Mélanie Jaeck

Robert Lifran

Hubert Stahn

Emergence of organic farming under imperfect competition: economic conditions and incentives

Keywords: agricultural inputs, organic farming, imperfect competition, technological choice, free entry, policy design JEL Classi…cation: Q12, L13

This article explores the economic conditions for the viability of organic farming in a context of imperfect competition. While most research dealing with this issue has adopted an empirical approach, we propose a theoretical foundation. Farmers have a choice between two technologies, the conventional one using two complementary inputs, chemicals and seeds, and the organic one only requiring organic seeds. The upstream markets are oligopolistic and the …rms adopt Cournot behavior. The game is solved backward. The equilibrium repartition of the farmers between both sectors is obtained by a free entry condition.Since multiple equilibria could exist, including the non emergence of organic farming, we spell out viability conditions for organic farming. Then, using an "infant industry" argument, we propose several public policy instruments able to support the development of organic farming, and assess their relative e¢ ciency. Results could be usefull to asses the conditions of emergence and viability of agricultural innovations in analogous contexts.

Introduction

Organic agriculture is de…ned by the International Federation of Organic Agriculture Movements (IFOAM) as "a production system that sustains the health of soils, ecosystems and people".

"It relies on ecological processes, biodiversity and cycles adapted to local conditions, rather than the use of inputs with adverse e¤ects. Organic agriculture combines tradition, innovation and science to bene…t the shared environment and promote fair relationships and a good quality of life for all involved". Thus, the ambition of organic farming is to accommodate agricultural production and the consumers'interest, by limiting the impact of agriculture on the environment. While some experts express doubt about the e¢ ciency of organic farming, several studies show sustained interest and willingness of consumers to pay for organic products (Boccaletti and Nardella, 2000, Dimitri and Richman, 2000, [START_REF] Batte | Putting their money where their mouths are : Consumer willingness to pay for multi-ingredient, processed organic food products[END_REF]. That paradox makes the study of the conditions of emergence of organic farming a real challenge for research [START_REF] Park | Supply and Demand Factors for Organic Produce[END_REF] . Most research dealing with this subject adopts an empirical approach, and focus mainly on farmers and farms characteristics [START_REF] Burton | Modelling the adoption of organic horticultural technology in the UK using Duration Analysis[END_REF], Wheeler, 2008, Wynen and Edwards, 1990). Very little research has been performed on the type of policy instruments able to enhance this emergence (Dimitri andOberholtzer, 2005, Eerola and[START_REF] Eerola | Voting for Environmental Policy under Income and Preference Heterogeneity[END_REF].

Of course, farms characteristics matter. Considering [START_REF] Kle¤er | Economic Performance and Energy Intensiveness on Organic and Conventional Farms in the Corn Belt: a preliminary comparison[END_REF], Oude Lansink et al. (2002), [START_REF] O¤erman | Economic performance of organic farms in Europe. Organic farming in Europe[END_REF], [START_REF] Mayen | Technology Adoption and Technical E¢ ciency: organic and conventional dairy farms in the United States[END_REF], we must conclude that the organic sector is less productive than the conventional one. As a consequence, the emergence of an organic sector is only possible if the price of the organic products are not too close to the conventional one and/or if there is some mechanism that compensates this productivity gap [START_REF] Mayen | Technology Adoption and Technical E¢ ciency: organic and conventional dairy farms in the United States[END_REF]. This is why we incorporate two basic features that are often associated to organic farming: A "learning-by-doing" process and the existence of a "niche market" for the corresponding products.

The …rst feature is borrowed from [START_REF] Hanson | Organic versus conventional grain production in the mid-Atlantic: An economic and farming system overview[END_REF], [START_REF] Martini | Yield increases during the organic transition: Improving soil quality or increasing experience?[END_REF] and [START_REF] Sipiläinen | Learning in organic farming-An application on Finnish dairy farms[END_REF]. It relies on the idea that the adoption of organic production requires speci…c knowledge or at least some early experiments performed by innovators. [START_REF] Sipiläinen | Learning in organic farming-An application on Finnish dairy farms[END_REF] estimate technical e¢ ciency of organic farming and its development over time in Finnish dairy farms. They conclude that "the average e¢ ciency at …rst decreases (when the conversion towards organic farming starts) but at a decreasing rate, and turns then after 6-7 years to an increase" suggesting "learning e¤ects related to the experience in organic farming".

It is well documented that some consumers are willing to pay more for organic food ( It is therefore quite obvious that organic farmers do not produce, say, for a worldwide market but address more local markets in which they meet speci…c consumers with a higher willingness to pay. But unfortunately, and contrary to the learning e¤ect, this additional pro…t opportunity decreases with the number of organic farmers because the quantity supplied to this "niche" market simply increases.

However, farming decisions are not only based on farm constraints and farmers' preferences [START_REF] Jaeck | Preferences, norms and constraints in farmers agro-ecological choices. Case study using choice experiments survey in the Rhone River Delta, France', Montpellier, LAMETA[END_REF] but rely also on the characteristics of the marketing channels in which the farm is involved. That encompasses the set of relationships with both upstream and downstream …rms. This literature underlines the oligopolistic and oligopsonistic structure of industrial food market, the implications in terms of price transmission along the marketing channel and the pro…t capture realized by the upstream …rms (see for instance McCorriston et al., 1998, Rogers and Sexton, 1994, Saitone, Sexton and Sexton, 2008, [START_REF] Weldegebriel | Imperfect Price Transmission : Is Market Power Really to Blame[END_REF].

In this paper we focus on the behavior of upstream the input providers who are usually recognized as acting as an oligopoly [START_REF] Fulton | Agricultural Biotechnology and Industry Structure[END_REF]Giannakas, 2001, Hayenga, 1998). This peculiar market structure is induced by the strategic behavior of upstream …rms, and their interest in merging or in vertical integration [START_REF] Fulton | Agricultural Biotechnology and Industry Structure[END_REF][START_REF] Johnson | Strategic Behavior and Consolidation in the Agricultural Biotechnology Industry[END_REF][START_REF] Shi | Bundling and Licensing of Genes in Agricultural Biotechnology[END_REF].

Moreover, [START_REF] Just | Multimarket Exploitation: The Case of Biotechnology and Chemicals[END_REF] show that the joint supply of complementary goods by a unique …rm will be larger than the one proposed when each of the two goods are supplied separately.

That arises because of the increasing cross marginal revenue. This is why we assume that the agricultural inputs suppliers propose seeds and chemicals simultaneously to all farmers. As a consequence, they have a great in ‡uence on the adoption of the technological package by the farmer. To be more precise, we present a model in which the two agricultural inputs: seeds and chemicals, are complementary, and are jointly sold by upstream …rms. For the conventional sector, …rms supply the two goods as a "bundle", as presented by [START_REF] Shi | An Analysis of Bundle Pricing: The case of the Corn Seed Market[END_REF], and Shi (2009), while, for the organic sector, they provide only speci…c seeds without chemicals.

Given this particular context of imperfect competition, our paper attempts to characterize the conditions of emergence and viability of organic farming. We propose a three step game.

In the …rst step the farmers choose their mode of production by implementing either organic or conventional farming. This choice is based on the comparison of the expected return of each technology and by the potential learning by doing e¤ect. Moreover, since there is free entry in each sector, an equilibrium distribution is reached as soon as no farmer wants to change his mode of production. The equilibrium that occurs at this stage provides some insights on the condition of the emergence of organic farming. In the second step, the input providers choose the amount of chemical-free seeds and quantity of the bundle of seeds and chemical they want to sell on these two input markets. The transactions on this two markets result from a Cournot equilibrium in which these downstream …rms take into account the pro…t they can capture from both sectors.

Finally, in step three, farming takes place and the products are sold either on a "niche" market for organic farmers or at the current worldwide price for conventional farming. Since we seek a Nash equilibrium we solve this game backwards.

By solving this game, we also gain more insights on the conditions of the emergence and the development of the new technology. This is why we also analyse the set of instruments a policy maker would implement to boost the emergence of organic farming. Supporting for organic farming emergence could arise from an argument that it is an "infant industry" to be protected from rent capture by upstream oligopoly trough their power market. Competition enhancing policy could also be invoked and social welfare enhancing arguments could legitimate the support of "environnemental friendly technologies" [START_REF] Eerola | Voting for Environmental Policy under Income and Preference Heterogeneity[END_REF]. However, imperfect competition places speci…c constraints on the design of the instruments. We will assume that the regulator cannot signi…cantly control the degree of competition among the upstream …rms, and that he will contemplate only "conventional" instruments : a tax on chemicals, subsidies to organic seeds, subsidies to the production of organic products and actions to speed up the learning process about the new technology.

As the imperfect competition context appears to be widespread in the agricultural sector, the conclusions of our study about the emergence of organic farming could be relevant for all situations where a new technology and the corresponding market compete with the conventional one.

The paper is organized as follows. In section 2, we present the model, its assumptions and solve the three step game. Section 3 is devoted to the study of quantity ‡ows, i.e. the production of the farmers and the equilibrium level of inputs, for a given distribution of the farmers between both sectors. In section 4, we study the condition of the emergence of organic farming by studying the properties of our free entry equilibrium. Section 5 addresses some public policy issues (subsidies for organic farmers, taxes on chemicals etc.) and their role in easing or blocking the emergence of organic farming. Section 6 contains concluding remarks. Proofs which are not central to the argument are relegated to an appendix.

The model

Consider an economy in which the agricultural sector is composed of two types of farmers. The …rst type, called conventional, produces a generic product dedicated to a large market. The second, called organic, produces a speci…c chemical-free product and targets a niche market. Both buy seeds or a bundle of seeds and chemicals from a small number m of upstream …rms which exert some market power. Within this structure, each farmer (within the total number N ) will choose either classical or organic farming. We denote by n the number of organic farmers.

Within the conventional sector, seeds and chemicals are complementary inputs. From that point of view, we assume that the upstream …rm typically sells, at price p b , a bundle in which there is a …xed proportion of chemicals to seeds. Hayenga (1998) presents a linked seed and chemicals market, and concludes that the strategy of input providers is "to tie the seed customer more closely to the chemical product". We also assume that the quantity of land is given, and that the farmer allocates all his working time to the agricultural activity. He is not constrained by water availability or others inputs. We can therefore reduce the production function to a unique input: the amount of conventional seeds s c . We denote this function by f (s c ) and assume as usual that this function is increasing and exhibits decreasing return to scale, i.e. f 0 (s) > 0 and f "(s) < 0, satis…es the Inada conditions, i.e. lim s!0 f 0 (s) = +1; lim s!+1 f 0 (s) = 0 and does not allow "free lunch", i.e. f (0) = 0. We also introduce two additional assumptions: the elasticity e f 0 (s) of f 0 remains bounded, the elasticity e f " (s) of f " is larger than 21 .

We …nally state that the output of the conventional sector is sold on a large, competitive and perhaps worldwide market, at a given price p c . This simplifying assumption gives us the opportunity to treat the conventional farmers as pure competitive players and to mainly focus on the interaction with their suppliers.

The organic sector, by contrast, does not use chemicals and uses chemical-free seeds at price p s . We again assume that the production function of this sector depends only on the amount of seeds used s o . This production function is quite the same as the one for the conventional sector, in the sense that, without chemicals, the marginal productivity is reduced by some factor 2 [0; 1], and is given by f (s o ). Consistent with Rouvière and Soubeyran (2011), the emergence of organic production is constrained by two balancing e¤ects, a "learning-by-doing" e¤ect, and a "niche market" e¤ect".

The "learning-by-doing" e¤ect implies that the productivity gap between both sectors is decreasing with the number n of farmers who adopt organic production. In other word, we assume that (n) is increasing with the number n of adoptors. Moreover it also seems quite reasonable to assume that marginal contribution of a new entrant is decreasing with the number of participants in the organic sector, and perhaps even disappears if all farmers choose organic farming. In other words, we assume2 that 0 (n) > 0, "(n) < 0 and lim n!N 0 (n) = 0. We nevertheless maintain the idea that (N ) < 1: organic farming remains less productive than conventional farming. Thus, this cannot justify the emergence of an organic agricultural sector per se.

The "niche market" e¤ect is related to the fact that some consumers are willing to pay more for organic products. Since we essentially concentrate on the supply side we do not explicitly model this behavior. We simply assume that the price p(n) at which farmers sell their organic products depends on the number of adoptors and is, at least for the …rst mover, attractive enough, i.e. p(0) > p c . However, because we wish to capture the idea that we are on a "niche market", we also assume that this potential advantage decreases with the number of farmers producing organic products, and even at a increasing rate. For this reason, we require p 0 (n) < 0, p"(n) < 0 and that lim n!0 p 0 (n) = 0:

Following Fulton and Giannakas (2001) and [START_REF] Hayenga | Structural change in the biotech seed and chemical industrial complex[END_REF], the upstream input providers, indexed by j = 1; : : : ; m, are assumed to wield signi…cant market power on the input markets. We distinguish two markets: one for organic seeds and one for bundles of seeds and chemical since these two inputs enter in a …xed proportion in the conventional production function. Each …rm delivers both inputs by taking as given the quantities provided by the other …rms. We denote by s o j and s c j the the amount of organic seeds and of the bundle chosen by …rm j. We …nally assume that these two goods are produced at a constant marginal cost, and we denote by c 0 and c b respectively for the organic seeds and the bundle of seeds and chemicals.

Moreover we assume that c o < c b , which means that the production cost of organic seed is lower than the production cost of a bundle composed of conventional seeds and chemicals. This gives, of course, a competitive advantage to organic farming, but one has to keep in mind that these input providers do not sell their product at the marginal cost : they try to capture a part of the farmers'pro…ts.

The timing of the game in this Cournot Oligopoly context is quite usual. As in a standard entry model, farmers decide …rst wether they want to produce organic products or develop a conventional farming activity. Since entry is free, this choice is simply driven by the comparison of the expected pro…ts of moving from one sector to another. In the second step, the upstream …rms choose their optimal supply for both kind of seeds, having in mind that they deliver a bundle of seeds and chemicals to the conventional farmers and anticipate the impact of their strategic choice on the price of both products. In a third and …nal step, the conventional, as well as the organic farmers choose competitively the amount of organic seeds and the bundle of conventional seeds and chemicals they want to use. We seeking a subgame perfect equilibrium of this game. This allows us to identify the conditions inducing the existence of an organic farming sector, and to design the public policy rules supporting the development of organic farming.

The equilibrium of the inputs sector

In this section, we take the distribution of the farmers between the two sectors as given and look at the quantity of organic and conventional seeds that are traded. In other words, we focus on the last two steps of the game. This gives us the opportunity to compute the pro…ts realized by each player and to prepare the discussion on the emergence of an organic production sector.

Let us …rst begin with the competitive behavior of the two types of farmers. A standard pro…t maximizing condition tells us that each farmer purchases seeds until his marginal productivity is equal to the purchase price. Those conditions are written as:

( p(n) ( (n) f 0 (s o )) = p s p c f 0 (s c ) = p b (1)
respectively for organic and conventional farmers. Keeping in mind that all farmers are symmetric within each sector, we immediately obtain the following inverse demand functions:

P s (S 0 ; k(n)) = k(n) f 0 S o n (2)
P b (S c ; p c ) = p c f 0 S c N n (3)
where S 0 and S c stand for the aggregated demand for organic and conventional seeds and k(n

) := p(n) (n).
In the Cournot game context, the input providers set their optimal supply of organic seeds s j o and the bundle of conventional seeds and chemicals s j c in such a way to maximize their pro…ts. In other words, a Nash equilibrium of this game is given by: 8j = 1; : : : ; m

sj o ; sj c 2 arg max (s j o ;s j c) P s m X j=1 s j o ; k(n) ! c o ! s j o + P b m X j=1 s j c ; k(n) ! c b ! s j c (4)
This yields the following …rst order conditions:

8j = 1; : : : ; m 8 < : k(n) f " 1 n P m j=1 s j o s j o n + k(n) f 0 1 n P m j=1 s j o c o = 0 p c f " 1 N n P m j=1 s j c s j c N n + p c f 0 1 N n P m j=1 s j c c b = 0 (5)
Moreover, under the technical assumption (the elasticity e f " (s) of f " is larger than 2), these conditions are necessary and su¢ cient for optimality (see appendix A).

If markets clear at a Cournot equilibrium and farmers are symmetric within each sector, we can say that 1 n P m j=1 s j o and 1

N n P m j=1 s j c are, respectively, the amount of seed s o and s c used by an organic and a conventional farmer at the Cournot equilibrium. If we carry out this change of notation, we immediately observe from equation ( 5) that the equilibrium production levels are identical for each input provider and are given by:

8j = 1; : : : ; m s j o ; s j c = n f 0 (s o ) co k(n) f " (s o ) ! ; (N n) f 0 (s c ) c b pc f " (s c ) !! (6)
Summing up these quantities over all input providers and again making use of the previous market clearing conditions, a Cournot equilibrium of the input providing game can be obtained by simply solving for (s o ; s c ) the following system:

(

1 m f "(s o ) s o + f 0 (s o ) = co k(n) 1 m f "(s c ) s c + f 0 (s c ) = c b pc (7)
Under our assumptions, the conclusion follows: Lemma 1 This system has a unique solution for (s o ; s c ). Thus there exists a unique Cournot equilibrium of the input provider game.

The previous Lemma is a rather technical (but necessary) result on the existence and unique-ness of a solution. It allows us to fully characterize the quantities that are traded and even to construct the pro…t of the farmers and the input providers for any distribution of farmers between the organic and the conventional sectors.

In fact, by equation ( 7), we know that the equilibrium demand for seeds of an organic and a conventional farmer can be described by two functions,

(n) = (f 0 ) 1 co k(n) and s c c b pc = (f 0 ) 1 c b pc :
Recalling that each farmer behaves competitively by adjusting the marginal gain obtained from the seeds to its price (see equation 1), we can easily compute the pro…t of each type of farmers.

These pro…t functions are given by:

8 < : o (k(n); c o ; m) = k(n) h f (s) f 0 (s) sj so( co k(n) ;m) i = k(n) (s)j so( co k(n) ;m) c (p c ; c b ; m) = p c h f (s) f 0 (s) sj sc( c b pc ;m) i = p c (s)j sc( c b pc ;m) (8)
We also observe that pro…ts are non negative since for all neoclassical production functions f (s) the marginal productivity is always lower than the average productivity3 so that (s) := f (s)

f 0 (s) s 0.
In the same vein, we can also compute from lemma 1 the quantities of organic and conventional seeds sold by each input provider. By rearranging equation ( 6) these quantities are given by: 8 < :

s j o co k(n) ; m; n = n m s o co k(n) ; m s j c c b pc ; m; n = (N n) m s c c b pc ; m (9)
and the pro…t of each seed provider is given by:

(k(n); c o ; p c ; c b ; m; n) = 1 m 2 0 @ n k(n) h f " s o co k(n) ; m i s o co k(n) ; m 2 +(N n) p c h f " s c c b pc ; m i s c c b pc ; m 2 1 A (10)
Those results are, in some sense, usual. In fact, following Saitone, Sexton and Sexton (2008), we observe that the introduction of imperfect competition among the upstream sellers in the seed sector has important distributional impacts. Upstream market power (measured by the inverse of the number m of input providers) classically reduces the amount of seeds used in both the organic and the conventional sector (see lemma 2) with respect to a competitive situation. It also modi…es the pro…ts distribution because the input providers are able to capture a part of the pro…t of the farm contrary to a pure competitive situation in which constant returns to scale typically reduce their pro…t to zero. Of course, this e¤ect disappears when the number of input providers becomes large. In that case, the quantities traded converge to the competitive outcome (see lemma 2) and the pro…t of the input providers goes to zero (see equation 10).

The free entry equilibrium

We now move to the issue of the distribution of the di¤erent farmers between organic and conventional activity and to the conditions that ensure the emergence of organic farming as a plausible alternative to conventional agriculture. Moreover, this will give us the opportunity to assess, in the next section, some public policy intervention that sustain the development of organic farming.

The free entry equilibrium distribution

We must …rst de…ne an equilibrium concept in order to construct this distribution. Under free entry, the equilibrium distribution of farmers between both sectors is reached if no farmer (expecting higher return) is willing to move to the other sector. The free entry condition is quite simple to de…ne since the pro…t of the conventional farmer (see equation 8) is independent of the number of organic farmers. This means that an equilibrium distribution is reached for a n with the property that :

( o (k(n ); c o ; m) c (p c ; c b ; m) c (p c ; c b ; m) o (k(n + 1); c o ; m) (11)
In this case, no organic farmer is willing to turn into a conventional one and reciprocally no conventional farmer is willing to change his activity. If, for the sake of simplicity, we consider n as a continuous variable, this means that we have to …nd an n satisfying the following two properties:

( o (k(n ); c o ; m) = c (p c ; c b ; m) o (k(n ); c o ; m) is decreasing at n : (12)
If we want to study this equilibrium distribution, it becomes important to investigate the behavior of the pro…t of organic farmers when n changes. By computing this partial derivative, we obtain:

@ o @n = k 0 (n) " (s)j so( co k(n) ;m) d ds so( co k(n) ;m) ! @s o @ (c 0 =k(n)) c o k(n) # (13)
We also know that: We can therefore assert that the sign of @ o @n is the same as the sign of k 0 (n). In other words, the fact that o (k(n); c o ; m) is decreasing or not with n is essentially explained by the interaction of the progressive learning process (n) and the constant erosion of the advantage due to the "niche" market measured by p(n). If we now have in mind that the …rst e¤ect has decreasing return with the number of farmers who adopt organic farming, i.e. "(n) < 0; while the erosion of the niche bene…ts increases with the numbers of organic farmers, i.e. p 0 (n) < 0 and p"(n) < 0., we can expect that: Lemma 2 The pro…t function of an organic farmer is \-shaped in n. It is …rst increasing because of the gain from the learning process, then dominated by the losses induced by the erosion of the price in the niche market. The learning e¤ect works up to a critical number n max , while the erosion of the price will dominate after that number.

This \-shaped pro…t function has several consequences on the emergence of organic farming. First, if, at the critical number n max , this pro…t is lower than the returns obtained in the conventional sector, i.e.

o (k(n max ); c o ; m) < c (p c ; c b ; m) (14)
an organic farmer sector will never emerge. This situation occurs simply because the maximal gain induced by the learning process never compensate the productivity losses speci…c to organic farming. If condition 14 is not met, this nevertheless does not imply that organic farming occurs for sure. Because of this \-shaped property, it may happen that the pro…t of the …rst farmer which moves to organic farming remain lower than the returns of conventional agriculture, i.e.

o (k(1); c o ; m) < c (p c ; c b ; m) (15)
In this case, we typically have two equilibrium distributions :

one with both organic and conventional farming (or even only organic

farming if o (k(N ); c o ; m) is greater than c (p c ; c b ; m)).
one in which only conventional farming occurs simply because the …rst mover to organic farming, which bene…ts from any learning e¤ect, is not able to compensate his productivity losses by the additional gain induced by the "niche" market.

Finally if both conditions ( 14) and ( 15) are not satis…ed we can expect that there exists a unique

The emergence of organic farming under imperfect competition

In this subsection, we go a step further in the understanding of the emergence of organic farming by identifying su¢ cient conditions that bring together the costs of the up-stream seed providers, the prices for organic and conventional products and the learning by doing e¤ect.

Let us …rst start with the simplest case in which the organic farming sector has no competitive advantage with respect to the conventional one, i.e.

8n;

c o p(n) (n) > c b p c (16)
One may typically expect that organic farming never occurs. The mechanism which leads to this situation is however di¤erent from a competitive one and is driven by the equilibrium behavior of the up-stream imperfectly competitive …rms. Since they are aware of this competitive advantage, they a quantity strategy that favors conventional farmers. This follows directly from equation 7 and equations 2 3: under condition ( 16) and for any distribution n of the farmers between both sector, they are willing to sell fewer seeds at a higher price to organic farmers. This immediately reduces the pro…t opportunity of organic farming. Moreover, since we have assumed that c o < c b , equation 16 implies that 8n, p(n) (n) < p c , i.e. the combination of the learning-by-doing and the "niche" market e¤ect never reaches the price level for conventional products. This last drawback on the output market de…nitively relegates organic farming.

At contrario, is tempting to think that if the organic farmers have, for at least one distribution n, a competitive advantage then organic farming is a potential outcome. This intuition, due to imperfect competition, is again wrong. In this case, even if the up-stream seed providers now favor the organic farmers by selling more seeds at a lower price, they also capture a larger part of their pro…ts (see equation 10 by having in mind that e f " (s) > 2). From that point of view, we have, in our general setting 4 , to make sure the organic farmers can sell their products at a higher price on the "niche market" and/or bene…ts from the learning-by-doing e¤ect, i.e. 9n, p(n) (n) p c .

In other words, we can say :

Proposition 2 Let n max be de…ned as in lemma 2. Concerning the emergence of organic farming we can say that: Moreover, even if there exists a free entry equilibrium distribution that involves organic farmers (case ii and iii), there may also be another equilibrium with only conventional farming simply because there is no advantage for the …rst mover to organic farming. To rule out this case, we have to verify that o (k(1);

(i) if max n p(n) (n) 2 h 0; co c b p c h organic
c o ; m) c c b pc ; m .
The last remark of the previous proposition is linked to the fact that the pro…t of organic farmers is \-shaped. If the last condition is not met, the development of organic farming can therefore su¤er from a lack of coordination: a situation which has to be taken into account if a policy maker wants to support organic farming. But, before we move to this issue, we …rst present some properties of our equilibrium.

Equilibrium with organic farming

Since we are primarily interested in organic farming, we henceforth assume that

o (k(n max ); c o ; n max ; m) c (p c ; c b ; m)
in order to ensure there always exists a equilibrium that sustains organic farming; equilibrium can be characterized by three main equations

8 > > < > > : (s o ) := 1 m f "(s o ) s o + f 0 (s o ) = co p(n) (n) (s c ) := 1 m f "(s c ) s c + f 0 (s c ) = c b pc p(n) (n) (s o ) = p c (s c ) with (s c ) = f (s) f 0 (s)s
The two …rst equations summarize the optimal behavior of the providers (see equation 7) while the last one directly follows from the entry condition. Moreover we also recall that is decreasing while is increasing.

This gives us the opportunity to underscore several properties of an equilibrium with organic farming. From the entry mechanism, we can obviously deduce that the pro…ts will be the same for each kind of farmer. But this last condition also gives us the opportunity to go a step further in the characterization of an equilibrium. Let us …rst remember that we have assumed that the unit production cost of organic seeds is lower than the one of a bundle of seeds and chemical, i.e. c 0 < c b . We now assume that the price of conventional products is equal to the price of organic food de ‡ated by the productivity loses, , i.e. p c = p(n) (n). We immediately deduce from the optimal behaviors of the providers that they sell more seeds to the organic farmers simply because co p(n) (n) < c b pc and (s) is decreasing. But if we now add into the picture the fact that (s) is increasing, we cannot be at a free-entry equilibrium since the pro…ts of the organic farmers are higher than those of the conventional ones. This induces entry, and contributes to a decrease of p(n) (n). We can therefore claim that at equilibrium the price of organic products de ‡ated by the productivity loses, i.e. k(n) = p(n) (n) is always lower than the one for conventional products p c . By taking this result for granted, we can even deduce from the free entry condition and the fact that (s) is increasing, that that each organic farmer uses, at equilibrium and on a same plots of land, more seed than conventional farmer, i.e. s 0 s c : We can therefore state : Proposition 3 At a free-entry equilibrium involving organic farming, we observe that:

(i) the price of organic products de ‡ated by the productivity loses is lower than the price for conventional products i.e. p(n ) (n ) p c ;

(ii) Organic farming requires more seed per plots of land i.e. s o s c 5 Public policy supporting the emergence of organic farming As the main argument for support is of the "infant industry" type, we will focus here on the impact of policy instruments on the level of the organic production, and on the distribution of farms between two sectors. Using criteria like social welfare variation could be more appropriate in a stationary regime, but would requires information on consumer's surplus not accounted for in our model. It is however well-known that, under imperfect competition some instruments could prove to be ine¢ cient or even have perverse e¤ects. As a consequence we organize our discussion in two steps: we …rst present the main instruments and their basic e¤ects and in a second step we discuss their global impact on our virtual agricultural sector.

The set of candidate instruments

First it may happen that the externalities induced by the combination of the "learning-by-doing" process and the decreasing gain obtained from the "niche" market premium results in an equilibrium precluding organic farming. This situation could occur even if organic farming is viable simply because the pro…t of the …rst mover to organic farming is lower than the pro…t of a conventional farmer. There is, in other words, a coordination problem that bars organic farming.

This situation can nevertheless be overcome by improving the information on organic farming and by developing extension services and farmer associations. The idea is to reduce the inhibition of the …rst mover and to motivate enough farmers to move together toward organic farming in order to bene…t of a greater learning-by-doing e¤ect. We must nevertheless concede that the e¢ ciency of this policy is typically inversely related to the size of the smallest coalition that will realize a higher pro…t under organic farming i.e. made from the minimal number n min of agents such that

o (k(n min ); c o ; n min ; m) c (p c ; c b ; m) (17)
If a coalition composed of n min agents moves to organic farming, a natural entry mechanism guarantees that an equilibrium with organic farming will be selected. That argument is typically one of the type of "supporting infant industry".

As soon such an equilibrium occurs, a public agency can also decide to support organic farming by using more standard instruments. We can for instance consider taxation on chemicals, a subsidy for chemical free seeds, a subsidy for organic food production or eventually an improvement of the learning curve. These di¤erent instruments can by easely introduced in our model: a tax per unit of chemical sold modi…es the unit production cost c b of the bundle by some (which of course takes into acount the proportion of chemicals in the bundle) since this tax is indirectly paid by input providers.

a subsidy per unit of chemical-free seed bought by organic farmers reduces their cost. This is measured by p s the price of organic seed (see equation 1). But inspecting equations ( 2) and ( 4), this reduces the inverse demand function by . As a consequence, this is formally equivalent to reducing the unit production cost c o by , a subsidy per unit of organic food provides a new return for organic farmers that can be added to the price p(n) in equation ( 2) impacts of changes in the learning curve are more di¢ cult to capture: we simply state that this curve shifts up by some so that we replace (n) by (n) +

By recalling equations ( 7) and [START_REF]De…nition of Organic Agriculture[END_REF] and by assuming that we look at a new equilibrium in the neighborhood of a interior equilibrium distribution (i.e. n < N ), the e¤ect of these policy instruments can be measured by applying the implicit function theorem to:

8 > > < > > : (s o ) := 1 m f "(s o ) s o + f 0 (s o ) = co (n) 0 (s c ) := 1 m f "(s c ) s c + f 0 (s c ) = c b + pc (n) (s o ) = p c (s c ) (18) with (n
; ; ) = (p(n) + )( (n) + )
and we can show by computation that:

Proposition 4 If we consider an equilibrim in the neighborhood of an interior distribution of the farmers between both sectors, we can summarize the e¤ect of the di¤erent policy instruments on the production level of the di¤erent types of farmers and on the equilibrium distribution between both sectors in the following table:

Tax on chemim ic a ls (@ ) Subs. for chemi- Seeds used by an organic farmer (@s 0 ) 0 (sc) (so)

D 0 (sc) (n; ; ) < 0 (so) D (n; ; ) > 0 0 0
Bundles.used by a conventional farmer (@sc)

1 0 (sc) pc < 0 0 0 0
Distribution of the farmers

(@n) 0 (sc) 0 (so) D 0 (sc) @n (n; ; ) > 0 0 (so) D @n (n; ; ) > 0 (n) + @n (n; ; ) > 0 p(n) + @n (n; ; ) > 0 with D = [ 0 (s o ) (s o ) 0 (s o ) (s o )] < 0

E¤ects of the di¤erent policy instruments

We …rst examine a tax on chemicals. This instrument has a very indirect and contrasted e¤ect on organic farming. Taxing chemicals increases the price of seeds/chemicals bundle by (see equation

1): @p b @ = @ @ (p c f 0 (s c )) = p c f "(s c ) @s c @ > 0
since the non-competitive seeds providers simply try to maintain their margins. This reduces the pro…ts of the conventional farmers and provides incentives to move to organic farming. But entry in the organic sector reduces the bene…ts expected from this "niche" market and when an equilibrium with organic farming occurs, this last e¤ect dominates the potential gain from learning-by-doing. This means that even if the taxation of chemicals increases the number of organic farmers ( @n @ > 0 in the previous table), it also contributes to decrease the pro…t of each unit of production simply because the free-entry mechanism stops only as the pro…t in the two sectors are the same. Subsidies for chemical free seeds have a more clear-cut e¤ect on organic farming. If this subsidy is paid to the organic farmers, the seeds providers will have an incentive to decrease their margin on chemical-free seeds in order to sell more to each farmer and therefore capture a part of this subsidy. This is why the price for organic seeds decreases by (see equation 1):

@p o @ = @ @ ( (n; ; ) f 0 (s o )) = @ (n; ; ) @n @n @ f 0 (s o ) + (n; ; ) f "(s o ) @s o @ = f (s o ) f "(s o ) D < 0
Since a remaining share is left to the organic farmers, this sector becomes more attractive and entry occurs ( @n @ > 0 in the previous table) until this additional gain is eliminated. We therefore end up with more organic farmers who earn the same pro…t as in the situation without subsidies. This last point is essentially due to the fact that input providers do not change their strategy on the conventional seed market since @s c @ = 0 and therefore @p b @ = 0

Even if some conventional farmers move to the organic sector and reduce the demand for conventional seeds, these input providers will bene…t from the capture of the subsidy given to these new organic farmers.

The two last instruments (subsidies for organic production or investments that improve the learning-by-doing) have rather similar e¤ects that are mainly explained by the free entry assumption and the existence of imperfect competition on the input markets. Adjustments that are induced by such policies are quite simple : subsidies to organic production or investment in learningby-doing improve the pro…tability of organic farming since (n; ; ) = (p(n) + )( (n) + ) increases. So if nothing else changes, this induces entry until (n; ; ) comes back to its initial level in order to equalize pro…ts in both sectors. This can of course be easily checked:

d (n; ; ) d = @ (
n; ; ) @n @n @ + @ (n; ; ) @ = @ (n; ; ) @n

( (n) + ) @ n (n; ; ) + (n) + = 0
From that point of view, most of the public spending goes to the development of organic farming since the input providers do not adjust their margin behavior in order to capture the additional pro…t, contrary to the previous case. To understand why the input providers do not try to capture a part of these subsidies, we revisit the basic equations that specify their behavior (equation 7).

We immediately observe that these …rms do not adapt their behavior on the conventional seeds market so that the pro…t of each of these farmers remains constant. For organic seeds, the story is quite di¤erent since equation 7 says that

1 m f "(s o ) s o + f 0 (s o ) = c o (n; ; )
This means that if no entry occurs, any increase of (n; ; ) due to additional subsidies increases the quantity of seeds sold to each farmer in order to capture a part of this subsidy. But when entry occurs and when the distribution is close to an equilibrium with organic farming, we know that (n; ; ) decreases so that the input providers have now an incentive do decrease s o . This mechanism works until the initial level of traded seeds is reached.

Concluding remarks

In this paper, we have examined the viability conditions of organic farming under an imperfect competition. While most of the research dealing with the issue of organic farming has adopted an empirical approach and focused on farmers and farms characteristics, we rely instead on a theoretical approach. Our model is based on a set of six major assumptions : (i) the farmers are homogenous in all respects, (ii) they are free to adopt conventional or organic farming (iii) they face an oligopolistic seeds and chemical industry which provides both chemicals free seed or a bundle of seed and pesticides (iiii) the existence of a niche market e¤ect (v) a learning-by-doing process for organic farmers which partially compensates the technical gap between organic and conventional technologies and (vi) pure competition on the market for conventional products.

By using free-entry conditions and backward solving of the game, we have been able to spell out precise conditions for the emergence of organic farming. These conditions depend on the degree of competition among the upstream …rms, the cost of production of chemicals-free seeds and the cost of production of the bundle of seeds and chemical for conventional production, and the prices for both products. We then examined the impacts of several policy instruments to support the emergence of organic production. They are motivated by environmental considerations and by the desire to protect an "infant industry". We examined four plausible instruments and analysed their impact on the level of production and on the distribution of farms between organic and conventional production. We introduced a tax on the bundle of chemicals and seeds, a subsidy for chemical free seeds, a subsidy to the organic production and support of the learning by doing process. All the selected instruments increase the share of organic sector but have di¤erent impacts on pro…ts distribution . Indeed, because most of the farmers in developped countries face concentrated agro-bussiness …rms, our results have some degree of generality. This general framework could be used to examine the conditions of emergence and of di¤usion of several innovations under imperfect competition. However, some assumptions are speci…c to the game theoretical approach. For instance, we assumed that all the farmers are homogenous, and only motivated by the same goal, pro…t maximization. The choice of technology by farmers could be also driven by individual preferences. Moreover, our approach has focused on the supply side, and does not explore the demand side in much detail. Eventually this choice could be updated to account for a quickly growing demand for organic products.

A = k(n) f (3) 1 n P m j=1 s j o s j o n 2 + (1 + 1 n ) f (2) 1 n P m j=1 s j o B = p c f (3) 1 N n P m j=1 s j c s j c (N n) 2 + 1 + 1 N n f (2) 1 N n P m j=1 s j c
where f (n) stands for the n th derivative. Now remember that under market clearing the amount of seeds used by an organic farmer is s o = 1 n P m j=1 s j o , the same being true for conventional farming hence s c = 1

N n P m j=1 s j c . If we carry out this change of variables and introduce e f " (s) := f (3) (s) s f (2) (s) the elasticity of f ", the previous Hessian becomes:

H = 2 6 6 4 k(n) n f (2) (s o ) e f " (s 0 ) s j o n s 0 + 1 + n 0 0 p c N n f (2) (s c ) e f " (s c ) s j c (N n) s c + 1 + N n 3 7 7 5
If both diagonal terms are negative, H is negative de…nite. Since f (2) (s) < 0, it remains to check that

8 > > < > > : e f " (s 0 ) s j o n s 0 + 1 + n > 0 e f " (s c ) s j c (N n) s c + 1 + N n > 0 (19)
This result is of course obvious when e f " (s) 0. So let us consider the case in which e f " (s) < 0. Now let us …rst observe that at an optimal strategy of a Cournot player markets always clear. We can therefore say that n s 0 and

(N n) s c are

B Proof of Lemma 1

Let us de…ne (s; m; K) := 1 m f "(s) s + f 0 (s) K. It is easy to observe that:

@ (s; m; K) @s = 1 m f (3) (s) s + f (2) (s) 1 + 1 m = 1 m f (2) (s) (e f " (s) + 1 + m) < 0 since f (2) (s) < 0; e f " (
s) > 2 and m 1. Moreover we notice that:

lim s!0 (s; m; K) = f 0 (s) 1 m e f 0 (s) K f 0 (s) = +1 since lim s!0 f 0 (s) = +1
and e f 0 (s) remains bounded.

lim s!+1 (s; m; K) = K since lim s!+1 f 0 (s) = 0.
We can therefore state that there exists a unique solution in s (m; K) to (s; m; K) = 0 and the lemma is obtained by applying the preceding argument to each equation of system (7)

C Proof of Proposition 1

Let us come back to the de…nition of (s; m; K) given in lemma 1. If we now apply the implicit function theorem, we immediately observe that : @s (m; K) @m = 1 m 2 f "(s) @ (s;m;K) @s > 0 and @s (m; K) @K = @ (s; m; K) @s 1 < 0 which proves the …rst part of the proposition. Let us now push m to in…nite, the equation (s; m; K) = 0 simply becomes f 0 (s) K = 0 since e f 0 (s) is bounded. Hence s = (f 0 ) 1 (K).

D Proof of Lemma 2

The proof of Lemma 2 is immediate. Remember that the sign

@ o (k(n); c o ; m) @n = sign (k 0 (n)). So let us study k(n) = p(n) (n)
. By computation we obtain:

k 0 (n) = p 0 (n) (n) + p(n) 0 (n) (20) k"(n) = p " (n) (n) + 2 p 0 (n) 0 (n) + p(n) "(n) (21)
Now let us observe that:

k"(n) < 0, since we have assumed that p 0 (n) < 0 and p " (n) < 0, 0 (n) > 0 and "(n) < 0.

lim n!0 k 0 (n) > 0 because lim n!0 p 0 (n) = 0 and 0 (n) > 0 lim n!N k 0 (n) < 0 because lim n!N 0 (n) = 0 and p 0 (n) < 0

We conclude that there exists a unique n 0 verifying k 0 (n 0 ) = 0, and therefore such that @ o (k(n 0 ); c o ; m) @n = 0.

F Proof of Proposition 4

Let us recall that the outcome of our model can be reduced to three equations : the modi…ed …rst order conditions of the input providers, i.e. equations [START_REF] Escalante | Good Growing: why Organic Farming Works[END_REF] and the free entry condition, i.e. equation [START_REF]De…nition of Organic Agriculture[END_REF]. These equation, after the introduction of the di¤erent policy arguments are sumerized in equation [START_REF] Mayen | Technology Adoption and Technical E¢ ciency: organic and conventional dairy farms in the United States[END_REF]. However to simpli ‡y the notations let us introduce (s) = 1 m f "(s) s + f 0 (s), (s) = f (s) f 0 (s) s and . (n; ; ) = (k(n) + (n) + p(n)) . We can even notice that (i) 0 (s) < 0 see lemma 1, (ii) 0 (s) = f "(s) s > 0 and (iii) @ n (n; ; ) < 0 by construction. This last point requires an additional comment. In the comparative static excercice we are looking at what happen in a neighborhood of an equilibrium wich has the property that n 2 ]0; N [ and that all policy argument are set to 0. So by construction at the equilibrium @ n (n; ; ) < 0, and since we apply the Implicite Function Theorem (IFT) from a local point of view, we can choose the neighborhoods such that @ n (n; ; ) < 0 at the new equilibrium. Now let us observe that the determinant of @ (so;sc;n) given by: det @ (so;sc;n) = 0 (s c ) @ n (n; ; ) 0 (s o ) (s o ) 0 (s o ) (s o ) < 0

Being non-zero, we can therefore apply the IFT and we know that @ ( ; ; ) (s o ; s c ; n) = @ (so;sc;n) 1 @ ( ; ; ) (at least locally). Moreover it is a matter of fact to check that:

@ (so;sc;n) 1 = 1 D 2 6 6 4
(s o ) pc 0 (sc) (so) (n; ; ) 0 (sc) (so) (n; ; ) 0 D 0 (sc) 0 0 (s o ) (n; ; ) @n (n; ; ) 0 (so)pc 0 (sc) 0 (sc) @n (n; ; ) 0 (so) @n (n; ; ) Since (s); 0 (s); (s); (n) > 0 and 0 (s); @ n (n; ; ) < 0 at an equilibrium, we can conclude that: sign @ ( ; ; ; ) (s o ; s c ; n) =

  Batte et al., 2007, Krystallis and Chryssohoidis, 2005, Yiridoe et al., 2005, Boccaletti and Nardella, 2000, Gil et al., 2000, ...).

  s o co k(n) ; m and s c c b pc ; m ; which relate the quantity of seeds used in each sector to the number m of input provider and to the relative pro…tability of each sector measured by the ratio of the cost over the price (but taking into account the learning-by-doing e¤ect). We can even observe: Proposition 1 The equilibrium quantities of seeds in the organic sector, s o co k(n) ; m and in the conventional one, s c c b pc ; m used by a representative farmer are decreasing respectively with the ratio co k(n) and c b pc , and increasing with the degree of competition measured by the number m of input providers. Moreover as m ! 1; these quantities converge toward the competitive equilibrium quantities given respectively by s o co k

(s) 0

 0 since the marginal productivity is lower than the average one, by proposition 1, s o co k(n) ; m is decreasing with (c 0 =k(n)) and, by computation d ds = f "(s) s 0

  distribution of the farmers between both sectors (or only organic farming if o (k(N ); c o ; m) c (p c ; c b ; m)).

c a l fre

  e se e d s (@ ) Subs. for organ ic p ro d u c ts (@ ) Improving learn in g by d o in g (@ )

[ 31 ]

 31 Wheeler, S. A. 2008. What in ‡uences agricultural professionals'view towards organic agriculture? Ecological Economics 65 : 145 154: [32] Wynen, E. and Edwards G. 1990. Towards a comparison of chemical-free and conventional farming in Australia. Australian Journal of Agricultural Economics 34(1) : 39 55: [33] Yiridoe, E. K., Bonti-Ankomah, S. and Martin, R. C. 2005. Comparison of consumer perceptions and preference toward organic versus conventionally produced foods: A review and update of the literature. Renewable Agriculture and Food Systems 20 : 193 205: APPENDIX A The su¢ cient conditions for optimality Let us observe that the Hessian matrix of the pro…t function is given by H = "

  Moreover, since k"(n) < 0, o (k(n 0 ); c o ; m) is \-shaped.E Proof of Proposition 2(i) Assume that max n p(n) (n) < co c b p c , This means that 8n, co p(n) (n) > c bpc and we can deduce from Lemma 1 that 8n, s o co k(n) ; m < s c c b pc ; m . If we now remember that (s) := (f (s) f 0 (s) s) is increasing since 0 (s) = f "(s) s, we can say that 8n, ; m . Now remember that c o < c b , this implies, in case (i), that 8n, p(n) (n) < p c . It remains to mix these two observations in order to say that:8n; o (k(n); c o ; m) = p(n) (n) s o c o k(n) ; m < p c s c c b p c ; m = c (p c ; c b ; m)It is impossible to observe an equilibrium distribution which involves organic farming.(ii) if max n p(n) (n) 2 ]p c ;1[ and since c o < c b , we can say that 8n, co p(n) (n) < c b pc . With the same arguments as in point (i) and by simply reversing the inequalities we can conclude that 8n; o (k(n); c o ; m) > c (p c ; c b ; m), i.e. organic farming always dominates conventional agriculture. (iii) if none of these conditions is satis…ed, organic farming occurs if and only if o co k(nmax) ; n max ; m c c b pc ; m because 0 is \-shaped with respect to n.

  Now let build the function:(s o ; s c ; n; ; ; s; ) = (s o ) c o (n; ; ) , (s c ) c b + p c ; (n; ; ) (s o ) p c (s c )And since an equilibrium is given by (s o ; s c ; n; ; ; s) = 0, let us apply the IFT. By a simple exercice of compu-tation and by bearing in mind that (s o ) = co (n) ; we observe that :@ (so;sc;n) ; ) 0 (s o ) p c 0 (s c ) @ n (n; ; ) (s o ) ) + ) (s o ) (p(n) + ) (s o )

0

  0 (s o ) (s o ) 0 (s o ) (s o ) < 0We therefore obtain that: @ ( ; ; ; ) (s o ; s c ; n) = 1 (so) 0 (sc) 0 (sc) @n (n; ; ) 0 (so) @n (n; ; ) ( (n)+ ) D @n (n; ; ) (p(n)+ ) D @n (n; ; )

  the aggregated quantities of the two kinds of seeds that are supplied, so that are market shares which belong by construction to [0; 1]. Moreover n and N n are both greater than 1 otherwise one sector would not be activated. Finally remember that we have assumed that e f " (s) > 2. If we make use of the three remarks, it immediately follows that conditions[START_REF] Martini | Yield increases during the organic transition: Improving soil quality or increasing experience?[END_REF] holds

	s j o n s 0	and	s j o (N n) s c

In our vertical structure the demand of inputs is linked to the marginal productivity. These assumptions therefore help to control the …rst and the second order conditions of the optimization problem of the input providers. These restrictions are typically met by any iso-elastic production function.

For simplicity, we consider n as a continuous variable.

This directly follows from the absence of "free lunch" (i.e f (0) = 0) and the concavity of f .

Since we work with a general production function we can only give su¢ cient condition. A more precise threshold could be computed with a constant elasticity production function.


Mohamed Belhaj
email: mohamed.belhaj@centrale-marseille.fr.

Nataliya Klimenko
email: nataliya.klimenko@univ-amu.fr.

On the role of external nancing costs in optimal investment decisions

Keywords: real options, investment intensity, debt issuance costs, credit constraints JEL classication: G31, G32, G33

This paper brings into focus a link between the investment andnancing decisions of a rm which has an access to costly debt nancing. Our analysis shows that lump-sum debt issuance costs play a prominent role in a determination of the optimal investment strategy. Faced with larger lump-sum debt issuance costs, a rm will optimally set up a higher-scale investment project in order to "compensate" deadweight nancing costs by higher return. Moreover, in the presence of lump-sum debt issuance costs, the optimal investment scale ofnancially constrained rms exhibits an inverted U-shaped relationship with the rm's borrowing capacity, so that relatively more/less constrained rms will realize smaller investment projects, whereas rms with an intermediate borrowing capacity will undertake larger investment.

Introduction

Recognizing the fact that the Modigliangy and Miller irrelevance theorem (1958) is not robust in the context of any frictions, a corporate nance literature doesn't cease looking for a consistent explanation of the connection between nancing and investment decisions. Given a substantial dierence between the costs of external and internal funding caused by information asymmetry problems and eventual transaction costs, a question of particular interest is how the optimal choicees of investment time and the scale of investment are aected by the tightness of nancial constraints?

While many studies unanimously advocate for the U-shaped relationship between the investment time and the tightness of nancial constraints (see, for instance, Belhaj and Djembissi (2009), Hirth and Uhrig-Homburg (2010), [START_REF] Wong | On the neutrality of debt on investment intensity[END_REF], Shibata and Nishihara (2012)), there is a lack of consensus concerning the impact of nancial constraints on the optimal choice of investment scale. Earlier theoretical works [START_REF] Fazzari | Financing Constraints and Corporate Investment[END_REF], Hoshi et al. (1988), [START_REF] Whited | Debt, liquidity constraints, and corporate investment: evidence from panel data[END_REF], Gilchrist and Himmelberg (1995), Hubbard (1998))

argue that a sensitivity of investment expenditures to cash ows are much stronger for the rms which are likely to be nancially constrained. This fact suggest about a negative monotonic relationship between the tightness ofnancial constraints and the scale of investments chosen by rms. [START_REF] Kaplan | Investment-Cash Flow Sensitivities Provide Useful Measures of Financing Constraints?[END_REF] provide a conicting evidence, showing that less nancially constrained rms exhibit higher investment cash-ow sensitivities. Recent studies of [START_REF] Cleary | The U-shaped Investment Curve: Theory and Evidence[END_REF] and [START_REF] Guariglia | Internal nancial constraints, external nancial constraints, and investment choice: Evidence from a panel of UK rms[END_REF] report a Ushaped relation between investment size and internal funds, explaining this form by a trade-o between a higher risk of default and a higher expected investment return of the levered rm. A theoretical model of [START_REF] Wong | On the neutrality of debt on investment intensity[END_REF] contributes to this puzzle, showing that a negative eect of higher default risk and a positive eect of higher investment return resulted from larger investment will oset each other. As a result, the optimal investment scale will be independent on the rm's credit constraints, remaining equal to that of the all-equity nanced rm.

However, it seems to be misleading to discuss the optimal choice of in-vestment scale without taking into account external nancing costs. Allowing for external nancing costs and the dynamic aspect of investment, our model uncovers an inverted U-shaped relationship between the scale of investment and the rm's borrowing capacity. A striking evidence we obtain is that this relationship emerges only in the presence of lump-sum debt issuance costs. Without lump-sum debt issuance costs, a scale of investment would be completely independent on the rm's borrowing capacity, so that a neutrality of debt in investment intensity established by [START_REF] Wong | On the neutrality of debt on investment intensity[END_REF] would hold even under signicant variable debt issuance costs.

It is worth noting that several recent studies have attempted to integrate external nancing costs into the analysis of optimal investment decisions. [START_REF] Lyandres | Costly External Financing, Investment Timing, and Investment-Cash Flow Sensitivity[END_REF] documents a hump-shaped relationship between the magnitude of external nancing costs and investment, explaining this form by the interplay between two opposite eects of external nancing costs on investment timing. Indeed, external nancing costs reduce the attractiveness of current investment, but at the same time also reduce the value of the option to wait, whereas a scale of investment is found to be monotonically increasing on the investment time. The study of [START_REF] Hirth | Investment timing when external nancing is costly[END_REF] examines the impact of variable external nancing costs on the optimal investment time in the context of a xed-scale investment model. They nd that low-liquidity rms will delay investment to avoid present nancing costs, whereas high-liquidity rms will speed up investment to avoid even higher nancing costs in future. They associate earlier investment with a higher investment volume, arguing that the optimal investment scale would exhibit a reversed U-shaped pattern when plotted against liquid funds. Nishihara and Shibata (2012) allow for two levels of investment scale and show that relatively low and relatively high levels of cash reserves imply larger investment scale, whereas the intermediate levels of cash correspond to the lower investment scale.

While the above mentioned studies use the rm's current level of cash holdings as a proxy for nancial constraints and consider a discrete choice of investment scale, we conduct our analysis in the context of explicit credit constraints, allowing for a continuous choice of investment scale. Indeed, in practice rms often face a quantity credit rationing, so that borrowed funds represent only a fraction of a required investment amount, whereas a remaining part has to be nanced by internal funds. Such a representation of credit constraints makes it possible to capture endogenous changes in credit capacity following the changes of investment scale.

The rest of the paper is organized as follows. Section 2 presents the model. Section 3 describes the optimal investment decisions of an all-equity nanced rm. In Section 4 we examine the optimal nancing and investment decisions of a levered rm when it faces debt issuance costs. Section 5 concludes.

The model

We consider an owner-managed rm protected by limited liability and endowed with a perpetual investment option. An investment project can be undertaken at the irreversible cost I(q) which is positively related with investment intensity, q > 0. Moreover, we assume that I (q) > 0, I(0) ≥ 0 and I (0) = 0. Investment intensity aects the scale of stochastic earnings before interest and taxes (EBIT), qX t , where X t denotes EBIT per unit of investment intensity and follows a stochastic process:

dX t = µX t dt + σX t dW t , (1)
where W = {W t , F t , 0 ≤ t < ∞} is a standard Brownian motion on a probability space (Ω, F, P). Investors are risk neutral and discount the future at a constant rate r > µ.

The rm operating prots are taxed at the rate θ, so that equity holders collect a residual operating cash-ow (1 -θ)qX t .

A rm has an access to the external nancing. In order to nance a part of investment costs and to benet from tax shields, at the investment date τ the rm's equity holders can raise an amount b ≤ ψI(q) of perpetual debt, where ψ reects an exogenously given borrowing capacity of the rm. The remaining fraction of investment costs, I(q) -b > 0, is nanced by equity capital. 1 However, issuing debt is costly because of information asymmetries inherent to external nancing. As [START_REF] Belhaj | Optimal investment under credit constraint[END_REF], we consider a linear form of debt issuance costs, K(b) = kb + K 0 , with both variable and xed components. 2A debt contact implies that a constant coupon c will be continuously paid to debt holders until the rm's default, which occurs when shareholders cease injecting cash in order to serve the debt. The optimal default trigger maximizing the rm's equity value is given by the standard formula:

x L (q) = β 1 β 1 -1 r -µ r c q ≡ δ c q (2)
where

β 1 < 0 is a root of σ 2 /2β(β -1) + µβ = r.
Given that after the default the rm will be run by new owners as an unlevered concern, we model bankruptcy costs as a fraction γ of the allequity nanced rm's value.

Equityholders optimally decide about the timing, the scale and thenancial structure of investment, maximizing ex-ante equity value under the exogenously given credit constraint. For any coupon c and investment intensity q, let D(X τ , q, c) denote debt value at the investment time τ , given that the rm will be liquidated at the time τ L = inf{t ≥ 0 s.t X t = x L (q)}. Since, by the absence of arbitrage, we have b = D(X τ , c, q), the equity holders' maximization program can be formalized as follows:

Sup τ,q,c<∞ E e -rτ (V (X τ , q, c) -I(q) -kD(X τ , q, c) -K 0 ) s.t. D(X τ , q, c) ≤ ψI(q) (3)
where V (X τ , q, c) denotes a value of the rm at the investment time. 33 Investment decisions of an all-equity nanced rm: a benchmark Consider rst a benchmark case where ψ = 0, so that the investment project is fully nanced by equity and the rm never goes bankrupt. In this case the equity holders' problem takes the following form:

Sup τ,q<∞ E e -rτ (1 -θ)qX τ r -µ -I(q) = Sup x e I ,q ((1 -θ)νqx e I -I(q)) X t x e I β 2 (4
) where x e I is the investment trigger such that τ = inf{t ≥ 0 s.t X t = x e I },

β 2 > 0 is a root of σ 2 /2β(β -1) + µβ = r and ν = (r -µ) -1 .
The corresponding optimal investment intensity satises:

I (q e ) = β 2 β 2 -1 I(q e ) q e (5)
whereas the optimal investment trigger is given by:

x e I = β 2 β 2 -1 I(q e ) q e 1 (1 -θ)ν (6)
The optimal investment intensity in [START_REF] Guariglia | Internal nancial constraints, external nancial constraints, and investment choice: Evidence from a panel of UK rms[END_REF] is chosen in a way to balance the marginal investment cost and the average investment cost adjusted by the option value multiple β 2 /(β 2 -1). Thus, the optimal investment scale of an all-equity nanced rm is constant and completely independent on the optimal choice of investment time. [START_REF] Fazzari | Financing Constraints and Corporate Investment[END_REF] Investment and nancing decisions of a levered rm

The case of non-binding credit constraints

First, we are going to consider a solution of the maximization problem in the case when ψ is large enough, so that the rm can raise a required amount of debt without binding credit constraints. In this case, the credit constraint in (3) can be omitted and the program can be resolved in two steps: (i) rst, for any xed investment parameters q and x I such that τ = inf{t ≥ 0 s.t X t = x I }, we dene the optimal coupon c * (q, x I ); (ii) then, the problem is solved for the remaining parameters of investment policy and the optimal coupon is recovered.

For any given q and x I , the optimal c * (q, x I ) maximizing the rm's value net of the variable debt issuance costs is given by:

c * (q, x I ) = h δ qx I , (7)
where

h = θ -k (1 -β 1 )(θ -k) -β 1 (1 -θ)(γ + k(1 -γ)) 1 -β 1 (8)
Given the optimal coupon c * (q, x I ), the maximization problem of equity holders can be rewritten as follows:

Sup x I ,q<∞ ((1 -θ)νqx I + (θ -k)hνqx I -I(q) -K 0 ) X t x I β 2 (9)
The solution of the problem is given by the optimal investment trigger [START_REF] Sarkar | Optimal size, optimal timing and optimal nancing of an investment[END_REF] and the optimal investment intensity q f which satises the following equation:

x f I = β 2 β 2 -1 I(q) + K 0 q 1 (1 -θ)ν + (θ -k)hν
q f I (q f ) = β 2 β 2 -1 (I(q f ) + K 0 ) (11)
The optimal coupon will be given by c f = c * (q f , x f I ).

Lemma 1 q f is increasing on K 0 .

Let f 1 (q) and f 2 (q, K 0 ) denote respectively the left and the right side of the equation [START_REF] Whited | Debt, liquidity constraints, and corporate investment: evidence from panel data[END_REF]. Both f 1 (q) and f 2 (q, K 0 ) monotonically increase on q. Given that f 2 (0, K 0 ) > f 1 (0), the equation ( 11) has a unique solution q * (K 0 ). Since ∂f 2 (q,K 0 ) ∂K 0 is increasing on K 0 , we conclude that q * (K 0 ) is increasing on K 0 .

Thus, similar to the benchmark case with all-equity nancing, the optimal investment intensity is set independently on both the optimal investment time and the rm's borrowing capacity, remaining constant. 4 However, under non-zero xed debt issuance costs, the levered rm will realize higher investment as compared to the all-equity nanced rm. At the same time, given higher investment intensity, the investment trigger of the levered rm would be lower than the investment trigger of the equity-nanced rm when xed debt issuance costs are quite small.

To illustrate the impact of debt issuance costs on the rm's investment and nancing decisions in the context of non-binding credit constraints, we resort to numerical simulations (see Table 1.a-1.b in Appendix B) and obtain the following evidence:

• The optimal investment intensity, q f , and thus the optimal scale of investment, I(q) f , are independent on k but increase on K 0 , since the rm will tend to "compensate" higher lump-sum debt issuance costs by a higher asset return.

• The optimal investment trigger, x f I , is increasing on both k and K 0 . Indeed, a higher investment trigger reduces a current discounted value of xed debt issuance costs, increasing a value of the rm's option.

• The optimal coupon value, c f , and thus the rm's leverage is increasing with K 0 , since the rm will tend to reduce the average cost of debt issuance by raising more debt. However, the amount of debt decreases on k, since the variable debt issuance cost reduces the value of tax shields. As a result, the rm's leverage ratio D(x f I , q f , c f )/I(q) f tends to zero when k → θ.

• The value of investment option calculated at the current moment of time is decreasing on both k and K 0 .

The case of binding credit constraints

Consider now the optimal investment and nancing decisions of the rm in the context of binding credit constraints. For any given investment intensity q and coupon rate c, the binding credit constraint provides the following investment trigger:

x I (q, c) = 1 -ψr I(q) c 1 -(1 -γ)(1 -θ)νδr 1 β 1 x L (q), (12)
where x L (q) is given by (2). Then, the equity holders' problem (3) can be rewritten as follows:

Sup q,c (V (q, c) -I(q)(1 + kψ) -K 0 ) X t x I (q, c) β 2 , ( 13
)
where V (q, c) is a rm value under the investment threshold x I (q, c).

The optimal investment and nancial strategies are determined by the optimal values of q and c which solves the above problem.

In order to track the impact of debt issuance costs on the investment intensity, we use the Lagrangian method to obtain the following optimality condition:

qI (q) = β 2 β 2 -1   I(q) + K 0 1 -λψ x I X 0 β 2 -1   , (14)
where X 0 is a current value of the state variable and λ > 0 is the optimal Lagrange multiplier, independent on X 0 .

Proposition 1 Given non-zero xed debt issuance costs, the optimal investment intensity of a nancially constrained rm will be aected by its borrowing capacity.

Numerical simulations suggest the existence of an inverted U-shaped relationship between the optimal investment intensity q and the rm's borrowing capacity ψ that also reects the rm's leverage (see Table 2.c in Appendix B). In other words, a rm with relatively low or relatively high borrowing capacity will set up smaller investment project, whereas a rm with intermediate borrowing capacity will undertake larger investment project. In fact, larger investment scale implies two opposite eects on the rm's perspectives. On the one hand, larger investment implies a higher return, which would increase the equity holders' ability to serve the debt and thus would reduce default risk. On the other hand, under the binding credit constraint, larger investment is associated with a larger volume of debt, thereby, increasing default risk. Thus, when a rm's borrowing capacity is relatively low, the rst eect dominates, so that the optimal investment scale is increasing with ψ. However, as ψ reaches some critical level, the second eect becomes stronger, so that the optimal scale of investment goes down. It is important to note that xed debt issuance costs play a crucial role in the trade-o between two described eects. In fact, in the presence of xed debt issuance costs, larger investment will reduce a marginal cost of debt issuance, which makes the rst eect more pronounced. Without xed debt issuance costs, two eects would oset each other and the optimal investment intensity would be constant, as was shown by Wong (2010). 5Expression (14) also shows that, in contrast to the case of the non-binding credit constraint, the optimal choice of investment intensity will be aected not only by xed debt issuance costs, but by variable costs as well. This is due to the fact that a choice of investment intensity under the binding credit constraint becomes inseparable from the choice of investment trigger, whereas the latter is aected by variable debt issuance costs. Numerical simulations show that the optimal investment intensity is decreasing with k.

Conclusion

In this paper we contribute to the ongoing investigations of the link between a rm's investment and nancing decisions in the context of external nancing costs. We show that, for a rm which is not nancially constrained, a choice of investment intensity will be unrelated to the rm's debt capacity and will remain constant. However, in the presence of non-zero lump-sum debt issuance costs, a nancially constrained rm will realize larger investment projects for intermediate levels of credit capacity and will undertake investment projects of lower scales when credit capacity is relatively low/high. Table 1.b Table 2.a ψ = 0.7, k 0 = 0.1 K 0 q x I c I(q) D(x I , q, c) F 0 1,8566

K 0 = 5 k q f x f I c f I(q f ) D(x f I , q f , c f ) ρ F 0 2,

K 0 = 5 ,

 5 k = 0 and dierent values of ψ ∈ [0.1, 1].

Table 2 .

 2 a displays simulation results for ψ = 0.7, k = 0.1 and dierent values of K 0 . Table2.b displays results obtained for ψ = 0.7, K 0 = 5 and dierent values of k < θ. Table2.c contains simulation outcomes for

	1253 5,3258 6,9924 58	70,8100	1,2209 4,1413
	0,02 2,1253 5,3769 6,4445 58	66,5943	1,1482 4,0716
	0,04 2,1253 5,4250 5,8118 58	61,3926	1,0585 4,0077
	0,06 2,1253 5,4691 5,0799 58	54,9698	0,9478 3,9504
	0,08 2,1253 5,5080 4,2318 58	47,0209	0,8107 3,9008
	0,1 2,1253 5,5405 3,2467 58	37,1443	0,6404 3,8603
	0,12 2,1253 5,5645 2,0987 58	24,8034	0,4276 3,8307
	0,14 2,1253 5,5780 0,7563 58	9,26800	0,1598 3,8143
	B.2. Optimal investment and nancing decisions when the credit
	constraint is binding

In the case when I(q) -b < 0 (which is possible only if ψ > 1), equity value at the investment time increases by b -I(q).

Issuing equity might be costly as well. However, it is commonly recognized that the cost of external nancing exceed the cost of internal nancing, so we assume the latter to be zero in order to focus on the impact of external nancing costs.

For the expressions of contingent claims used in the model see Appendix A.

This is in line with the[START_REF] Almeida | Financial Constraints and Investment -Cash Flow Sensitivities: New Research Directions[END_REF] who suggest that the investment of unconstrained rms is insensitive to cash ow, taken the latter as a proxy for credit constraints.

Examining a simultaneous choice of investment timing, investment scale and leverage in the presence of variable investment costs,[START_REF] Sarkar | Optimal size, optimal timing and optimal nancing of an investment[END_REF] nds that, generally, investment scale is rst decreasing and then increasing on the amount of debt raised, but for high levels of a tax rate and bankruptcy costs this pattern is reversed. However, his model doesn't allow for credit rationing and lump-sum nancing costs.

Appendix A

A.1. Equity value The rm's equity value is given by:

where β 1 < 0 is a root of σ 2 /2β(β -1) + µβ = r and the optimal liquidation rule x L (q) such that E x L (X t , q, c) = 0 is given by:

Debt value

The value of the rm's debt is given by:

A.3. A rm's value

The value of the rm is given by the sum of equity and debt:

We use the following parameter values: market interest rate r = 8%, the expected growth rate of operating prot µ = 1%, operating prot volatility σ = 30%, the default cost coecient γ = 0.3, the tax rate θ = 0.15, the investment function I(q) = 10 + 5q 3 . A current value of the state variable is taken as X 0 = 1.

The optimal investment strategy in the benchmark case (all-equitynancing) is given by q e = 1.8566 and x e I = 4.2582.

B.1. Optimal investment and nancing decisions when the credit constraint is not binding Table 1.a displays simulation results for k = 0.1 and dierent values of K 0 . Table 1.b displays results obtained for K 0 = 5 and dierent values of k < θ. We denote F the current value of the rm investment option, whereas ρ = D(x I , q, c)/I(q) represents the rm's leverage ratio. Table 1.a