fully agree).

Results

36 subjects have realized our experiment ranging from 16 to 57 years old (M = 29, SD = 11.7) with 42% women and 58% men. 39% of respondents were having a strong background in informatics. All variables have normal distributions allowing the use of parametric tests. Scale reliability is evaluated with Cronbach's α. All scales range from acceptable to good with Cronbach's α ranging from 0.61 to 0.81. Table 2 presents Pearson and partial correlations between perceived embodiment variables Int and Pro with Humanness. The three variables correlate positively and significantly. When controlled by Humanness, Int and Pro are no more significantly correlated. Table 3. shows multivariate linear regressions with humanness variables in predicting perceived embodiment variables. Such analysis enables to understand the unique influence of each predictor taken into account the variance of the others.

A low level of multicollinearity was present among the six predictors and the two perceived embodiment variables (Variance Inflation Factor < 2.5). Multivariate linear regressions are done twice: first without and then with the three interaction terms (Rea*Ant, Rea*Sim, Ant*Sim).

Table 1. List of items and the reliability of each scale.

Scale Items

Realism

The avatar has a natural appearance α = 0,71

The avatar seems to be real The avatar seems to be fake * The avatar has an artificial appearance * Anthropo-

The avatar seems to be human morphism

The avatar behave as a human being α = 0,76

The avatar seems to be different from a human* The avatar behave differently than a human being * Similarity

The avatar and me are alike inside α = 0,71

The avatar and me have common characteristics

The avatar and me are different * The avatar and me are dissimilar */** Humanness The twelve items from Realism, Antrhopoα = 0,80 morphism and Similarity Intention-

The avatar has intentions ality

The avatar could want to do something α = 0,61

The avatar thinks Projection

The avatar reflects me α = 0,81

The avatar represent my person I am feeling at the avatar's place * These items results were inverted ** This item was removed from the scale due to bad fit Significant predictors are highlighted in Table 3. Model 1.2 explains 45,6% of the Intentionality variation and model 2.2 explains 61,4% of the Projection variation. *. Significance at the 0.05 level

Discussion

In our case study, the Humanness of the embodied entity is positively associated with Intentionality (the scale associated with the perception of a conversational entity) as well as Projection (the scale associated with the perception of an avatar). Both are coherent results regarding the literature. The positive association between Intentionality and Projection appears to be mostly mediated by the humanness level. An interesting finding is the absence of correlation between them when controlled by humanness: attributing intentions to the embodied entity is not contradictory with self projecting in it.

A question for future researches is to look at this pattern of relations with a more extreme avatar experience, for example by increasing the coupling between the user and the virtual body. In this case, increasing the conversational approach by autonomous behaviours might impair the avatar experience.

For both perceived embodiment scales, the three Humanness subscales with their interaction terms explain a large part of their variances in the regression analysis. This result confirms the importance of these three dimensions as well as their complementarities. Looking at subscales of Humanness, it appears that both scales (Int and Pro) are differently predicted by Humanness. Intentions attribution is associated with the more explicitly related to Humanness scale (Ant). The interaction term Rea*Sim shows the relative importance of Similarity as a moderator, as shown by Vugt et al [VKH*07]. This means that giving a human shape and behaviors to the entity alone is not a guarantee of the intentions attribution: the perceived similarity should be high as well. The Projection scale is associated with Similarity as well as two interaction terms: Rea*Sim and Ant*Sim. To foster the avatar perception, increasing the similarity level should go along with increasing the entity realism and decreasing its human attributes. To enable to operationalize these results as guidelines for virtual embodiment design, the next step is to identify the embodied entity attributes (shapes, behaviours, controllers, etc.) influencing these self-reported dimensions. Taken together, these results outline the necessity to overcome the traditional dichotomy in embodiment studies.

Owning our body and seeing other's bodies is a real life case which is no longer the rule in immersive virtual words. This is even true for robot in real life [START_REF] Rich | Robots and Avatars as Hosts, Advisors, Companions, and Jesters[END_REF]. This study shows that these two experiences are not contradictory (one is not inhibiting the other). It highlights the need to study these hybrid experiences as a combination of both approaches.

The authors wish to thank Saint Gobain Recherche which has supported this research. They provided the industrial use case as well as the 3D environment associated with. Special thanks to Frédéric Achard who followed the project for his patience and insightful feedbacks.

Figure 1 .

 1 Figure 1. (a) User in the immersive environment. (b) Left: High anthropomorphic character, Right:Low anthropomorphic character. (c) Top: Exocentric point of view, Down: Egocentric point of view. (d) Ray to point on the 2D mini-map to displace the character.

Table 2 .

 2 Pearson correlations and partial correlations between perceived embodiments and humanness.

	Pearson	Hum	Int	Pro
	Humanness	r	1	,478 *	,620 *
		p		,003	,000
	Intentionality r		1	,492 *
		p			,002
	Projection	r			1
		p
	Partial (controlled by Hum)	Int	Pro
	Intentionality r	1	,284
		p		,098

Table 3 .

 3 Multivariate linear regressions.

			Intentionality			Projection
		model 1.1 model 1.2 model 2.1 model 2.2
	R²	0,341	0,456	0,458	0,614
		β	p	β	p	β	p	β	p
	Rea	-,115 ,46 -,083 ,60 ,117 ,41 ,234 ,09
	Ant	,425* ,02 ,375* ,04 ,165 ,31 ,080 ,72
	Sim	,278 ,11 ,294 ,08 ,527* ,00 ,518* ,00
	Rea*Ant			-,358 ,12			-,036 ,85
	Rea*Sim			,470* ,05			,395* ,05
	Ant*Sim			-,281 ,07			-,312* ,02