Recently, iron catalysts, and more particularly organophosphorus based ones, have emerged in this field exhibiting high activity and selectivity.

This review provides an overview of the recent developments in the area of iron-catalyzed hydrometalation of olefins and alkynes. More particularly, it will report iron-catalyzed hydrophosphination, hydrosilylation, hydroborylation and hydromagnesiation reactions promoted by organophosphorus-iron catalysts.

Organophosphorus and Iron Catalysis: Good

Partners for Hydrometalation of Olefins and Alkynes

Duo Wei, Christophe Darcel* Univ Rennes, CNRS, ISCR -UMR 6226, F-35000 Rennes, France, e-mail: christophe.darcel@univ-rennes1.fr ABSTRACT: The last decades have seen an impressive development of iron complexes involving organophosphorus ligands applied in homogeneous catalyzed hydrometalation of olefins and alkynes. Two main topics will be covered in this review: (i) an overview of the achievements in the area of iron-catalyzed hydrophosphination, and then (ii) hydrosilylation, hydroborylation and hydromagnesiation reactions promoted by catalysts based on organophosphorus ligands and iron.

The fine development of low toxic, sustainable and highly efficient transition metal catalysts is one of the hottest research topics in modern synthetic chemistry. Nevertheless, this field was mainly dominated by precious metals for decades. Their substitution by more benign first row ones is thoroughly desired and nowadays is undoubtedly one of the meaningful challenges. As the most abundant transition metal in the Earth's crust, the application of iron in homogeneous catalysis is a highly valuable target for various organic transformations, [START_REF] Bauer | Iron Catalysis in Organic Synthesis[END_REF] including the chemoselective reduction and hydrofunctionalization of unsaturated compounds. [START_REF] Castro | When Iron Met Phosphines: a Happy Marriage for Reduction Catalysis[END_REF][START_REF] Wei | Iron Catalysis in Reduction and Hydrometalation Reactions[END_REF][START_REF] Greenhalgh | Iron-Catalyzed Hydrofunctionalisation of Alkenes and Alkynes[END_REF][START_REF] Chen | Recent Advances in Hydrometallation of Alkenes and Alkynes via the First Row Transition Metal Catalysis[END_REF][START_REF] Bolm | Iron-Catalyzed Reactions in Organic Synthesis[END_REF][START_REF] Plietker | Iron Catalysis: Fundamentals and Applications[END_REF][START_REF] Junge | Homogeneous Catalysis Using Iron Complexes: Recent Developments in Selective Reductions[END_REF][START_REF] Riener | Chemistry of Iron N-Heterocyclic Carbene Complexes: Syntheses, Structures, Reactivities, and Catalytic Applications[END_REF][START_REF] Ollevier | Enantioselective Iron Catalysts[END_REF][START_REF] Nakajima | Hydrosilylation Reaction of Olefins: Recent Advances and Perspectives[END_REF][START_REF] Xinwei | Progress in Iron Complexes-Catalyzed Organic Reactions[END_REF][START_REF] Renaud | Recent Advances in Iron-and Cobalt-Complex-Catalyzed Tandem/Consecutive Processes Involving Hydrogenation[END_REF] A c c e p t e d M a n u s c r i p t

Hydrophosphination

Selective formation of P-C bonds for the preparation of organophosphorus derivatives is still a topical challenging area of research despite the universality of phosphines in synthetic and catalytic methodologies, [START_REF] Castro | When Iron Met Phosphines: a Happy Marriage for Reduction Catalysis[END_REF][START_REF] Fleming | Primary Phosphine Chemistry[END_REF][START_REF] Nell | Reactivity, and Coordination Chemistry of Secondary Phosphines[END_REF][START_REF] Bezzenine-Lafollée | First-Row Late Transition Metals for Catalytic Alkene Hydrofunctionalisation: Recent Advances in C-N, C-O and C-P Bond Formation[END_REF][START_REF] Delacroix | Hydrophosphination of Unactivated Alkenes, Dienes and Alkynes: a Versatile and Valuable Approach for the Synthesis of Phosphines[END_REF] which usually exhibited an impressive functional group tolerance in comparison to classical routes. [START_REF] Wauters | Preparation of Phosphines through C-P Bond Formation[END_REF] Gaumont and Taillefer reported an interesting regioselective hydrophosphination of styrenes (Scheme 1): [START_REF] Routaboul | Iron-Salt-Promoted Highly Regioselective α and β Hydrophosphination of Alkenyl Arenes[END_REF] Indeed, the regioselectivity of the α-or β-phosphinylation of styrenes with diphenylphosphine can be reached by selecting the iron salt (FeCl 3 or FeCl 2 , respectively, 30 mol%). The borane adducts were isolated with 5-87% yields, after reaction with BH 3 •SMe 2 . The difference of regioselectivity may be due to the difference of Lewis acidity between Fe salts, the Fe(III)-salt generating a polarized and intramolecularly stabilized π-complex. Webster then described Salen-type complexes, Fe-1 [START_REF] Gallagher | Room Temperature Hydrophosphination Using a Simple Iron Salen Pre-catalyst[END_REF][START_REF] Gallagher | A Study of Two Highly Active, Air-Stable Iron(III)-μ-Oxo Precatalysts: Synthetic Scope of Hydrophosphination using Phenyl-and Diphenylphosphine[END_REF][START_REF] Espinal-Viguri | Iron Catalysis for the Synthesis of Ligands: Exploring the Products of Hydrophosphination as Ligands in Cross-coupling[END_REF] and the porphyrin μ-oxo complex Fe-2 [START_REF] Gallagher | A Study of Two Highly Active, Air-Stable Iron(III)-μ-Oxo Precatalysts: Synthetic Scope of Hydrophosphination using Phenyl-and Diphenylphosphine[END_REF] in the hydrophosphination of styrene, vinylpyridine and acrylate derivatives performed using Fe-1 or Fe-2 (0.5 mol%) at r.t. leading selectively the anti-Markovnikov adducts (35-98% yields, Scheme 2). Webster also described the iron(II)-β-diketiminate Fe-3 (5 mol%) as a suitable catalyst for the hydrophosphination of activated alkenes at 70 o C leading to the corresponding tertiary phosphines with an anti-Markonikov selectivity (49-94% yields). [START_REF] King | Catalytic Dehydrocoupling of Phosphines Using β-Diketiminate Iron(II) Complexes[END_REF] Notably, less active allylbenzene and 1hexene gave low amounts of phosphines. Additionally, styrene poorly reacted with hindered phosphines or monophosphines.

Waterman found that [CpFe(CO) 2 ] 2 Fe-4 (5 mol%) was efficient for the hydrophosphination of styrene and acrylic derivatives with Ph 2 PH (1 equiv.) under visible light irradiation (LED bulb, A c c e p t e d M a n u s c r i p t hn>500 nm) at 25-28°C (2-99% conv.). [START_REF] Pagano | Visible Light Photocatalysis Using a Commercially Available Iron Compound[END_REF] Under these conditions, styrene led to only 11% conversion. Starting with Fe-1 and Fe-2, the double olefin hydrophosphination reaction of substituted styrenes (4 equiv.) with PPhH 2 led to the corresponding tertiary phosphines with 62-96% yields (Scheme 3). [START_REF] Gallagher | A Study of Two Highly Active, Air-Stable Iron(III)-μ-Oxo Precatalysts: Synthetic Scope of Hydrophosphination using Phenyl-and Diphenylphosphine[END_REF] Interestingly, a two-step sequence was developed to prepare a racemic unsymmetrical tertiary phosphine: the first catalyst free step was performed with H 2 PPh and styrene at 80°C leading to secondary phosphines which then reacted with 4-chlorostyrene using Fe-5 (0.5 mol%) (Scheme 4).

Scheme 3. Double hydrophosphination of styrenes

Scheme 4. Preparation of unsymmetrical phosphines.

R PPhH

Fe-5 (0. Hydrophosphination of alkynes could be also performed. The first catalytic double hydrophosphination of terminal arylalkynes was reported by Nakazawa (Scheme 6) by reaction with diarylphosphines catalyzed by CpFe(CO) 2 Me (Fe-6, 5 mol%). [START_REF] Kamitani | Regioselective Double Hydrophosphination of Terminal Arylacetylenes Catalyzed by an Iron Complex[END_REF] Notably, the reaction proceeded regioselectively with various secondary arylphosphines (2 equiv.) and arylalkynes at 110°C to give 1,2-bisphosphinoethane derivatives. In contrast, n-(cyclo)hexyl-, benzylacetylene, dicyclohexylphosphine and di-tert-butylphosphine did not undergo hydrophosphination. Using [CpFe(CO) 2 ] 2 (5 mol%, Fe-4), Waterman performed the double hydrophosphination of terminal arylalkynes conducting the reaction with diphenylphosphine under visible-light or thermal conditions (Scheme 6). [START_REF] Ackley | Visible-light and Thermal Driven Double Hydrophosphination of Terminal Alkynes Using a Commercially Available Iron Compound[END_REF] The corresponding products were isolated in 66-99% yields.

The mono-hydrophosphination of terminal arylalkynes with secondary phosphines were also selectively performed by Nakazawa using Cp*Fe(CO)(py)(Me) (Fe-7, 10 mol%) at 110°C (Scheme 7a), yielding to (Z)-vinylphosphines. [START_REF] Itazaki | Synthesis of Vinylphosphines and Unsymmetric Diphosphines: Iron-catalyzed Selective Hydrophosphination Reaction of Alkynes and Vinylphosphines with Secondary Phosphines[END_REF] The authors proposed a mechanism, involving a Cp*Fe(CO)(PPh 2 ) active species and the insertion of the coordinated C≡C bond into the Fe-P bond, controlling the regioselectivty. Additionally, using CpFe(CO) 2 Me Fe-6 (5 mol%), the hydrophosphination of vinyldiarylphosphines led to unsymmetrical 1,2-bis(phosphino)ethanes bearing two different phosphine moieties (Scheme 7b). Webster also reported the preparation of vinyl phosphines from terminal (hetero)arylalkynes using Fe-3. Remarkably, the nature of the solvent was crucial on the regioselectivity: Markovnikov-selective products gem-vinylphosphines were obtained at 50°C in benzene, whereas Z-selective-anti-Markovnikov products were obtained at 70°C in dichloromethane (Scheme 8). [START_REF] King | Markovnikov versus anti-Markovnikov Hydrophosphination: Divergent Reactivity Using an Iron(II) β-Diketiminate Pre-Catalyst[END_REF] Even if not fully rationalized, preliminary mechanistic studies suggested that the regioselectivities may be linked to the oxidation state of the active species [Fe(II) vs Fe(III)] and the mode of the C-P bond-forming process. Even if iron-catalyzed hydrophosphination is a growing area of research, with nice chemoselectivities and original regioselectivities, it is still described with limited examples of organophosphorus moieties (mainly -PPh 2 and -PHPh) and the stereoselective preparation of chiral phosphines is still a challenging point to be solved.

Hydrosilylation

The catalyzed hydrosilylation of olefins is an important research area in academy and industry for the production of alkylsilanes, starting materials for high value silicon-based materials. [START_REF] Greenhalgh | Iron-Catalyzed Hydrofunctionalisation of Alkenes and Alkynes[END_REF][START_REF] Du | Advances in Base-Metal-Catalyzed Alkene Hydrosilylation[END_REF] A c c e p t e d M a n u s c r i p t Thus, there are numerous methods to perform hydrosilylation using noble transition metal complexes such as Speier [START_REF] Speier | Homogeneous Catalysis of Hydrosilation by Transition Metals[END_REF] With iron, the pioneering report published in 1962 by Nesmeyanov described the reaction of alkenes with hydrosilanes in the presence of Fe(CO) 5 or colloidal iron catalysts, leading to mixtures of alkyl-and vinyl-silanes. [START_REF] Nesmeyanov | Substitution, and Telomerization Reactions of Olefins in The Presence of Metal Carbonyls or Colloidal Iron[END_REF] Using Fe 2 (CO) 9 , the reaction of styrene with triethylsilane led specifically to (E)-Ph-CH=CH-SiEt 3 , resulting from dehydrogenative hydrosilylation. [START_REF] Kakiuchi | Completely Selective Synthesis of (E)β-(Triethylsilyl)styrenes by Fe 3 (CO) 12 -Catalyzed Reaction of Styrenes with Triethylsilane[END_REF] During the last decades, iron catalyzed hydrosilylations of olefins have seen an increasing interest. Noticeably, iron in association with organophosphorus ligands led to efficient catalysts. [START_REF] Castro | When Iron Met Phosphines: a Happy Marriage for Reduction Catalysis[END_REF][START_REF] Zhang | Iron-catalyzed Hydrosilylation Reactions[END_REF] Based on the breakthrough made by Chirik with bisiminopyridine-Fe complexes (e.g. Fe-8) highly efficient in olefin hydrosilylation, [START_REF] Tondreau | Iron Catalysts for Selective Anti-Markovnikov Alkene Hydrosilylation Using Tertiary Silanes[END_REF][START_REF] Bart | Preparation and Molecular and Electronic Structures of Iron(0) Dinitrogen and Silane Complexes and their Application to Catalytic Hydrogenation and Hydrosilation[END_REF] mixed structures bearing iminopyridine or bipyridine moieties and organophosphorus motifs were designed. Thus, Huang described pincer complexes having a phosphinite-iminopyridine skeleton and demonstrated their efficiency in chemoselective hydrosilylation of functionalized olefins. [START_REF] Peng | Phosphinite-Iminopyridine Iron Catalysts for Chemoselective Alkene Hydrosilylation[END_REF] Indeed, the catalyst (1.0 mol%) generated from (Fe-9a) and NaBHEt 3 hydrosilylated numerous olefins with PhSiH 3 or Ph 2 SiH 2 (1.0 equiv., 23°C, 3h). This catalytic system exhibited good functional tolerance (ketones, esters, amides). Interestingly, no reaction occurred with internal alkenes and styrenes (Scheme 9). It should be underlined the divergent selectivity in the challenging hydrosilylation of unsaturated ketones (e.g. 5-hexen-2-one). Whereas with Chirik's complex Fe-8, the ketone was selectively reduced with diphenylsilane, the chemoselectivity was reversed using Fe-9a with the hydrosilylation of the C=C bond. Although the Fe-9a/NaBEt 3 H system exhibited good chemoselectivity, these PONN ligands were prone to degradation via P-O bond cleavage. Thus, Huang prepared more robust phosphineiminopyridine catalysts (Fe-9b, Scheme 9), [START_REF] Du | Base-Metal-Catalyzed Regiodivergent Alkene Hydrosilylations[END_REF] and evaluated them in hydrosilylation of alkenes leading to linear products in good yields and regioselectivities (linear/branched up to >99:1).

Notably, lowering the catalytic loading to 0.02 mol% of Fe-9b/NaBEt 3 H (1:4), the hydrosilylation of 1-octene with PhSiH 3 exhibited a TON=3950, which was significantly more active than Fe-9a. Similarly, Rauchfuss reported catalytic systems from phosphine-iminopyridine complexes (e.g.

Scheme 10. PNN-pincer-iron complexes for hydrosilylation of alkenes

Fe-10) [START_REF] Gilbert-Wilson | Phosphine-Iminopyridines as Platforms for Catalytic Hydrofunctionalization of Alkenes[END_REF] and NaBEt 3 H (0.1 mol%) for selective anti-Markovnikov addition of 1-octene or vinylpentamethyldisiloxane with diphenylsilane. Fe-11a was also employed for the hydrosilylation of 1-octene with Ph 2 SiH 2 (Scheme 10). [START_REF] Basu | Fe and Co Complexes of Rigidly Planar Phosphino-Quinoline-Pyridine Ligands for Catalytic Hydrosilylation and Dehydrogenative Silylation[END_REF] Additionally, hydrosilylation was performed with triethoxysilane at 55 o C, with a lower selectivity due to isomerization to 2-octene.

Similarly, Kamitani described an even more active pre-catalyst (PNN)Fe(CH 2 SiMe 3 ) 2 Fe-11b for hydrosilylation of terminal alkenes. Remarkably, hydrosilylation of 1-octene with phenylsilane led to the anti-Markovnikov phenylsilyloctane with TON and TOF up to 480,000 and 10,000 h -1 , respectively, (catalytic loading as low as 0.0001 mol%). [START_REF] Kamitani | Development of Activator-free Iron Pincer Complexes for Alkene Hydrosilylation and Elucidation of Its Activation Mechanism[END_REF] Notably, these values are the highest Vinylsilanes are versatile synthetic building blocks in molecular synthesis. [START_REF] Luh | Synthetic Applications of Allylsilanes and Vinylsilanes[END_REF] One of the efficient access to such compounds is the hydrosilylation of alkynes. Using [Fe(H)(CO)(NO)(PPh 3 ) 2 ] (Fe-13), Plietker performed stereoselective hydrosilylations of internal alkynes to vinylsilanes (Scheme 11). [START_REF] Belger | Aryl-Aryl Interactions as Directing Motifs in the Stereodivergent Iron-Catalyzed Hydrosilylation of Internal Alkynes[END_REF] Of particular interest, the nature of the hydrosilane had a crucial effect on the selectivity E vs Z. Indeed, with Fe-13 (1 mol%) and trimethylamine (50 mol%), diphenylacetylene reacted with a primary silane to conduct selectively to Z-vinylsilanes (E/Z up to 1/20; yields up to 98%). With tertiary silanes, the reverse stereoselectivity was obtained (E/Z ratio of >20:1 and yields up to 97%). The regioselectivity can also be managed by the nature of internal alkynes: for example, E-vinylsilane was generated from dialkylacetylene whatever the hydrosilane [PhSiH 3 or Ph(Me)(CH=CH 2 )Si-H]. Despite the progress made and the huge variety of silanes used, in terms of the application for the production of organosilanes in fine and bulk synthesis, iron is still overshadowed by the traditional noble-metal catalysts in terms of reactivity, selectivity and diversity.

Scheme 11. Stereoselective reduction of internal alkynes

Hydroborylation

Organoborane derivatives are of importance in synthesis, with an increasing use since the discovery of Suzuki cross-coupling reactions. Among the main methodologies of preparation, the catalyst-free or metal-catalyzed selective hydroboration of alkenes and alkynes are usually efficient and sustainable pathways. [START_REF] Burgess | Transition-Metal Promoted Hydroborations of Alkenes, Emerging Methodology for Organic Transformations[END_REF][START_REF] Miyaura | Diboration, Silylboration, and Stannylboration[END_REF] In the area of transition metal catalyzed hydroboration, by contrast with organoborane, organoboronate is an interesting class of compounds. [START_REF] Doucet | Suzuki-Miyaura Cross -Coupling Reactions of Alkylboronic Acid Derivatives or Alkyltrifluoroborates with Aryl, Alkenyl or Alkyl Halides and Triflates[END_REF][START_REF] Hall | Boronic Acids: Preparation, Applications in Organic Synthesis and Medicine[END_REF][START_REF] Jana | Advances in Transition Metal (Pd,Ni,Fe)-Catalyzed Cross-Coupling Reactions Using Alkyl-organometallics as Reaction Partners[END_REF][START_REF] Miyaura | Palladium-Catalyzed Cross-Coupling Reactions of Organoboron Compounds[END_REF] Nowadays, Rhodium is far away the metal of choice for this reaction, [START_REF] Brown | Modern Rhodium-Catalysed Organic Reactions[END_REF][START_REF] Coyne | The Development and Application of Rhodium-Catalyzed Hydroboration of Alkenes[END_REF] and examples with iron are scarce. Following the pioneering results of Ritter on diene, [START_REF] Wu | Iron-Catalyzed 1,4-Hydroboration of 1,3-Dienes[END_REF] Huang reported an efficient catalyst, prepared from the bipyridyl-phosphine complex (Fe-14, 0.25-5.0 mol%) and NaHBEt 3 (0.75-15 mol%), for the hydroborylation of terminal alkenes with HBpin (0.5 equiv.) at r.t.

leading to alkylboronates in 62-96% yields (Scheme 13). 60

Scheme 13. Hydroborylation of terminal alkenes

H-Bpin

Fe-14

Rauchfuss reported another efficient catalyst prepared from phosphine-iminopyridine complex (Fe-10, 0.1 mol%) and NaHBEt 3 (0.25 mol%) for the hydroborylation of 1-octene with HBpin (1.0 equiv.) at r.t. (Scheme 13). [START_REF] Gilbert-Wilson | Phosphine-Iminopyridines as Platforms for Catalytic Hydrofunctionalization of Alkenes[END_REF] Using a N-phosphinoamidinate complex (Fe-15), the isomerization/hydroborylation of internal alkenes leading to linear terminal alkyl diazaborolanes was reported (Scheme 15). Interestingly, 1,3-dimethyl-1,3-diaza-2-boracyclopentane gave better results than HBpin. 61

+ N N Fe P N(SiMe3)2 H B N N n R n B [Fe] N N
Nishibayashi demonstrated that the diphosphinopyrrolide complex (Fe-16, 1.0 mol%) was active in hydroboration of terminal alkynes with HBpin (1.1 equiv.) at r.t. leading selectively to the Evinylboronates in 84-94% yields (Scheme 16). [START_REF] Nakajima | Hydroboration of Alkynes Catalyzed by Pyrrolide-Based PNP Pincer-Iron Complexes[END_REF] Nevertheless, the reactivity was inhibited using hindered alkynes, and lowered with internal alkynes. Using 0. Interestingly, the modification of the structure of the iron complex have a strong influence on the transformation stereoselectivity as described by Kirchner using the [Fe(PNP)(H) 2 (η 2 -H 2 )] complex (Fe-17, 0.4-4.0 mol%): the hydroboration of terminal alkynes with pinacolborane (1.1 equiv.) at r.t. conducted to vinylboronates with high Z-selectivity (Scheme 17). [START_REF] Gorgas | Yet Highly Reactive Nonclassical Iron(II) Polyhydride Pincer Complexes: Z-Selective Dimerization and Hydroboration of Terminal Alkynes[END_REF] The transformation was then rationalized via a Fe-C/H-B σ-bond metathesis followed by the selective syn insertion of the resulting coordinated alkynylboronate into the formed iron-hydride bond, thus explaining the (Z)-stereoselectivity. [START_REF] Gorgas | Iron (II) Bis (acetylide) Complexes as Key Intermediates in the Catalytic Hydrofunctionalization of Terminal Alkynes[END_REF] Another attractive pathway to access to vinyl boronate esters is the catalytic dehydrogenative borylation of styrenes, [START_REF] Geier | Dehydrogenative Borylation: the Dark Horse in Metal-Catalyzed Hydroborations and Diborations?[END_REF] scarcely reported at iron. Ge 66 described such transformation using Fe(PMe 3 ) 4 (Fe-18, 3.0-5.0 mol%), in the presence of mono-and di-substituted vinylarenes, HBpin (0.83 equiv.) and norbornene (2.0 equiv.) at 50°C. The resulting E-vinyl-boronate esters were obtained in 60-94% yields with a good group tolerance (imines, amino, boronate, Ph 2 P…).

Scheme 17. Z-Hydroboration of alkynes

The use of norbornene as hydrogen acceptor was crucial for the reaction selectivity (Scheme 18). 66 Simultaneously, our group has shown that PCP-iron-hydride complexes (Fe-19-20, 5 mol%), were selective for the dehydrogenative borylation of styrene, [START_REF] Jiang | Direct Synthesis of Dicarbonyl PCP-Iron Hydride Complexes and Catalytic Dehydrogenative Borylation of Styrene[END_REF] using HBpin (1 equiv.), without hydrogen scavenger at r.t., under UV-irradiation, leading to the vinylboronate ester with up to 80% yield (Scheme 18). [START_REF] Jiang | Direct Synthesis of Dicarbonyl PCP-Iron Hydride Complexes and Catalytic Dehydrogenative Borylation of Styrene[END_REF] Noticeably, the performance obtained by Nishibayashi (TON up to 700) is among the best obtained at iron and has to be placed in perspective with the one reported by Thomas using a bis(imimo)pyridine iron catalyst (TOF up to 30,000 h -1 ). [START_REF] Greenhalgh | Chemo-, Regio-, and Stereoselective Iron-Catalysed Hydroboration of Alkenes and Alkynes[END_REF] Nevertheless, one notable limitation of the known iron catalysts is the lack of catalytic activity for hydroboration of internal alkenes and alkynes.

A c c e p t e d M a n u s c r i p t

Hydromagnesiation

Grignard reagents are one of the class of organometallic derivatives of great importance in molecular synthesis thanks to their availability, ease of handling and impressive reactivity. [START_REF] Silverman | Handbook of Grignard Reagents[END_REF] They are obtained (i) by mixing the alkylhalide with magnesium or (ii) by magnesium-halide exchange of a organohalide with AlkylMgX. [START_REF] Ziegler | Improving the Halogen-Magnesium Exchange by using New Turbo-Grignard Reagents[END_REF] Noteworthy, the hydromagnesiation may provide an interesting alternative to the traditional techniques. It is thus consisting in a Grignardalkene/alkyne exchange via a metal catalyzed β-elimination affording the alkyl or vinyl Grignard reagents. [START_REF] Sato | The Preparation of Grignard Reagents via the Hydromagnesation Reaction and Their Uses in Organic Synthesis[END_REF][START_REF] Greenhalgh | Iron-Catalyzed Hydromagnesiation of Olefins[END_REF] At iron, after the pioneering contributions of Cooper, Finkbeiner, and Kochi, [START_REF] Greenhalgh | Iron-Catalyzed Hydromagnesiation of Olefins[END_REF][START_REF] Finkbeiner | Synthetic Applications of the Titanium-Catalyzed Exchange of Olefins with Grignard Reagents[END_REF][START_REF] Tamura | Iron Catalysis in the Reaction of Grignard Reagents with Alkyl Halides[END_REF][START_REF] Tamura | The Reaction of Grignard Reagents with Transition Metal Halides: Coupling, Disproportionation, and exchange with Olefins[END_REF] Shirakawa and

Hayashi described the hydromagnesiation of olefins with a Grignard reagent using a catalytic system associating tributylphosphine (10 mol%), FeCl 3 (2.5 mol%) and CuBr (5 mol%) (Scheme 19). [START_REF] Shirakawa | Iron-Copper Cooperative Catalysis in the Reactions of Alkyl Grignard Reagents: Exchange Reaction with Alkenes and Carbometalation of Alkynes[END_REF] The exchange between terminal olefins and cyclopentylmagnesium bromide led to linear Grignard reagents in high yields, and cyclopentene, which could not be hydromagnesiated, thus making the reaction irreversible, and the cyclopentylmagnesium bromide, the reagent of choice.

The obtained 1-alkylmagnesium halides could then react with various electrophiles. The alkylmagnesiation was also extended to alkynes. Using a similar catalytic system (5 mol% Fe(acac) 3 , 10 mol% CuBr, 40 mol% PBu 3 ), arylmagnesium halides (2 equiv.) could react with internal alkynes at 60°C to produce after hydrolysis the trisubstituted olefins in 36-90% yields A c c e p t e d M a n u s c r i p t and E/Z-selectivity up to 97:3. [START_REF] Shirakawa | Arylmagnesiation of Alkynes Catalyzed Cooperatively by Iron and Copper Complexes[END_REF] To promote the reaction of 1-alkyl Grignard reagents with internal alkynes, a modified catalytic system was used [Fe(acac) 3 (5 mol%), CuBr (10 mol%), PBu 3 (20 mol%), and TMEDA] leading to β-alkylvinylmagnesium reagents, then the corresponding tri-substituted alkenes in 57-93% yields after methanolysis (Scheme 20). [START_REF] Shirakawa | Iron-Copper Cooperative Catalysis in the Reactions of Alkyl Grignard Reagents: Exchange Reaction with Alkenes and Carbometalation of Alkynes[END_REF] By contrast, dialkylacetylenes were unreactive.

Scheme 20. Alkylmagnesiation of alkynes

1 equiv. R 3 R 2 + H R 3 R 2 R 1 MgBr R 3 R 2 R 1 R 3 H R 2 R 1 +
Using this iron/copper catalytic system, a sequential reaction involving the alkene-Grignard exchange then the alkylmagnesiation of internal alkynes were conducted by adding 1phenyloctyne to the reaction mixture obtained from 1-hexene and cyclopentylmagnesium bromide and 79% of the tri-substituted alkene was isolated (Scheme 21). It was proposed that iron promoted the hydrometalation step, and the copper assisted the transmetalation step from iron to magnesium. 76 Overall, these hydromagnesiation reactions paved the way to extend this strategy for the easy in-situ preparation of other organometallic reagents from zinc, aluminum... In summary, among the variety of ligands associated to iron able to perform efficiently catalytic transformations, the combination with organophosphorus, mainly arylphosphines, provides versatile catalysts, particularly for hydrometalation of olefins and alkynes. The fine design of the organophosphorus iron complexes led to divergent chemo-, regio-or stereo-selectivities in comparison with other families of iron catalysts, as highlighted in hydrosilylation and hydroborylation areas. A special focus was also made on iron-catalyzed hydrophosphination leading selectively to alkyl-and alkenyl-phosphines. Such reaction constitutes an interesting and powerful tool for the synthesis of organophosphorus derivatives, even if the diversity of the phosphorus substituents has still to be enlarged. Iron-catalyzed hydromagnesiation is another exciting area of research; even if discovered in the 60's, it has emerged since only less than a decade.

There are still interesting challenges associated to iron-catalyzed hydrometalation, including hydromagnesiation, hydrozincation or hydroalumination leading to the preparation of synthetically valuable organometallic reagents. The chiral version of hydroborylation, hydrosilylation and hydrophosphination is another quite virgin area to be developed. The study of iron-phosphine based catalysts is thus believed to be a promising area of research and more new reactivity should be reported in the near future. joined Prof. P. Knochel's group in Marburg (Germany). He was then appointed at the Université de Cergy-Pontoise, and at the Université de Bourgogne, Dijon as associate professor, and finally at the Université de Rennes 1 since 2007 as full professor. His current research interests focus on homogeneous catalysis using non noble metals, with particular emphasis on well-defined iron catalysts.

A c c e p t e d M a n u s c r i p t Abstract graphic

R 1 R 2 R 1 R 4 R 2 R 3 R 1 R 4 R 3 Y R 2 R 1 R 2 Y R 1 R 4 R 2 MgX R 3 R 1 R 2 MgX R 3

Scheme 1 .

 1 Scheme 1. Regioselective α-and β-hydrophosphinations of alkenylarenes

Scheme 2 .Fe- 2 2 Fe- 3 2 Fe- 1

 222321 Scheme 2. Hydrophosphination of alkenes with diphenylphosphine leading to R-CH 2 -CH 2 -PPh 2

Scheme 7 .

 7 Scheme 7. Synthesis of Z-vinylphosphines and unsymmetric diphosphines

Scheme 8 .

 8 Scheme 8. Markovnikov vs anti-Markovnikov hydrophosphination

Scheme 9 .

 9 Scheme 9. Unsymmetrical pincer-iron complexes for hydrosilylation of terminal alkenes leading

  A c c e p t e d M a n u s c r i p t

A c c e p t e d M a n u s c r i p t observed for iron

  catalyzed hydrosilylation of olefins, even if far away for the efficiency of platinum based Karstedt catalyst.

Scheme 14 .

 14 Scheme 14. Mechanism for hydroborylation of alkenes.

Scheme 15 .

 15 Scheme 15. Isomerization/hydroborylation of internal olefinsA c c e p t e d

  , 2017: Fe-17 (0.4-4.0 mol%), HBpin (1.1 equiv.) C 6 D 6 , r.t., 3-16 h,13 examples, 87-96% yields, E:Z : 29:71-1:99 A c c e p t e d M a n u s c r i p t

Scheme 18 .

 18 Scheme 18. Dehydrogenative borylation of alkenes

Scheme 21 .

 21 Scheme 21. One-pot synthesis of tri-substituted alkenes

  s c r i p t

Scheme 6. Double hydrophosphination of terminal arylalkynes

	R 2 equiv. + R ' 2 PH	[Fe]	R	'	R 2 P	PR' 2	Nakazawa, 2012 10 examples, 27-95% yield neat, 110 o C, 72 h CpFe(CO)2 Me, Fe-6 (5 mol%)	Waterman, 2018 [CpFe(CO)2]2 Fe-4 (5 mol%) h n ( l irr > 500 nm), r.t. or 110 o C, neat, 1-36 h 11examples, 66-99% yield

  [H 2 PtCl 6 ]•6H 2 O and Karstedt 36-37 [Pt 2 (Me 2 SiCH=CH 2 ) 2 O] 3 catalysts.

  1 mol% of Fe-16, at 60°C, TONs up to 710 were obtained.

	Scheme 16. Hdroboration of alkynes
							P t Bu 2
	R	H + HBpin	[Fe]	R	BPin	N	Fe	H
	Nishibayashi, 2017: Fe-16 (1.0 mol%), HBPin (1.1 equiv.) hexane, r.t., 16 h, 15 examples, 21-94% yields	P t Bu 2 Fe-16

ACKNOWLEDGMENT

We gratefully acknowledge funding from the University de Rennes 1, the Centre National de la Recherche Scientifique (CNRS), the CNRS Federation Increase.

AUTHOR INFORMATION

Corresponding Author

*E-mail: christophe.darcel@univ-rennes1.fr


Damien Besancenot
email: damien.besancenot@parisdescartes.fr

Karine Lamiraud
email: lamiraud@essec.fr.

Radu Vranceanu
email: vranceanu@essec.edu

A MODEL FOR DUAL HEALTHCARE MARKET WITH CONGESTION DIFFERENTIATION

Keywords: D83, I11, I18 Dual market, Congestion, Regulation, Balance billing

The French market for specialist physician care has a dual legal structure: physicians must exclusively work in sector 1 and charge regulated fees or in sector 2, where they can freely set their fees. Patient out-ofpocket payments in sector 2 are partially covered by private insurance. The primary di¤erentiating factor between both sectors is the number of patients per specialist, which in turn directly a¤ects the overall quality of the service provided. We built an equilibrium model to analyse both specialists'decisions about which sector to work in, and patients' choice of physician and therefore sector. More speci…cally, the model allowed us to study the e¤ect of changes in prices and economy-wide patient-to-specialist ratios on pro…ts and patients'utility associated with the services provided in each sector.

Introduction

In many countries, national healthcare systems fully cover the cost of basic care while patients make out-of-pocket payments for any incremental costs related to non-basic care (e.g. additional quality of service, choosing a speci…c physician) [START_REF] Einav | Paying on the margin for medical care: Evidence from breast cancer treatments[END_REF]. In this paper, we focus on France, a country where the above-mentioned principle is embedded in its dual legal market structure for specialized outpatient healthcare services.

Since 1980, self-employed specialist physicians in France have been able to choose between practicing in 'sector 1'(which is heavily regulated) or 'sector 2'(where regulation is much looser) of the dual market for specialized outpatient healthcare. In both sectors, self-employed specialists are paid on a fee-for-service basis. The essential di¤erences between both concern the setting of fees and the levels of coverage of these fees provided by basic and complementary health insurance.

The former is provided under France's National Health Insurance scheme and covers everyone living in the country, while the latter is provided by private health insurers. Speci…cally, sector 1 specialists agree to provide medical services at …xed regulated prices, while those working in sector 2 can choose to set their fees above the regulated price. According to Article 53 of the Public Health Law, French physicians opting to work in sector 2 should set their fees with "a degree of tact and moderation". Fees charged by sector 1 and sector 2 specialists totalled 11.9 billion euros in 2017, of which 1.7 billion euros were balance billing amounts charged by sector 2 specialists.

Basic insurance reimburses 70% of the regulated price (i.e., the price charged by sector 1 physicians) minus a one-euro co-payment. For those who have complementary insurance, the di¤erence (i.e., 30% co-insurance in basic insurance) is fully covered but not the one-euro co-payment. In 2018, over 95% of the French population were covered by complementary insurance (DRESS, 2019). 1 Complementary insurance covers the same services covered by basic insurance and can reimburse co-payments/co-insurance in basic insurance except for the one-euro co-payment. How- 1 In this paper, we will assume that 100% of the French population is covered by complementary insurance.

1 Figure 1: Payment structure of the sector 2 fee ever, complementary insurance contracts di¤er greatly with respect to the coverage they o¤er for balance billing amounts (i.e., amounts above the regulated price). Two types of complementary contracts exist: corporate collective contracts (provided by …rms for their employees) and individual contracts (negotiated on an individual basis, mostly by self-employed people). In general, corporate contracts provide greater coverage of balance billing amounts. Half of the individual contracts taken out do not cover balance billing at all (DRESS, 2019). Accordingly, patients who consult sector 1 physicians have very low co-payments (only 1 euro), while generally their counterparts consulting sector 2 physicians have to pay larger out-of-pocket payments. Figure 1 provides an example of reimbursement for the cost of a specialist visit in sectors 1 and 2. Based on survey data from 2012, Dormont and Peron (2016) estimated out-of-pocket payments for sector 2 amounted to 439 euros per patient per year.

Patients (irrespective of the type of complementary contract they take out) can freely choose between consulting a specialist working in sector 1 or in sector 2. Why then would anyone pay an out-of-pocket fee for a consultation in sector 2, when consulting a specialist in sector 1 is e¤ectively free (except for the one euro co-payment)? The di¤erentiating factor is the quality of the service provided, de…ned broadly by [START_REF] Glazer | Should physicians be permitted to balance bill patients[END_REF] as: "any costly non-price attribute of health care that a¤ects patients' valuation, including dimensions of convenience, comfort, communication about medical conditions, and other factors... In concrete terms, it is easiest to think of quality as the time a physician spends with a patient to conduct a procedure". This de…nition -which re ‡ects the healthcare situation in France -does not suggest a di¤erence in quality of medical care provided per se, rather substantial di¤erences in the non-medical dimensions of the service and patient-perceived quality of care [START_REF] Calcoen | Supplementary physicians'fees: a sustainable system?[END_REF]. In particular, it has been shown that sector 2 specialists perform longer consultations (+4% in per-patient consultation time) than their sector 1 counterparts, all other things being equal [START_REF] Drees | La durée des séances des médecins généralistes[END_REF]. This superior quality of service is to a large extent related to the number of consultations to be made each day [START_REF] Bardey | Competition in two-sided markets with common network externalities[END_REF]), which we interpret in this paper as a proxy for the degree of congestion in a specialist's medical practice. Congestion increases a specialist's workload and stress, overstretches material resources and time, and ultimately leads to reduced patient satisfaction [START_REF] Wilson | The relationship between consultation length, process and outcomes in general practice: a systematic review[END_REF][START_REF] Hutton | Do longer consultations improve the management of psychological problems in general practice? A systematic literature review[END_REF]. Specialists must choose between sector 1 and sector 2 when setting up their practice and this choice is irrevocable. Hence, any change in the proportion of physicians working in both sectors is slow to occur as it depends on physicians retiring and how many new physicians enter the work market. In 2018, 47% of all specialists providing outpatient services in France were operating in sector 2 (compared with 43% in 2014). Why would any specialist choose to work in the heavily regulated sector 1, when working in sector 2 a¤ords them much more freedom in terms of setting prices? Because the number of patient per physician is larger in sector 1 compared to sector 2, which might o¤set the price e¤ect. Furthermore, physicians in sector 1 get a subsidy for their contribution to their pension scheme, while sector 2 physicians do not. Also, access to sector 2 is restricted to physicians with a proven high level of skill in their speciality, as attested by experience and/or prestigious degrees.

In this paper, we develop a simple two-sector model to analyze the e¤ects of recent French healthcare policies on the patient-to-specialist ratio in each sector, on specialists'pro…t and patients' utility. As mentioned above, the fees charged by physicians and out-of-pocket payments paid by patients di¤er between the two sectors. An original contribution of our analysis is the endogenous determination of the patient-to-physician ratio in each sector. More speci…cally, the equilibrium of our model is de…ned as a situation in which physicians and patients optimally choose their preferred sector, based on information available to them about the number of patients per specialist and about consultation fees. Accordingly, the number of patients per physician becomes the unique outcome for these choices (and must be correct in equilibrium). Patients who opt for sector 1 specialist services will only have to bear the high congestion 'cost'. Those who opt for sector 2 have a lower congestion cost, yet must make out-of-pocket payments (albeit small thanks to reimbursement). We assume that patients are heterogenous with respect to the coverage of the sector 2 extra-payment. In our model, physicians can move from one sector to another until pro…ts are equalized between both. We analyse the consequences of varying the regulated price in sector 1 for a speci…c service, of setting a higher price for the same service in sector 2, and of the real-world increase observed in the patient-to-physician ratio in France in recent years, thanks to an ongoing government policy to increase the number of physicians trained every year. Some of our model's assumptions are restrictive. First, we assume that any price increase in the regulated sector (i.e., sector 1) will be covered in full by both basic and complementary insurance.

While this is plausible for relatively small variations in the existing (low) price, it would not be sustainable for a much higher price increase. Second, we assume that all physicians in sector 2 charge the same price, and that this price is exogenous. This assumption is justi…ed by the fact that the recent regulatory changes in France have strived to encourage sector 2 physicians to cap the fees they charge for services, through two main mechanisms. The …rst concerns additional tax incentives o¤ered by the National Health Authority to those who voluntarily agree to cap their fees at no more than twice the regulated sector fee. An initial step in this direction was the introduction in 2012 of the Contract for Access to Care (Contrat d'Accès aux Soins (CAS)), which was then followed by the Controlled Pricing Practices Option (Option de Pratique Tarifaire Maîtrisée (OPTAM)) in 2017. On average, the maximum fee for a service charged by sector 2 specialists under OPTAM cannot exceed twice the regulated sector 1 fee. The second mechanism introduced is that complementary insurance contracts can no longer reimburse the full di¤erence between the regulated sector 1 fee and the sector 2 specialist's fee if the latter has not signed up to OPTAM.2 More speci…cally, complementary insurance reimbursement cannot exceed 100% of the regulated fee when the service is provided by a non-OPTAM physician, while this value is 200% of the regulated fee when care is provided by an OPTAM member. 3 4 The literature on health economics has already explored the topic of price and quality di¤erentiation of the health care service in the "balance billing"setting, whereby physicians can charge an extra fee on top of the statutory fee. Early analyses of di¤erentiated pricing represented the physician as a monopolist providing a service of homogenous quality who could price discriminate between patients who had di¤erent levels of willingness to pay [START_REF] Mitchell | Physician behavior under the Medicare assignment option[END_REF]Zuckerman and Houlihan, 1991;[START_REF] Savage | An analysis of the General Practice Access Scheme on GP incomes, bulk-billing and consumer copayments[END_REF]. In those models, balance billing could only increase the pro…ts of physicians at the expense of patients. [START_REF] Feldman | Competition among physicians, revisited[END_REF] argued that a monopolist physician who is subject to restrictions on balance billing would consciously or unconsciously negatively alter the quality of the service they provide to patients. [START_REF] Glazer | Should physicians be permitted to balance bill patients[END_REF] analysed monopolistic competition between two physicians who could engage in price and quality di¤erentiation. They revealed the existence of a positive net fee that maximizes social welfare, and showed that restrictions on balance billing applied in the US in the late 1980s insurance is 70% of the regulated price minus a one-euro co-payment, i.e., 16.50 euros (calculation: 25*0.70 -1 euro). Any complementary insurance contract will reimburse the remaining amount minus the one euro co-payment, i.e. 7.50 euros. Let us assume that a sector 2 physician charges 40 euros for the same consultation. If s/he is an OPTAM subscriber, his/her patient will be reimbursed on the basis of the regulated price (25 euros). Hence reimbursement by basic insurance is still 16.50 euros. Under the OPTAM agreement for sector 2, complementary insurance reimburses 200% of the regulated price, i.e., 32.50 + 16.50 = 49 which is >40, hence complementary insurance reimburses 40-16.50-1= 22.50 euros). Therefore, the total out-of-pocket payment (i.e., including the co-payment) is one euro. If the sector 2 physician has not signed up to OPTAM, reimbursement for patients with basic insurance is still 70% of the regulated price minus the one euro co-payment. In this case, the considered regulated price is 23 euros (instead of 25 euros for OPTAM subscribers), hence basic insurance reimburses 15.10 euros (calculation: 23*0.70 -1). Complementary insurance reimburses 6.90 euros (calculation: 23*1 -15.10 -1). The …nal patient out-of-pocket payment (including the co-payment) is 18 euros (calculation: 40 -15.10 -6.90).

would reduce quality of care for all patients, irrespective of whether they paid an extra fee or not. [START_REF] Kifmann | Balance billing: the patients' perspective[END_REF] used a similar framework to show that patient's welfare is higher under the balance billing system. [START_REF] Jelovac | Physicians' balance billing, supplemental insurance and access to health care[END_REF] found that balance billing might restrict access to care for patients with heterogeneous health coverage, and therefore constituted a suboptimal policy. [START_REF] Gravelle | Competition, prices and quality in the market for physician consultations[END_REF] developed a classic model of price and quality di¤erentiation à la Salop which is very suitable to assess the deregulated Australian market. They also used an empirical test to show that reduced competition leads to higher prices, yet has a negligible e¤ect on quality.5

Main assumptions

Using our two-sector model, we studied how physicians, who supply the medical service, and patients, who demand the service, interact in France's dual market for specialised outpatient care.

Physicians can choose to work either in the regulated sector (sector 1) or in the sector allowing for balance billing (sector 2). In each sector, the care o¤er has two dimensions: i) the consultation fee and ii) the overall quality of the service provided in the broadest sense (i.e., waiting times, consultation duration, etc.), as it is assumed that the actual quality of medical care provided does not di¤er between the two sectors.

Let p 1 be the fee for a medical consultation in sector 1 with p 1 0: This fee is regulated. Let p 2 be the fee for the same consultation in sector 2, with p 2 > p 1 . We consider a situation in which the fee p 2 is also …xed by health care authorities. 6The overall quality of the encounter between the physician and the patient depends on the number of patients a physician must see every day. As mentioned in the introduction, having a high patient-to-physician ratio (or high 'congestion) stretches material resources and the physician's own time, translating into shorter consultations and consequently reduced patient satisfaction with the care provided.

Let M denote the total number of specialists working in France. They choose their sector of activity in order to maximize their pro…t. In equilibrium, is the share of physicians practicing in sector 1, and ( 1) is the share of physicians practicing in sector 2.

Let P denote the total number of patients in France. They choose between consulting a sector 1 or sector 2 specialist, and seek to maximize their utility associated with the healthcare consultation provided. At equilibrium, a fraction of the patients will choose to consult a physician in sector 1, while (1 ) of the patients will opt for sector 2. Patients in both sectors are distributed homogeneously among all physicians.

The number of patients per physician in each sector is therefore:

n 1 = vP M = k (1)
n 2 = (1 )P (1 ) M = (1 ) 1 k; (2)
with the patient-to-specialist ratio k = P=M . The ratio can evolve for structural reasons, such as population ageing which might increase P , and policies that would regulate the numbers of physicians (e.g., abolishing existing quotas on higher education admissions or on immigrant specialists). [START_REF] Bardey | Competition in two-sided markets with common network externalities[END_REF] point out that many scholars in healthcare and education assume that the quality of the service provided by a hospital or university depends upon provider/patient or teacher/student ratio. In this line of reasoning, we assume that, in each sector j; the monetary cost of patient discomfort associated with congestion is a linear function of the number of patients per specialist:

c j = n j ; with > 0 and j = (1; 2) (3)
Within the French context, the sector 1 fee p 1 is fully reimbursed. However, in the more expensive sector 2, p 2 is only partially covered by health insurance contracts

In sector j with j = (1; 2); for a patient i; the health insurance coverage (i.e., both for basic and complementary contracts) covers a monetary amount R i j with:

R i j = 8 > > < > > :
p 1 for j = 1

r i for j = 2 (4)
Although everyone in France has complementary insurance (the assumption made in our model),

there is nonetheless a great deal of heterogeneity in the extent of coverage for balance billing expenses between contracts for complementary insurance [START_REF] Drees | Les Dépenses de Santé en 2018, collection Panoramas de la Drees[END_REF]. We model this heterogeneity by assuming that reimbursements r i are uniformly distributed over the interval [p 1 ; p 2 ]:7

In this two-sector market, a patient i must choose between two alternative care o¤ers (re ‡ecting both sectors 1 and 2), each being characterized by two dimensions: {out-of-pocket payments and congestion costs}:

8 > > < > > :
O¤er sector 1 : (0; c 1 )

O¤er sector 2 : (p 2 r i ; c 2 )

(5)

Patient utility:

Denoting the monetary value of healthcare by u, a patient i who accepts the o¤er (p j ; c j ) has a net utility U i j :

U i j = u (p j R i j ) c j ; (6)
The patient is aware of the o¤ers in the two sectors. S/he selects a physician in a speci…c sector in order to maximize his/her utility U i :

Physician pro…t:

In their professional activity, each doctor bears a cost per patient q. To keep the analysis as simple as possible, we do not take into account the …xed cost of setting the practice in a given sector. Pro…ts are therefore proportional to the net reward of a consultation, and to the number of patients who choose them. Pro…ts (per doctor) in the two sectors are:

8 > > < > > :
Sector 1:

1 (p 1 ; c 1 ) = (p 1 q) P M = k (p 1 q)
Sector 2: 2 (p 2 ; c 2 ) = (p 2 q) (1 )P

(1

)M = k (p 2 q) (1 )
(1 )

(7)
3 The two-sector equilibrium

In equilibrium, patients maximize their utility under the assumption that physicians have made an optimal choice with respect to their choice of sector of practice, and physicians choose between sector 1 and sector 2 under the assumption that patients have made an optimal choice with respect to consulting a sector 1 or sector 2 specialist. Equilibrium in this market is an allocation ( ; ); with 2 [0; 1] and 2 [0; 1]:

Comparative statics allow us to study the impact of changes in exogenous variables on the equilibrium endogenous variables. Yet moving from one equilibrium to another requires that patients switch from one sector to another, and that the distribution of physicians between the two sectors is reallocated until their pro…ts once again balance each other. While patients can move without friction, physicians cannot. More speci…cally, changes in physician distribution are much slower, as they are determined by the number of older physicians going into retirement, and the number of new younger generations of physicians arriving into the work market. Accordingly, moving from one equilibrium to another is a lengthy process.

The demand for health services (short-term changes)

We …rst study the short-term changes to patient distribution in each sector, with a constant share of physicians in sector 1 ( ): Discarding the speci…c cases of corner solutions with only one optimal sector, we then search for the indi¤erent patient between the two sectors. Let r be the "threshold" reimbursement for this patient, as de…ned by:

u(p 1 ; c 1 ) = u(p 2 ; c 2 ) , u c 1 = u p 2 + r c 2 (8)
All patients with r < r prefer sector 1 over sector 2 while those with r > r prefer sector 2.

The share v of the patient population which prefers sector 1 is thus equal to

v = Pr[r < r ],
and given that r is uniformly distributed over the interval [p 1 ; p 2 ]; we have: v = r p1 (p2 p1) ; or:

r = p 1 + v (p 2 p 1 )
: If we substitute this expression in the previous de…nition of r ; we obtain:

u c 1 = u p 2 + p 1 + v (p 2 p 1 ) c 2 (9) , c 1 c 2 = (1 v ) (p 2 p 1 ) (10)
, k v k (1 v ) (1 ) = (1 v ) (p 2 p 1 ) (11)
, v = (p 2 p 1 ) (1 ) + k (p 2 p 1 ) (1 ) + k 1 : (12)
Note that as p 2 > p 1 ; Eq. ( 10) implies c 1 > c 2 and, by de…nition of c 1 and c 2 (Eq. 1 and Eq. 2)

that v > irrespective of the values of the parameters. Note also that, for p 1 = p 2 ;

= .

Finally, the de…nition of v reveals an increasing relation between v and :

@v @ = k 2 (p 2 p 1 ) (1 ) 2 + k (p 2 p 1 ) (1 ) + k 1 2 > 0: (13)
The short-term e¤ects of changes in price or the patient-to-physician ratio on v are:

@v @p 1 = (1 ) k 1 1 (p 2 p 1 ) (1 ) + k 1 2 < 0 (14) @v @p 2 = (1 ) k 1 1 (p 2 p 1 ) (1 ) + k 1 2 > 0 (15) @v @ k = [(p 2 p 1 ) (1 )] 1 1 (p 2 p 1 ) (1 ) + k 1 2 < 0: (16)
Unsurprisingly, an increase in the fee in one sector increases the attractiveness of the other sector.

Note that an increase in p 1 does not change the price paid in sector 1, as the fee is completely reimbursed by the combination of public and complementary insurance (except the one euro copayment). However, as p 2 remains constant, the rise in p 1 reduces the out-of-pocket payment for sector 2 patients and therefore contributes to the attractiveness of this sector. Note also that an increase in the patient-to-physician ratio, k contributes mainly to the growth in the number of sector 2 patients. More speci…cally, an increase in k increases the spread between the two congestion costs. Because c 1 rises faster than c 2 more patients are willing to consult a sector 2 physician.

In the Appendix we show that the optimal proportion of patients in sector 1, v , is the outcome of a convergent adjustment process. If the proportion of patients in sector 1 is larger than v ; for instance v > v , it can be shown that the utility of the "switching patient" is higher in sector 2 than in sector 1. As individuals in the left neighborhood of this patient move to sector 2, v declines until the "switching patient" becomes indi¤erent between the two sectors. The opposite process occurs for v < v :

3.2 The supply for health services (long-term changes)

We next study the physician's decision regarding which sector to work in. This decision is based on maximizing his/her pro…t. Physicians enter the sector that brings them the highest pro…ts until pro…ts in the two sectors balance each other out, 2 = 1 . For a given set of prices (p 1 ; p 2 ), patients-to-physician ratio (k) and a given proportion of patients in sector 1 (v), the equilibrium proportion of physicians in sector 1 is the implicit solution to:

1 = 2 (17) , k (p 1 q) = k (p 2 q) (1 ) (1 ) (18)
,

(1 ) = (p 1 q) (p 2 q) (1 )
This leads to the explicit form:

= (p 1 q) (1 ) (p 2 q) + (p 1 q) ; with 2 [0; 1] (20)
with d =dv > 0:8

We can check that is the outcome of a convergent process of physician allocation between the two sectors. When the proportion of doctors in sector 1 is higher than the optimal proportion

; pro…ts are higher in sector 2 than in sector 1, and therefore the share of doctors in sector 1 must decrease in favour of sector 2. In the opposite case, when the proportion is lower than , we have 1 > 2 , and the share of doctors in sector 1 will increase and will tend towards .

The equilibrium of the dual health care market

The equilibrium proportions of physicians and patients ( ; ) must simultaneously satisfy equations ( 12) and ( 20).

If we now substitute [(1 v )= ( 1)] for its expression in Equation ( 19) in Equation ( 12), we obtain:

, k v k v (p 1 q) (p 2 q) = (1 v ) (p 2 p 1 ) (21) , v = (p 2 q) (p 2 q) + k 2 [0; 1] (22)
Solving the equations for and ; we obtain the equilibrium values:

= (p 2 q) [(p 1 q) k] [ k (p 1 q) + (p 2 q) [(p 1 q) k]] (23) = (p 1 q) k (p 1 q) (24)
Note that and v are lower than 1. If allowed by the organization of the healthcare market, sector 2 will always remain the optimal choice for some physicians.

On the other hand, some physicians will continue to choose sector 1 only if the price in sector 1 is large enough:

> 0 , p 1 > q + k; (25)
otherwise all physicians will choose to work in sector 2 ( > 0 , v > 0).

Hereafter, in order to rule out any possibility of a trivial equilibrium, we consider the condition in equation ( 25) is ful…lled.

For the above-de…ned long-run equilibrium values of and ; we obtain the sector-speci…c congestion cost c 1 and c 2 , and the pro…ts per physician 1 = 2 = : Furthermore, for a patient who does not change sector, his/her utility is: u 1 (resp u 2 ):

c 1 = k (p 2 q) (p 1 q) (p 1 q k) (p 2 q) + k (p 1 q) (26) c 2 = k (p 1 q) 2 (p 1 q k) (p 2 q) + k (p 1 q) (27) = k (p 2 q) (p 1 q) 2 (p 1 q k) (p 2 q) + k (p 1 q) (28) u 1 = u c 1 (29) u 2 = u p 2 + r c 2 ; with r 2 [p 1 ; p 2 ] (30)
Finally, from a utilitarian perspective, the total utility of the patients is the sum of the utilities of all patients, of whom a share v opt for sector 1 while the rest (1 v ) opt for the sector 2:

U = U 1 + U 2 ; with 8 > > < > > : U 1 = P v ( u c 1 ) U 2 = R p2 r [ u (p 2 r) c 2 ] f (r) P dr = P v ( u c 1 ) + P (1 v ) ( u c 2 ) 1 2 (p 2 p 1 ) (1 ) (31)
where P is the patient population. For this analysis we make the assumption that P is constant (accordingly, changes in k are the consequence of the number of physicians varying).

Properties of the solution

In this section we analyze the consequences of changes in three variables representative of key healthcare policies:

-a change in the regulated price, p 1 ;

-an exogenous variation in the sector 2 price, p 2 ; -a change in the economy-wide patient-to-physician ratio, k = P=M .

The essential endogenous variables are:

-the frequencies of patients and physicians in sector 1, and v , (and the complementary proportions in sector 2)

-the sector-speci…c patient-to-physician ratio and the related congestion costs, c 1 and c 2 :

-the physician pro…ts, identical in the two sector, 1 = 2 = :

-the utility of the representative patient in the two sectors, u 1 and u 2 .

In terms of patient utility, we will …rst consider only the utility of 'incumbent' patients, i.e.

patients who do not change sector. Obviously, these are patients relatively far from the indi¤erent patient r : If parameters change, some individuals (in the neighborhood of r ) will move from one sector to another. This more complex overall variation in utility is analysed later in this article. Over the long term, an increase in the regulated price p 1 attracts both more physicians and, counterintuitively, more patients to this sector. Changes in the equilibrium proportions of patients and physicians in sector 1 are:

d dp 1 = (p 1 q) (p 1 q) + k (p 1 q) 2 = k (p 1 q) 2 > 0 (32) dv dp 1 = (p 2 q) ( k) 2 [ k (p 1 q) + (p 2 q) (p 1 q k)] 2 > 0: (33)
Using this, we can evaluate whether the congestion cost will decrease. Indeed:

dc 1 dp 1 = [ k(p 2 q)] 2 [((p 1 q) k) (p 2 q) + k (p 1 q)] 2 < 0: (34)
With fewer physicians and fewer patients in sector 2, the evolution of the congestion cost in sector 2 depends on the model's parameters. We determine:

dc 2 dp 1 = k (p 1 q) (p 1 q k) (p 2 q) k (p 2 p 1 ) [(p 1 q k) (p 2 q) + k (p 1 q)] 2 : (35)
The sign of this derivative is the sign of expression

[(p 1 q k) (p 2 q) k (p 2 p 1 )] : If we de…ne p = q(p 2 q) + k(2p 2 q) (p 2 q + k) ; ( 36
)
then it turns out that:

8 > > < > > : dc2 dp1 0 for p 1 p dc2 dp1 > 0 for p 1 > p : (37)
We argued that a two-sector equilibrium exists only for p 1 > q + k (condition 25). Given the sectorial constraint p 2 > p 1 (see above), we can check that (q + k) < p < p 2 : This means that the turning point is well located within the interval of de…nition of p 1 :

As long as p 1 is initially relatively small, an increase in this price will reduce the congestion cost in sector 2; however, this e¤ect is reversed for a relatively large p 1 : The congestion cost in sector 2 is the lowest when p 1 = p1 :9

The e¤ect of a higher p 1 on the variation in pro…ts is related to the variation of the congestion cost in the sector 2:

d dp 1 = k(p 2 q) (p 1 q) (p 1 q) (p 2 q) k (2p 2 q p 1 ) [(p 1 q k) (p 2 q) + k (p 1 q)] 2 : (38)
We can easily check whether d dp1 > 0 if dc2 dp1 > 0 and d dp1 < 0 if dc2 dp1 < 0:

The economic mechanisms driving these changes are relatively complex. We remind the reader that sector 1 patients do not pay the fee p 1 , as it is assumed that this cost is fully covered by basic and complementary insurance. However, a larger p 1 reduces the out-of-pocket payment for sector 2 patients, and consequently attracts patients to this sector. In turn, this induces a rise in the congestion cost in sector 2, c 2 . At the same time, as p 1 increases, the value which ensures the long-term equality of pro…t increases (see Eq. 32); at a constant ; this implies that 1 > 2 :

In the long term, all other things being equal, higher pro…ts in sector 1 attract more physicians to this sector, d dp1 > 0: The congestion cost c 1 continues to decline, attracting more patients to sector 1, which limits and may even o¤set the initial increase in the congestion cost c 2 .

The long term e¤ect of a higher p 1 on pro…ts per physician (identical in the two sectors) is

undetermined. An increase in p 1 prompts more physicians and patients to choose the sector 1.

The e¤ect on pro…ts may be positive or negative, depending on the initial situation, and the elasticity of sector choice of the two agents. If following the increase in p 1 there are less patients per physician in sector 2 (the pro…t margin in sector 2 being constant), pro…ts per physician decline. On the other hand, if there are more patients per physician in sector 2, the pro…t per physician increases.

Our analysis therefore highlights that an increase in the regulated price p 1 does not mechanically lead to higher pro…ts for the medical profession. This result indicates that there is no simple solution to the question of how to increase physicians'incomes.

In terms of utility change (for incumbent patients), for a sector 1 patient we have:

du 1 dp 1 = dc 1 dp 1 > 0: (39)
Because sector 1 patients do not pay the price p 1 ; and bene…t from a lower congestion cost as p 1 increases, the utility of the patient who remains with a physician in sector 1 will increase as long as p 1 increases.

In sector 2, we study the variation in the utility of a patient who does not change sector when p 1 changes. The utility of this patient is u 2 = u (p 2 r) c 2 : His/her variation in utility therefore depends only on the variation in congestion cost, the latter being related to the derivatives in Eq.

(37) as follows:

du 2 dp 1 = dc 2 dp 1 (40)
The utility …rst increases in p 1 , reaching a maximum when p 1 = p1 ; and then declines.

When considering the total utility U; any changes in parameters lead to a complex e¤ect, as both individual utilities and the numbers of patients in each sector may change. For instance, if fewer patients remain in sector 2, total utility in sector 2 can decline even if individual utility increases. To address this complexity, we present here a numerical simulation here, representing total utility in sector 1, in sector 2 and overall, when p 1 varies. Invariant parameters are u = 100; q = 20; p 2 = 75; k = 8; = 1=2. When p 1 varies between ( k + q) = 24 and p 2 ; increases in p 1 lead to an increase in the sector 1's overall utility.

For p 1 < p = 27:45; the utility of an incumbent patient in sector 2 increases with p 1 . However, the simulation shows that the overall utility of the patients in the sector 2 systematically falls as the sector shrinks. Not only doctors leave the sector 2, but so do some patients.

Overall utility nonetheless increases in p 1 .

Changes in p 2

As mentioned in the introduction, recent French government regulations have strived to encourage sector 2 physicians to cap their fees. A decisive step was taken in 2017 with the implementation of the OPTAM agreement (Option Pratique Tarifaire Maitrisée). Physicians who join this scheme commit themselves to charging a maximum fee of twice the regulated price for a speci…c service.

As of September 2019, 30% of sector 2 physicians had signed up to this agreement.

The e¤ects of these regulations on changes in p 2 can be analysed in the same way we analysed changes in p 1 .

From Equation ( 15) we see that in the short term, any price increase reduces the number of patients in sector 2 and increases sector 1's congestion cost, while pro…ts increase in both sectors:

in sector 1 this increase is due to the higher number of patients per physician, while in sector 2 it stems from the rise in physician fees. Changes in pro…ts are equal in both sectors. Accordingly, changes in p 2 do not alter the distribution of physicians in the two sectors.

In the long term, Eq. ( 24) reveals that the equilibrium number of physicians in sector 1 is not related to p 2 :

d dp 2 = 0: (41)
Long-term e¤ects are therefore similar to short-term e¤ects.

The e¤ect of an increase in p 2 on the number of patients migrating to sector 1 (i.e., leaving sector 2) is positive.

dv dp 2 = (p 1 q k) k (p 1 q) [ k (p 1 q) + (p 2 q) [(p 1 q) k]] 2 > 0 (42)
Since is not a¤ected, this implies a higher congestion cost in sector 1, and an increase in the congestion cost c 1 :

dc 1 dp 2 = [ k (p 1 q)] 2 [((p 1 q) k) (p 2 q) + k (p 1 q)] 2 > 0 (43)
There is a reduction in c 2 :

dc 2 dp 2 = k (p 1 q) 2 (p 1 q k) [(p 1 q k) (p 2 q) + k (p 1 q)] 2 < 0 (44)
And the pro…t per physician goes up: The variation in utility of the sector 1 incumbent patients depends on the change in congestion costs; it lowers when the price p 2 increases because sector 2 patients ‡ee to sector 1.

d dp 2 = (p 1 q) dv dp 2 > 0 (45)
du 1 dp 2 = dc 1 dp 2 < 0: (46)
For the sector 2 individual who has to pay the out-of-pocket payment (p 2 r); the utility variation is a combination of a higher price, and a lower congestion cost. Under our assumptions, the negative price e¤ect prevails over the later:

du 2 dp 2 = 1 dc 2 dp 2 < 0 (47)
The overall e¤ect on utility of a change of p 2 is illustrated by the following numerical simulation, representing utility in both sectors when p 2 varies. As in the previous simulation, invariant parameters are u = 100; q = 20; k = 8; = 1=2: Furthermore, we assume that p 1 = 25.

The increase in p 2 brings about a decline in overall utility in sector 2. However, unlike the …ndings from the previous analysis (for p 1 ), it leads to an increase in overall utility of the patients in sector 1, because this sector is growing. However, even if the changes in both sectors o¤set each other, total utility declines when p 2 increases.

To summarize, an increase in p 2 will push up the out-of-pocket payment and prompt patients with less generous health insurance coverage to migrate to sector 1. This will reduce congestion in sector 2, which in turns limits the outward ‡ow. With the additional patients (who have migrated from sector 2), the congestion cost in sector 1 increases, which makes sector 1 less attractive. One would expect that the increase in p 2 would attract physicians to sector 2. However, with our linear function, this "intensive margin" e¤ect is o¤set by the lower number of patients (a reduction in the extensive margin).

Pro…ts increase in sector 1 because the number of patients per physician increases. In sector 2, pro…ts increase despite the reduced number of patients per doctor, because the margins are now bigger.

The utility of incumbent patients declines in both sectors, driven by a higher congestion cost in sector 1, and by the higher out-of-pocket payment not fully compensated by the lower congestion cost in sector 2. Overall utility decreases slightly as p 2 increases.

Changes in k

Changes in the economy-wide patients-to-physician ratio, k = P=M; are partially the result of structural changes. In particular, population aging in the developed world is contributing to increase P; the number of persons needing an increased amount of healthcare services. Policies that allow for an increase in the number of practicing physicians by increasing the number of physicians trained in higher education institutions and easing immigration restrictions on healthcare professionals -can increase M (and reduce k):

In France, successive governments from 1970 onward adopted a numerus clausus (i.e., …xed number or quota) approach to medical education. However, over time this led to a shortage of physicians in many specialist areas, with a peak shortage in the mid-1990s. Since then, numbers of new trainee physicians have increased every year, but the same logic of education quotas remains.

From 2020-2021, this policy of education quotas will be replaced with more ‡exible management of the numbers of places in higher education for medical-related …elds. 10 This change will be tantamount to an increase in M , and a reduction in k.

In the following analysis, we will focus on the e¤ect of a reduction in the more general variable ( k):

@v @ k = [(p 2 p 1 ) (1 )] 1 1 (p 2 p 1 ) (1 ) + k 1 2 < 0 (48)
In our model, a reduction in k (or in k) leads to a short term decrease in the number of patients in sector 2. In the long term, this leads to a decrease in pro…ts in this sector and makes the sector less attractive for physicians.

d d k = 1 (p 1 q) < 0: (49)
As congestion increases in sector 2 (because of fewer physicians), this leads to an increase in the number of patients in sector 1, and the number of patients in sector 2 decreases.

dv d k = (p 2 q) (p 1 q) 2 [ k (p 1 q) + (p 1 q k) (p 2 q)] 2 < 0: (50)
From the two previous equations, we can infer that a change in prompts the number of physicians and patients in a given sector to move in the same direction.

The congestion cost decreases in both sectors (there are fewer patients per doctor on average).

dc 1 dk = [(p 1 q) (p 2 q)] 2 [((p 1 q) k) (p 2 q) + k (p 1 q)] 2 > 0 (51) dc 2 dk = (p 1 q) 3 (p 2 q) [(p 1 q k) (p 2 q) + k (p 1 q)] 2 > 0: (52)
Yet the change in the congestion cost is lower in sector 2 than in sector 1 ( dc2 dk < dc1 dk ):

With fewer patients per doctor, pro…ts in the healthcare sector decline: To summarize, a reduction in the number of patients per physician k will push down congestion costs in both sectors. However, the congestion cost declines relatively less in sector 2. Pro…ts decrease given the lower number of patients per physician, while the utility of incumbent patients increases.

d 1 dk = (p 1 q) 3 (p 2 q) 2 [((p 1 q) k) (p 2 q) + k (p 1 q)] 2 > 0 (53)
In the current context where the French government continues to declare its desire to attract more talent into the medical profession, any reduction in pro…t could hamper achieving this goal.

Conclusion

This paper introduces a two-sector equilibrium model to investigate the dual-legal structure of the French market for specialized outpatient services. In the model, the two sectors di¤er in terms of fees, insurance coverage, and, as an original contribution to the literature, number of patients a physician meets over a standard period of working time. The number of patients per physician in each sector is endogenously determined, as the related congestion cost enters the utility function of the patient. Although our model focusses on the speci…c case of France, it can be applied to any healthcare system with a balance billing design where full reimbursement is granted for a basic service, and where out-of-pocket payments allow the patient to bene…t of a better non-medical quality of care (e.g., a private room in hospital, longer consultations, reduced waiting times).

Our model presents a unique, stable, non-trivial equilibrium de…ned as the equilibrium proportion of physicians and patients in each sector. While many results re ‡ect what intuitive reasoning would suggest, some are more challenging to understand.

For instance, over the long term, and as long as p 1 < p; an increase in the regulated price p 1 cuts congestion in both sectors due to the reallocation of physicians and patients (the economywide patient-to-physician ratio being invariant). Patient utility in both sectors increases while the physician pro…ts decrease, meaning that the global cost of healthcare also decreases.

From a policy perspective, a rise in p 1 (as long as p 1 < p) brings about greater patient welfare at a lower social cost. Furthermore, an increase in the regulated price (p 1 ) fosters the size of the regulated sector (sector 1) relative to sector 2. An increase in the regulated price attracts not only more physicians but also more patients to sector 1. However, the e¤ect of an increase of p 1 on a physician's income (in both sectors) depends on the speci…c parameters of the problem. A situation where a higher price for the service leads to lower physician income in the long term cannot be ruled out. The French government, which continues to indicate its desire to attract more talent to the medical profession, should keep this possibility in mind.

In our model, a change in p 2 has no e¤ect on the relative number of physicians in the two sectors. For physicians in sector 2, the increase in p 2 leads to higher physician pro…ts per patient, but as patients leave sector 2 for sector 1, pro…ts for physicians in the latter increase in the same proportions. As pro…ts remain the same in both sectors, physicians have no incentive to change sector. This result is driven by our study assumptions and should not be considered general.

However, the model emphasizes the existence of two opposite e¤ects on a physician's choice of sector, in the case of a change in the unregulated price p 2 : Even in more general contexts, these two opposite e¤ects would still be at work, explaining why a change in p 2 could only have a second order e¤ect on the distribution of physicians among the two sectors. One important policy implication is that the recent steps by the French government to encourage physicians to cap sector 2 service fees, might have only a limited impact on the number of physicians deciding to work in that sector.

Finally, increasing the number of available physicians by training more of them (i.e., a lower k)

would increase the number of physicians who decide to work in sector 1. One might expect that this increase would either have a neutral impact on the sector sizes or that new physicians would be more attracted by the higher fees (and therefore perceived net higher pro…ts) in sector 2. In our model, both the proportions of patients and physicians in sector 1 increase when the overall number of physicians increases. Congestion declines in the two sectors, but this decline is faster in sector 1 than in sector 2. Furthermore, reforms leading to an increased number of physicians in turn increase patient utility and reduce pro…ts. It is likely that the medical profession will support moderate increases in the number of physicians proportional to the increase in the number of patients, but will resist larger increases. Certainly, at some point, the planned increase in the numbers of trained physicians in France might enter con ‡ict with the same government's 2020 promise to raise incomes in the medical profession.

A Appendix

A.1 Long-term partial derivatives

The analysis is carried out under the condition p 1 > (q + k):

dp 1 dp 2 d( k) d k (p1 q) 2 > 0 0 1 ( p1
q) < 0 dv (p2 q)( k) 2 [ k(p1 q)+(p2 q)(p1 q k)] 2 > 0 (p1 q k) k(p1 q) [ k(p1 q)+(p2 q)[(p1 q) k]] 2 > 0 (p2 q)(p1 q) 2 [ k(p1 q)+(p1 q k)(p2 q)] 2 dc 1

[ k(p2 q)] 2 [((p1 q) k)(p2 q)+ k(p1 q)] 2 < 0 [ k(p1 q)] 2 [((p1 q) k)(p2 q)+ k(p1 q)] 2 > 0 [(p1 q)(p2 q)] 2 [((p1 q) k)(p2 q)+ k(p1 q)] 2 dc 2 k (p 1 q) (p1 q k)(p2 q) k(p2 p1)

[(p1 q k)(p2 q)+ k(p1 q)] 2 ? 0 k(p1 q) 2 (p1 q k) [(p1 q k)(p2 q)+ k(p1 q)] 2 < 0 (p1 q) 3 (p2 q) [(p1 q k)(p2 q)+ k(p1 q)] 2 d k(p 2 q) (p 1 q) (p1 q)(p2 q) k[2p2 q p1] [(p1 q k)(p2 q)+ k(p1 q)] 2 ? 0 (p1 q) dv dp2 > 0 (p1 q) 3 (p2 q) 2 [((p1 q) k)(p2 q)+ k(p1 q)] 2

@v @

ss 0 ; =0 = (p 2 p 1 ) + k k > 1 (68) @ @v nn 0 ;v=0 = (p 1 q) (p 2 q) , @v @ nn 0 ; =0 = (p 2 q) (p 1 q) (69) @v @ nn 0 ; =0 < @v @ ss 0 ; =0 (p 2 q) (p 1 q) < (p 2 p 1 ) + k k k < (p 1 q) , > 0 (A.70)

Figure 2 :

 2 Figure 2: Utility with respect to p 1

Figure 3 :

 3 Figure 3: Utility with respect to p 2

Figure 4 :

 4 Figure 4: Utility with respect to k

Figure 5 :

 5 Figure 5: Equilibrium proportions of patients and physicians.

Table 1

 1 summarizes the main relationships between the variables (explicit expressions are displayed in Table2in the Appendix).

	d	dv	dc 1 dc 2	d		du 1 du 2
	dp 1 +	+		;if p 1 < +;if p 1 > p p	;if p 1 < +;if p 1 >	p p	+	+;if p 1 < p ;if p 1 > p
	dp 2 0	+	+		+
	dk		+	+	+
				Table 1: Long-term e¤ects
	4.1 Changes in p 1

This is part of recent French regulation on 'responsible'insurance contracts. In total, 90% of existing insurance contracts are certi…ed as 'responsible'. However, some private insurance contracts do not cover in full even some of the moderate OPTAM fees.

As of September 2019, 17000 specialist physicians (i.e., 30% of sector 2 physicians) had joined the OPTAM agreement. Furthermore, the share of balance billing in total fees in sector 2 dropped from 35% in 2011 to 31.5% in 2018 (DRESS, 2019).

Let us take the example of a consultation whose regulated (sector 1) price is 25 euros. Reimbursement by basic

Other papers recognize that in many countries, physicians must charge uniform tari¤s and focus on this quality di¤erentiation issue(Godager et al., 2009; Brekke et al., 2010;[START_REF] Brekke | Quality competition with pro…t constraints[END_REF].

In this simple model, the specialist's pro…t monotonically increases in p 2 . In France, recent regulatory changes have striven to encourage specialists in sector 2 to voluntarily set an upper limit on p 2 of twice the regulated fee for the same service.

The assumption of a uniform distribution allows us to obtain closed-form solutions. In any case, the structure of the problem would not change even if a more realistic distribution were assumed.

Note that = 0 if = 0 and = 1 if = 1: In both cases, patients and physicians are clustered in a single sector. The case = 0 (respectively = 1) implies that no patient is interested in any sector 2 o¤er (respectively sector 1 o¤er) and, as the resulting pro…t 2 (resp 1 ) is zero, no physician chooses to work in sector 2 (respectively, sector 1).

This shortest waiting time is not necessarily socially optimal, given that a higher p 1 could further reduce waiting time in sector 1.

0 Press release of the French Government, May 18,

A.2 Existence of a stable, non-trivial solution

The equilibrium proportion of patients v (and the implicit compensation r of the "switching" patient) was de…ned by:

Leading us to write:

with v (0) = 0. Let us denote this curve by ss 0 (see Figure 4).

We show that v ( ) is concave and increasing in :

To study convergence, let us consider a desequilibrium situation in which for an allocation of v patients in sector 1 and (1 v) in sector 2, the "switching patient" (with a reimbursement r)

has a larger utility in sector 2 than in sector 1; there must then be some individuals in his "left neighborhood" that could switch to sector 2 as well, i.e., the frequency of patients in sector 1 should decline. We check that u c

Patients must move to sector 2; the frequency of patients in sector 1 declines, until v ( ) is reached.

We turn now to the pro…t indi¤erence condition.

For a given v; the equilibrium frequancy of physicians is a function:

with (0) = 0: Let us denote by nn 0 the curve (v):

We can show it is increasing and concave:

To study convergence, let us assume that the proportion of physicians in sector one is larger than the equilibrium proportion, > : Then 1 < 2 , thus declines, to converge towards .

The problem has a non-trivial solution if ss 0 and nn 0 cross for v > 0 and > 0. We show that such a solution exists and it is unique.

Both ss 0 and nn 0 have 0 as the origin. In the (O ; Ov) plane, ss 0 is concave, while nn 0 is convex. We can check (see Figure 5) that the two curves cross once if, at the origin, the slope of ss 0 is smaller than the slope of nn 0 : This can be easily veri…ed:

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