Muriel Fadairo
email: muriel.fadairo@univ-st-etienne.fr

Cintya Lanchimba
email: cintya.lanchimba@epn.edu.ec

Miguel Yangari
email: miguel.yangari@epn.edu.ec

Optimal Monetary Provisions and Risk Aversion in Plural Form Franchise Networks A Model of Incentives with Heterogeneous Agents

Keywords: Franchising, dual distribution, royalty rate, commission rate, moral hazard JEL codes: L14, D82 Franchising, dual distribution, royalty rate, commission rate, moral hazard. JEL codes: L14, D82

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Introduction

This paper deals with the contractual design in franchise networks. Franchising can be defined as a vertical contractual relationship between two independent firms, an upstream unit, the franchisor, and a downstream unit, the franchisee. With the contract, the franchisor allows the franchisee to use his brand name, business format and know-how, in exchange for compensation, royalties and/or up-front fee. This organizational form is present in many economic sectors, and widespread internationally.

Most of the franchised networks are mixed systems where company-owned units coexist with franchised units 1 . For example, statistics from the Bond's guide 2 highlight the predominance of mixed systems in North American franchising (74% in 2014). A similar situation is observed in other countries like Brazil (77% of the franchised networks in 2012 would be mixed networks, according to the Brazilian franchising association), France (with 65% of mixed networks in this country, according to the the most recent statistics of the National institute of statistics and economic studies, in 2007), or Venezuela (with 67% of mixed networks in 2012 according to the Venezuelan chamber of franchises). In addition, [START_REF] Lafontaine | Targeting managerial control: evidence from franchising[END_REF] provided evidence that dual distribution is not a transitory phenomenon, as the proportion of company-owned outlets remains relatively stable in mature franchised chains.

Consequently, dual distribution is the focus of an important stream of literature, in economics and strategic management (e.g. [START_REF] Gallini | Dual Distribution and Royalty Fees in Franchising[END_REF] Contractual design is another important issue regarding franchising [START_REF] Lafontaine | Agency theory and franchising: some empirical results[END_REF][START_REF] Brickley | Royalty Rates and Upfront Fees in Share Contracts: Evidence from Franchising[END_REF][START_REF] Vazquez | Up-front franchise fees and ongoing variable payments as substitutes: An agency perspective[END_REF] ; Gonzalez-Diaz and Solis-Rodriguez, 2012a;

1 "Mixed networks", "plural form" organization or "dual distribution" are used as synonymous and refer to the simultaneous presence of both franchised and company-owned outlets in the same network. 2 The Bond's Franchise Guide is the main source of secondary data regarding franchising in the U.S.A and Canada.

Indeed, the royalty rate is a determinant monetary provision, defining the "sharecontract", that is the conditions under which the profit generated by the decentralized vertical structure is shared between the franchisor and the franchisee. For this reason, this contractual provision plays a central role as incentive device.

However, despite the importance of mixed networks and of monetary devices in franchise contracts, these subjects are the matter of two separate fields of literature.

The aim of this article is to fill this gap. We provide a model of monetary provisions in the case of a plural form franchise network. This paper builds upon Bhattacharyya and Lafontaine (1995)'s model, which remains a central reference in the literature dealing with monetary devices in franchise contracts. The two authors study formally the franchisor-franchisee relationship as a two-sided moral hazard problem, and define the optimal contract based on the royalty rate.

In this theoretical framework, we introduce the assumption of a mixed network.

The incentive problem concerns then two types of agents, the manager of a companyowned unit and the franchisee. We study how the franchisor can contractually encourage both types of agents to achieved the best non-observable e↵ort, in a risk environment associated to the variation of sales at the retail unit level. The agents are distinguished by di↵erent levels of risk aversion.

In this paper we do not aim at providing an exhaustive theory of the dual system, but we assume it as given. We consider a situation where a firm has conceived a new concept. Before delegating part of the business, it has developed his activity through an integrated retail outlet. Once the product has matured in the market, the firm becomes a franchisor, delegating part of the retail activity to an independent unit -the franchisee. Hence, at a certain point of time, the sales of the structure are undertaken by both company-owned and independent retail units. Within this dual structure, we are interested by the determination of the optimal monetary provisions.

At this regard, we make use of the agency theory and study an environment where the volume of sales of each unit is a↵ected by a non-verifiable e↵ort and by a random component relating to the environmental risk. Since the e↵ort level cannot be made part of the contract, the equilibrium monetary provisions will aim at providing incentives to the retailing units, and to the franchisor, to exert valuable e↵orts.

We demonstrate that at the equilibrium of monetary provisions regarding the independent outlet and the company-owned unit -respectively, the royalty rate and the commission rate -both monetary devices are interdependent and negatively related.

The remaining of the paper is organized as follows. After a short literature review (section 2), we present the formal model (section 3). We then proceed to the formal analysis (section 4). In order to study the interdependencies between the rates and how the equilibrium monetary provisions di↵er from optimality, we begin by introducing a benchmark situation where the provision of e↵ort can be part of the contract designed by the franchisor. Our conclusion sums up the main findings and provides suggestions for further research (section 5). All the proofs are relegated to the appendix section.

Related Research

Given that this paper investigates monetary provisions in dual distribution with a double-sided moral hazard environment, it is related to three streams of literature:

(a) agency theory, and more precisely moral hazard theory of franchising, (b) plural form distribution networks, (c) contractual design, and more precisely monetary provisions in franchise contracts.

The first stream of literature is the main theoretical background in the study of franchising. Indeed, franchising represents a rich context for investigating interorganizational precepts and phenomena, due to its particular structure and its behavioral aspects. The nature of franchising allows for di↵erent vertical relationships inside the network. The relevance of agency theory in the study of vertical restraints is now widely accepted on the basis of initial contributions of Mathewson andWinter (1984, 1985) and Rey and Tirole (1986a, b). The agency framework has generated a large amount of literature dealing with franchising (e.g. Lafontaine and Slade, 2007 ; [START_REF] Barthélémy | Agency and Institutional Influences on Franchising[END_REF][START_REF] Etro | Endogenous market structures and contract theory: Delegation, principal-agent contracts, screening, franchising and tying[END_REF] ; Gonzalez-Diaz and Solis-Rodriguez, 2012b ; Agency literature of franchising examines the bilateral contract relationship between an upstream firm and a retailer within a distribution network. The contract thus creates an agency relationship because the upstream firm gives the downstream party a mandate for the commercial exploitation of its brand. The principal (the upstream firm, i.e. the franchisor) designs the contract, and the retailer may either accept or reject the contract. Contracts are uniform within the same network; the analysis refers thus to the representative contract. A moral hazard emerges downstream, as the retailers e↵ort a↵ects the profit function of the principal, who cannot observe this e↵ort. However, franchising being based on renting the upstream firm's brand name, a moral hazard can also be considered upstream since retail sales depend on the franchisor's e↵ort in promoting the brand and network reputation.

The downstream firm (the franchisee) cannot fully observe the e↵ort made by the franchisor while it su↵ers the consequences.

The second stream includes studies that account for the contract-mix (franchised and company-owned units) in the same network. Since [START_REF] Caves | Franchising: Firms, Markets and intangible assets[END_REF], capital constraint is a main argument to explain the presence of franchised units in distribution networks. Franchisees are seen as financial and human capital providers, enabling a fast and wide development of the network. In addition with the resourcebased view, dual distribution is explained in the framework of agency theory, in terms of monitoring costs, signaling, or complementarity between the two types of units.

With the unilateral moral-hazard framework, that is moral hazard on the downstream side, an essential determinant of the extent of franchising in the network is the cost of monitoring the downstream units and maintaining control over the system. Control cost rises disproportionately with the increasing growth of the system, especially into more distant territories. For each new retailer, the upstream firm faces a choice about the way it will expand the network: company-owned unit versus independent retailer with a vertical contract as franchising. This choice reflects a trade-o↵ between incentive and control. In the moral hazard context, higher incentives come from independent retailers, but better control is possible with inte-grated units. The upstream choice concerning each downstream unit is motivated by the local conditions regarding the monitoring cost. Thus the contract-mix in the network is the result of localized decisions. This view finds empirical support in the econometrical literature on franchising data [START_REF] Brickley | The Choice of Organizational Form: The Case of Franchising[END_REF]Brickley,1999;[START_REF] Arrunada | Contractual allocation of decision rights and incentives: the case of automobile distribution[END_REF][START_REF] Barthélémy | Agency and Institutional Influences on Franchising[END_REF]. Other interesting explanation of dual distribution is proposed by [START_REF] Gallini | Dual Distribution and Royalty Fees in Franchising[END_REF] in terms of signal. In a context of information asymmetry relating to the value of the upstream firm's brand name, these authors consider two types of franchisors (good type versus bad type), and demonstrate formally that good type franchisors can signal their type, and therefore provide relevant information to potential future franchisees. The signal devices are organizational forms that make the franchisor's revenue highly dependent upon the performance of the business concept ; like dual distribution. [START_REF] Lafontaine | Contractual Arrangements as Signaling Devices: Evidence from Franchising[END_REF] using US data and the predictions deriving from Gallini and Lutz (1992)'s seminal theoretical model of signal in franchising demonstrated that signaling theory is not quite appropriate to study franchising.

Finally, plural form networks are analyzed based on the complementarity between franchised and company-owned outlets. More than the result of localized choices relating to monitoring costs, or the result of signaling issues, dual distribution can be a global strategy of the upstream firm, which maximizes the synergies between the two types of retail units. In line with this view, [START_REF] Bai | Contract Mixing In Franchising as a Mechanism for Public-Good Provision[END_REF] adapt the [START_REF] Holmstrom | Multi-task principal-agent analyses: linear contracts, asset ownership and job design[END_REF]'s multitask model to study retailing as a two-task activity requiring i) an e↵ort to maintain the brand name value and ii) an e↵ort to sale. They argue that franchisees are better in performing the second task as their revenue depends on sale results, while the managers of company-owned units have a fixed wage. On the other hand, company-owned units are better in maintaining the brand value. Drawing thus attention on the complementarities between the two types of retail units, [START_REF] Bai | Contract Mixing In Franchising as a Mechanism for Public-Good Provision[END_REF] analyze their coexistence in the network as a strategy which allows the franchisor to take advantage of the special feature of each type of unit. [START_REF] Sorenson | Finding the right mix: Franchising, organizational learning and chain performance[END_REF] The third stream of literature considers monetary clauses and more particularly the share-parameter, that is the royalty rate. As discussed earlier, past research have focused on pure franchise systems.

In a context of unilateral moral hazard, that is on the franchisee side, the status of total residual claimant is the most e↵ective incentive mechanism for the downstream firm. In this case, the share-contract includes an up-front fee and no royalties. Once the entry fee is paid, the retailer captures the totality of the results of its sale e↵ort.

Yet, royalties are common in distribution contracts. In the agency framework, the presence of royalties has two justifications. The first one, initially proposed by [START_REF] Martin | Franchising and risk management[END_REF], concerns the need to insure the downstream firm against risk, namely, the hazard on the level of the final demand. Here the share contract defined by the royalty rate corresponds to a level of risk sharing.

The second justification concerns incentives in the context of bilateral moral hazard. Then, the franchise contract also has to contain an incentive mechanism for the upstream firm. The two-sided moral hazard requires ongoing payments to the franchisor to motivate its e↵orts in promoting the network reputation throughout the duration of the relationship. This bilateral moral hazard situation is the primary theoretical justification for profit-sharing contracts in distribution. This theory has developed since the seminal article of [START_REF] Mathewson | The economics of franchise contracts[END_REF], who proposed the first formal analysis of franchise agreements in the framework of agency theory. This notion was augmented by [START_REF] Lal | Improving channel coordination through franchising[END_REF], and by [START_REF] Bhattacharyya | Double-Sided Moral Hazard and the Nature of Share Contracts[END_REF]. The explanation of royalties in terms of bilateral moral hazard finds support in the econometric literature on franchise data [START_REF] Lafontaine | Agency theory and franchising: some empirical results[END_REF]; Agrawal and Lal, 1995; [START_REF] Brickley | Royalty Rates and Upfront Fees in Share Contracts: Evidence from Franchising[END_REF][START_REF] Vazquez | Up-front franchise fees and ongoing variable payments as substitutes: An agency perspective[END_REF].

Our paper targets at conciliating both last presented strands of the literature, studying optimal monetary provisions in a dual distribution context. Assuming as given the plural form system, we analyze how those provisions are determined, and their possible interdependence relationship.

3 The Model

Baseline statements

We consider a two-sided moral hazard model of intra-brand competition with an exogenous downstream market structure, composed by a franchisee, denoted by (f ), and a company-owned unit denoted by (m) 3 . Territorial exclusivity is a↵orded to each retail unit. Hence there is no direct interaction between the two outlets. We further assume that the retailing units are price takers. They can only influence demand or the volume of sales trough a non verifiable e↵ort. The network organizational structure is represented by Figure 1.

Our objective is to determine the optimal monetary provisions of the system, more precisely, the optimal royalty and commission rates.

Franchise Chain

Operated by a franchisor (p)

Franchise-Unit

Operated by a franchisee (f ) Incentives depend on:

• Royalty rate (r) • Fee (F )

Owned-Unit

Operated by a manager (m) Incentives depend on:

• Commission Rate (y)

• Wage (w)

We model a static game with the following general scheme. First, the principal, i.e. the franchisor, designs a specific contract for each retail unit. The agents, i.e.

the franchisee and the manager, decide whether or not to accept the contracts. The game ends if none of them accepts.

If the contracts are accepted, all the players, i.e. the principal and the two agents, provide a non-verifiable e↵ort a↵ecting the volume of sales on the market.

The e↵ort levels are chosen given the contracts that have been signed. However, for each player, the provided e↵ort level is a free decision since e↵ort is not a contracted variable. Once e↵orts had been made, the nature determines the sate of the world, which is defined as a random shock a↵ecting the demand realization in the two market-segments independently. Finally, the game outcome is achieved and the payo↵s are executed according to the terms of the contracts.

As the e↵ort levels cannot be part of the contract because they are not observable, and due to the random realization of demand, the franchisor's problem is to design a contract providing e↵ort incentives for all the players. The time-schedule of the game is summarized in Figure 2.

The franchisor designs the contract

Risk and e↵ort issues

We assume that the principal and the agents are risk-averse, with constant risk aversion. Their utility is expressed by u(I) = exp( ⇢I), where I denotes the income and ⇢ is the coe cient of absolute risk aversion. In this context of risk aversion, the objective of each player is to maximize the certainty equivalent income:

CE = E(I) R (3.1)
where E(•) is the expectation operator and R denotes the risk premium.

Because the franchisor has constant absolute risk aversion, the risk premium equals:

R p = ⇢ 2 V ar(I) ( 3 . 2 )
where ⇢ denotes the coe cient of absolute risk aversion, and V ar(I) the variability of income.

Considering that the franchisee is less risk averse than the manager, in all the cases, we assume that ⇢ m(m) > ⇢ f (M ) , i.e. the minimum manager's coe cient of risk aversion is greater than the highest franchisee's coe cient of risk aversion.

Each retailer sells the franchisor's products. The franchisor produces at no cost.

We normalize the selling price at 1. To generate sales at the outlet level, each retailer has to undertake a costly e↵ort e 0. The franchisor's e↵ort increases sales of both retail units simultaneously. In addition with the e↵ort of each player, the volume of sales is a↵ected by a random component ✏, that is a demand shock that none of the players is able to control. This uncertain environment precludes each player from inferring the e↵ort levels of the other players from the observed volume of sales and gives scope to the moral hazard problem.

Sales volumes, respectively for the franchisee and the manager, are then: where ↵, and are exogenous demand parameters representing the e↵ort influence on sales volumes. The sales of each outlet are only a↵ected by the outlet e↵ort (franchisee or manager's e↵ort) and by the franchisor's e↵ort which benefits equally to both retail units. Hence, we assume that there is no e↵ort externality between the outlets, while the franchisor's e↵ort to promote the network works as a public good in the system. The random variable " j follows a normal distribution " j v N 0, ✓ 2 j for j = f, m. In addition, we assume that the cost of e↵ort is increasing, convex, and the same for all the players:

S f (e f , e p ) = ↵e f + e p 2 + " f (3.
C(e) = e 2 2 (3.5)
Introducing the cost of e↵ort in equation (3.1), we obtain the certainty equivalent for each player:

CE i = E(I i ) R i e 2 i 2 for i = p, m, f (3.6)
Regarding the parameters of the model, we make the following assumptions that will be used for the analysis.

Assumption 1. The parameter of lost aversion for the franchisee is such that

⇢ f (M ) ⇢ f (m) = (⇢ f (M ) +⇢p) 2 6
> 2 3 ⇢ p . Assumption 2. The parameter of lost aversion for the manager is such that

⇢ m(M ) ⇢ m(m) = (⇢ m(M ) +⇢p) 2 6
> 2 3 ⇢ p . In order to ensure an equilibrium, we assume the following: Assumption 3.

((⇢ p + ⇢ f (M ) )✓ 2 f + 2 + ↵ 2 ) 2 = 6(⇢ f (M ) ⇢ f (m) )✓ 2 f (↵ 2 + ⇢ p ✓ 2 f e m 2 )
Assumption 4.

((⇢ p + ⇢ m(M ) )✓ 2 m + 2 + 2 ) 2 = 6(⇢ m(M ) ⇢ m(m) )✓ 2 m ( 2 + ⇢ p ✓ 2 m e f 2 ↵ )
The aim of the franchisor is to design contracts which incite the agents to make costly e↵ort maximizing their certainty equivalent.

As demonstrated by [START_REF] Bhattacharyya | Double-Sided Moral Hazard and the Nature of Share Contracts[END_REF], the analysis can be restricted to linear contracts without loss of generality. In addition, evidence regarding relatively simple and often linear payment rules is rather extensive.

Assumption 5. Contracts have the following form: {F, x}, where F is a fixed payment and x is an output-based rate.

Using linear contracts, the risk premium for three players is defined as follows:

R f = 1 2 (r⇢ f (M ) + (1 r)⇢ f (m) )V ar((1 r)✓ f ) ( 3 . 7 ) R m = 1 2 ((1 y)⇢ m(M ) + y⇢ m(m) )V ar(y✓ m ) ( 3 . 8 ) R p = ⇢ p 2 V ar(r✓ f + (1 y)✓ m ) ( 3 . 9 )
Hence, the risk premium associated with the franchisee depends on the coe cient of risk aversion, which can vary between the minimum "⇢ f (m) " and the maximum "⇢ f (M ) ", the percentage of revenue assigned to him (1 r), and the variance in sales denoted by ✓ f . Similarly, the manager's risk premium depends on the coe cient of risk aversion, which can vary between "⇢ m(m) " and "⇢ m(M ) ", the percentage of revenue assigned to him (y), and the variance in sales ✓ m . Lastly, the franchisor's risk premium (r, (1 y)) is based on the two percentages that he receives, and on the variance in sales of both retailing units.

By reformulating the equations for the certainty equivalent, we obtain for the franchisor:

CE p = r ✓ ↵e f + e p 2 ◆ + (1 y) ✓ e m + e p 2 ◆ + F e 2 p 2 w ⇢ p 2 r 2 ✓ 2 f + (1 y) 2 ✓ 2 m + 2Cov(✓ f r, ✓ m (1 y) (3.10)
The first and the second parts of the expression represent the income provided by the franchisee and by the manager, depending on their e↵ort and on the franchisor's e↵ort. The last component is the cost for the franchisor, composed of the risk premium and the cost of e↵ort.

The certainty equivalent for the franchisee, depending negatively on the royalty rate (r), the fixed fee (F ) he has to pay to the franchisor, the risk premium and the cost of e↵ort, is reformulated as follows:

CE f = (1 r) ✓ ↵e f + e p 2 ◆ F e 2 f 2 1 2 (r⇢ f (M ) + (1 r)⇢ f (m) )(1 r) 2 ✓ 2 f (3.11)
Finally, the certainty equivalent for the manager, depending positively on the wage (w) and the commission rate (y) received from the franchisor, and negatively on the risk premium and the cost of e↵ort is:

CE m = w + y ✓ e m + e p 2 ◆ e 2 m 2 1 2 ((1 y)⇢ m(M ) + y⇢ m(m) )y 2 ✓ 2 m ) (3.12)
4 Analysis

The franchisor's problem

In franchise systems, the franchisor is typically responsible for promoting the chain.

Franchisees and managers are responsible for managing the outlets on a day-today basis. The downstream and upstream e↵orts a↵ect the outlet performance.

However, the e↵ort levels are not easily monitored.

On this basis, the franchisor's problem can be expressed as follows:

M ax [r,y,w,F,ep] CE p = r ✓ ↵e f + e p 2 ◆ + F | {z } Income from franchised unit + (1 y) ✓ e m + e p 2 ◆ w | {z } Income from company-owned unit e 2 p 2 |{z} Cost ⇢ p 2 V ar(r✓ f + (1 y)✓ m ) | {z }
Franchisor's risk aversion , subject to:

(i) CE m = w + y ✓ e m + e p 2 ◆ e 2 m 2 1 2 ((1 y)⇢ m(M ) + y⇢ m(m) )y 2 ✓ 2 m 0 ( IR m ) (ii) CE f = (1 r) ✓ ↵e f + e p 2 ◆ F e 2 f 2 1 2 (r⇢ f (M ) + (1 r)⇢ f (m) )(1 r) 2 ✓ 2 f 0 (IR f ) (iii) d (CEp) de p = r 2 + (1 y) 2 e p = 0 (IC p ) (iv) d (CEf ) de f = (1 r)↵ e f = 0 (IC f ) (v) d (CEm) de m = y e m = 0 (IC m )
Conditions (i) and (ii) are the participation constraints. These conditions must be met for the manager and the franchisee accept the contracts. We normalize to zero the outside option of both agents. The last three conditions are the incentive constraints a↵ecting the e↵ort levels. Since e↵ort variables in the model are continuous, we use the first-order approach.

In the following, we focus first on the case of symmetric information, to highlight in a second step the specific features of the case with asymmetric information.

Case 1: Symmetric information

In case of symmetric information, i.e. when all the players have the same information, both before establishing the contract and during the relationship, e↵orts are observable and contractable. Then, the optimal monetary provisions depends only on risk aversions, as expressed by proposition 1:

Proposition 1. When all relevant information is verifiable, optimal monetary provisions depend only on the degree of risk aversion of the players. Thus,

r ⇤ = 1 s 2⇢ p 3(⇢ f (M ) ⇢ f (m) ) y ⇤ = s 2⇢ p 3(⇢ m(M ) ⇢ m(m) )
Because the e↵ort level is part of the contract designed by the franchisor, the distribution of risk between the players depends only on the relative absolute risk aversions. Then, optimality requires that lower is the risk aversion of a player, compared to the others, higher is the risk he bears. Since in the specific case of symmetric information, there is no need to provide incentives to induce e↵ort, the franchisor's maximization problem is only based on the participation constraints (Appendix A).

The royalty and commission rates act as insurance devices. If the franchisor's risk aversion increases, he will prefer a safer payment mechanism to extract the surplus of the franchised unit. Therefore, he will set a higher up-front fee and a lower royalty rate, as the rate depends on uncertain sales. In addition, to extract income from the company-owned unit, the franchisor will prefer to pay a lower wage with a higher commission rate. If the franchisee's risk aversion increases, he will prefer a payment mechanism based on a higher royalty rate and a lower up-front fee. This is a way to share with the franchisor the risk related to sales. Finally, if the manager's risk aversion increases, he will prefer a higher fixed wage in addition with a lower commission rate related to the uncertain sales on the downstream market.

Case 2: Asymmetric information

We focus now on the case where e↵orts are not observable and they can not be part of the contracts. In this case, our game is subject to moral hazards, and the contracts designed by the franchisor have to provide incentives in order to promote the three players' e↵ort. The correlation between the events of the two segment-market is set at zero 4 .

In this case, the optimal monetary provisions do not only depend on the relative risk aversions, and have to vary according to the obtained outputs. Therefore, the franchisor's maximization problem depends now on the participation and on and the incentive constraints (Appendix B). We obtain the following proposition: Proposition 2. With non contractible e↵orts and uncorrelated events Cov(✓ f , ✓ m ) = 0, the monetary provisions are interdependent, negatively related, and equal to:

r ⇤ = 1 v u u t 2(↵ 2 + ⇢ p ✓ 2 f em 2 ) 3(⇢ f (M ) ⇢ f (m) )✓ 2 f y ⇤ = s 2( 2 + ⇢ p ✓ 2 m e f 2↵ ) 3(⇢ m(M ) ⇢ m(m) )✓ 2 m
As in the previous case, the royalty and commission rates depend on the relative absolute risk aversion of the three players. However, because of the bilateral moral hazard, they now depend also on the e↵orts provided by the agents and by the franchisor.

Thus, if the manager's e↵ort generates more sales, the principal must incite this e↵ort with a higher commission rate. The same reasoning applies for the franchisee.

In this case, the incentive mechanism is a reduction of the royalty rate. Finally, the incentive devices for the franchisor's e↵ort are a higher royalty rate and a lower commission rate. This generates an ine cient distribution of risk, but allows for promoting incentives for the players to make greater e↵ort.

An interesting result is that the monetary provisions are interrelated, even if the demand shocks on the segment markets are not correlated. This result is explained by the franchisor's e↵ort, acting as a public good for both retailing units 5 . 4 This implies that:

R p = ⇢p 2 V ar(r✓ f +(1 y)✓ m ) = ⇢p 2 (r 2 ✓ 2 f +(1 y) 2 ✓ 2 m +2Cov(✓ f r, ✓ m (1 y)) y Cov(✓ f r, ✓ m (1 y)) = 0 5
Complementary results available upon request show that with correlated events Cov(✓ f , ✓ m ) 6 = 0, the optimal provisions are interdependent but higher than in the case with no correlation.

Conclusion

Existing literature on franchising has extensively studied the presence of mixed distribution networks. However, despite the importance of monetary provisions in franchise contracts, their definition in the case of plural form networks had not been addressed.

In this paper, we focus more precisely on the "share parameters" in integrated (company-owned retail outlet) and decentralized (franchised outlet) vertical relationships, respectively the commission rate and the royalty rate.

We develop an agency model of payment mechanism in a two-sided moral hazard context, with one principal and two heterogenous agents distinguished by di↵erent levels of risk aversion. We define the optimal monetary provisions, and demonstrate that even in the case of segmented markets, with no correlation between demand shocks, the two rates (commission rate, royalty rate) are negatively interrelated.

Two limitations of the model can provide avenues for future research. First, the market structure of our model is exogenous. The contractual choices are thus studied independently of the organizational choices. Yet, for example, the mix level in the network, that is the proportion of company-owned retail units, may influence the payment mechanisms. Second, we assume here that there is no externality between the retail outlet e↵orts. However, demand interdependencies coming from the e↵ort provided by the di↵erent retail units may influence the determination of monetary provisions in distribution contracts.

We anticipate that future research will benefit and hopefully build on the approach presented in this paper.

Appendix A

Proof of proposition (1): We consider here a context of symmetric information where e↵ort is observable and can be part of the contract o↵ered by the franchisor.

In this case, the reduced problem is as follows:

M ax [r,y,w,F,ep,e f ,em] CE p = r ✓ ↵e f + e p 2 ◆ + F | {z } Income from franchised unit + (1 y) ✓ e m + e p 2 ◆ w | {z } Income from company-owned unit e 2 p 2 |{z} Cost ⇢ p 2 r 2 ✓ 2 f + (1 y) 2 ✓ 2 m + 2Cov(✓ f r, ✓ m (1 y)) | {z }
Franchisor's risk aversion , subject to:

(i) CE m = w + y ✓ e m + e p 2 ◆ e 2 m 2 1 2 ((1 y)⇢ m(M ) + y⇢ m(m) )y 2 ✓ 2 m 0 ( IR m ) (ii) CE f = (1 r) ✓ ↵e f + e p 2 ◆ F e 2 f 2 1 2 (r⇢ f (M ) + (1 r)⇢ f (m) )(1 r) 2 ✓ 2 f 0 (IR f )
The Lagrangian of the problem is:

◆ = r ⇣ ↵e f + ep 2 ⌘ + (1 y) ⇣ e m + ep 2 ⌘ + F e 2 p 2 w ⇢p 2 r 2 ✓ 2 f + (1 y) 2 ✓ 2 m 1 ⇣ e 2 m 2 + 1 2 ((1 y)⇢ m(M ) + y⇢ m(m) )y 2 ✓ 2 m w y ⇣ e m + ep 2 ⌘⌘ 2 ⇣ F + e 2 f 2 + 1 2 (r⇢ f (M ) + (1 r)⇢ f (m) )(1 r) 2 ✓ 2 f (1 r) ⇣ ↵e f + ep 2 ⌘⌘ .
We then proceed to calculate the first order conditions of the Lagrangian.

With respect to wages and to the up-front fee, we obtain respectively:

1 = 1, (6.1) 2 = 1 (6.2)
With respect to both Lagrange multipliers, we get the following conditions:

w + y ✓ e m + e p 2 ◆ e 2 m 2 ((1 y)⇢ m(M ) + y⇢ m(m) ) y 2 ✓ 2 m 2 = 0 (6.3) (1 r) ✓ ↵e f + e p 2 ◆ F e 2 f 2 (r⇢ f (M ) + (1 r)⇢ f (m) ) (1 r) 2 ✓ 2 f 2 = 0 (6.4)
Therefore, the franchisee's and manager's individual rationality constraint must be binding, which implies that there is no rent left downstream. By calculating the first-order condition of the Lagrangian with respect to the e↵ort of the principal we obtain:

r 2 + (1 y) 2 e p 1 ✓ y 2 ◆ 2 ✓ (1 r) 2 ◆ = 0 (6.5)
and substituting by equations (6.1) and (6.2) into (6.5), we get:

r 2 + 2 y 2 e p + y 2 + 2 2 r = 0 ! e p =
This expression indicates that the e↵ort of the franchisor is such that the marginal cost equals the marginal benefit. This last e↵ect is represented by the e↵ectiveness that the e↵ort has on the demand to both units. By the same procedure, we obtain that the e↵ort for the manager and the franchisee are equal to:

e m = ; e f = ↵
Regarding the first order condition with respect to the royalty rate we obtain:

↵e f + e p 2 ⇢ p ✓ 2 f r 2 ✓ 1 2 (⇢ f (M ) ⇢ f (m) )(1 r) 2 ✓ 2 f (r⇢ f (M ) + (1 r)⇢ f (m) )(1 r)✓ 2 f + ↵e f + e p 2 ◆ = 0
Substituting with the expressions obtained previously, we get:

3(⇢ f (M ) ⇢ f (m) )(1 r) 2 2(⇢ f (M ) + ⇢ p )(1 r) + 2⇢ p = 0 (6.6)
In accordance with Assumption 1, we know that ⇢

f (M ) ⇢ f (m) = (⇢ f (M ) +⇢p) 2 6
, then

⇣q 3(⇢ f (M ) ⇢ f (m) )(1 r) p 2⇢ p ⌘ 2 = 0 (6.7) Therefore, r ⇤ = 1 s 2⇢ p 3(⇢ f (M ) ⇢ f (m) ) (6.8)
The same calculations are applied for the commission rate.

We get: We first characterize the monetary provisions, before proving points i) and ii) of proposition 2.

The Lagrangian is equal to: We then proceed to calculate the first order conditions of the Lagrangian.With respect to wages and to the up-front fee, we obtain respectively:

◆ = r ⇣ ↵e f + ep
1 = 1 (6.12) Regarding the first order condition with respect to the royalty rate we obtain:

2 = 1 (6.
↵e f + e p 2 ⇢ p r✓ 2 f + 3 2 4 ↵ (6.25) 2 ✓ 1 2 (⇢ f (M ) ⇢ f (m) )(1 r) 2 ✓ 2 f (r⇢ f (M ) + (1 r)⇢ f (m) )(1 r) 2 ✓ 2 f + ↵e f + e p 2 ◆ = 0
Substituting equation (6.13), (6.20) and (6.24) into (6.25), we get:

r(⇢ p ✓ 2 f + 2 + ↵ 2 ) 1 2 (⇢ f (M ) ⇢ f (m) )(1 r) 2 ✓ 2 f +(r⇢ f (M ) + (1 r)⇢ f (m) )(1 r)✓ 2 f + (y + 1) 2 = 0
Then:

3(⇢ f (M ) ⇢ f (m) )(1 r) 2 2(⇢ p ✓ 2 f + 2 + ↵ 2 + ⇢ f (M ) ✓ 2 f )(1 r) + 2( y 2 + ⇢ p ✓ 2 f + ↵ 2 ) = 0 (6.26)
Substituting equation (6.18) into (6.26), in accordance with Assumption 3, we get:

r ⇤ = 1 v u u t 2

  ; Bai and Tao, 2000 ; Sorenson and Sorensen, 2001 ; Srinivasan, 2006 ; Dant et al., 2008 ; Bürkle and Posselt, 2008 ; Perrigot et al., 2009 ; Hendrikse and Jiang, 2011 ; Perryman and Combs, 2012).

  support the idea of complementaries into the dual networks, while Mitsubishi et al. (2008), Perrigot et al. (2009), Hendrikse and Jiang (2011), show that dual distribution, as a governance form, is more e cient than the others.

Figure 2

 2 Figure 2

3 )

 3 S m (e m , e p ) = e m + e p 2 + " m (3.4)

( e m + e p 2 ) 1 ✓ 1 2 ( 2 m( 6 , then ⇣q 3 (⇤ 6 . 2

 211226362 + ⇢ p (1 y)✓ 2 m ⇢ m(M ) + ⇢ m(m) )(y 2 ✓ 2 m ) + ((1 y)⇢ m(M ) + y⇢ m(m) )y✓ m(M ) ⇢ m(m) )y 2 2(⇢ m(M ) + ⇢ p )y + 2⇢ p = 0 (6.9)In accordance with Assumption 2, we know that⇢ m(M ) ⇢ m(m) = (⇢ m(M ) +⇢p) 2 ⇢ m(M ) ⇢ m(m) m(M ) ⇢ m(m) ) (6.11) Appendix BProof of proposition (2): We consider here a context of two-sided moral hazard, where e↵orts are not observable and cannot be part of the contract. The reduced problem is similar to case 1, but we introduce now the incentive constraints:

2 f 2 + 1 2 ( 2 ✓ 2 f(1 r) ⇣ ↵e f + ep 2 ⌘⌘ 3 (e p r 2 (1 y) 2 4

 222222322 y)⇢ m(M ) + y⇢ m(m) )y 2 r⇢ f (M ) + (1 r)⇢ f (m) )(1 r) (e f (1 r)↵)5 (e m y ) .

13 )

 13 With respect to all Lagrange multipliers, we get the following conditions:w + y( e m + y)⇢ m(M ) + y⇢ m(m) f (M ) + (1 r)⇢ f (m) )(1 r) 2 ✓ 2 f = 0 (6.15)

(↵ 2 + ⇢ p ✓ 2 f em 2 ) 3 ( 2 f( 6 . 27 )

 2232627 ⇢ f (M ) ⇢ f (m) )✓The same calculations are applied for the commission rate.(M ) + ⇢ m(m) )y 2 ✓ 2 m + ((1 y)⇢ m(M ) + y⇢ m(m) )y✓ 2

We focus on the "representative" retail units, assuming that if more outlets are present in the network, all the franchised outlets are similar and all the integrated outlets are similar.

Therefore, the franchisee's and manager's individual rationality constraint must be binding, which implies that there is no rent left downstream.

(r + (1 y)) e p 2 = e p (6.16)

(1 r)↵ = e f (6.17)

The e↵ort provided by each agent and by the principal is equal to their incentives and to the impact of their e↵ort on the downstream sales. By calculating the firstorder condition of the Lagrangian with respect to the e↵ort of the principal we obtain

Substituting by equations (6.12) and (6.13) and (iii) into (6.19), we get:

The Lagrangian with respect to the manager's e↵ort indicates that:

(1 y) 1 (e m y ) 5 = 0 (6.21)

Substituting equation (6.12) into (6.21), we get:

From the Lagrangian with respect to the franchisee's e↵ort we get:

Substituting equation (6.13) and (iv) into (6.23), we obtain: 4 = r↵ (6.24)

Appendix C

Summary results -Comparative statics for di↵erent fundaments of the economy:

(1 y) " (1 y) "

r " "

(1 y) "

↵ " r # "

(1 y) # r " (1 y) # (1 y) " r #


Stefan Ankirchner
email: s.ankirchner@uni-jena.de

Christophette Blanchet-Scalliet
email: christophette.blanchet@ec-lyon.fr

Anne Eyraud-Loisel
email: anne.eyraud-loisel@univ-lyon1.fr

We consider the problem of how to optimally close a large asset position in a market with a linear temporary price impact. We take the perspective of an agent who obtains a signal about the future price evolvement. By means of classical stochastic control we derive explicit formulas for the closing strategy that minimizes the expected execution costs. We compare agents observing the signal with agents who do not see it. We compute explicitly the expected additional gain due to the signal, and perform a comparative statics analysis.

Introduction

For many companies it is part of day-to-day business to build up and close large asset positions on financial markets. For example, whenever a fund modifies its investment strategy, it will reduce the position of some of its assets, while enlarging the holdings of other ones. Energy companies have to unwind long positions of power and buy the commodities they need for the power generation.

Selling or buying a large amount of an asset in short time usually entails a price impact. This is why in practice financial institutions, from now on referred to as agents, frequently unwind large positions by splitting them into smaller parts and closing them successively. Spreading orders over time implies the price impact to be smaller.

Agents closing a large asset position sometimes have additional information about the future price. The extra information can be based for example on assessments of market analysts. Trading houses have teams of analysts constantly observing markets. The analysts provide market assessments or even forecasts that are incorporated in the company's trading decisions.

The main aim of the paper is to study the value of a market assessment before it is revealed. To put it differently, we look at the expected additional value of market expertise. To this end we introduce into our model an expert who obtains a signal about the asset price at time T , the time up to which the position has to be closed. If the expert passes the knowledge on to the agent having to close the position, then we say that the agent is informed; else she is non-informed. We use the technique of filtration enlargements for modeling the information flow of the informed agent.

By comparing the optimal execution strategy of a non-informed agent with the one of an informed agent, one can derive the additional gain due to the signal. In order to obtain a closed form formula for the additional gain, we work in a stylized model. We assume that the price signal is the asset price at T disturbed by an independent centered Gaussian noise. Morever, we suppose that any transaction has a linear absolute temporary (abbreviated by LAT in the pioneering paper [START_REF] Bertsimas | Optimal control of execution costs[END_REF]) impact on the asset's price. The fundamental (i.e. non-influenced) price process is assumed to be a Brownian motion, complemented by a drift. We suppose that the agent aims at maximizing the expected proceeds (resp. minimizing expected costs) from closing a position.

We characterize optimal closing strategies by using standard stochastic control methods. The simple model set-up allows to obtain explicit formulas for the value functions and the optimal controls, and hence for the expected additional gain due to the signal. We find that the expected additional gain due to the signal does not dependent on the initial price and on the agent's initial position size. The gain from the signal is only determined by three factors, namely the signal's noise, the volatility and the liquidity of the asset. We perform a comparative statics analysis of the additional gain to distill the precise dependence on these three factors.

We consider also the case where the signal reveals the asset's exact fundamental price at T , i.e. where the signal is not distorted by noise. We show that in this case the additional gain is finite. The market would admit arbitrage if there were no market frictions. The price impact entailed by any trading implies that the gain from exactly knowing the fundamental price at a future date is only finite.

The value of a price signal has so far been studied mainly within utility maximization models. In [START_REF] Karatzas | Anticipative portfolio optimization[END_REF] the authors calculate, also by employing filtration enlargements, the expected additional logarithmic utility of an investor possessing inside information. They do not consider market frictions and hence obtain that the additional utility is infinite if the exact asset's price at T is known to the investor. The model of [START_REF] Karatzas | Anticipative portfolio optimization[END_REF] has been put forward in many succeeding papers, e.g. in [START_REF] Amendinger | A monetary value for initial information in portfolio optimization[END_REF], [START_REF] Ankirchner | The Shannon information of filtrations and the additional logarithmic utility of insiders[END_REF], [START_REF] Ankirchner | Financial markets with asymmetric information: information drift, additional utility and entropy[END_REF].

As an auxiliary step in the calculation of the additional gain we compute first the signal's a posteriori value. After receiving the signal, the agent perceives the price dynamics with an additional linear drift. Therefore, we shortly discuss also the liquidation problem under directional views. In Section 1 we summarize findings on optimal liquidation in a Bachelier model with a linear drift.

So far the liquidation literature has only briefly analyzed the impact of market opinions on trading strategies. Almgren & Chriss [START_REF] Almgren | Optimal execution of portfolio transactions[END_REF] calculate optimal deterministic liquidation strategies, allowing for directional views. They assume that the agent's objective is to minimize a weighted sum of the mean and the variance of the proceeds. It is remarkable that the optimal strategy from [START_REF] Almgren | Optimal execution of portfolio transactions[END_REF] maximizes CARA utility -not only among all deterministic, but even among all predictable trajectories. This is shown in [START_REF] Schied | Optimal basket liquidation for CARA investors is deterministic[END_REF] for a time continuous version of the Almgren & Chriss model. The paper [START_REF] Schied | A control problem with fuel constraint and Dawson-Watanabe superprocesses[END_REF] also studies the influence of price drifts on optimal liquidation strategies. A general semimartingale perspective is taken, leading to a more abstract representation of liquidation strategies in terms of conditional expectations.

The paper is organized as follows: In Section 1 we set up the model framework and provide optimal position strategies within a Bachelier-type model with a linear price drift. In Section 2, we estimate the value of additional information from an agent's perspective before the information is revealed. Section 3 performs a comparative statics analysis.

1 Closing positions in a Bachelier model with drift

The model set-up

Consider an agent who has to unwind a position of X 0 ∈ R shares of an asset until a time horizon T > 0. We assume that the fundamental asset price is a drifted Brownian motion satisfying the SDE

dS t = a(t, S t )dt + σdW t ,
where σ > 0 is a constant volatility, a : [0, T ] × R → R a measurable drift function and W a Brownian motion on a filtered probability space (Ω, F, (H t ) t∈[0,T ] , P). We assume that there exists C ∈ R + such that |a(t, s)| ≤ C(1 + s) for all t ∈ [0, T ] and s ∈ R + . We interpret the drift as the agent's directional view or extra information about the future price evolution. Moreover, we assume that all prices are forward prices so that no discounting is needed. The filtration H represents the agent's information.

A closing strategy (or simply strategy) of a position x ∈ R at time t ∈ [0, T ) is a (H t )-predictable strategy ξ = (ξ u ) satisfying T t ξ u du = x. We interpret ξ t as the selling rate at time t ∈ [0, T ]. Given ξ, the total position at time t ∈ [0, T ] is given by

X t = X 0 - t 0 ξ s ds.
Notice that X T = 0, i.e. the position is closed at T .

For technical reasons we impose the following integrability condition on the closing strategies: a strategy (ξ u ), resp. its associated position process (X t ), is called admissible if

(A1) the process ξ is L 2 -integrable, i.e E( T 0 ξ 2 u du) < ∞, (A2) the family X * t T -t 2 0≤t≤T
is uniformly integrable, and lim t→T X 2 t T -t = 0, a.s. We denote by A H (t, x) the set of all admissible closing strategies of x at t.

We suppose that any transaction entails a price impact that is linear with respect to the selling rate. Moreover, the impact is assumed to be absolute and only instantaneous. Selling at a rate of ξ t is thus possible only at the realized price of St = S t -ηξ t ,

where η > 0 is the price impact parameter. The final revenues (possibly negative) of the liquidation operation when selling at a rate (ξ t ) t∈[0,T ] are given by

R T = T 0 ξ u Su du.
Notice that by the product formula we have

R T = T 0 ξ u S u du -η T 0 ξ 2 u du = X 0 S 0 + T 0 X u a(u, S u )du + T 0 X u σdW u -η T 0 ξ 2 u du. (1)
Assumption (A2) guarantees that a position process X, associated to an admissible strategy ξ, is square integrable, and thus taking expectations in [START_REF] Almgren | Optimal execution of portfolio transactions[END_REF] we get

E(R T ) = X 0 S 0 + E T 0 (X u a(u, S u ) -ηξ 2 u )du.
We assume that the agent aims at maximizing the expected value of the final revenues. More precisely the target function is given by

J(t, x, s, ξ) = E T t X u a(u, S u ) -ηξ 2 u du X t = x, S t = s , for (t, x, s) ∈ [0, T ] × R × R.
The value function is defined by

V (t, x, s) = sup ξ∈A H (t,x) J(t, x, s, ξ). (2)
Notice that the value function can be interpreted as expected execution costs.

Remark 1 If the impact of the liquidation operation on the price dynamics is not only instantaneous, but lasts in the considered period, one can add to the model a so-called perpetual impact factor, depending on the total amount of the position closed up to time t. The form of the realized price dynamics is then

St = S t -η t -c(X 0 -X t ),
where c is the permanent impact factor. The final revenues in this case are given by

R T = X 0 S 0 + 1 2 cX 2 0 + T 0 X u a(u, S u )du + T 0 X u σdW u -η T 0 ξ 2 u du. (3)
The only difference with Equation ( 1) is the constant term 1 2 cX 2 0 , which does not influence the optimization. From a mathematical point of view the problem is identical.

Optimal closure for linear price drifts

If the price drift coefficient is linear, then the value function turns out to be a quadratic form of both the position size and the price. One can determine the value function and the optimal position process in closed form by using classical stochastic control techniques.

The Hamilton-Jacobi-Bellman Equation associated to the control problem (2) is given by

-V t -a(t, s)V s - 1 2 σ 2 V ss -a(t, s)x -sup ξ∈R [-ξV x -ηξ 2 ] = 0, (4)
with the singular terminal condition

lim t↑T V (t, x, s) = 0, if x = 0, -∞, if x = 0. ( 5
)
The first order condition implies that the supremum on the left hand side of ( 4) is attained by ξ * = -Vx 2η . We obtain a simplified HJB equation

-V t -a(t, s)V s - 1 2 σ 2 V ss -a(t, s)x - V 2 x 4η = 0. ( 6
)
From now on, we suppose that the price drift coefficient is an affine linear function of the form a(t, s) = α(t) + β(t)s, where α and β are functions on [0, T ]. Notice that this implies

S t = h(0, t)S 0 + t 0 h(r, t)α(r)dr + t 0 h(r, t)σdW r ,
where h(r, t) = e t r β(s)ds . Moreover, the value function is a quadratic function of the price and the remaining position, as stated in the following theorem:

Theorem 1 Assume that a(t, s) = α(t) + β(t)s, where α and β are bounded. Then the value function satisfies

V (t, x, s) = b(t)x 2 + c(t)xs + d(t)s 2 + e(t)x + f (t)s + g(t), (7)
where b, c, d, e, f and g are deterministic functions with explicit expressions provided in the appendix. The optimal position trajectory is given by

X * t = T -t T X 0 + 1 2η t 0 [c(u)S u + e(u)] T T -u du . (8)
Theorem 1.2 can be derived from results in [START_REF] Schied | A control problem with fuel constraint and Dawson-Watanabe superprocesses[END_REF]. Nevertheless, we present in the appendix a simple, direct and self-contained proof, based on classical verifications arguments (in contrast to variational arguments used in [START_REF] Schied | A control problem with fuel constraint and Dawson-Watanabe superprocesses[END_REF]).

Informed and non-informed agents

Suppose that there is an expert, e.g. a market analyst or an insider, who has obtained a signal about the asset price at time T . In this section we aim at quantifying the value of the signal from the perspective of the agent having to close the asset position.

For simplicity we suppose in the following that the price process is a Brownian motion without drift; more precisely S t = σW t . This means that S is a martingale with respect to (F W t ), the filtration generated by W . We model the signal as a random variable G = S T + N , where N is independent of the price process and normally distributed with mean zero. Since G is Gaussian, it is equivalent to the signal sent from a price S T , where T ≥ T . One can interpret the difference σ 2 (T -T ) as the variance of the signal's noise.

If the expert discloses the signal to the agent, then we say that the agent is informed. In this case the agent's information flow can be modeled as the following initial enlargement of the Brownian filtration:

G t = F W t ∨ σ(S T ), 0 ≤ t ≤ T.
In case the expert does not pass on the signal, we say that the agent is noninformed. The information flow of the agent is then represented by the natural filtration (F W t ). The value function of the informed agent, conditional to S T , is given by

V I (t, x, s) = sup ξ∈A G (t,x) E T t X u a(u, S u ) -ηξ 2 u du X t = x, S t = s, S T , (9)
and the value function of the non-informed agent by

V N (t, x, s) = sup ξ∈A F (t,x) J(t, x, s, ξ). (10)

A priori and posteriori signal value

One can show that the price dynamics under (G t ) satisfy

dS t = σdW G t + σ S T -S t T -t dt, (11)
where W G is a Brownian motion with respect to (G t ) (see e.g. [START_REF] Mansuy | Random times and enlargements of filtrations in a Brownian setting[END_REF]). The drift in the (G t )-dynamics ( 11) is linear with

α(t) = S T T -t and β(t) = -1 T -t .
If T > T , then α and β are bounded. To put it differently: once the signal is revealed, the informed agent perceives the price with a linear drift satisfying the assumptions of Subsection 1.2. Therefore, we can use Theorem 1 to calculate, conditional to S T , the a posteriori expected execution costs. The a priori costs are given as the expectation of the a posteriori costs.

The next theorem provides the value functions in closed form for both the informed and non-informed agent.

Theorem 2 Let T > T . The value function of the informed agent is a quadratic function as in [START_REF] Bertsimas | Optimal control of execution costs[END_REF]

with coefficients b I (t) = -η 1 T -t c I (t) = - 1 2
T -t T -t d I (t) = 1 48η (T -t) 3 (T -t) 2 e I (t) = 1 2
T -t T -t S T f I (t) = 1 η S T T -t 1 8 (T -T )(T -t) - 1 24 (T -t) 2 + 1 24 (T -T ) 3 T -t g I (t) = S 2 T η 1 12 (T -T ) - 1 16 (T -T ) 2 T -t - S 2 T η 1 8 (T -T ) T -t T -t - 1 24 (T -t) - 1 48 (T -T ) 3 (T -t) 2 + σ 2 48η (T -T ) 3 T -t - 3 2 (T -T ) 2 + 3(T -T ) 2 ln T -t T -T + σ 2 48η -3(T -T )(T -t) + 1 2 (T -t) 2 .
The value function for the non-informed agent is given by V N (0, x, s) = -η x 2 T . Proof The result follows from Theorem 1 and straightforward calculations.

The expected additional gain

We define the difference E V I (0, s, x) -V N (0, s, x) as the expected additional gain of the agent having access to the signal. The next result provides an explicit formula for additional gain.

Theorem 3 Let T > T . The expected additional gain is given by

E V I (0, s, x) -V N (0, s, x) = σ 2 16η (T -T ) 2 ln T T -T -T T + 3 2 T 2 . ( 12
)
The additional gain does not depend on the size of the initial position x. This is because a priori the direction of the signal is not clear. The signal can entail a positive or a negative drift. If the agent has a long position in the asset, then a negative drift entails lower expected revenues and a positive drift yields higher expected revenues. In average, this effect cancels out so that the expected additional gain from the signal is independent of the position size.

Observe further that the additional value does also not depend on the price level s. The reason is that in the underlying Bachelier model the size of the price increments do not depend on the starting price.

In the following we denote the expected additional gain by ∆(T, T ) = E V I (0, s, x) -V N (0, s, x) . Sometimes we write ∆(T, T , σ, η) in order to stress its dependence on σ and η.

The following lemma will be necessary to prove Theorem 3.

Lemma 1

E t 0 W T -W u T -u du 2 = 2t(1 + ln(T )) -2T ln(T ) + 2(T -t) ln(T -t) (13)
Proof The product formula applied to the (G t )-semimartingales

X t = ln(T -t) and Y t = W T -W t , t ∈ [0, T ), implies - t 0 W T -W u T -u du = ln(T -t)(W T -W t ) -ln(T )W T - t 0 ln(T -u)dW u , (14)
and hence

E t 0 W T -W u T -u du 2 = ln 2 (T -t)(T -t) + ln 2 (T )T 2 + t 0 ln 2 (T -u)du -2 ln(T -t) ln(T )(T -t) -2 ln(T ) t 0 ln(T -u)du.
A straightforward simplification of the integrals leads to Equation [START_REF] Karatzas | Anticipative portfolio optimization[END_REF].

Proof (Proof of Theorem 3) By Theorem 1 the optimal strategy of the informed agent satisfies

ξ * t = X * t T -t - 1 4η
T -t T -t (S T -S t ),
and the optimal position trajectory is given by

X * t = T -t T x + 1 4η T t 0 S T -S u T -u du .
The martingale property of the price process implies that the value function satisfies

V I (0, x, s) (15)
= -η

x 2 T + E T 0 3 2 X * t S T -S t T -t - 1 16η t 0 S T -S u T -u du 2 - 1 16η (T -t) 2 (T -t) 2 (S T -S t ) 2 dt Observe that E(X * t (S T -S t )) = (T -t) 1 4η E (S T -S t ) t 0 S T -S u T -u du = (T -t) 1 4η t 0 E [(S T -S t ) ((S T -S t ) + (S t -S u ))] T -u du = σ 2 4η (T -t)(T -t) ln( T T -t ). ( 16
)
Moreover, by Lemma 1,

E t 0 S T -S u T -u du 2 = σ 2 [2t(1 + ln(T )) -2T ln(T ) + 2(T -t) ln(T -t)]. (17)
Combining Equation ( 16) and ( 17) with ( 15) yields

E[V I (0, x, s)] = -η x 2 T + 3σ 2 8η T 0 (T -t) ln( T T -t )dt - σ 2 8η T 0 [t(1 + ln(T )) -T ln(T ) + (T -t) ln(T -t)]dt - σ 2 16η T 0 (T -t) 2 (T -t) dt. ( 18
)
Notice that

T 0 (T -t) ln(T -t)dt = 1 2 (T -T ) 2 ln(T -T ) + T T ln(T ) - 1 2 (T ) 2 ln(T ) + 1 2 T T - 3 4 T 2 and T 0 (T -t) ln(T -t)dt = - 1 2 (T -T ) 2 ln(T -T ) + 1 4 (T -T ) 2 + 1 2 (T ) 2 ln(T ) - 1 4 (T ) 2 and T 0 (T -t) 2 (T -t) dt = (T -T ) 2 ln T T -T -2(T -T )T + T T - 1 2 T 2 .
A straightforward calculation shows that (18) simplifies to

E[V I (0, x, s)] = -η x 2 T + σ 2 16 (T -T ) 2 ln T T -T -T T + 3 2 T 2 .
Remark 2 One can alternatively calculate the additional utility by computing the expectation of the coefficients e I (0), f I (0) and g I (0). By simplifying terms one obtains again formula [START_REF] Ihara | Information theory for continuous systems[END_REF].

The expected additional gain ∆(T, T ) converges to a finite value as T ↓ T . If T = T , then the market would admit arbitrage if there was no price impact. It has been shown that an informed investor can achieve infinite expected utility in a frictionless market (see e.g. [START_REF] Karatzas | Anticipative portfolio optimization[END_REF] and [START_REF] Grorud | Insider trading in a continuous time market model[END_REF]). In our model, in contrast, the price impact excludes arbitrage and implies that the expected additional gain doesn't become infinite when T is equal to T .

Notice that if we choose T = T , then the drift in the (G t )-price dynamics [START_REF] Hillairet | Comparison of insiders' optimal strategies depending on the type of side-information[END_REF] is not bounded, and hence the assumptions of Theorem 1 are technically not satisfied. Nevertheless, one can show that the result applies also to this particular case. To this end one needs to make sure that the candidate for the optimal control is admissible.

Proposition 1 Suppose that T = T . Then the expected additional gain of the informed agent is given by

E[V I (0, s, x) -V N (0, s, x)] = σ 2 32η T 2 . ( 19
)
The optimal strategy is admissible and the associated position process satisfies

X * t = T -t T x + 1 4η T t 0 S T -Su T -u du .
Proof The first expression is obtained by taking the limit in ∆(T, T ) as T ↓ T .

To prove the admissibility, notice first that using Equation ( 14) for any p > 2, there exists C p such that

E | t 0 S T -S u T -u | p ≤ C p    ln p (T -t)(T -t) p/2 + ln p (T )T p/2 + T 0 ln 2 (T -u)du p/2    This shows that t 0 S T -Su T -u du 2
is uniformly integrable. We further obtain that

X * t T -t 2 , 0 ≤ t ≤ T , is uniformly integrable and lim t→T X * t 2
T -t = 0, a.s. Moreover the process ξ * is squared integrable.

Comparative Statics

We next analyze the impact of the model parameters on the additional gain. We start with the dependence on the signal quality.

Sensitivity with respect to the signal noise

If the noise of the signal increases, then the additional revenues of the informed agent decrease. This is indeed confirmed by the next result.

Lemma 2

The expected revenues from additional information decrease as T increases, i.e. the mapping

f (x) = (x -T ) 2 ln x x-T -T x + 3 2 T 2 is decreasing on [T, ∞). Moreover, f (T ) = 1 2 T 2 and lim x→∞ f (x) = 0. Proof Notice that f (x) = 2(x -T ) ln x x-T + T 2 x -2T and f (x) = -2 ln 1 - T x - 2T x - T 2 x 2 .
Since the logarithm is analytic on the open interval (0, 2), we further have for

x > T f (x) = 2 T x + 1 2 T 2 x 2 + 1 3
T 3 x 3 + • • • - 2T x - T 2 x 2 = 1 3
T 3 x 3 + • • • ≥ 0.
Consequently f is increasing on [T, ∞). Besides observe that f (T ) = -T , and

lim x→∞ f (x) = 2(x -T ) T x + 1 2
T 2 x 2 + 1 3
T 3 x 3 + • • • - 2T x - T 2 x 2 = 0,
which, together with the monotonicity of f , implies f ≤ 0 on [T, ∞). The function f , therefore, is decreasing in x.

Sensitivity with respect to the time horizon

The additional gain increases when the time horizon T increases, while T stays constant. Indeed, a straightforward computation shows that

∂ 2 ∆ ∂T 2 = 2 ln T T -T ≥ 0 for all T ∈ [0, T ].
Hence the first derivative is increasing. Since ∂∆ ∂T (0, T ) = 0, the first derivative is non-negative and hence ∆ is increasing in T .

The increase in expected revenues has three reasons: first the signal becomes more valuable as the difference between T and T decreases (information effect); second there is more time for spreading orders over time and hence one can reduce trading costs (liquidity effect); finally the variance of the price over the trading period increases (variance effect).

We next aim at analyzing the three effects separately. The additional revenues depend linearly on the volatility squared. We can thus eliminate the variance effect by making σ 2 inversely proportional to T . We define the variance corrected gain by

l(T, x) = ∆(T, x, σ/ √ T , η) = σ 2 16η (x -T ) 2 T ln x x -T -x + 3 2 T , for 0 ≤ T ≤ x.
We next aim at analyzing the part of revenue increase that goes back to the liquidity effect. To this end we simultaneously change T and T such that the information content of the signal remains the same. We appeal to the notion of mutual information for measuring the information content of the signal.

Recall that the mutual information between two normally distributed random variables X and Y is given by I(X, Y ) = - 1 2 ln(1 -corr 2 (X, Y )) (see e.g. [START_REF] Ihara | Information theory for continuous systems[END_REF]). In particular, for any δ > 0 we have I(S T , S T +δ ) = 1 2 ln T +δ T . For γ > 0 the mutual information I(S T , S (γ+1)T ) = 1 2 ln (1 + γ) does not depend on the time horizon T . We can thus interpret h(T ) = l(T, (1 + γ)T ) as a variance and information (v&i) corrected gain function. The next proposition shows that the v&i corrected gain increases as the time horizon increases. The reason is that the additional time for trading allows to reduce liquidity costs and to make more use of the information advantage.

Proposition 2 (The liquidity effect) Let γ > 0. The v &i corrected gain function h is linear, increasing and satisfies h(0) = 0.

Proof Note that 16η σ 2 h(T ) = γ 2 T ln 1 + γ γ -γT + 1 2 T = γ 2 1 γ - 1 2
1 γ 2 + 1 3 1 γ 3 - 1 4 1 γ 4 + • • • -γ + 1 2 T = γ 2 1 3 1 γ 3 - 1 4 1 γ 4 + • • • T,
which shows that h is non-negative and linearly increasing in T .

Finally we turn to the information effect. By scaling the volatility with 1/ √ T and the price impact parameter with T , we obtain a variance and liquidity (v&l) corrected gain function

(x) = 1 2 σ 2 16η . Proof k (x) = 2T 3T -2x x 4 ln( T T -x ) -6T x 3 + 1
x 2 is positive and k (0) ≤ 0, which implies the first statement. The second is straightforward to show.

Conclusion

The paper studies the optimal liquidation problem under additional information. The kind of additional information chosen here is modeled via an initial enlargement of filtration, sometimes referred to as strong initial information (see for example [START_REF] Amendinger | Additional logarithmic utility of an insider[END_REF], [START_REF] Grorud | Insider trading in a continuous time market model[END_REF], or [START_REF] Eyraud-Loisel | Backward stochastic differential equations with enlarged filtration. option hedging of an insider trader in a financial market with jumps[END_REF] for an introduction into the subject. See also [START_REF] Hillairet | Comparison of insiders' optimal strategies depending on the type of side-information[END_REF] and [START_REF] Baudoin | Modeling anticipations on financial markets[END_REF] for a presentation of other types of additional information). Since the price dynamics under the enlargement have a drift that is linear with respect to the price, we obtain the optimal liquidation strategy in closed form. For any kind of additional information resp. filtration enlargement under which the drift is linear, one can derive explicitly the additional gain by using Theorem 1. For example the additional information studied in the paper of Corcuera et al. [START_REF] Corcuera | Additional utility of insiders with imperfect dynamical information[END_REF] (strong noisy information, represented by a signal plus a decreasing noise) leads to a linear drift, and hence fits to our model. Acknowledgement The authors thank anonymous referees for very useful comments.

T t β(u) T -u T h(0, u)du, d hom (t) = exp - t 0 2β(u)du d(t) = d hom (t) T t c 2 (u) 4ηd hom (u) du e(t) = T T -t T t (c(u) + 1)α(u) T -u T du, f hom (t) = exp - t 0 β(u)du f (t) = f hom (t) T t 1 2η c(u)e(u) + 2α(u)d(u) 1 f hom (u) du g(t) = T t e 2 (u) 4η + α(u)f + σ 2 d(u) du, with t ∈ [0, T ].
We remark that Theorem 1 can be derived from Theorem 2 in [START_REF] Schied | A control problem with fuel constraint and Dawson-Watanabe superprocesses[END_REF]. The proof given below is completely different though, using classical verification arguments.

Proof (Proof of Theorem 1) Let w(t, x, s) = b(t)x 2 + c(t)xs + d(t)s 2 + e(t)x + f (t)s + g(t). We first show that the value function satisfies V ≤ w. Notice that w is a solution of the HJB Equation ( 4) and satisfies the terminal condition [START_REF] Ankirchner | Financial markets with asymmetric information: information drift, additional utility and entropy[END_REF]. This follows from the fact that the coefficients satisfy the following ODEs (-a(u, S u )X u + ηξ 2 u )du = J(t, x, s, ξ).

-b t - 1 η b 2 = 0 -c t - 1 η bc -βc -β = 0 -d t - 1 4η c 2 -2βd = 0 -e t - 1 η be -αc -α = 0 -f t - 1 2η ce -2αd -βf = 0 -g t - 1
Moreover, since also

X 2 t T -t t∈[0,T )
is uniformly integrable and lim t→T X 2 t T -t = 0, we have lim τ →T E[w(τ, X τ , S τ )] = 0. Inequality (21), therefore, implies w(t, x, s) ≥ J(t, x, s, ξ). Taking the supremum over all admissible controls, one has V (t, x, s) ≤ w(t, x, s).

Secondly, we show that the control (ξ * t ) t∈[0,T ] is admissible. Using the majoration ( 20 21) by choosing ξ = ξ * . This proves that J(t, x, s, ξ * ) = w(t, s, x). Thus the proof is complete.

Proposition 3 (

 3 k(y) = ∆(y, T , σ/ √ y, ηy), defined for all y ∈ [0, T ]. The function k describes the gain that exclusively goes back to the additional information, as T approaches T . It is, as expected, increasing: The information effect) The v &l corrected gain function k increases superlinearly on [0, T ] and satisfies lim x↑T k

4η e 2 - 1 2 σ 2 w

 2122 αf -σ 2 d = 0.Since the functions α and β are bounded, there exists a constant C ∈ R + such that|c(t)| + |d(t)| + |e(t)| + |f (t)| + |g(t)| ≤ C(T -t) (20) for all t ∈ [0, T ]. Moreover, we have |b(t)| ≤ C 1 T -t . Let ξ ∈ A(t,x) be an arbitrary admissible control and let X be its associated position process. Let τ < T . Itô's formula impliesw(τ, X τ , S τ ) = w(t, x, s) + τ t ss (u, X u , S u )du + M τ + τ t [w t (u, X u , S u ) -w x (u, X u , S u )ξ u + a(u, S u )w s (u, X u , S u )]du,where M s = s t w s (u, X u , S u )σdW u . As (X t ) t∈[0,τ ] is L 2 -bounded and all functions b, c, d, e, f, g and their derivatives are bounded on [t, τ ], M is a strict martingale on [t, τ ]. Taking expectations, therefore, leads toE(w(τ, X τ , S τ )) = w(t, x, s) + E τ t w t -w x ξ + aw s + 1 2 σ 2 w ss (u, X u , S u )du ≤ w(t, x, s) + E τ t (-a(u, S u )X u + ηξ 2 u )du . (21)As ξ is square integrable (Condition (A1)). This further implies that we have lim

  ) on the coefficients c, b and e, one can show that there exists a constant C such that|[c(u)S u + e(u)]| T (T -u) ≤ C(|S u | + 1)for all u ∈ [0, T ]. With[START_REF] Corcuera | Additional utility of insiders with imperfect dynamical information[END_REF] we obtain that|X * t | ≤ C(T -t)(1 + t 0 |S u | du) and hence Condition (A2) is satisfied. Condition (A1) is a consequence of ξ 2 t ≤ C(b(t) 2 X 2 t + b(t)X t ) ≤ C.Equality holds in Inequality (

The first author gratefully acknowledges the financial support by the Ecole Centrale de Lyon during his visit in March 2012.

Appendix

The coefficients of the value function in Theorem 1 are given by b