T Astruc

CONNECTIVE TISSUE: STRUCTURE, FUNCTION, AND INFLUENCE ON MEAT QUALITY

Collagen It is the main component of connective tissue. There are many types of collagen, of which the main ones in muscle are type I and type III, which are the focus of this article. In muscle, collagen forms the endomysium, perimysium, and the epimysium.

Collagen content With approximately 25-35% of the whole body protein content, collagen is the most abundant protein in mammals. It generally constitutes 1-2% of muscle tissue but may represent up to 6% of the weight in high tendinous muscles. Collagen cross-linking The individual protein subunits of collagen are assembled together by covalent intermolecular cross-links, which increase the tensile strength of collagen. The number of cross-links generally increases with the age of the animals. Collagen solubility While heating, molecular chains of collagen separate and solubilizes to gelatin. The solubilization of collagen decrease when the number of cross-links increase. Proteoglycans These are heavily glycosolated proteins that are a filler substance between cells that regulate movement of molecules through the matrix and affect activity and stability of proteins and signaling molecules.

Introduction

Connective tissue is produced by fibroblasts and is the most abundant protein in mammals, making up approximately 30% of the total protein content. When mineralized, it is found in skeletal tissues including bone, teeth, and cartilage. The nonmineralized soft connective tissue is found in skin, tendon, ligament, adipose tissue, blood and lymphatic tissue, and the connective fiber framework in muscles. It contains a mixture of polysaccharides and fibrous proteins. The most abundant protein in connective tissue is collagen, but it also contains various other proteins (such as elastin, fibronectin, and laminin) and proteoglycans in varying proportions, depending on the organ. Connective tissue represents a small proportion of skeletal muscle and is mainly found in extracellular matrix of muscle fibers. However, the meat is the result of transformation of the muscle after slaughter of the livestock animal. Connective tissue content and composition significantly affects the quality of meat and meat products, particularly texture and water-holding capacity. For this reason, numerous scientific studies have been led to characterize its structure, organization, and composition in order to understand its role in determining meat quality.

Distribution, Composition, and Structure of Muscle Connective Tissue

Muscle connective tissue content varies from 1.5 to 15% of dry matter depending on muscle and muscle function.

Intramuscular connective tissue is principally comprised of collagen and elastin proteins within a proteoglycan matrix. Muscle elastin content is approximately 0.1-0.2% of dry matter, except in semitendinosus and latissimus dorsi muscles, where it reaches approximately 2%. The vast majority of this connective tissue is composed of collagen in amounts ranging from 1.5% to approximately 10% of dry weight. Collagen I and III are the predominant types in muscle, but collagens IV, V, VI, XII, XIV, XV, and XIX are also present in minor quantities (Figure 1).

Collagen is composed of three polypeptide alpha chains with a simple repeating primary structure Gly-X-Y, where X is often proline and Y is often hydroxyproline. The three chains are assembled together to form a triple helix of tropocollagen. The tropocollagen molecules are assembled together to form fibrils that are then assembled to form collagen fibers (Figure 2).

The fibrous collagen is cross-linked by a mechanism based on aldehyde formation from lysine or hydroxylysine side chains. Fibrils and fibers are stabilized by intermolecular lysine-derived cross-links formed between lysine aldehyde or hydroxylysine aldehyde and hydroxylysine to form an aldimine or oxo-imine bond. These bonds are replaced during aging by stable multivalent mature cross-links, which are thought to link microfibrils and increase matrix stability. Thus, higher collagen cross-linking translates into higher mechanical strength.

From a structural standpoint, intramuscular connective tissue is divided into three main structures (see Figure 3).

Epimysium is a layer of dense irregular connective tissue that surrounds whole skeletal muscle. It protects muscles from friction against other muscles and bones. The epimysium is generally separated from the body of the muscle and need not be considered as a factor in meat texture, except in some meat products such as dry and cooked ham where muscles are intact.

Perimysium is a layer that separates bundles of muscle fibers within the muscle. It represents approximately 90% of total muscle connective tissue, where it is organized as a network of interconnected segments that vary extensively according to muscle type, species, age, and region in the muscle. The perimysium is usually arranged in large circular or pentagon-shaped fascicles several millimeters in diameter, with the smallest bundles, or primary perimysium, grouped within larger bundles of thicker perimysium, the secondary fascicles, which may be further organized by tertiary fascicles. Perimysium composition is highly variable and can change with nutrition and exercise. Strength muscles tend to have more collagen than postural muscles have, and collagen from old animals has significantly more cross-links than collagen from young ones. Endomysium surrounds each muscle fiber. It is composed of basal lamina proteins, proteoglycans, and laminin, plus collagens I, III, and IV, and is very similar in all muscle types and even across species.

'In vivo' Function of Connective Tissue

Skeletal muscle connective tissue forms a network that plays a dynamic role in muscle differentiation and development. It serves as a supportive structure in skeletal muscle.

Connective tissue adheres to myofibers and fiber bundles, providing a scaffold that maintains muscle structure during contraction, and acts as an interface allowing fiber bundle sliding. Connective tissue provides the cell-to-cell connections both between individual muscle cells and between muscle cells and neighboring small blood vessels and nerves. It gives coherence and mechanical strength, functions as an elastic, stress-tolerant system, and distributes the forces of muscular contractions in both muscle and tendon. The muscle connective tissue participates in cell growth and tissue regeneration after damage.

The organization of muscle into fascicles separated by the perimysium reflects the need to accommodate shape changes as the muscle contracts, which is achieved by allowing fascicles to slide past each other. Functionally, different muscles have very different requirements in terms of accommodating the shear strains that necessarily occur as the muscle contracts and changes shape, which explains why the amounts and distribution of perimysial connective tissue varies so widely between functionally different muscles.

Connective Tissue Properties Related to Meat Texture

The content, nature, and heat solubility of collagen are key factor of meat tenderness.

Technological processes help to reduce the impact of connective tissue on the texture of meat products. Mincing, for example, can decrease the sensation of toughness in meat, whereas cooking, by solubilizing collagen, significantly reduces the mechanical strength of the connective tissue. Acidic marinating can also partially degrade the connective tissue, which may tenderize meat. However, some meats, like roasts and grilled meat, are consumed after maturation and shortterm cooking, in which case cooking temperature at core of the meat piece remains too low to solubilize the collagen. As collagen content is relatively unaffected by the action of proteases during maturation, it ultimately defines the basal hardness of raw or undercooked meat. Thus, the less tender muscles, like pectoralis profundus, generally have high collagen concentrations and a high number of cross-links, whereas a tender muscle like Longissimus thoracis has a lower collagen concentration (1.86%) and fewer cross-links. It is assumed that collagen concentration and number of cross-links have a cumulative effect, leading to a negative impact on meat tenderness. Collagen type also plays a role in meat tenderness. Particularly, type-III collagen, which is more susceptible to proteases, is associated with tenderness, whereas type-XII and type-XIV collagen are thought to decrease total collagen solubility and therefore overall meat tenderness.

Method of Measuring Meat Texture

An important consideration when determining the connective tissue properties tied to texture is the texture measurement method used. In evaluating overall texture quality, and especially the contribution made by connective tissue, the best tests include adhesion force between fibers, peak shear force minus initial yield, and compression, which correlates well to sensory chewiness and texture. For example, old animals show increased adhesion force, decreased collagen solubility, and tougher meat. Understanding the mechanical properties of meat requires an assessment of large amounts of information about sample muscle fiber orientation, cooking temperature, fiber size, and the amount of connective tissue. Compression tests at 20-80% of the initial height of the sample with fibers perpendicular and parallel to the test tool can be used to estimate the relative contribution of each myofiber and connective tissue to the texture. These types of tests have shown that amount of connective tissue goes a long way toward explaining the between-breed and between-muscle differences in texture.

Sensory evaluation of the contribution of connective tissue to texture is not only the most sensitive and accurate but also the most laborious method. Results using this method follow the same trend as mechanical measures, indicating that collagen quantity is not a major factor in animals of the same age when comparing the same muscle type. Collagen content does explain some of the differences in sensory texture between different types of beef muscles, with the psoas and longissimus having low collagen content and tender texture, whereas biceps and pectoralis having high collagen content and tough texture. In general, pork has similar connective tissue properties to beef, with a weak but significant relationship between hydroxyproline content and sensory tenderness, connective tissue, and flavor.

Biological Factors, Connective Tissue, and Meat Tenderness

In general, the content or quality of the connective tissue varies with biological factors.

Species

Structural characteristics are similar from one species to another. Longissimus muscle from young cattle, sheep, and pigs do not differ in collagen content and solubility, indicating that longissimus has similar collagen properties across species.

Breed

Textural differences can be related to connective tissue properties as well as aging potential. For example, Bos indicus breeds have a high collagen content, low collagen solubility, and a tough texture. However, a comparison between B. indicus and Bos taurus crosses did not find significant differences in collagen content or solubility, which may suggest that connective tissue only makes a significant contribution to texture in purebred B. indicus lines. Similarly, a comparison of six European and African breeds found little difference in collagen content and solubility. Compared with normal animals, double-muscled cattle have less collagen, higher collagen solubility, and low raw meat texture scores but similar cooked texture scores. Bos taurus breeds that mature late have high collagen solubility, and the rheology of the connective tissue is variable in raw samples but not in cooked ones. In conclusion, it seems that no consistent differences amongst breeds for texture-related connective tissue properties are evidenced.

Sex

The sex of animal could have an effect on the connective tissue content and quality. Comparisons on animals at the same age beyond puberty show that meat from uncastrated males is less tender than meat from castrated males, which, in turn, is less tender than meat from females. These variations can be partly explained by differences in collagen solubility. Thus, the lower collagen content in muscle from heifers at 13 and 24 months compared with young bulls is a key determinant of differences in tenderness between the two types of animals. Moreover, the meat of females is softer than that of bulls, as it has a lower collagen content and higher collagen solubility.

Age

Tenderness is highest in meat from very young animals but subsequently declines with age and physical activity, along with an increase in the amount of collagen and its degree of complexity. The increase in toughness is more linked to a reduction in collagen solubility than an increase in collagen content. The water solubility of collagen under the action of heat decreases gradually with increasing age, with the result that the meat progressively becomes less tender. These changes are usually attributed to a gradual increase in the number of stable intermolecular cross-links.

Many studies have shown a progressive age-related decrease in muscle collagen solubility and tenderness, with little or no change in total collagen content. Although most of these studies were done on ruminants, the pattern is a general feature of connective tissue. The change in solubility results in increased meat toughness in old animals and is the major factor shaping the contribution of connective tissue to meat quality. Very young animals have more collagen and elastin than mature animals have, but veal and lamb have a very soluble collagen and consequently a very tender meat. There is no evidence that elastin per se is associated with texture, despite the fact that its content varies.

Several studies on the effect of animal age on connective tissue-related meat texture showed that from 1 to 60 months, the textural properties of meat were significantly related to collagen content and its solubility. In the same studies, comparisons of muscle types showed that all 12 types were acceptably tender in animals of approximately 24 months of age or less. Myofiber toughness did not change markedly after 24 months of age, but the connective tissue toughness increased markedly in 10 of the 12 meat types in animals aged 3 years upward. In addition, an increased cross-sectional area of perimysium is observed with age. Perimysium becomes either thicker or has more branched bundles, provided no change in content occurs. Increasing age also results in firmer fiber adhesion by connective tissue. Furthermore, fiber adhesiveness is lost at 60 °C in meat from young animals but not old animals. This may be the property linking collagen solubility to cooked toughness, as it is also related to sensory chewiness.

Muscle Type and Connective Tissue Organization

Collagen and elastin contents differ severalfold between different muscle types. In general, the force muscles such as the biceps have more collagen than postural muscles such as the psoas have. In addition, perimysium volume and organization vary greatly between muscle types (see Figure 4). In the 1930s and 1940s, structural classification of muscle suggested that muscle type differences in texture were correlated to connective tissue content and organization. Muscle grain and muscle fascicles are known to vary significantly between muscle types. Fascicles vary in size, from approximately 1-10 mm, and shape and are hard to visualize and measure. A larger bundle size of fibers ensheathed in primary perimysium would presumably translate into a firmer meat texture, especially when the perimysium is thick and has extensive collagen cross-links that will not melt at cooking temperatures. Fascicle size is correlated to sensory tenderness and shear force texture. Thickness of the perimysium has been shown to be correlated to shear force in chicken and pork. Thicker perimysium melts more slowly and thicker regions have a different composition such as more type-I collagen, more heat-stable elastin, or more cross-links. However, veal also has thicker perimysium than that from mature animals, but the meat stays tender due to its high collagen solubility and small fiber size, which rules out a systematic direct relationship between perimysium thickness and texture. As stated above, collagen can only influence shear force if it has more cross-links and is in greater quantity.

There have been attempts to link perimysium organization to meat quality, and a relationship has been found. Several research teams have used a qualitative grading system to show that perimysium organization varies with muscle type, species, and age. Semiquantification by image analysis on digital images coming from magnetic resonance imaging and histology techniques show that tough muscles have smaller fiber bundles.

The last property to discuss in this section is myofiber adhesion to perimysium and fiber-fiber adhesions. These are the most fragile structures in cooked meat. The initial fracture plane is usually at the endomysium-perimysium junction. When stress is applied parallel to the fiber plane, the fracture occurs in the endomysium, whereas when stress is applied perpendicular to the fiber plane, the fracture occurs at the perimysium-endomysium junction.

Postmortem Changes in Connective Tissue

There are changes in connective tissue that could play a role in tenderness and thus require further study.

Endomysium

It has been shown that the endomysium starts to detach from fibers as early as 6 h postmortem, and at least half of the endomysium is detached within 24 h.

The detachment of the endomysium may be related to the very rapid drop in shear force observed soon after rigor and may be a general aspect shaping the development of texture. The detachment of endomysium in callipyge sheep, even though the meat is very tough, demonstrates that it is marginally involved in meat texture properties. Nevertheless, endomysium detachment from sarcomeres remains a necessary first step for texture development. However, mechanical measures show that lateral adhesion of fibers by endomysium is stable postmortem and accounts for approximately a tenth of the longitudinal force of fibers. The contribution of endomysium to meat texture is, therefore, stable postmortem and very minor compared with the contribution of myofibers.

Perimysium

Perimysial collagen is degraded during storage. Transmission and scanning electron microscopy studies show that the structural integrity of the intramuscular connective tissue decreases during postmortem aging. The major postmortem change in perimysium is that myofibers separate from perimysium within 6 h. Perimysium seems to be the structure most vulnerable to meat shearing action. It has also been shown that the isometric tension of intramuscular collagen decreases at 21 days postmortem in beef. Furthermore, the breaking strength of the perimysial connective tissue in raw beef decreases during postmortem aging. The thermal shrinkage temperature of bovine intramuscular collagen decreases by 7-8 °C within 7 days postmortem. These structural changes are strongly related to the mechanical strength of meat, as demonstrated by shear measurements on raw muscle or uncooked intramuscular connective tissue structures. Proteoglycan degradation occurs at the same time frame as structural changes in connective tissue (Figure 5), but further studies are required to better understand the role of proteoglycans in texture.

Changes in Connective Tissue with Cooking

Elastin is very heat stable, and its properties are generally unaffected by cooking. However, elastin content in the perimysium is generally low (0.1-0.2% of the total connective tissue) and its impact on meat texture is, therefore, considered as not significant. However, elastin may contribute to resistance to shear in the semitendinosus muscle, where elastin accounts for up to 40% of total connective tissue.

In contrast, collagens and proteoglycans are unstable to heat, and their properties are affected by cooking. The thermal denaturation of proteoglycans does not play a direct structural role in mechanical terms, but it can play a role in the thermal stability of collagen due to molecular interactions with collagen fibers.

Although, connective tissue makes the largest contribution to texture in raw or undercooked meat, myofibers make the largest contribution to texture in cooked meat. Cooking at temperatures of 60-70 °C causes gelation of most of the perimysium. The temperature for structural changes tends to vary between studies: Some authors report beef perimysium gelation at 70 °C and above, whereas others cite 60-63 °C. Using isolated fibers and small muscle samples it was demonstrated that perimysium changes its thermal mechanical property at 50-60 °C, before the myofibers toughen. These changes are mostly due to collagen melting. The contribution of connective tissue to meat texture declines as temperature increases.

Although thermal stability is well established as the major property determining the role of connective tissue in cooked meat texture, it is less clear how structure and composition variability affect the structural changes involved. Perimysium attributes like thickness are highly variable in cooked meat. Furthermore, elastin distribution as well as collagen type varies greatly within small regions of perimysium. The relationship of connective tissue structure and composition to thermal and mechanical properties of meat requires further study in order to better evaluate the specific influence of collagen type, elastin, or perimysium thickness.

When heated at more than 60-65 °C, the collagen molecule changes from its native helical ordered state to a randomly coiled structure, and fibrous collagen is converted from a fibrillar to a rubber-like amorphous structure. The destruction of hydrogen bonds in the molecule is accompanied by shrinkage of the collagen fibers. If collagen is free to shorten, it shrinks to one-quarter of its original length. The contraction of collagenous networks during cooking is a major mechanism by which collagen influences meat texture. The thermal contraction of collagen fibers and fibrils starts with a free contraction, after which there is a forced contraction where collagen fibers and fibrils apply pressure on muscles fibers and muscle fiber bundles. The more the collagen fiber contracts during heating, the lower its mechanical strength. As the thermal contraction of meat collagen fibers is limited by muscle fibers, the resistance of muscle fibers or muscle fiber bundles influences the ultimate elastic modulus of the collagen fibers. Moreover, the nature of the cross-links determines contraction strength, contraction amplitude, collagen solubilization, and the final mechanical properties of the connective tissue after heating. The thermal stability of collagen is related to the presence of heat-stable mature cross-links, which increase with age and contribute to toughness. At 70 °C, 42% of veal collagen is soluble compared with 2% solubility in beef from 10-year-old animals, and the thermal shrinkage temperature is 55 °C for veal and 70 °C for beef from 10-year-old animals. Collagenase digests 21% of the collagen in veal, whereas the percentage is approximately 10% in 10-year-old animals. These results indicate that the age-related toughness of cooked meat is directly associated with the change in collagen thermal stability.

The role of cross-links in meat quality is less clear in sameage animals. Stable cross-links do vary significantly by muscle type, with sheep longissimus having few and biceps many. But the tenderest muscle, the psoas, has a high number of cross-links, yet it remains tender even at normal sarcomere lengths. These results suggest that both high collagen content and cross-link formation are needed for connective tissue in order to be able to influence toughness, as is the case in biceps muscle. Between-muscle differences in cross-link quantity have also been reported for beef, sheep, and goat. Several studies have failed to demonstrate a relationship between number of cross-links and collagen solubility. In addition, numerous studies have failed to find a relationship between cross-links and shear force in same-age animals, including two studies that measured five different types of cross-links. Consequently, a relationship can be ruled out between these cross-link types and shear force, especially as meat from callipyge sheep has relatively few cross-links but is still extremely tough. Therefore, it is not yet clear which cross-links are responsible for the age-related change in meat texture.

One hypothesis is that heat stability is the result of collagen interaction with decorin, which is found at every D-line (every 64 nm) along the collagen fibril, is heat stable, and cross-links collagen fibers. There is evidence that decorin is stable postmortem, and there is little evidence that it has mechanical properties. However, decorin has not been yet sufficiently investigated to determine its importance in connective tissue stability at cooking temperatures, despite the fact that this may prove an important factor.

Conclusion

Skeletal muscle connective tissue is a supportive matrix composed of various fibrous proteins, primarily collagen. Many molecular interactions converge to stiffen the matrix, whose properties vary with the physiological and biological characteristics of source animals. The quantity, quality, and organization of this connective tissue depend on the muscle and its function in the body. Connective tissue changes as the animal matures, and the quantity and degree of cross-linking in muscle connective tissue are generally related to the basic toughness of animal meat. As it is impossible to fully control the characteristics of the muscle tissue, several strategies are employed in order to meet consumer expectations. The first is to sort the muscles according to their position on the carcass and their function, both of which are typically associated with connective tissue characteristics. Strength muscles are generally richer in cross-linked collagen and are oriented toward the production of minced meat or for longterm moist cooking processes to dissolve the collagen and tenderize the meats. Postural muscles are generally less collagen rich, and some are used as meat to grill. Therefore, it is possible to optimize meat tenderness by selecting breeds known for their low connective tissue content or by using young animals with lightly cross-linked collagen. Research to date has provided some insight to help elucidate the role and function of connective tissue in the muscle, to determine its composition and organization from the tissue level down to molecular level, and to assess its contribution to the organoleptic properties of the meat end product. Although the role of connective tissue in the mechanical properties of meat is generally understood, the properties of meat textures are complex and multifactorial, resulting from the behavior of connective tissue, muscle fibers, and interactions between these two muscle components. The future development of innovative technological processes that can degrade or even partially dissolve the connective tissue could improve the price and market value of those meat cuts known for their high mechanical strength.

Finally, food science research to date has mainly focused on connective tissue to understand its role in shaping meat texture, and it is only recently that the focus has shifted to also look at the nutritional qualities of meat. Although connective tissue by itself is not known for its nutritional value, its structural organization at different levels in whole tissue muscles may play a role in the distribution of small molecules of nutritional interest during various processes such as cooking, mincing, and cutting, thus resulting in a loss of micronutrients into exudates and/or cooking juices. The endomysium and perimysium envelopes are also more likely to limit the accessibility of digestive enzymes (pepsin, trypsin, and chymotrypsin) to the high nutritional value myofibrillar proteins. Such transformation processes, especially cooking, dissolve all or part of collagen depending on its degree of cross-linking and on the heating conditions. However, there is still a lack of data on the role played by the overall whole muscle tissue in the transfer of micronutrients and small molecules, and on the bioaccessibility of digestive enzymes in myofibrillar proteins.

Figure 2 Figure 1

 21 Figure 2 Collagen fibers (bovine semitendinosus muscle). (a) Collagen fibers seen in perimysium. (b) Longitudinal section of collagen fibers. Myo, myofibrils; Pm, perimysium.

Figure 3

 3 Figure 3 Transversal section of porcine longissimus thoracis muscle. Ad, Adipocyte; Em, endomysium; Fi, muscle fiber; Pm, perimysium.

Figure 4

 4 Figure 4 Differences in structural organization of pig muscles. (a) Masseter, (b) semitendinosus, and (c) longissimus thoracis. Connective tissue is stained in red and muscle fibers in yellow. The content and distribution of connective tissue depends on the muscle. Longissimus thoracis, which have less thicker perimysium than semitendinosus and masseter muscles, is more tender. Fiber size is smaller in masseter than in semitendinosus and longissimus thoracis.

Figure 5

 5 Figure 5 Proteoglycan involvment on intramuscular connective tissue changes during postmortem ageing. IMCT, Intramuscular connective tissue; PGs, Proteoglycans. Reproduced from Nishimura, T., 2010. The role of intramuscular connective tissue in meat texture. Animal Science Journal 81, 21-27, with permission from Japanese Society of Animal Science, John Wiley & Sons Ltd.

Encyclopedia of Meat Sciences, Volume 1 doi:10.1016/B978-0-12-384731-7.00186-0

Connective Tissue: Structure, Function, and Influence on Meat Quality


Bocar Samba Ba
email: bocar-samba.ba@univ-montp1.fr

Phosphorus Conservation, Eutrophication Reduction and Social Welfare Improvement: Taxation of Extracted Phosphorus or Subsidy of Recycled Phosphorus?

Keywords: Environmental Tax, Subsidy, Phosphorus, Eutrophication, Recycling, Competition "à la Stackelberg". ----

This paper aims at analyzing the role of an environmental tax or an environmental subsidy as instruments for preserving phosphate reserves, for improving water quality by reducing eutrophication, and for increasing social welfare. Toward these goals, we use a duopoly model "à la Stackelberg", assume the presence of a benevolent government that takes into account the bene…cial e¤ect of recycling in the social welfare function and refunds the revenue of the tax to the society. First, we …nd that taxing extracted phosphorus or subsidizing recycled phosphorus contributes to the postponement of the depletion of the resource and to the reduction of eutrophication. Second, we …nd that taxing extracted phosphorus reduces consumers'surplus, whereas subsidizing recycled phosphorus increases it. Third, we show that the subsidy set by the regulator is higher than the marginal bene…t of recycling, whereas the level of the tax with respect to the marginal social damage of pollution is ambiguous. Fourth, we state that the tax and the subsidy increase social welfare. Fifth, by way of comparison, we …nd that if the regulator aims at saving phosphorus, reducing eutrophication and improving social welfare simultaneously, subsidizing recycled phosphorus is the best policy, but if he aims only at saving phosphorus and reducing eutrophication, taxing extracted phosphorus is more optimal than subsidizing recycled phosphorus.

Introduction

A basic economic insight is that a competitive economy, under ideal conditions, will generate a socially e¢ cient or a Pareto optimal allocation of private goods, meaning that it is not possible to reallocate resources in such a way that everyone becomes better o¤ [START_REF] Sandmo | Environmental Taxation and Revenue for Development[END_REF]. One element of the ideal conditions requirement is the absence of external e¤ects. In other words, if one agent generates externalities, the allocation is no longer socially optimal if the market is not regulated. Externalities may be both positive or negative. In this paper, we focus on both types of externalities because extracted phosphorus creates eutrophication phenomenons by polluting water, whereas recycling prevents phosphorus from polluting water. Accordingly, extraction generates a negative externality, while recycling generates a positive externality. Since [START_REF] Pigou | The Economics of Welfare[END_REF], it is well known that negative externalities caused by pollution would be internalized by the market if polluters paid a tax equal to the marginal social cost of polluting emissions [START_REF] Nimubona | The pigovian Tax Rule in the Presence of an Eco-Industry[END_REF], while several economists stress that it would be desirable to subsidize a polluter in order to induce him to abate pollution or to subsidize green products, which generate a positive externality.

In this paper, the polluter is associated with a …rm that extracts phosphorus, whereas the environmentally friendly …rm refers to the recycler, as mentioned above. It is widely recognized that unrecycled phosphorus1 pollutes waters. In fact, primary 2 phosphorus ends up into the water due to water run-o¤, soil erosion, drainage from agricultural land, excreta from livestock, municipal and industrial e-uents. Consequently, this situation creates eutrophication phenomenons. As stated above, recycling 3 will reduce water pollution by preventing phosphorus from ending up into the water. In addition to the reduction of eutrophication, recycling contributes to create energy and to save phosphorus. For instance, [START_REF] Li | Chemestry: Reuse Water Pollutants[END_REF] argue that the implementation of a factory in China (in Chongqing town), which treats wastewater and sludge, allows to save 67; 000 KWh per day and enables to generate 1 tonne of phosphorus per day. Recall that the term "Eutrophication" refers to an unwanted explosion of living aquatic-based organisms in lakes and estuaries that results in oxygen depletion, which can destroy an aquatic ecosystem [START_REF] Liu | Global Phosphorus Flows and Environmental Impacts from a Consumption Perspective[END_REF]. Signi…cant eutrophication took place in the 1950s in the Great Lakes of North America, in Cayuga Lake, which is in Central New York [START_REF] Jacobs | Internalizing Externalities of Phosphorus Discharges from Crop Production to Surface Water: E-uent Taxes versus Uniform Reduction[END_REF], in the Poyang Lake watershed that is in China [START_REF] Deng | Analysis of the Trade-o¤ between Economic Growth and the Reduction of Nitrogen and Phosphorus Emission in the Poyang Lake Watershed, China[END_REF], in the Norfolk broads of United Kingdom [START_REF] Philipps | A large Scale Field Experiment in the Control of Eutrophication in the Norfolk Broads[END_REF]. It has been also prevalent in many other lakes and estuaries around the world (International Lake Environment Committee Foundation, 2003).

One way to reduce 4 eutrophication consists of taxing virgin phosphorus or subsidizing recycled phosphorus, as mentioned above. This would diminish (increase) extracted phosphorus (recycled phosphorus) and would reduce (boost) extracted phosphorus (recycled phosphorus). Owing to the strategic substitutability of both types of phosphorus, the increase of one triggers the decrease of the other, and vice versa. Aiming at reducing eutrophication, taxation of extracted phosphorus has been applied in some countries, including the United States of America 5 (see [START_REF] Jacobs | Internalizing Externalities of Phosphorus Discharges from Crop Production to Surface Water: E-uent Taxes versus Uniform Reduction[END_REF][START_REF] Shakhramanyan | The Impacts of Higheer Mineral Phosphorus Prices and Externality Taxation on the Use of Organic Phosphorus Sources in US Agriculture[END_REF] and China. As well as reducing pollution of waters, taxation or subsidy would contribute to prolong the lifetime of phosphorus. The exhaustion 6 of which is predicted in a near future. Although subsidy can be seen as a cost for the government in question, taxation would give him the opportunity to collect some funds in order to …nance several goals (see [START_REF] Gersbach | Emission Taxes and Optimal Refunding Shemes[END_REF].

The consideration of positive or negative external e¤ects in the decisions of production of the …rms and in the social welfare function generates a number of interesting questions. Does the tax or the subsidy contribute to the prolongation of the lifetime of phosphorus and the reduction of water pollution ? Is the level of the tax (or of the subsidy) set above or below the marginal social damage of pollution (the marginal social bene…t of recycling) ? What is the e¤ect of each of these policies on consumers'surplus and on social welfare ? Is taxation more optimal in saving phosphorus and reducing eutrophication than subsidy ? What is the best policy in terms of the improvement of social welfare ? In the present paper, we address these and related questions.

In connection with the questions above, the aim of this paper is …rst to analyze the e¤ect of the tax or the subsidy on the depletion of phosphorus and on water pollution. Second, we will see what is the level of the tax or of the subsidy, respectively with respect to the marginal social damage of pollution and with respect to the marginal social bene…t of recycling. Third, we aim at comparing the two policies in terms of optimality. Toward these ends, we use a duopoly model in which two …rms compete "à la Stackelberg" for two consecutive steps. In the …rst step, …rm A chooses the quantity it extracts, whereas in the second step …rm B chooses the quantity it recycles. At the very begining of the game, we assume the presence of a benevolent 7 government which sets the level of the tax or that of the subsidy, and both …rms produce accordingly.

Summarizing some of our …ndings, we can state the following e¤ects. First, we …nd that taxing extracted phosphorus or subsidizing recycled phosphorus contributes to delay the depletion of the resource and to reduce eutrophication. Second, we …nd that taxing extracted phosphorus reduces consumers'surplus, whereas subsidizing recycled phosphorus increases it. Third, we show that the subsidy set by the regulator is higher than the marginal bene…t of recycling, whereas the level of the tax with respect to the marginal social damage of pollution is ambiguous. Fourth, we show that the tax and the subsidy increase social welfare. Fifth, by way of comparison, we …nd that if the regulator aims at saving phosphorus, reducing eutrophication and improving social welfare simultaneously, subsidizing recycled phosphorus is the best policy, but if he aims only at saving phosphorus and reducing eutrophication, taxing extracted phosphorus is more optimal than subsidizing recycled phosphorus.

This paper is based on several strands of the literature: the …rst one is related to the analysis of the e¤ect of taxation on environmental pollution and on social welfare. Recalling that the pigovian conclusion that the level of the tax must equal the marginal social cost of polluting emissions is made within a context of perfect competition. In fact, when the market is imperfectly competitive, the tax should be set lower than the marginal social cost of pollution, because it trades o¤ the desire to provide incentives for abatement and the necessity to prevent a greater contraction of output [START_REF] Nimubona | The pigovian Tax Rule in the Presence of an Eco-Industry[END_REF], see also [START_REF] Levin | Taxation within Cournot Oligopoly[END_REF]). [START_REF] Buchanan | External Diseconomies, Corrective Taxes, and Market Structure[END_REF] echoes the previous conclusion and gives the following intuition: setting a tax lower than the marginal social damage solves the tendency of imperfectly competitive …rms to underproduce. The same conclusion is highlighted by [START_REF] Barnett | The Pigovian Tax Under Monopoly[END_REF] in the case where the elasticity of demand is …nite. Nevertheless, there are other authors who show that the corrective tax can be higher than the marginal social cost. Using a Cournot duopoly model, [START_REF] Simpson | Optimal Pollution Taxation in a Cournot Duopoly[END_REF] …nds that the optimal tax rate may exceed the marginal damage and provides the following intuition: "Cournot duopolists will not, in general, produce their outputs e¢ ciently. To the extent that a pollution tax may shift production output from the less e¢ cient …rm to its more e¢ cient rival, higher tax rates may be called for". David and Sinclaire-Desgagné (2005) state that, under some conditions, an optimal emission tax should be set higher than the marginal social cost of pollution. The intuition underlying this idea is that imperfect competition between environment …rms results in abatement prices larger than the marginal social cost of abatement; emission taxes must then be raised in order to make polluters reduce their emissions su¢ ciently. [START_REF] Barnett | The Pigovian Tax Under Monopoly[END_REF] …nds that the tax can be higher than the marginal social damage if demand is more price elastic. Using a Bertrand duopoly model with di¤erentiated products, [START_REF] Kurtyka | The Switching E¤ect of Environmental Taxation within Bertrand Di¤erentiated Duopoly[END_REF] state that the corrective tax can be higher than the marginal damage. The intuition behind their result is that heavy taxation of the polluting good will encourage consumers to switch toward the green product.

In spite of the lack of consensus on the level of the tax with respect to the marginal social damage in ‡icted by the polluter, the overall conclusion of this line of research is that taxation reduces pollution. However, there are some situations where the e¤ect of the environmental tax may induce an undesirable e¤ect. [START_REF] Levin | Taxation within Cournot Oligopoly[END_REF] analyzes this issue within a Cournot oligopoly model. He assumes that a tax is imposed on each seller at a uniform rate per unit of output and shows if …rms are su¢ ciently di¤erent, pollution increases in the tax rate. The intuition he provides is that if …rms are asymmetric and one taxes them symmetrically, there is a chance that the heavy polluter will bene…t, resulting in the rise of pollution 8 . Even if our two …rms are symmetric in terms of costs of production, which are zero, we show, contrary to [START_REF] Levin | Taxation within Cournot Oligopoly[END_REF], that the tax always reduces pollution.

It is noteworthy to mention that several authors argue that taxation reduces social welfare. [START_REF] Buchanan | External Diseconomies, Corrective Taxes, and Market Structure[END_REF] stresses that, under an imperfectly competitive organization 9 , the corrective tax will often reduce 10 than increase welfare. He shows that any levy of a per unit tax on the monopolist's output will decrease total welfare. [START_REF] Barnett | The Pigovian Tax Under Monopoly[END_REF] echoes the same conclusion. In contrast to them, we …nd that the corrective tax can increase social welfare. Using a Cournot Oligopoly model, Ushio 11 (2000) highlights that the tax reduces social welfare if the market structure is symmetric or if the inverse demand function is linear or convex. Albeit the fact that the inverse demand function is linear in our model, social welfare is improved here by the implementation of the taxation scheme.

The second strand of the literature concerns the relationship between a subsidy and pollution control. [START_REF] Baumol | The Theory of Environmental Policy[END_REF] argue that although a subsidy will tend to reduce the emission of the …rm, it is apt to increase the emissions of the industry beyond what they would be in the absence of the …scal incentives. [START_REF] Mestelman | Production Externalities and Corrective Subsidies: A General Equilibrium Analysis[END_REF] uses a general equilibrium model and analyzes the e¤ects of taxes and subsidies in a competitive economy which is characterized by a production externality. He shows that the use of a subsidy is ine¢ cient. Mestelman (1984) shows that a pollution tax is consistently preferred to a subsidy by majority of individuals. Diamond and Mirrlees (1971) …nd that the e¢ cient combination of abatement and outpout requires the pigovian tax alone, because the pollution abatement subsidy distorts the price the input used for reducing pollution, resulting thus in a non-achievement of e¢ ciency. [START_REF] Fredriksson | Environmental Policy Choice: Pollution Abatement Subsidies[END_REF] highlights that pollution abatement subsidies are ine¢ cient instruments for pollution control. He stresses as follows the reasons for which subsidies reduce social welfare. He considers the benchmark as the social optimum situation where the government sets a tax and not a subsidy and argues that if pollution is decreasing in the subsidy rate, the subsidy bene…ts the environmentalists. Also, the industrialists always gain from receiving the subsidy. Conversely, the remaining groups in society pay a share of the subsidy, but derive not utility from it. Even if aggregate payo¤s of the industrialists and the government rise, total welfare declines when one moves away from the social optimum. In contrast to Diamond and Mirrlees (1971), [START_REF] Mestelman | Production Externalities and Corrective Subsidies: A General Equilibrium Analysis[END_REF], and [START_REF] Fredriksson | Environmental Policy Choice: Pollution Abatement Subsidies[END_REF], we show that the subsidy is e¢ cient in pollution control in the sense that it leads to the reduction of pollution. Unlike [START_REF] Fredriksson | Environmental Policy Choice: Pollution Abatement Subsidies[END_REF], we show clearly that the subsidy improves social welfare even if one moves away from a situation where no tax is applied. Contrary to Mestelman (1984), we …nd that a subsidy is preferred to a tax in terms of the improvement of social welfare.

Clearly, the innovation introduced in this paper is that it is one of the few, if not the only, which takes simultaneously into account the damage of pollution and the bene…t of a non-polluting good/resource in a social welfare function. This consideration is notably due to the speci…city of the considered resource that presents a variety which is not polluting, at least in the short run. To the best of our knowledge, there is not another paper which makes this consideration. In terms of results, the main contributions are as follows. First, in contrast to the earlier literature which states that the corrective tax always reduces social welfare (See for instance [START_REF] Buchanan | External Diseconomies, Corrective Taxes, and Market Structure[END_REF] and [START_REF] Barnett | The Pigovian Tax Under Monopoly[END_REF]), the present paper shows that taxation improves social welfare. Second, some authors (see for instance Simpson 12 (1995), David and Sinclaire-Desgagné (2005), etc.) have shown that the optimal pollution tax on an imperfectly competitive industry is higher than the marginal damage if and only if …rms are asymmetric in terms of costs of production. In the present paper, we …nd that, even if …rms are symmetric, the tax may be higher than the marginal damage. Third, in contrast to the bulk of the literature which prefers absolutely taxation to subsidy, we show that the subsidy is more optimal than the tax in terms of the improvement of social welfare.

The remainder of the paper is as follows. The next section introduces the concept of "eutrophication". Section 3 presents the model and the results. The main conclusions and some further research lines are given in section 4 and all proofs are relegated to the appendix in section 5.

Eutrophication

Bodies of water can be categorized as being in one of two states on the basis of their nutrient content. Low nutrient oligotrophic waters are clear and have relatively little animal and plant life, whereas the high nutrient content of eutrophic waters encourages the development of fauna and ‡ora [START_REF] Salerno | The Political Economy of Shallow Lakes[END_REF]. Eutrophication denotes the enrichment in nutrients of lakes and rivers that leads to this state of abundant life and therefore sounds like a positive development for a natural habitat. This enrichment can disrupt the natural balance of the natural system and lead to a complete transformation of the habitat [START_REF] Ricklefs | Ecology[END_REF]. The new altered state is often characterized by rapid plant and algae growth. When the density of the vegetation becomes such the ecosystem can no longer support it, it dies and begins to decay [START_REF] Salerno | The Political Economy of Shallow Lakes[END_REF]. Since the rate of decomposition enhances, the process consumes so much oxygen that …sh and other aquatic animals su¤ocate [START_REF] Ricklefs | Ecology[END_REF]. In addition, the growth of non-toxic algae results in shade and an rise of the water pH 13 , which then favors the abundance of the cyanobacteria or blue-green algae, a bacterium that can produce lethal toxins (Sche¤er, 1998). Algae can also a¤ect treatment of water for potable supply, by blocking …lters or passing through them causing bad odeur and taste [START_REF] Collingwood | A Survey of Eutrophication in Britains and its E¤ect on Water Supplies[END_REF].

Eutrophication manifests in four stages: (i) Increasing pollution: phosphorus ends up into waters, due to water run-o¤, soil erosion, etc. At the beginning, the oxygen content favors aquatic life. Fish are not a¤ected.

(ii) Algae growth: phosphorus leads to the development of algae which consume so much oxygen. the oxygen content increases at the surface of the waters but diminishes signi…cantly in the depths of the waters. Some species die.

(iii) Anaerobic decomposition: sediments rich in organic matter accumulate more. Aerobic bacteria multiply in order to degrade organic matter and consume oxygen. The oxygen content is strongly weakened on the whole water column.

(iv) Extreme degradation of the environment dystrophy stage : The oxygen content has signi…cantly fallen. There is an absence of oxygen in the aquatic environment. The depletion of oxygen favors the formation of sulfuric acid and ammoniac in the water, leading to the death of …sh. At this stage, there is a health risk for fauna and for humanity that use this water, because some cyanobacteria produce toxins.

The model

The economy we consider consists of a benevolent government, consumers and two …rms, named …rm A and …rm B . Firm A holds phosphate rocks, extracts them and transforms them into phosphorus which is used as a fertilizer. This, phophorus is what is, widely, commercialized. Firm B, after consuming phosphorus, recycles it and sells it. Therefore, both …rms compete. Note that it is technically impossible to recycle the whole phosphorus extracted previously, resulting in r < q. Phosphorus which is directly extracted from phosphate rocks is called primary phosphorus and that which is recycled is called secondary or green phosphorus. As mentioned above, the primary phosphorus is considered as a polluting resource [START_REF] Cordell | Towards Global Phosphorus Security: A Systems Framework for Phosphorus Recovery and Reuse Options[END_REF], because it ends up into the water due to water run-o¤, soil erosion, etc. and, consequently, creates eutrophication phenomenons. Recycling of phosphorus prevents it from ending up into the waters, therefore, reduces water pollution (Weikard and ). Another way or an additional means of reducing water pollution would consist of applying a tax to the primary phosphorus or subsidizing recycled phosphorus. Thus, since taxation of virgin phosphorus or subsidy of recycled phosphorus discourages the extraction and encourages the recycling activity, each of these policies reduces eutrophication phenomenons. We assume the presence of a benevolent government that applies this taxation or subsidizes the recycler. In order to compare these two policies, we assume that the benevolent government implements both policies separately. This analysis could be extended to an international scale if there was a supranational government that regulates pollution.

Both …rms compete with the quantities of phosphorus they put in the market. Let q denote the quantity which is extracted by Firm A and r be the quantity that is recycled by …rm B. For the sake of simplicity, we assume that the inverse demand function is linear and is given by p(Q) = a Q, where Q is the total quantity of phosphorus which is sold and a may be interpreted as the size of the market or, the maximum price at which phosphorus can be sold or also as a choke price [START_REF] Sweeney | Handbook of Natural Resource and Energy Economics[END_REF][START_REF] Baksi | Endogeneous Consumer Participation and the Recycling Problem[END_REF]. We, also, consider only one tax applied to the polluting resource instead of having another tax applied to the emissions (for this case, see Cremer and Firouz, 2003). It is also considered that there are no extraction and recycling costs. We, also, assume that the tax-revenue ( q) is refunded to the society.

The timing of the game between the regulator and the …rms can be described as follows. In the …rst step, the regulator sets the level of the tax or the level of the subsidy and refunds all the revenue of the tax to the society (in the case of taxation). In the second step, …rm A chooses the quantity it extracts. In the third step, …rm B chooses the amount it recycles, after consuming phosphorus.

Phosphorus conservation and eutrophication reduction

Since [START_REF] Pigou | The Economics of Welfare[END_REF], it is well known that taxation is a means of reducing pollution. Recent studies have also shown that subsidy can yield a similar result, at least within a static context. In this section, we aim at investigating the e¤ects of each of these policies on the reduction of eutrophication. As the reduction of eutrophication coincides here with the decrease of extracted phosphorus, reaching the former goal enables to delay the depletion of phosphorus. First, we will analyze the e¤ect of the tax on the lifetime of phosphorus and on the reduction of eutrophication. Second, the same issue will be explored within the context of a subsidy.

Taxation of extracted phosphorus

As stated above the timing of the game can be described as follows. In the …rst step, the regulator sets the level of the tax rate and refunds the whole revenue of the tax to the society. In the second step, …rm A chooses the quantity it extracts. In the third step, …rm B chooses the amount it recycles. Knowing the level of the tax charged by the benevolent government, each …rm maximizes its own programme. In order to …nd the subgame-perfect Nash equilibrium of the whole game, we will solve the latter by backward induction. Thus,

Step 2: The pro…t maximization for …rm B is:

max r>0 B = (a q r)r (1)
s:t: r < q (2)
Condition ( 2) indicates the fact that recycling is never complete (see [START_REF] Weikard | The Distribution of Phophorus between Rich and Power Countries: The E¤ect of Recycling[END_REF]. The …rst-order condition for the programme above is given by:

r(q) = a q 2 (3)
Step 1: Since the game is solved by backward induction, …rm A inserts the reaction function of …rm B in its decision of production and maximizes the following programme:

max q>0 A = (a q 1 2 (a q))q q (4)
Solving the two previous programmes yields the optimal quantities q( ) and r( ) whose the comparative statics are summed up through the following proposition:

Proposition 1 The tax levied by the authority decreases (increases) extracted phosphorus (recycled phosphorus). Formally, we have:

(i) @ q( ) @ < 0 (5) (ii) @ r( ) @ > 0 (6)
Proof. See Appendix II

The comparative static results deriving from proposition (1) are quite intuitive. The point (i) states that the environmental tax curbs the quantity which is extracted directly from phosphate rocks. This decreasing e¤ect will delay the exhaustion of phosphorus and enables, therefore, the resource to be saved. In fact, the imposition of a tax will make extracted phosphorus more expensive in that the extractor will set higher price, leading consumers to switch towards recycled phosphorus which remains cheaper. Thus, this environmental tax creates a switching e¤ect which consists of boosting the recycling activity, resulting in @ r( ) @ > 0. Since the primary resource is polluting, its decrease reduces environmental pollution. The environmental pollution reduction is also strengthened by recycling. Indeed, as mentioned above, recycling prevents phosphorus from ending up into the waters, avoids, therefore, eutrophication phenomenons.Thus, the tax plays a twofold role.

For a = 4, this result is illustrated through the following …gure: This …gure shows that the tax reduces extracted phosphorus while it boosts recycled phosphorus. As r < q, the graphic is valid only if 2 (0; 2 3 ). Otherwise, the recycling curve is above the extraction curve (one can also verify it through (63) in appendix II). For this level of the tax, pollution diminishes but does not entirely disappear. This …gure indicates also that taxation reduces polluting phosphorus from 2 to 4 3 . In addition, it increases green phosphorus from 1 to 4 3 . The variation of extracted phosphorus is given by 2 3 , whereas the variation of recycled phosphorus is 1 3 . Clearly, we can state that the tax reduces consumers' surplus in the sense that its decreasing e¤ect is higher than its increasing e¤ect. Now, let us analyze the impact of the subsidy on the optimal quantities.

Subsidy of recycled phosphorus

We now turn to the equilibrium outcome under the presence of a subsidy. The timing of the game can be described as follows. In the …rst step, the regulator sets the level of the subsidy rate s that it levies from the society and that it pays to the recycling …rm. In the second step, …rm A chooses the quantity it extracts. In the third step, …rm B chooses the amount it recycles. Knowing the level of the subsidy paid by the benevolent government, each …rm maximizes its own programme. As in the case of taxation, we solve the game by backward induction. Thus,

Step 2: The pro…t maximization for …rm B is:

max r>0 B = (a q r)r + sr (7)
s:t: r < q (8)

The …rst-order condition deriving from the programme above is then given by:

r(q) = a q + s 2 (9)
Step 1: Firm A inserts the best-reply function of …rm B in its decision of production and maximizes the following programme:

max q>0 A = (a q 1 2 (a q + s))q (10)
Solving the two previous programmes yields the optimal quantities r( ) and q( ) whose the comparative statics are summed up through the following proposition:

Proposition 2 The subsidy increases (decreases) recycled phosphorus (extracted phosphorus). Formally, we have:

(i) @ r(s) @s > 0 (11)
(ii) @ q(s) @s < 0

Proof. For the detail of calculations, see appendix III

The intuition underlying proposition (2) can be explained as follows. Subsidizing recycled phosphorus increases the pro…t of the recycling …rm, resulting in the rise of the quantity it recycles. Since recycled phosphorus and extracted phosphorus are strategic substitutes, the increase of the former induces mechanically the slowdown of the latter.

For a = 4, the previous result is depicted through the following …gure: Figure 2 shows that the subsidy increases recycled phosphorus, whereas it reduces extracted phosphorus. As r < q, the graphic is valid only if s 2 (0; 4 5 ). Otherwise, the recycling curve is above the extraction curve. Through the …gure, it appears also that the subsidy reduces eutrophication and contributes to delay the depletion of the resource (in that it reduces extracted phosphorus from 2 to 8 5 ) but it does not eliminate the former and will not enable to avoid the exhaustion of phosphorus in the sense that extraction still occurs. The …gure shows also that the subsidy increases recycled phosphorus from 1 to 8 5 . The variation of extracted phosphorus is 0:4 and that of recycled phosphorus is 0:6. Since the increasing e¤ect is higher than the decreasing e¤ect, the implementation of subsidy scheme increases consumers'surplus.

The two previous propositions highlight that both the tax on extracted phosphorus and the subsidy on recycled phosphorus yield the same results in qualitative terms. The question which obviously arises is whether or not they yield identical results in quantitative terms. In other words, which policy is more optimal in terms of phosphorus saving and in terms of the reduction of eutrophication ? In the next section, we will address this related issue.

Tax on extracted phosphorus or subsidy on recycled phosphorus ?

In order to know which policy is more optimal in terms of phosphorus saving and in terms of the reduction of eutrophication, we will see the extent of the variation of the optimal quantities. In other words, does the tax reduce (increase) extracted phosphorus (recycled phosphorus) more than the subsidy ?

To answer this question, let us consider the two following cases: case 1: Taxation: the variation of the optimal quantities in moving from the benchmark (zero-tax scenario) to the situation where the market is regulated (presence of a pigovian tax) is:

I For extracted phosphorus:

z}|{ q = (13)
I For recycled phosphorus:

+ z}|{ r = 1 2 (14)
For the proof of the calculus above, see appendix II. It is straightforward to see through [START_REF] Deng | Analysis of the Trade-o¤ between Economic Growth and the Reduction of Nitrogen and Phosphorus Emission in the Poyang Lake Watershed, China[END_REF] and ( 14) that the tax decreases extracted phosphorus more than it increases recycled phosphorus. This explains why the total quantity sold by the industry decreases due to the introduction of this taxation scheme, resulting therefore in the decline of consumers'surplus. case 2: Subsidy: in this case, the variation of the optimal quantities in moving from the benchmark (zero-subsidy scenario) to the situation where the market is regulated (presence of a subsidy) is:

I For extracted phosphorus:

z}|{ q s = 1 2 s (15)
I For recycled phosphorus:

+ z}|{ r s = 3 4 s (16)
For the proof of the calculus above, refer to appendix III. Conditions ( 15) and ( 16) show clearly that the subsidy increases recycled phosphorus more than it decreases extracted phosphorus. That is why the subsidy increases the total quantity sold by the two …rms, resulting then in the rise of consumers'surplus.

Let us turn now to the comparison of the two policies: to get a clearer picture, we sum up (13), ( 14), ( 15) and ( 16) in the following table :   policies tax subsidy q 1 2 s r In order to obtain a standard for a comparison, we will set the tax rate ( ) equal to the subsidy rate (s) as in [START_REF] Ballard | The Marginal E¢ ciency Effects of Taxes and Subsidies in the Presence of Externalities: A Computational General Equilibrium Approach[END_REF]. This choice can be justi…ed as follows. Since both policies are implemented separately, the government can use the whole revenue of the tax in order to subsidize the recycling …rm, under the assumption that the implementation of the tax scheme is earlier than that of the subsidy scheme. The analysis of the results of this table yields the upcoming proposition.

Proposition 3 If the amount of the tax per unit of pollution is equal to the amount of the subsidy per unit of the recycled product:

(i) The tax is better than the subsidy in terms of the reduction of eutrophication and in terms of the postponement of the depletion of extracted phosphorus. Formally, we have:

z}|{ q > z}|{ q s (17)
(ii) Conversely, if the government aims only at boosting the recycling activity, it is more optimal to subsidize recycled phosphorus than to tax extracted phosphorus. Formally, we have:

+ z}|{ r s > + z}|{ r (18)
Point (i) of proposition (3) states that the reduction of extracted phosphorus is rather greater in the situation where taxation is applied than in the case where the subsidy scheme is implemented. Equivalently, this means that taxation of virgin phosphorus is more optimal than subsidy of recycled phosphorus in terms of the reduction of eutrophication and in terms of the postponement of the depletion of extracted phosphorus. A possible explanation for this scenario is that taxation reduces directly extracted phosphorus by increasing its price, whereas subsidy of recycled phosphorus reduces it indirectly by boosting recycling, …rst, before discouraging extraction, accordingly, due to the strategic substitutability of both types of phosphorus. In addition, by subsidizing recycled phosphorus, the rise of recycling cannot exceed a certain threshold, due to the fact that recycling is never complete, i.e. r < q. Such a constraint indicates that the enhancement of recycled phosphorus in the subsidy rate is limited. Consequently, this will limit the reduction of extracted phosphorus, whereas in the case of taxation, the latter can decrease continuously in the tax rate.

The intuition behind point (ii) is as follows. Subsidizing recycled phosphorus gives more power to the recycling …rm than taxing extracted phosphorus in the sense that it increases directly its revenue. Even if taxation enhances its revenue by augmenting the quantity it recycles, due to the strategic substitutability, everything happens as if the revenue that it earns directly in the case of the subsidy is higher than that it earns indirectly in the case of taxation.

Comparison of taxation and subsidy in terms of social welfare improvement

In this section, we will explore whether taxing extracted phosphorus is more optimal than subsidizing recycled phosphorus or not in terms of social welfare.

Taxation of extracted phosphorus

Consider that the market is regulated, i.e. a tax is applied to extracted phosphorus and assume that the revenue of the tax is returned to the society and the positive externality generated by recycling is taken into account by the regulator. To maximize welfare by taxation, the benevolent government must …nd some tax rate per unit of pollution, which will maximize:

w( ) = CS( ) + A ( ) + B ( ) D(q( )) + B(r( )) + q( ) (19)
We use a linear inverse demand function p(Q) = a Q, a linear damage function, i.e. D(q) = "q and a linear bene…t function, i.e. B(r) = r (where is the marginal bene…t of recycling). For the sake of simplicity and in order to focus only on the level of the tax ^ with respect to the marginal damage of pollution ", let us assume that = 1. After making some simplications, the social welfare function writes:

w( ) = r( ) "q( ) + 1 2 aq( ) + ar( ) 1 2 (r( )) 2 (20)
Inserting in [START_REF] Pigou | The Economics of Welfare[END_REF] the optimal quantities established in appendix II and making some simplications yield:

w( ) = 4 2 [2 (2a 21") + 1] + a[15a 8 (2" 1)] 32 (21)
It has been clearly shown in appendix II that w( ) > 0. Clearly, it appears that the marginal damage of pollution decreases social welfare, in the sense that @w( ) @" = 1 2 a < 0, because < 1 6 a, as shown by equation (63) established in appendix II. This result is quite intuitive. Before establishing the …nal result in this case, let us make the following assumptions: Assumption 1.

a 4 8 < " < a 3 6 (22) Assumption 2. a > 3 (23)
Assumption 1 garantees to have a positive optimal tax rate ^ , a postive quantity q and the condition r < q, whereas assumption 2 garantees to have " > 0. Solving @w( ) @ = 0, yields the results summarized in the following preliminary result: Lemma 4 The government selects an equilibrium pollution abatement tax ^ and an optimal social welfare ŵ^ which satisfy:

(i) ^ = 4" (a 4) 2 (24)
(ii) ŵ^ = 4" 2 2 (a 2) " + a 2 + 1 2 (25)
Proof. See appendix II Under ( 22) and ( 23), point (i) states that the level of the tax with respect to the marginal social damage is ambiguous and depends on the level of pollution. In fact, if a 4 8 < " < a 4 6 , the tax is set below the marginal social damage of pollution. Conversely, if a 4 6 < " < a 3 6 , the tax is set above the marginal social damage of pollution. In connection with some purposes of this paper, which consist of saving phosphorus and of reducing eutrophication, it is reasonable for the benevolent government to set the corrective tax above the marginal social damage. This will lead the extracting …rm to underproduce more, resulting in the rise of recycled phosphorus that he aims also at fostering. In contrast to the conventional wisdom that the tax is set above the marginal damage only and only if the two …rms have di¤erent costs of production, we show that the tax may be higher than the marginal damage even if …rms are symmetric.

In order to explore the e¤ect of the tax on social welfare, derivation of the latter with respect to the tax rate yields:

dw( ) d = a + 2 8" 4 8 (26)
From ( 25) and given our assumptions, we can state the following result:

Proposition 5 The optimal tax set by the regulator increases social welfare. If the government has the incentive to deviate from the optimal tax, for whatever reason, we have: Proof. See appendix III The general intuition underlying the increase of social welfare in the tax rate is the following. Owing to the tax applied by the benevolent government, two e¤ects working in opposite directions emerge. On the one hand, the tax enhances the cost incurred by the extracting …rm. This curbs the equilibrium extracted quantity, which in turn reduces consumers'surplus and the pro…t of the extracting …rm. On the other hand, the tax increases the quantity which is recycled by the recycling …rm, resulting in the increase of the pro…t of the latter. As mentioned above, this result can be explained by the fact that extracted and recycled phosphorus are strategic substitutes. Then, the decline of the extracted quantity leads, mechanically, the recycling …rm to increase the quantity it recycles. The rise of recycled phosphorus may improve the consumers' bene…ts compared to the situation where the whole demand was met by one supplier. As well as improving quantitatively social welfare, the increase of the recycled quantity improves it qualitatively because it prevents phosphorus from ending up into the waters. Consequently, it contributes to the improvement of the quality of the waters, thanks to the reduction of the eutrophication phenomenons. The enhancing e¤ect of the tax on the recycling …rm's pro…t and the improvement of the quality of the waters outweigh the decreasing e¤ect of the tax on the sum of consumers'surplus and the extracting …rm's pro…t, resulting in higher social welfare.

(iii) If ^ 1 < < ^ ,

Subsidy of recycled phosphorus

Consider that the market of recycled phosphorus is regulated, i.e. the recycling …rm receives a subsidy rate s from the government. This rate of the subsidy comes obviously from the budget of the society which is managed by the government. Thus, the amount of the subsidy sr(s) will be removed from the social welfare function. To maximize welfare with respect to the subsidy, the government must …nd some subsidy rate s per unit of recycled product, which will maximize:

w(s) = SC(s) + A (s) + B (s) D(q(s)) + B(r(s)) sr(s) (29)
We use a linear inverse demand function, a linear damage function, i.e. D(q) = "q (where " is the marginal damage of pollution) and a linear bene…t function, i.e. B(r) = r (where is the marginal bene…t of recycling). For the sake of simplicity and in order to focus only on the level of the subsidy ŝ with respect to the marginal bene…t of recycling , let us assume that " = 1. After making some simplications, the social welfare function writes:

w(s) = (a s 2) q(s) + (2a r + 2 ) r(s) 2 (30)
Inserting in [START_REF] Ridder | Risks and Opportunities in the Global Phosphate Rock Market, Robust Strategies in Times of Uncertainty[END_REF] the optimal quantities given in appendix III yields:

w(s) = s 2 + 2[a + 4 (3 + 2)]s + a[15a + 8 ( 2)] 32 (31)
We have clearly proved in appendix III that w(s) > 0. Clearly, it appears that the marginal bene…t of recycling increases social welfare, in the sense that @w(s) @ = a+3s 4 . This result is quite intuitive. Solving @w(s) @s = 0, yields the results summarized in the following lemma: Lemma 6 The government selects an equilibrium pollution abatement subsidy rate ŝ and optimal social welfare ŵŝ which satisfy:

(i) ŝ = a + 12 + 8 (32)
(ii) ŵŝ = [2a + 3 (3 + 4)] + a 2 + 4 2 (33)
Proof. See appendix III Under our assumptions, we show that the optimal level of the subsidy rate is higher than the marginal bene…t generated by recycling. The intuition is similar to that established in the case of taxation. In fact, the benevolent government sets the level of the subsidy above the marginal bene…t of recycling in order to boost the recycling activity. The rise of recycled phosphorus induces the decline of extracted phosphorus, resulting then in the postponement of the depletion of the resource and on the reduction of eutrophication.

Given our assumptions, we are able to establish the following proposition:

Proposition 7 The optimal subsidy rate set by the regulator increases social welfare. If the government has the incentive to deviate from the optimal subsidy, for whatever reason, we have: (iii) If ŝ1 < s < ŝ, social welfare increases in the subsidy rate, resulting in:

dw(s) ds > 0 (34)
(iv) Conversely, if ŝ < s < ŝ2 , social welfare decreases in the subsidy rate, resulting in:

dw(s) ds < 0 (35)
Proof. See appendix III Proposition [START_REF] Cogoye | A Model of Eco-E¢ ciency and Recycling[END_REF] indicates that the subsidy improves social welfare. Owing to the subsidy paid by the benevolent government to the recycling …rm, two e¤ects working in opposite directions emerge. In fact, the subsidy enhances the pro…t of the recycling …rm, resulting in the rise of recycled phosphorus. Since recycled phosphorus and extracted phosphorus are strategic substitutes, this leads to the decline of extracted phosphorus. But as the increasing e¤ect is higher than the decreasing e¤ect (i.e. r s > q s ), the total quantity of the industry increases in subsidy, resulting in the rise of consumers' surplus. As well as improving quantitatively social welfare, the increase of the recycled quantity improves it qualitatively because it prevents phosphorus from ending up into the waters. Consequently, it contributes to the improvement of the quality of the waters, thanks to the reduction of the eutrophication phenomenons. The increasing e¤ect of the subsidy on the sum of consumers'surplus and the recycling …rm's pro…t outweighs the decreasing e¤ect of the subsidy on the extracting …rm's pro…t, resulting in higher social welfare.

Comparison of both policies

Clearly, we observe that taxation of extracted phosphorus and subsidy of recycled phosphorus play the same role, in that both improve social welfare. The remaining question consists naturally of investigating what policy is more optimal in quantitative terms. In order to compare both policies, let us assume that " = = 1. Then, optimal social welfares are respectively given by: I In the case of taxation:

ŵ^ = a 2 2a + 9 2 (36)
I In the case of subsidy:

ŵŝ = a 2 + 2a + 25 2 (37)
The comparison of ( 36) and ( 37) yields the next proposition:

Proposition 8 If the regulator has the incentive to maximize social welfare, subsidy is the best policy. Formally, we have:

ŵŝ > ŵ^ (38)
Proof. See appendix III

As a conclusion, we can highlight that the subsidy is more optimal than the tax from a social welfare standpoint.

Conclusion

In the search for an e¢ cient control for negative externalities, economists, generally, use taxes or/and subsidies as instruments of pollution reduction. This paper analyzes separately the e¤ect of each of these two environmental policies on the lifetime of phosphorus, on the reduction of eutrophication phenomenons and on the improvement of social welfare. Considering two …rms, a polluter and a green …rm, competing "à la Stackelberg" and assuming the presence of a benevolent government that taxes the polluter or subsidizes the recycler, we state the following results. First, we show that taxing polluting phosphorus (subsidizing recycled phosphorus) permits to reach this target, in the sense that each of them enables to reduce eutrophication phenonemons. Second, we …nd that taxation or subsidy allows for saving this resource by fostering the recycling activity. Indeed, the latter contributes to postpone the extraction of primary phosphorus. Third, we show that the subsidy set by the regulator is higher than the marginal bene…t of recycling, whereas the level of the tax with respect to the marginal social damage of pollution is ambiguous. Fourth, we show that the tax and the subsidy increase social welfare. Fifth, by way of comparison, we …nd that if the regulator aims at saving phosphorus, reducing eutrophication and improving social welfare simultaneously, subsidizing recycled phosphorus is the best policy, but if he aims only at saving phosphorus and reducing eutrophication, taxing extracted phosphorus is more optimal than subsidizing recycled phosphorus.

We may extrapolate by concluding that the implementation of a speci…c policy depends on the power of the agent in terms of lobbying. Also, it is not unreasonable to suspect that the government will have the incentive to implement subsidy rather than taxation in order to enjoy the favour of consumers, on the eve of the elections.

Since, in this paper, we have addressed the issues by considering taxation and subsidy separately, the …rst natural challenge for the future consists of investigating whether the combination of both policies will yield the same results or not.

The second challenge is to set up a dynamic model which allows to take into account the problem that entry causes on pollution. In fact, several authors have stated that subsidy gives to the potential entrants the incentive to enter the market. They have argued that global pollution increases in the subsidy rate even if the individual pollution may decrease. It is noteworthy to mention that these studies are based on the assumption that subsidized …rms are polluters. Since in this paper we consider that the subsidized …rm is green in that its activity reduces pollution, we may expect that the subsidy will continue to reduce pollution even if we consider a dynamic contex.

Another interesting issue is related to the level of the subsidy that the government can set to maintain the polluter in the market or to drop it out of the market. As subsidy reduces pollution and improves social welfare, higher is the level of the subsidy, lower is the level of pollution and higher is social welfare. One can think that there may be a level of subsidy, named s (higher than the optimal level ŝ obtained when pollution occurs) which will drop the polluter out of the market. Then, if s < ŝ, the government gives to the polluter more importance 14 . If ŝ < s < s, the polluter remains on the market but its importance reduces with respect to the previous situation. If s ŝ, the polluter is brought out of the market and only the green …rm produces. Accordingly, there is no pollution and social welfare increases due to the high level of the subsidy.

For the sake of simplicity, we have not addressed the problem as the issue of an exhaustible resource. Imposing a capacity constraint to the resource would be interesting and is another challenge for the future.

Appendix

5.1 Appendix I: the market is unregulated Detailed calculus for the optimal quantities: the timing of the game between the extractor (…rm A) and the recycler (…rm B) is as follows. In the …rst step, …rm A extracts a quantity of phosphorus q > 0. After observing the level of …rm A's extraction, …rm B recycles a quantity r > 0, in the second step. We solve this game by backward induction. The programmes of the two …rms are:

Step 2: …rm B maximizes

max r>0 B = (a q r)r (39)
s:t: r < q (40)

The lagrangian for this programme is de…ned as follows:

L(r; ) = (a q r)r + (q r) (41)
The …rst-order conditions are given by: dL(r; ) dr = a q 2r = 0 (42) (q r) = 0; 0 (43) I case 1: from (43), we know if = 0, r < q, then (42) writes:

a q 2r = 0 (44)
which results in: r(q) = a q 2 (45)

I case 2: from (43), we know if > 0, r = q. It is impossible to have r = q because recycling is never complete. Then, remains always positive and the best-reply function of the recycling …rm is then given by (45).

Step 1: the programme of …rm A is:

max q>0 A = (a q 1 2 (a q))q (46)
The …rst-order condition is then given by:

1 2 a q = 0 (47)
Which results in the following optimal quantities:

q = 1 2 a (48)
I case 2: from (55), we know if > 0, r = q. This is impossible because recycling is never complete. Then, remains always positive and the best-reply function of the recycling …rm is then given by (57).

Step 1: the programme of …rm A is de…ned as follows:

max q>0 A = (a q 1 2 (a q))q q (58)
The …rst-order condition for this programme yields the optimal extracted quantity:

q( ) = 1 2 a (59) With dq( ) d = 1 (60)
And the optimal recycled qunatity:

r( ) = a + 2 4 (61) With dr( ) d = 1 2 (62)
Since > 0, q( ) 0 and q( ) < r( ) if and only if:

0 < 1 6 a (63)
Under our assumptions, (63) explains why in …gure 1 the graphic is valid only if < 2 3 . The total quantity sold par the two …rms is:

Q( ) = 3a 2 4 (64)
It is straightforward to see that

d Q( ) d = 1 2 (65)
(65) clearly signals that the tax decreases consumers's surplus in that it reduces the global quantity.

Variation of the quantities

q = q( ) q = (66) r = r( ) r = 1 2 (67)

Detailed calculus for the optimal welfare

Proof of lemma 5: we assume that the benevolent regulator refunds the revenue of the tax to the society, takes into account the external e¤ects (positive for r and negative for q) in the damage function . The tax maximizes the following social welfare function which is the di¤erence between the sum of producer's pro…ts and consumer's surplus and any technological external costs which are not accounted for in …rm A's pro…ts :

w( ) = CS( ) + A ( ) + B ( ) D(q( )) + B(r( )) + q( ) (68)
We use a linear inverse demand function p(Q) = a Q, a linear damage function, i.e. D(q) = "q and a linear bene…t function, i.e. B(r) = r (where is the marginal bene…t of recycling). For the sake of simplicity and in order to focus only on the level of the tax ^ relatively to the marginal damage of pollution ", let us assume = 1. After making some simplications, the social welfare function writes:

w( ) = r( ) "q( ) + 1 2 aq( ) + ar( ) 1 2 (r( )) 2 (69)
Inserting the optimal quantities in the social welfare function and making some simplications yield:

w( ) = 4 2 [4a 42" + 1] + a[15a 8 (2" 1)] 32 (70)
Derivation of the social welfare function with respect to the tax yields the …rst-order condition below:

8" + 4 a 2 = 0 (71)
Which, in turn, yields the optimal level of the tax given by:

^ = 4" a 4 2 (72)
Inserting the optimal tax in (59) and (61) yields the following optimal quantities: q = a 4" 2

And r = 2" + 1 > 0 (74)

q > 0 if " < a 2 4 (75)
Notice that, for technical reasons, recycling is never complete, i.e. r < q. Then,

" < a 3 6 (76)
The combination of (75) and (76) gives:

" < a 3 6 (77)
" must be positive. Then a > 3

As ^ > 0, we have:

" > a 4 8 (79)
The combination of (77) and (79) yields:

a 4 8 < " < a 3 6 (80)
Let us show that the above interval is not empty, i.e. a 4 8 < a 3 6 . This holds if and only if

a > 1 2 (81)
This is true under (78). Now let us investigate whether the tax is above or below the marginal damage. To do so, we proceed as follows:

^ > " if and only if

" > a 4 6 (82)
But since a 4 6 > a 4 8 , the position of the tax with respect to the marginal damage is ambiguous:

(i) If " > a 4 6 , as seen above, the tax is higher than the marginal social damage, resulting in:

^ > " (83)
(ii) Conversely, if " < a 4 6 , the tax is lower than the marginal social damage, resulting in:

^ < " (84)
In contrast to the conventional wisdom that the tax is set above the marginal damage only and only if the two …rms have di¤erent costs of production, we show that the tax may be greater than the marginal damage when …rms are symmetric (point (i)). Now, let us show that w( ) > 0. Solving w( ) = 0 with respect to gives the two following roots:

^ 1 = ^ z }| { 4" a 4 2 2 p 4" 2 2 (a 2) " + a 2 + 1 < ^ (85) ^ 2 = ^ z }| { 4" a 4 2 + 2 p 4" 2 2 (a 2) " + a 2 + 1 > ^ (86)
Since w( ) is a second degree equation in , w( ) > 0 if ^ 1 < ^ < ^ 2 , which is true. Consequently, w( ) > 0. Inserting the optimal level of the tax in (70) yields the following optimal social welfare:

ŵ^ = M z }| { 4" 2 2 (a 2) " + a 2 + 1 2 (87)
For warrantly reasons, let us verify that M > 0. Since M = 0 is a second degree equation in ", let us calculate the discrminant. Calculations yield:

= 4a (3a + 4) < 0 (88)
Then, M is positive because it has the same sign as 4. Hence, ŵ^ > 0.

Let us now investigate the e¤ect of the marginal damage on the social welfare. Thus,

d ŵ^ d" = 4" a + 2 < 0 (89)
Resulting in:

" < a 2 4 (90)
This result holds under (75). Hence, as one may expect, the marginal social damage of pollution decreases social welfare. Now let us investigate the e¤ect of the tax on social welfare. It is obvious that for ^ , social welfare is optimal and is then given by (87). In order to see the e¤ect of the tax on social welfare, derivation of (70) with respect to the tax rate yields:

dw( ) d = a + 2 8" 4 8 (91)
Our calculations show clearly that dw( )

d > 0 if ^ 1 < < ^ .
In this situation the tax increases social welfare, whereas the latter reduces in the tax rate if ^ < < ^ 2 , in that dw( ) d < 0 in such a case. One can reasonably expect that the regulator will never set a tax which is di¤erent from the optimal tax, i.e. 6 = ^ but if he has the incentive to deviate from the optimal tax for whatever reason, the tax rate can increase or decrease social welfare.

Appendix III: Subsidizing recycled phosphorus

In this section, we assume that virgin phosphorus is not taxed. Only recycled phosphorus can be subsidized. We distinguish the following cases: 5.3.1 case 1: benchmark: there is no subsidy

The programmes of the two …rms are given by:

Step 2: pro…t of …rm B max r>0 B = (a q r)r (92) s:t: r < q (93) r(q) = a q 2 (94)

Step 1: pro…t of …rm A max q>0 A = (a q 1 2 (a q))q (95) Step 2: the programme of …rm B is de…ned as follows: max r>0 B = (a q r)r + sr (98) s:t: r < q (99)

q = 1 2 a (96)
The reaction function deriving from this programme is: r(q) = a q + s 2 (100)

Step 1: the programme of …rm A is de…ned as follows: max q>0 A = (a q 1 2 (a q + s))q (101)

The optimal quantity resulting from this programme is:

q s = a s 2 (102)
With

dq s ds = 1 2 (103)
And the optimal quantity recycled by …rm B is given by:

r s = a + 3s 4 (104)
With

dr s ds = 3 4 (105)
q S > 0 and r S < q S imply:

a > 5s (106)
The total quantity is then given by:

Q s = 3a + s 4 (107)
With @Q s @s = 1 4 (108)

Variation of quantities

q s |{z} = q s q = 1 2 s (109)

r s |{z} + = r s r = 3 4 s (110)
But r s > q s (111)

Resulting in:

Q > 0 (112)
Or in:

dQ s ds > 0 (113)
The subsidy increases the recycled quantity whereas it decreases the extracted quantity. But, the increasing e¤ect is higher than the decreasing e¤ect, resulting in the rise of the total quantity (q + r) with respect to the increase of the subsidy. Consequently, the subsidy increases consumers'surplus.

Comparison of social welfares

Implementation of subsidy scheme: Proof of lemma 6: the social welfare function is given by: w(s) = CS + A + B D(q) + B(r) sr (114)

We use a linear inverse demand function, a linear damage function, i.e. D(q) = "q (where " is the marginal damage of pollution) and a linear bene…t function, i.e. B(r) = r (where is the marginal bene…t of recycling). For the sake of simplicity and in order to focus only on the level of the subsidy ŝ relatively to the marginal bene…t of recycling , let us assume " = 1. After making some simplications, the social welfare function writes: w(s) = (a s 2) q(s) + (2a r + 2 ) r(s) 2 (115a) Inserting q(s) and r(s) in (115a) yields: Derivation the social welfare function with respect to the subsidy s yields the …rst-order condition below:

w(s) = s 2 +
a s + 12 + 8 = 0 (118)
Which results in the following optimal level of the subsidy:

ŝ = a + 12 + 8 (119)
Let us explore if a + 12 + 8 > . This results in: a + 11 + 8 > 0, which is true. Then, ŝ >

The optimal subsidy rate is above the marginal bene…t of recycling. It is straightforward to show that the marginal bene…t increases the subsidy. This results in:

dŝ d = 12 (121)
Let us show that w(s) > 0. Solving w(s) = 0 with respect to s gives the two following roots: From (125), it is easy to show that dw(s) ds > 0 if ŝ1 < s < ŝ, meaning that the subsidy increases social welfare in such a situation, whereas the latter decreases in the subsidy rate if ŝ < s < ŝ2 . As in the case of taxation, it is not obvious that the regulator will set a subsidy rate di¤erent from the optimal subsidy. But if, for a whatever reason, he has the incentive to increase or to decrease the subsidy with respect to the optimal subsidy, social welfare may increase or decrease in the subsidy rate.

ŝ1 = ŝ z }| { a + 12 +
Let us now turn to the comparison of taxation and subsidy: for a standart of comparison, assume that = " = 1. Then, optimal social welfares are respectively given by: I In the case of taxation:

ŵ^ = a 2 2a + 9 2 (126)
I In the case of subsidy:

ŵŝ = a 2 + 2a + 25 2 (127)
It is straightforward to observe that:

ŵŝ > ŵ^ (128)
The result established in (128) is the proof of proposition 8.

Figure 1 :

 1 Figure 1: E¤ect of the tax on the optimal quantities Legend: Red: Curve of extraction of phosphorus Green: Curve of recycling of phosphorus

Figure 2 :

 2 Figure 2: E¤ect of the subsidy on the optimal quantities

2 :

 2 Implementation of a subsidy scheme Proof of proposition 2

  Seyhan, 2009; Cordell and al., 2011; Cogoye, 2009; Beir and Girmens, 2009; Ridder and al., 2012

  social welfare increases in the tax rate, resulting in:

		dw( ) d	> 0	(27)
	(iv) Conversely, if ^ <	< ^ 2 , social welfare decreases in the tax rate,
	resulting in:
		dw( ) d	< 0	(28)

  Since w(s) is a second degree equation in s, w(s) > 0 if ŝ1 < ŝ < ŝ2 , which is true. Consequently, w(s) > 0. Inserting ŝ in (116) yields the next optimal social welfare:

						z	ŝ }|	{
				dw(s) ds	=	a + 12 + 8 s 16	(125)
				8 4 p	a 2 + 2a + 9 2 + 12 + 4 < ŝ	(122)
	And	ŝ2 =	ŝ }| a + 12 + 8 + 4 z { p	a 2 + 2a + 9 2 + 12 + 4 > ŝ	(123)
			ŵŝ =	(2a + 3 (3 + 4)) + a 2 + 4 2	> 0	(124)

There are other elements like nitrogen, carbon and trace, which create eutrophication[START_REF] Lee | Role of Phosphorus in Eutrophication and Di¤use Source Control[END_REF].

We distinguish primary phosphorus to secondary phosphorus (which is also called green phosphorus). The former refers to extracted phosphorus, whereas the latter corresponds to recycled phosphorus. It is taken as green phosphorus because it reduces water pollution.

It is noteworthy to mention that if recycled phosphorus ends up into the water it yields the same e¤ect which is triggered by extracted phosphorus. But to focus on the bene…t of recycling in the reduction of eutrophication, we assume, in the world of this model, that recycled phosphorus does not end up into the water after its consumption.

The policy of reduction of eutrophication took place in many countries, including the United States and Canada, via the the 1978 Great Lakes Water Quality Agreement which states that the e-uents from 400 municipal treatment plants discharging to the lakes should contain a maximum of 1:0 mg litre 1 of phosphorus in the upper lakes and 0:5 mg litre 1 of phosphorus in the lower lakes (Harrington-Hughes, 1978), China and United Kingdom[START_REF] Philipps | A large Scale Field Experiment in the Control of Eutrophication in the Norfolk Broads[END_REF].

With the aim of reducing eutrophication of Cayuga Lake (United States),[START_REF] Jacobs | Internalizing Externalities of Phosphorus Discharges from Crop Production to Surface Water: E-uent Taxes versus Uniform Reduction[END_REF] compare an e-uent tax policy with a uniforme reduction policy. They stress that it is less costly to reduce eutrophication by taxing e-uents than by aiming at reducing it uniformely. For instance, they estimate that the reduction of phosphorus discharge at a level of 10 percent costs 32; 065$ in the case of the tax on e-uents, whereas it costs 37; 177$ in the case of the uniforme reduction policy.

[START_REF] Cordell | The Story of Phosphorus: Global Food Security and Food for Thought[END_REF] highlight that phosphate reserves may be depleted in 50 100 years.

We prefer to keep the term "benevolent" in the sense that the government cares about the whole society, which consists here of the two …rms and consumers.

Also, he argues that if output and pollution levels are positively correlated, then sharp concavity in the demand curve, given by P 00 < 0, will result in increased pollution. This curvature leads small …rms to cut output and large …rms to increase output as a result of the tax, which can lead to increased pollution even though industry output falls.

Speci…cally, under a monopolistic organization. 1 0 He highlights that if the industry generating the external diseconomy is competitively organized, the corrective tax can be unambiguously hailed as welfare-improvement.1 1 Note that he indicates that the tax increases social welfare if the market structure is asymmetric and the inverse demand function is su¢ ciently concave at the equilibrium.

[START_REF] Beir | Alcoa Re-revisited: Recycling, Market Power and Environmental Policy[END_REF] We have used a quantity competition model like him and …nd the same result but under opposite assumptions.

[START_REF] Baksi | Endogeneous Consumer Participation and the Recycling Problem[END_REF] The pH measures the acidity or the basicity of a solution. A solution with a pH of 7 is considered to be neutral. If pH < 7, the solution is considered to be acid and basic if pH > 7 (https://fr.wikipedia.org/wiki/Potentiel_hydrog%C3%A8ne).

[START_REF] Ballard | The Marginal E¢ ciency Effects of Taxes and Subsidies in the Presence of Externalities: A Computational General Equilibrium Approach[END_REF] This can be the case because the polluter is very strong in terms of lobbying. The government subsidizes only in order to reduce pollution slightly.

[START_REF] Baumol | The Theory of Environmental Policy[END_REF] It is noteworthy to mention that the timing of the game between the regulator and the …rms is as follows. In the …rst step, the regulator sets the level of the tax. In the second step, …rm A extracts q > 0. In the third step, …rm B recycles r > 0.

The total quantity put in the market is given by

Appendix II: Detailed calculus when the market is regulated

In this section, we assume that only virgin phosphorus is taxed. The possibility of subsidizing recycled phosphorus is not taken into account here. It will be considered later.

Detailed calculus for the optimal quantities

Proof of proposition 1: Under the tax applied by the regulator 15 , each …rm maximizes its own payo¤. We solve this game by backward induction. Thus,

Step 2: the programme of …rm B is de…ned as follows:

s:t: r < q (52)

The lagrangian for this programme can be established as follows:

The …rst-order conditions are given by: dL(r; ) dr = a q 2r = 0 (54)

I case 1: from (55), we know if = 0, r < q, then (54) writes:

a q 2r = 0 (56) which results in: r(q) = a q 2 (57)