Haiqin Du

Fernando A De Oliveira

Lindomar J C Albuquerque

Guillaume Tresset

Ewa Pavlova

Cécile Huin

Philippe Guégan

Fernando Carlos Giacomelli
email: fernando.giacomelli@ufabc.edu.br

Polyglycidol-Stabilized Nanoparticles as a Promising Alternative to Nanoparticle PEGylation: Polymer Synthesis and Protein Fouling Considerations

We herein demonstrate the outstanding protein-repelling characteristic of starlike micelles and polymersomes manufactured from amphiphilic block copolymers made by poly(butylene oxide) (PBO) hydrophobic segments and polyglycidol (PGL) hydrophilic outer shells. Although positively charged proteins (herein modeled by lysozyme) may adsorb onto the surface of micelles and polymersomes where the assemblies are stabilized by short PGL chains (degree of polymerization smaller then 15), the protein adsorption vanishes when the degree of polymerization of the hydrophilic segment (PGL) is higher than ~ 20, regardless the morphology. This has been probed by using three different model proteins which are remarkable different concerning molecular weight, size and zeta potential (bovine serum albumin -BSA, lysozyme and immunoglobulin G -IgG). Indeed, the adsorption of the most abundant plasma protein (herein modeled as BSA) is circumvented even by using very short PGL shells due to the highly negative zeta potential of the produced assemblies which presumably promotes protein-nanoparticle electrostatic repulsion. The highly negative zeta potential, on the other hand, enables lysozyme adsorption and the phenomenon is governed by electrostatic forces as evidenced by isothermal titration calorimetry.

Nevertheless, the protein coating can be circumvented by slightly increasing the degree of polymerization of the hydrophilic segment. Notably, the PGL length required to circumvent protein fouling is significantly smaller than the one required for PEO. This feature and the safety concerns regarding the synthetic procedures on the preparation of poly(ethylene oxide)-based amphiphilic copolymers might make polyglycidol a promising alternative towards the production of non-fouling particles.

INTRODUCTION

Recently, great concerns have been invoiced regarding the safe use of nanotechnology-based platforms for healthcare thereby calling for a deeper understanding of the processes occurring at the interface between nanomaterials and biological systems. The nanoparticle-based therapeutic agents are prone to interaction with biomolecules whenever circulating in the bloodstream. The proteins adsorption onto the surface of nanoparticles and the self-development of protein coronas has been evidenced to introduce several shortcomings towards the application of nanoparticles in medicine. The presence of protein coronas usually results in the inability of the cells to encounter naked particles [START_REF] Gaspar | Nanoparticles: Pushed off Target with Proteins[END_REF][START_REF] Müller | The Transferability from Animal Models to Humans: Challenges Regarding Aggregation and Protein Corona Formation of Nanoparticles[END_REF][START_REF] Ke | A Decade of the Protein Corona[END_REF] impacting on their targeting ability, toxicity, cellular uptake, circulation lifetime, signaling, therapeutic effects, intracellular transport and biodistribution. [START_REF] Obst | Protein Corona Formation on Colloidal Polymeric Nanoparticles and Polymeric Nanogels: Impact on Cellular Uptake, Toxicity, Immunogenicity, and Drug Release Properties[END_REF][START_REF] Caracciolo | Biological Identity of Nanoparticles In Vivo : Clinical Implications of the Protein Corona[END_REF] The emergence of protein coronas onto the surface of colloidal materials is influenced by their structural features (morphology, size, shape, chemical nature, surface charge) [START_REF] Lynch | The Nanoparticle-Protein Complex as a Biological Entity; a Complex Fluids and Surface Science Challenge for the 21st Century[END_REF] and by the protein-particle chemical compatibility. [START_REF] Lundqvist | The Nanoparticle Protein Corona Formed in Human Blood or Human Blood Fractions[END_REF] The protein adsorption is usually governed by intermolecular forces including van der Waals interactions, hydrogen bonding, electrostatic and hydrophobic effects. [START_REF] Mahmoudi | Emerging Understanding of the Protein Corona at the Nano-Bio Interfaces[END_REF][START_REF] Nel | Understanding Biophysicochemical Interactions at the Nano-Bio Interface[END_REF] The strength of the interactions determines the formation, chemical nature, binding affinity and binding ratio of adsorbed biomacromolecules.

Although the emergence of protein coronas may bring benefits to dispersed nanomaterials (particularly with regard to cytotoxicity), [START_REF] Wang | Exploring 'New' Bioactivities of Polymers at the Nano-Bio Interface[END_REF][START_REF] Monopoli | Biomolecular Coronas Provide the Biological Identity of Nanosized Materials[END_REF] for most of the applications they are considered artifacts that have to be at least substantially reduced. [START_REF] Nel | Understanding Biophysicochemical Interactions at the Nano-Bio Interface[END_REF][START_REF] Pearson | Biomolecular Corona on Nanoparticles: A Survey of Recent Literature and Its Implications in Targeted Drug Delivery[END_REF][START_REF] Docter | The Nanoparticle Biomolecule Corona: Lessons Learned -Challenge Accepted?[END_REF][START_REF] Caracciolo | Liposome-Protein Corona in a Physiological Environment: Challenges and Opportunities for Targeted Delivery of Nanomedicines[END_REF] Typically, the presence of highly hydrophilic polymer chains at the interfaces remarkably reduces interparticle attractive forces thereby reducing protein adsorption.

The hydrophilic polymer poly(ethylene glycol) -PEG -is typically used for such purpose. Nevertheless, it has been highlighted relevant shortcomings in this regard: i) PEG can trigger the activation of the complement system [START_REF] Hamad | Ethylene Glycol)s Generate Complement Activation Products in Human Serum through Increased Alternative Pathway Turnover and a MASP-2-Dependent Process[END_REF] , iii) its antibodies (anti-PEG) are known to be produced by the immune system thereby reducing the therapeutic efficacy of PEG-based nanomedicines [START_REF] Zhang | Anti-PEG Antibodies in the Clinic: Current Issues and beyond PEGylation[END_REF][START_REF] Kierstead | The Effect of Polymer Backbone Chemistry on the Induction of the Accelerated Blood Clearance in Polymer Modified Liposomes[END_REF] , iv) PEGylation cannot completely prevent protein adsorption. [START_REF] Schöttler | Protein Adsorption Is Required for Stealth Effect of Poly(Ethylene Glycol)-and Poly(Phosphoester)-Coated Nanocarriers[END_REF] Accordingly, there is a growing body of novel polymers being investigated as suitable replacements to PEGylation.

In this framework, linear polyglycidol (PGL) is herein investigated as one of the alternatives. The main chain of polyglycidol is similar to PEG, except that each repeating unit of polyglycidol contains an additional hydroxymethyl group. Looking at the synthetic point of view, well-defined PGL are often prepared via anionic ring opening polymerization (AROP) of glycidyl ethers combined with a deprotection step. [START_REF] Faye | β-Cyclodextrin-Based Star Amphiphilic Copolymers: Synthesis, Characterization, and Evaluation as Artificial Channels[END_REF] The common used glycidyl ethers are tert-butylglycidyl ether (tBuGE) and ethoxyethyl glycidyl ether (EEGE) due to the facility of deprotection. [START_REF] Gervais | Synthesis of Linear High Molar Mass Glycidol-Based Polymers by Monomer-Activated Anionic Polymerization[END_REF][START_REF] Misaka | Synthesis of End-Functionalized Polyethers by Phosphazene Base-Catalyzed Ring-Opening Polymerization of 1,2-Butylene Oxide and Glycidyl Ether[END_REF][START_REF] Backes | Synthesis and Thermal Properties of Well-Defined Amphiphilic Block Copolymers Based on Polyglycidol[END_REF] Besides, there have been recently a lot of efforts towards finding efficient organo-catalysts for controlled polymerization of glycidyl ethers and other 1,2-subsituted epoxides by AROP. Amongst than, the tBuP4 is so far the preferred organic super base with low nucleophilicity. The tBuP4-catalysed AROP was evidenced to provide controlled and metal-free production of narrowly-distributed polyepoxides under mild conditions. [START_REF] Dentzer | Phosphazene-Promoted Metal-Free Ring-Opening Polymerization of 1,2-Epoxybutane Initiated by Secondary Amides[END_REF] The polymerization of various epoxide monomers including 1,2-alkene epoxides such as butylene oxide (BO), propylene oxide (PO) [START_REF] Misaka | Synthesis of End-Functionalized Polyethers by Phosphazene Base-Catalyzed Ring-Opening Polymerization of 1,2-Butylene Oxide and Glycidyl Ether[END_REF][START_REF] Isono | Synthesis of Star-and Figure-Eight-Shaped Polyethers by t -Bu-P 4 -Catalyzed Ring-Opening Polymerization of Butylene Oxide[END_REF] and glycidyl ethers such as allyl glycidyl ether (AGE) [START_REF] Kwon | Well-Defined Functional Linear Aliphatic Diblock Copolyethers: A Versatile Linear Aliphatic Polyether Platform for Selective Functionalizations and Various Nanostructures[END_REF] and EEGE [START_REF] Misaka | Synthesis of End-Functionalized Polyethers by Phosphazene Base-Catalyzed Ring-Opening Polymerization of 1,2-Butylene Oxide and Glycidyl Ether[END_REF][START_REF] Kwon | Well-Defined Functional Linear Aliphatic Diblock Copolyethers: A Versatile Linear Aliphatic Polyether Platform for Selective Functionalizations and Various Nanostructures[END_REF] has been successfully achieved by using the such synthetic strategy.

Additionally, tBuP4-catalyzed AROP shows living character allowing the sequential block copolymerization of two epoxide monomers such as PO/EEGE [START_REF] Dimitrov | Synthesis and Associating Properties of Poly(Ethoxyethyl Glycidyl Ether)/Poly(Propylene Oxide) Triblock Copolymers[END_REF] and decyl glycidyl ether/2-(2-(2-methoxyethoxy)ethoxy)ethyl glycidyl ether [START_REF] Isono | Synthesis of Linear, Cyclic, Figure-Eight-Shaped, and Tadpole-Shaped Amphiphilic Block Copolyethers via t -Bu-P 4 -Catalyzed Ring-Opening Polymerization of Hydrophilic and Hydrophobic Glycidyl Ethers[END_REF][START_REF] Satoh | Synthesis of Well-Defined Amphiphilic Star-Block and Miktoarm Star Copolyethers via t -Bu-P 4 -Catalyzed Ring-Opening Polymerization of Glycidyl Ethers[END_REF] for instance.

Taking into account the potential biomedical applications, PGL and PEO are reported to be similar regarding biocompatibility [START_REF] Gosecki | Its Derivatives, and Polyglycidol-Containing Copolymers-Synthesis and Medical Applications[END_REF] , and PGL-grafted surfaces were reported to resist protein adsorption [START_REF] Otulakowski | Micellisation of Polystyrene-b-Polyglycidol Copolymers in Water Solution[END_REF][START_REF] Dworak | Poly(2-Substituted-2-Oxazoline) Surfaces for Dermal Fibroblasts Adhesion and Detachment[END_REF] and biofilm growth. [START_REF] Lockhart | Glycidol) Coating on Ultrahigh Molecular Weight Polyethylene for Reduced Biofilm Growth[END_REF] Nevertheless, the protein adsorption onto PGL-stabilized nanoparticles (spherical surfaces) was never explored to the best of our knowledge. Therefore, we consider of due relevance the investigations regarding protein adsorption onto PGL-stabilized nanoparticles since such polymers can be thought as a promising solution to overcome the limitations of other polymers (including PEG).

Accordingly, these investigations were focused on amphiphilic copolymers having a polyglycidol (PGL) hydrophilic shell linked to poly(butylene oxide) (PBO) hydrophobic segments, either diblock, or triblock copolymers. We herein explored the syntheses and characterizations of such novel amphiphilic copolymers, and their selfassembly in aqueous solution. Subsequently, the protein-repelling character of the different morphologies produced was probed by using model proteins highly dissimilar with regard to size, molecular weight and isoelectric point.

MATERIALS AND METHODS

Syntheses and Characterization of the Diblock and Triblock Copolymers

Materials

The monomers butylene oxide (BO, 99%, Aldrich), ethoxyethyl glycidyl ether (EEGE, prepared from glycidol (96%, Aldrich) and ethyl vinyl ether (99%, Aldrich) were dried twice over calcium hydride (CaH 2 , ca. 93%, ACROS) via vacuum distillation and further stored in glovebox before use. Benzyl alcohol (BA, anhydrous, 99.8%, Sigma-Aldrich) and tBuP4 (~ 0.8 mol.L -1 in n-hexane, Sigma-Aldrich) were also stored in glovebox before use. CaH 2 , neutral alumina (Sigma-Aldrich), 1,4-benzene dimethanol (BDM, 99%, Aldrich), dichloromethane (DCM, VWR Chemical) and methanol (MeOH, VWR chemical) were used as received. Anhydrous toluene and THF for polymerizations were purified using an MBRAUN solvent purification system (MB SPS COMPACT) consisting of an activated alumina column and an activated copper catalyst column. Polymerizations were carried out in an MBRAUN stainless steel glovebox equipped with a gas purification system (molecular sieves and copper catalyst) and dry argon atmosphere (H 2 O < 0.5 ppm). The moisture content in the glovebox was monitored by an MB-MO-SE 1 moisture probe.

Syntheses of Diblock copolymers (PGL n -b-PBO m )

The diblock copolymers (PGL n -b-PBO m ) were prepared via tBuP4-catalyzed sequential AROP followed with an acidic deprotection. The typical procedure is described for PGL 50 -b-PBO 13 : tBuP4 (0.8 mol.L -1 in n-hexane, 125 μL, 0.1 L, 0.1 mmol, 0.5 equiv.) was added into a solution of BA (20.7 μL, 0.1 L, 0.2 mmol, 1 equiv.) and BO (0.1872 g, 2.6 mmol, 13 equiv.) in toluene (0.5 mL) to trigger the homopolymerization of BO. After stirring at 25 °C for 24 h, an aliquot of the reaction mixture was withdrawn and applied to 1 H NMR characterization to check the conversion of BO. After confirming the consumption of BO, the second monomer, EEGE (1.4016 g, 9.6 mmol, 47 equiv.) was added to start the copolymerization, and the reaction was carried out for 24h at 25 copolymers with different values of m and n were prepared following the same procedure but varying the BO/EEGE/BA feeding ratio.

Syntheses of Triblock Copolymers

(PGL n -b-PBO 2m -b-PGL n )
Similar synthetic procedure was applied to prepare the triblock copolymers. The typical procedure is described for PGL

Manufacturing of the Polymer Colloids

The manufacturing of the polymersomes and block copolymer micelles was conducted by thin-film hydration. The PGL n -b-PBO 2m -b-PGL n and PGL n -b-PBO m block copolymers (5.0 mg) were dissolved in glass vials using a good organic solvent for both blocks (ethanol or methanol depending on the PBO:PGL weight ratio). The organic solvent was left to evaporate for 72h resulting in polymer films sited at the bottom of glass vials. The polymer films were subsequently hydrated by the addition of 5.0 mL of

  6 2 q T k R B H 
distilled water thereby resulting in self-assembled structures. The solutions were subsequently centrifuged and-or filtered to remove dust and/or ill-aggregates.

Behavior of the Polymeric Assemblies in Protein Environments

The behavior of the polymeric assemblies in protein environments was evaluated firstly by monitoring changes in the size of the assemblies as a function of protein concentration by dynamic light scattering. The thermodynamic parameters related to the protein adsorption phenomena were determined through isothermal titration calorimetry (ITC). Furthermore, fluorescence and circular dichroism spectroscopy data were acquired in order to identify the protein quenching mechanism and conformational changes. All the details regarding the employed methods of characterization are detailed below.

Methods of Characterization

Dynamic Light Scattering: The dynamic light scattering (DLS) measurements were acquired using a ZetaSizer Nano ZS90. The averaged intensity autocorrelation functions were evaluated using non-negative least squares (NNLS) analysis implemented in the Zetasizer software therefore resulting in distribution of sizes. The hydrodynamic radius (R H ) of the nanoparticles was determined using the well-known

Stokes-Einstein relation with

D =  -1 q -2 : (1)
where k B is the Boltzmann constant, T is the absolute temperature, q is the scattering vector,  is the viscosity of the solvent and  is the mean relaxation time related to the

.... 2 + - ln = ) ( ln t μ t Γ C t g 4 2 2 2 4π λ N dc dn n = K A       2 ( ) 1 2 θwNPs w NPs Kc A c R M   ( . . ) PBO PBO PGL PGL water dn = n w n w n dc  
diffusion of the nanoparticles. The autocorrelation functions were also analyzed using the Cumulant method. [START_REF] Štěpánek | Dynamic Light Scattering: The Method and Some Applications[END_REF] (2

)
where C is the amplitude,  is the relaxation frequency (  and the parameter µ 2 is known as the second-order cumulant. The polydispersity index of the samples (PDI) was then computed as PDI = µ 2 / 2 .

Static Light Scattering:

The SLS data were acquired to determine the molecular weight (M w(NPs) ) and number of aggregation (N agg ) of the assemblies. The light scattering intensity was monitored at scattering angle  = 90 as a function of polymer concentration and the values of M w(NPs) were estimated by using the Debye plot:

(3) the concentration c is given in mg.mL -1 and K is the optical constant expressed by: (4) R θ (Rayleigh ratio) is the normalized scattered intensity (toluene was used as standard solvent), n is the refractive index of the solvent, N A is the Avogadro's number and dn/dc is the refractive index increment determined based on the refractive index increments of PGL and PBO blocks as well as their weight fraction (w) using Equation 5.

(5)

   3 2 f(ka) U E  ( ) ( ) w NPs agg w unimer M N M 
The refractive index of PBO (n PBO ) and PGL (n PGL ) are respectively 1.454 and 1.474 and the weight fractions are given in Table 1.

Taking into account the Equation 3, by plotting Kc/R θ vs. c at a given angle, the molecular weight is extracted from the reinverse of the intercept. The number of aggregation is determined as: (6) Electrophoretic Light Scattering: The ELS measurements (acquired using a Zetasizer Nano-ZS instrument) were used to determine the average zeta potential () of the nanoparticles. The values of electrophoretic mobility (U E ) were converted to values of -potential (mV) through the Henry's equation: (7) where is the dielectric constant of the medium and  is the viscosity. Furthermore, f(ka) is the Henry's function, which was calculated through the Smoluchowski approximation f(ka) = 1.5.

Transmission Electron Microscopy (TEM):

The morphology of self-assembled nanoparticles was also probed by transmission electron microscopy (TEM) using a Tecnai G2 Spirit Twin 120kV (FEI, Czech Republic) microscope. Two-μL, 0.1 L of the aqueous solutions were dropped onto a copper TEM microscopic grid (400 mesh) coated with thin, electron-transparent carbon film. The excess of solution was removed by touching the bottom of the grid with filtering paper. This fast removal of the solution was performed after 5 min of sedimentation in order to minimize oversaturation during the drying process. The assemblies were negatively stained with uranyl acetate (2 µL of 2 wt.% solution dropped onto not completely dried nanoparticles and removed after 30 s in the same way as described above). The samples were left to dry completely at environment temperature and then observed by TEM. Under these conditions, the micrographs displayed negatively stained background with bright nanoparticles.

Small-Angle X-ray Scattering (SAXS):

Isothermal Titration Calorimetry (ITC): The thermodynamic parameters of protein-nanoparticle interaction were determined by isothermal titration calorimetry (ITC) performed at 25 o C using a MicroCal iTC200 calorimeter. The reference cell was filled with water, the sample cell was filled with 2.0 mg.mL -1 of block copolymer solution and the syringe was filled with 1.6 mg.mL -1 of protein solutions. The titrations were performed by injecting the protein solutions in the sample cell every predetermined interval of time. The ITC raw data were integrated from a baseline to give the heat per injection as a function of the protein-to-nanoparticle weight ratio. The heat of dilution of protein solution was negligible, as determined in a blank experiment where protein solutions were injected into the sample cell containing only PBS buffer.

Fluorescence Spectroscopy: The fluorescence spectra were recorded on a Varian Cary Eclipse fluorescence spectrophotometer, in the range 300-450 nm after progressive additions of nanoparticles into the protein solutions, with excitation wavelength of 280 nm and using a quartz cell with 10.0 mm optical path at 296 K.

Circular Dichroism Spectroscopy ( CD ) : The CD data were acquired using a Jasco J-815 circular dichroism spectrometer and quartz cuvette with 1.0 mm optical path. The spectra was collected in the range 200-260 nm at room temperature and corrected by subtracting the solvent background.

RESULTS AND DISCUSSIONS

Syntheses and Characterization of the Block Copolymers

Taking into account the molar mass of the repeating units (74.0 and 72.0 g.mol -1 respectively for PGL and PBO) it has been designed the syntheses of block copolymers with distinct hydrophilic-to-hydrophobic weight ratios towards the manufacturing of different morphologies. The chemical structure of the amphiphilic block copolymers are given in Scheme 1 along to the synthetic procedures used in their manufacturing. ) is close to the theoretical value (M n(theo) ) calculated from the feeding ratio of BO/EEGE/BA (Row 2, Table 1). In addition, the GPC traces of PBO m (Figure S2a) and PEEGE n -b-PBO m (Figure S2b) were both unimodal with low dispersity. The results altogether demonstrate that the tBuP4-catalyzed sequential AROP is well controlled. To get PGL blocks, the acetal groups in EEGE units were cleaved under acidic conditions. The deprotection efficiency was confirmed by 1 H NMR (Figure S1c), as the characteristic peak belonging to methine protons in EEGE units (4.69 ppm) disappeared. In addition, the DPs of BO and EEGE in PEEGE n -b-PBO m were consistent with the DPs of BO and glycidol (GL) in PGL n -b-PBO m indicating the acidic conditions used are not harmful to the polyether backbone. Additionally, the GPC trace (Figure S2c) is unimodal and narrowly distributed with a shift in elution volume to lower molar mass as a results of the deprotection. Following the same procedure, although starting with different feeding ratios, the other PGL n -b-PBO m block copolymers could be synthesized (Row 1, Table 1). Altogether, the characterization results demonstrated the synthetic procedure leads to controlled diblock copolymer architectures.

Table 1. Characteristics of the block copolymers according to the weight fraction of PGL block (w PGL ).

Entry*

[BO]0/[EEGE]0/[I]0 [START_REF] Gaspar | Nanoparticles: Pushed off Target with Proteins[END_REF] Theoretical results [START_REF] Müller | The Transferability from Animal Models to Humans: Challenges Regarding Aggregation and Protein Corona Formation of Nanoparticles[END_REF] 1 H NMR results [START_REF] Ke | A Decade of the Protein Corona[END_REF] SEC results [START_REF] Obst | Protein Corona Formation on Colloidal Polymeric Nanoparticles and Polymeric Nanogels: Impact on Cellular Uptake, Toxicity, Immunogenicity, and Drug Release Properties[END_REF] Mntheo.

(kg/mol) ): [START_REF] Gaspar | Nanoparticles: Pushed off Target with Proteins[END_REF] Feeding ratio of monomers to initiator (BA for diblock copolymers or BDM for triblock copolymers); [START_REF] Müller | The Transferability from Animal Models to Humans: Challenges Regarding Aggregation and Protein Corona Formation of Nanoparticles[END_REF] Molecular weight (M n(theo) ) and weight fraction of PGL block(s) (w PGL,theo. ) calculated based on feeding ratio and monomer conversions (quantitative in all cases for both monomers); [START_REF] Ke | A Decade of the Protein Corona[END_REF]

Manufacturing and Characterization of the Block Copolymer Assemblies

The manufacturing and characterization of block copolymer self-assemblies was further investigated. This has been performed by using the resources of static dynamic light scattering (SDLS), electrophoretic light scattering (ELS), transmission electron microcopy (TEM) and small angle x-ray scattering (SAXS). The autocorrelation functions acquired by DLS (representative examples are given in Figure 1a) were mathematically treated by using the NNLS analysis resulting in single populations of relaxation times (representative examples are given in Figure 2b) meaning, accordingly, single populations of scattering particles. determine M w(NPs) and N agg (M w(NPs) /M w(polymer) ). The values of -potential were taken from ELS measurements. The whole set of structural parameters determined (or further calculated) are summarized in Table 2 and subsequently discussed. The values of R H are larger for polymersomes compared to star-like micelles.

This reflects the higher values of N agg of the hollow structures, but mainly the presence of different morphologies depending on the hydrophobic-to-hydrophilic weight ratios as confirmed by transmission electron microscopy and small angle X-ray scattering measurements. The whole set of TEM images are portrayed in Figure 2. The image reported in Figure 2 (a-c) highlights predominantly the presence of small micellar structures (~ 20-40 nm) in agreement with the size distributions obtained by dynamic light scattering as well as with the well-defined SAXS pattern given in Figure 1d (representatively for PGL 50 -b-PBO 13 ). The profile could be fitted by using the straightforward core-shell spherical form factor with R total = 6 nm. The micelles produced from PGL 11 -b-PBO 22 -b-PGL 11 are slightly larger reflecting the higher weight ratio of the hydrophobic PBO in the polymer chains thereby resulting in higher number of aggregation (Table 2). On the other hand, the images reported in Figure 2 (d-f inconclusive to some extent regarding the main morphology. Although we cannot identify the mixture of morphologies by using light scattering measurements, there could be identified spherical particles (probably polymeric vesicles) along to cylindrical structures by microscopy. [START_REF] Meins | Recent Trends in the Tuning of Polymersomes' Membrane Properties[END_REF] The values of -potential are highly negative regardless the morphology. Such feature, nevertheless, provided electrostatic stability to the assemblies as they are stable for months when stored at 4 o C. Although we might expect that nanoparticles made from neutral organic polymer would not have surface charge ( ~ 0 mV) all the selfassembled structures had a negative zeta potential. This observation suggests that anions from the aqueous phase preferentially accumulate at the mobile Stern layer. The negative -potentials appear to arise from the preferential accumulation of OH -ions at the interface between water and the organic shell. This indeed seems to be a more general phenomenon. We have particularly measured negative -potentials for many different non-ionic particles. [START_REF] De Castro | Understanding the Structural Parameters of Biocompatible Nanoparticles Dictating Protein Fouling[END_REF][START_REF] De Castro | Influence of Structural Features on the Cellular Uptake Behavior of Non-Targeted Polyester-Based Nanocarriers[END_REF][START_REF] Ribeiro | Biodegradable Nanoparticles as Nanomedicines: Are Drug-Loading Content and Release Mechanism Dictated by Particle Density?[END_REF]

Behavior in Protein Environments

In order to evaluate the behavior of the polymeric assemblies in protein environments, we previously estimated the molar concentration of each protein required to cover the produced micelles and polymersomes. The details and computations are given in the Supporting Information File. We did not proceed with such investigations for PGL 15 -b-PBO 44 -b-PGL 15 because of the existence of mixed morphologies (most probably vesicles and cylinders) thereby making unreliable such protocol for determining the required protein concentrations.

The values of N Protein reported in Table S1 are the molar ratios [protein]/[NPs] required for fully covering each soft spherical nanoparticle. Based on these numbers, we monitored R H as a function of protein concentration at least two times higher than N Protein (2N Protein ).

Indeed, the blood environment consists of over 3000 different proteins that are capable of interacting with nanomaterials producing the so-called protein-corona.

Nevertheless, there are major proteins that were already identified on an assortment of polymeric nanoparticles of different size, chemical make-up and surface properties.

Therefore, BSA, IgG and lysozyme were chosen as model proteins due to the pronounced differences in size, molecular weight and isoelectric point as given in Table S2.

The for IgG evidence that the size of the micellar assemblies remain essentially the same at the presence of the protein. The addition of IgG also induces the appearance of the light scattering contribution for free IgG (located at the left-hand side of the size distributions) thereby suggesting that the protein is not being adsorbed onto the surface of the polymeric assemblies. On the other hand, the data portrayed for lysozyme environment is remarkably different where one can notice the shift of the main particle population towards the right-hand side at the presence of the protein. This is a clear indication that protein adsorption is taking place. This methodological approach has been used to probe the adsorption of the model proteins at the surface of all produced self-assembled nanostructures and the quantitative results are all gathered in Table 3.

This table reports the size (R H ) variation for vesicles and micelles before and after the contact with the different model protein environments. The values of R H refer to R Hprotein-environment -R Hprotein-free and they were monitored in the presence of protein concentration two times higher than the estimated amount to cover the whole polymer assemblies (2N Protein ). We understand that a robust indication of protein adsorption is evidenced when the monitored size of the assemblies increases at least the dimension (D H = 2R H ) of the model protein. The hydrodynamic diameters of the proteins are given in Table S2 (Supporting Information File).

Table 3: Variation of the size of the nanostructures (R H = R Hprotein-environrment -R Hprotein-free ) considering the dimensions before and after the contact of the assemblies with the model proteins at 2N protein .

The Table 3 evidences that adsorption is protein-dependent with indications that electrostatic forces influence the phenomenon. Regardless the self-assembled structure, no evidences of BSA adsorption was observed. This is attributed to the highly negatively surfaces of the assemblies (~ -20 mV) and the isoelectric point of the BSA (pI ~ 4.9) which confers to the biomacromolecules residual negative charges at pH 7.

-2 ± 3 9 ± 2 2 ± 2 PGL 22 -b-PBO 13 -b-PGL 22 -1 ± 2 0 ± 1 3 ± 2 PGL 50 -b-PBO 13 0 ± 1 2 ± 1 1 ± 1                        t t t A t t t A cell t NM X M NK NM X M NK ΔHVHV NM = Q 4 1 1 1 1 2 2 A K RT = G ln   T G H = S    
then presumably enabling electrostatic attraction to the polymeric assemblies (that are negatively charged).

The thermodynamics of protein adsorption was further investigated by using the resources of isothermal titration calorimetry in order to provide a clearer scenario regarding the main forces driven the protein adsorption events. The Figure 5 portrays raw ITC data (left) and the values of energy per weight of injected protein as a function of protein-to-nanoparticle ratio (right) for the pairs where protein adsorption has been evidenced according to Table 3. Indeed, the raw ITC data already confirm protein adsorption since the process is accompanied by energy transfer. The resulting data set was fitted using the straightforward one set of binding site model. The molar enthalpy (H), stoichiometry of binding (N) that is, the number of protein molecules per nanoparticle and the equilibrium association constant K A for the adsorption of lysozyme and IgG onto the block copolymer nanoparticles was obtained whenever possible by fitting the experimental data with the following equation: (7) wherein Q is heat content after each injection, V cell is the cell volume and M t and X t are respectively the molar concentration of nanoparticles and protein (V cell, M t , X t are known parameters, Q is a measurable parameter and N, K A and H are fitting parameters).

Furthermore, the molar free energy change (G) and the molar entropy change (S) of the binding events were determined by using the following fundamental thermodynamic equations:

(8) (9) The thermodynamic parameters obtained from the fitting approaches are provided in Table 4. hydrogen bonding are usually the main operating forces when ΔH < 0 and ΔS < 0. Thus, the exothermic processes observed in the profiles suggest that the main force driven protein adsorption are note related to hydrophobic effects and they are then related to van der Waals, hydrogen bonding and-or electrostatic forces.

Indeed, the determined value of S suggests the electrostatic interactions dominate the phenomenon of lysozyme adsorption. On the other hand, the IgG adsorption result in very small (although negative) value of S accordingly suggesting van der Waals and hydrogen bonding as the main operating intermolecular interactions.

The surfaces charge of the assemblies is negative, and this supports the importance of electrostatic interaction regarding lysozyme adsorption. The nearly neutral surface charge of IgG at pH 7.4 (pI ~ 7.0) is probably the responsible, on the other hand, by the presence of van der Waals and hydrogen bonding as the main driven forces in such pair.

The binding affinity (K a ) is different and the values evidence that lysozyme binds stronger to the nanoparticles as compared to IgG. The data reported in Table 4 also highlight stronger lysozyme binding on the surface of the PGL 10 -b-PBO 88 -b-PGL 10 since the K a value is higher. Additionally, the lysozyme binding seems to be influenced by the PGL chain length since the binding affinity is weaker for assemblies with longer PGL-stabilizing shells.

Complementary, the ITC measurements were also used to confirm the nonexistence of protein adsorption events. In the supporting information file we provide ITC raw data for the titration of assemblies stabilized by short PGL stabilizing shells . Indeed, the same profiles were monitored for the pairs where protein adsorption has not been suggested by DLS (this is not shown here for brevity). The fluorescence spectroscopy data were used to determine the Stern-Volmer quenching constant (K SV )

for the nanoparticle-protein pairs where fluorescence quenching was evidenced.

(

) 10
being F 0 and F the values of fluorescence intensity in the absence and presence of nanoparticles. These results are portrayed in Figure 6 (c) and Table 5.

  °Cand further quenched by DI water. The reaction mixture was then concentrated giving a brown viscous liquid. The crude product was diluted with DCM and eluted through a short neutral alumina column to remove tBuP4. The eluent was concentrated giving the diblock copolymer PEEGE 47 -b-PBO 13 as a colourless viscous liquid. Then, the hydroxyls were deprotected in acidic conditions: PEEGE 47 -b-PBO 13 (Mn theo 4.5 kg/mol, 0.45 g, 0.1 mmol) was dissolved in MeOH (7.1 mL) and an amount of HCl solution (2 mol.L -1 in MeOH, 2.4 mL, 4.8 mmol) was added into the polymer solution. The mixture was stirred at room temperature for 4h before being neutralized by NaHCO 3 . The solid was removed by filtration and the filtrate was concentrated to give the deprotected diblock copolymer (PGL 50 -b-PBO 13 ) as a white waxy solid. The PGL n -b-PBO m block

1 .

 1 Scheme 1. Synthetic procedures of the amphiphilic block copolymers PGL n -b-PBO m (a) and PGL n -b-PBO 2m -b-PGL n (b). The hydrophobic PBO segment is given in blue.

  Degrees of polymerization of BO (DP BO ) and glycidol (DP GL ), molecular weight (M n(NMR) ) and weight fraction of PGL block(s) (w PGL,NMR ) calculated based on integral ratios in 1 H NMR spectrum; (4) Number-average molecular weight (M n(SEC) ) and polydispersity index (Ð) determined by SEC in DMF with PMMA as standard using RI detector. Similarly, the characterization of the intermediates and products in triblock copolymerization, including PBO 2m (homopolymerization mixture in fact), PEEGE n -b-PBO 2m -b-PEEGE n and PGL n -b-PBO 2m -b-PGL n by 1 H NMR (Figure S3, Supporting Informations) and GPC (Figure S4, Supporting Informations) evidences that the synthetic procedure by tBuP4-catalyzed sequential AROP and deprotection in acidic conditions can successfully be used in the synthesis of triblock copolymers. The characterization results of the triblock copolymers are compiled inTable 1 (from Row 3 to 7).

Figure 1 .

 1 Figure 1. Scattering characterization of PGL 50 -b-PBO 13 and PGL 15 -b-PBO 40 selfassemblies according to the legend: (a) autocorrelation functions monitored at 90 o ; (b) respective distributions of R H , (c-d) respective SAXS patterns for PGL 50 -b-PBO 13 and PGL 15 -b-PBO 40 according to the legend.

Figure 2 .

 2 Figure 2. TEM images for PGL11-b-PBO22-b-PGL11 (a), PGL50-b-PBO13 (b), PGL22-b-PBO13b-PGL22 (c), PGL10-b-PBO88-b-PGL10 (d), PGL15-b-PBO40 (e) and PGL20-b-PBO84-b-PGL20 (f).

  ) taken from PGL 15 -b-PBO 40 , PGL 10 -b-PBO 88 -b-PGL 10 and PGL 20 -b-PBO 84 -b-PGL 20 assemblies clearly confirm mainly the presence of vesicular structures with sizes no longer compatible to micellar objects. This has been also confirmed by SAXS measurements (as portrayed representatively for PGL 15 -b-PBO 40 in Figure 2d). The profile is typical of vesicular structures and it was fitted by using the bilayer vesicle form factor with the thickness of the hydrophilic PGL shell in contact with water (t h = 3 nm) and the thickness of hydrophobic PBO layer (t t = 6.6 nm) as adjustable parameters. These results are in full agreement with the well-known phase diagram for the self-assembly of amphiphilic block copolymers in solution since polymersomes are expected at high weight ratios of the hydrophobic (PBO) segment. The TEM image taken from PGL 15 -b-PBO 44 -b-PGL 15 (Figure S5 -Supporting Information File) on the other hand, is

Figure 4 (

 4 left) portrays representatively the autocorrelation functions and respective distributions of R H for PGL 10 -b-PBO 88 -b-PGL 10 polymersomes in PBS medium and at lysozyme environment ([lysozyme] = 2N lysozyme ). One easily sees that the mean decay of the autocorrelation function is shifted towards the right-hand side at lysozyme environment suggesting the protein adsorption onto the surface of PGL 10 -b-PBO 88 -b-PGL 10 assemblies. This is indeed confirmed by the distributions of R H .

Figure 4 .

 4 Figure 4. Dynamic light scattering data: (left) autocorrelation functions (top) and distributions of R H (bottom) for PGL 10 -b-PBO 88 -b-PGL 10 polymersomes in protein-free and at lysozyme environment (2N lysozyme ). (right) distributions of R H for PGL 11 -b-PBO 22 -

4 thereby

 4 enabling protein-nanoparticle electrostatic repulsions. Similarly, the IgG adsorption is avoided expect for the assembly with the shortest PGL stabilizing shell (PGL 10 -b-PBO 88 -b-PGL 10 ). Taking into account the isoelectric point of IgG (pI ~ 7.0), electrostatic repulsions may also contribute to the protein-repelling characteristic of the assemblies at least to some extent. Nevertheless, the particularly short PGL length in the PGL 10 -b-PBO 88 -b-PGL 10 assemblies is not enough to provide full protein repulsion. The lysozyme adsorption is observed in block copolymer assemblies made by PGL blocks smaller than DP PGL ~ 20. In such a case, size of the assemblies increases at the protein environment thus confirming protein adsorption. The lysozyme (D H = 4.2 nm; pI = 10.5) adsorption indeed suggest the role of electrostatic forces in the phenomena. The high isoelectric point of lysozyme confers to the protein residual positive charges at pH 7.4

Figure 5 .Table 4 . 11 PGL 15

 541115 Figure 5. (left) ITC raw data for the titration of lysozyme and IgG into PGL m -b-PBO n and PGL m -b-PBO n -b-PGL m block copolymer self-assemblies. (right) energy per weight of injected protein as a function of protein-to-nanoparticle weight ratio for the titration of self-assembled nanostructures according to the legend.

(

  PGL 10 -b-PBO 88 -b-PGL 10 , PGL 11 -b-PBO 22 -b-PGL 11 and PGL 15 -b-PBO 40 ) in the presence of different proteins (Figure S6). The results highlight negligible energy transfer and the overall set of data robustly confirm that the protein adsorption is polymer lengthdependent and that the smaller length of the PGL stabilizing shells (DP PGL ~ 20) is already sufficient to provide protein-repelling characteristic to the assemblies, regardless the morphology.Additionally, fluorescence spectroscopy data were acquired to evaluate possible protein conformational changes induced by the adsorption phenomena. The fluorescence spectroscopy is useful for probing the binding ability since the protein adsorption quenches the intrinsic fluorescence of different residues present in protein molecules such as tryptophan, tyrosine and phenylalanine.[START_REF] Hao | Fluorescence Quenching Study on the Interaction of Ferroferric Oxide Nanoparticles with Bovine Serum Albumin[END_REF] The Figure6(a) portrays the fluorescence spectra of lysozyme as a function of successive additions of PGL 10 -b-PBO 88 -b-PGL 10 . Similarly, the data for IgG as a function of successive additions of PGL 15 -b-PBO 40 is given in Figure 6 (b). In these representative pairs, in agreement with the previous light scattering and ITC data, the quenching process is clear only for the pair lysozyme-PGL 10 -b-PBO 88 -b-PGL 10 since the fluorescence emission of the protein is progressively quenched as a function of nanoparticle concentration. The fluorescence spectrum, on the other hand, remains the same when IgG is titrated with PGL 15 -b-PBO 40-

10 FFigure 6 .

 106 Figure 6. (left) Fluorescence spectra of lysozyme (a) as a function of successive additions of PGL 10 -b-PBO 88 -b-PGL 10 and IgG (b) as a function of successive additions of PGL 15 -b-PBO 40 . (right) Stern-Volmer (c) plots for the fluorescence quenching of lysozyme and IgG by different block copolymer assemblies, and circular dichroism spectra of lysozyme at the presence of PGL 10 -b-PBO 88 -b-PGL 10 assemblies at w/w ratio = 0.06 (d).

Table 5 .

 5 Stern-Volmer quenching constant determined for model proteins adsorbed onto the surface of different block copolymer assemblies as determined by fluorescence spectroscopy. Entry K SV Lysozyme PGL 10 -b-PBO 88 -b-PGL 10 1.07 x 10 9 PGL 15 -b-PBO 44 -bVolmer plot shows a good linear relationship as portrayed in Figure 6 (c). The data suggest the prevalence of one quenching mechanism (either static or dynamic quenching). The static quenching refers to the actual formation of groundstate complexes whereas the dynamic quenching is related to collisional encounters. The values of K SV are in the range ~ 10 7 -10 9 M -1 . The values of the quenching rate constant (K b ) were calculated as K b = K SV /  0 considering  0 ~ 10 -8 s -1 as an average value for fluorescence ? lifetime of proteins in the absence of quenchers. Accordingly, the values of K b are remarkably greater than the value for threshold diffusion collision rate constant of quenchers to proteins (2 × 10 10 M -1 s -1 ) thereby reflecting the involvement of static quenching by proper protein-nanoparticle interactions and ruling out collision effects since the linear profile suggest one dominating quenching mechanism. The absence of collision effects is indeed not surprising if one takes into account the low molar concentration of the nanoparticles. Overall, the fluorescence spectroscopy data agrees well with the DLS and ITC data confirming protein adsorption only at the surface of assemblies stabilized by short PGL. Complementary, circular dichroism spectroscopy data were acquired to probe conformational changes in the structure of the proteins induced by the adsorption phenomenon. The CD spectra are portrayed representatively in Figure 6 (d) for pure lysozyme and at the presence of PGL 10 -b-PBO 88 -b-PGL 10 polymeric assemblies. The lysozyme CD profile is characteristic of biomacromolecules with certain degree of -helix content as evidenced by the minima at 208 nm and 207 nm. One notices that the shapes and peak positions remained the same at the presence of the polymersomes highlighting that even when the adsorption takes place, it does cause major alterations in the conformation of the model proteins. This has also been evidence for other nanoparticle-protein pairs where the adsorption was confirmed by DLS, ITC and fluorescence spectroscopy.CONCLUSIONSWe investigated the protein-repelling features of star-like micelles and polymersomes produced from PGL m -b-PBOn n and PGL m -b-PBO n -b-PGL m block copolymers. The reported data evidences that protein adsorption is mainly influenced by the thickness of the stabilizing shells, regardless the morphology. The phenomenon is also influenced by the structural features of the proteins (essentially the isoelectric point). Whenever protein adsorption takes place, the phenomenon is exothermic and driven by electrostatic interaction (lysozyme) or hydrogen bonding and-or van der Walls interaction (IgG). The fluorescence spectroscopy measurements evidence that the binding is stronger on the surface of nanoparticles stabilized by shorter hydrophilic shells, although the phenomenon does not affect the secondary structure of the model proteins. Nevertheless, the most important finding of the current investigation is that the PGL hydrophilic polymer was evidenced to provide protein-repelling characteristic to polymeric assemblies even by using relatively short shells (DP PGL ~ 20) which is considerably shorter than the required of the well-known PEO -DP PEO (~ 100 units).

Table 2 .

 2 Scattering data for manufactured micelles and polymersomes as determined by scattering techniques.

	Copolymer	R H (nm) PDI M w(NPs) 10 7 g.mol -1	N agg	mV
			Polymersomes
	PGL 20 -b-PBO 84 -b-PGL 20	79.6	0.40	3.76	3723	-21.2
	PGL 10 -b-PBO 88 -b-PGL 10	85.9	0.28	15.10	19400	-10.3
	PGL 15 -b-PBO 40	67.2	0.26	4.89	12300	-24.3
	Vesicles,

Cylindrical and Star-Like Micelles PGL 15 -b-PBO 44 -b-PGL 15

		81.5	0.18	3.07	4040	-22.0
		Star-Like Micelles
	PGL 11 -b-PBO 22 -b-PGL 11	24.0	0.19	0.46	933	-19.0
	PGL 22 -b-PBO 13 -b-PGL 22	13.4	0.31	0.0668	155	-22.4
	PGL 50 -b-PBO 13	9.7	0.34	0.0362	77	-8.4

2 1

ACKNOWLEDGMENTS

These investigations were sponsored by FAPESP (Grant no. 2017/00459-4). F.C.G acknowledges the fellowship granted by FAPESP (Grant no. 2018/11038-2). H.D acknowledges the PhD scholarship offered by the China Scholarship Council (CSC).


Philippe Michel

Bertrand Wigniolle
email: wignioll@univ-paris1.fr.

Pareto-e¢ ciency and endogenous fertility: a simple model

Keywords: endogenous fertility, Pareto-e¢ ciency. JEL Classi…cation: D9, J13, and D61

This article is devoted to the study of the Pareto-e¢ ciency of the competitive equilibrium for a simple overlapping generations economy with endogenous fertility. For CES utility and production functions, it is proved that the economic properties of the economy are closely related to the two elasticities of substitution. First, the competitive equilibrium exists and is unique if the sum of the two elasticities is not smaller than one. Secondly, a set of parameters is provided such that the equilibrium is both in under-accumulation and ine¢ cient. Thirdly, a su¢ cient condition is proved that ensures that an equilibrium converging in under-accumulation is Pareto-e¢ cient: the sum of the two elasticities must not be greater than two.

Introduction

This work is devoted to the study of the Pareto-e¢ ciency of the competitive equilibrium for a simple overlapping generations economy with endogenous fertility.

Fertility is endogenous in our framework, and results from a choice made by each household. We assume that the number of children is an argument of the utility function of the parents and that each child entails a constant childrearing cost for his parents. Fertility behavior results from a rational choice, as consumption and saving choices. This framework is close to models used by [START_REF] Becker | An economic analysis of fertility[END_REF], [START_REF] Willis | A new approach to the economic theory of fertility behavior[END_REF], [START_REF] Razin | An intergenerational model of population growth[END_REF] and [START_REF] Eckstein | Endogenous fertility and optimal population size[END_REF]. It leads to economic properties, which are relevant when compared to empirical facts, as is shown in of Birdsall's survey (1988).

We consider a simple framework where all individuals of the same generation are identical. We will refer to the well-being of one generation to designate the utility of each member of this generation. A competitive equilibrium is said to be Pareto-e¢ cient if it is not possible to increase the well-being of one generation without decreasing the well-being of another.

The question of the Pareto-e¢ ciency of the competitive equilibrium is well-known when fertility is exogenous. Considering the standard overlapping generations model à la [START_REF] Diamond | National debt in a neoclassical growth model[END_REF], a su¢ cient condition ensuring the Pareto-e¢ ciency of the competitive equilibrium is that it converges towards a steady state such that the interest rate is greater than the growth rate. When there is no technical progress, the growth rate is equal to the growth rate of the population. When this su¢ cient condition holds, the competitive equilibrium is said to converge in under-accumulation. On the other hand, an equilibrium converging in over-accumulation (with a growth rate higher than the interest rate) is ine¢ cient. It is possible to increase the utility of one or many generations without a fall in the utility of other generations (De la [START_REF] De La Croix | A Theory of Economic Growth: Dynamics and Policy in Overlapping Generations[END_REF]).

These results have important consequences for economic policy. For example, in an economy converging in over-accumulation, it is possible to improve the welfare of all generations by inter-generational transfers from young to old people (a pay-as-you-go social security system, or the issuing of public debt, for instance).

What happens to these properties when fertility is endogenous? Before answering this question, the notion of Pareto-e¢ ciency needs a new de…nition able to deal with the endogenous fertility framework. Indeed, the basic notion is de…ned for a given set of agents, while endogenous fertility requires comparing allocations associated with di¤erent sets of agents. We use a simple criterion based on particular assumptions. First, we consider the fertility level as a continuous number. Secondly, we only focus on symmetric allocations for all agents belonging to the same generation, and we call the symmetric allocations satisfying the resource constraint of the economy RCallocations (for "Representative-Consumer" allocations). In this context, it is possible to refer to the utility of a generation as the utility level reached by each agent belonging to this generation. Thirdly, we introduce a concept called "Representative-Consumer"e¢ ciency (or RC-e¢ ciency), which refers to an RC-allocation, for which no other allocation exists that would lead to a higher level of utility for all generations with a strict improvement for at least one generation.

This article adds several results to a …rst one on the subject (Michel and Wigniolle (2006)). In this former article, we showed that over-accumulation remains ine¢ cient with endogenous fertility. We have also proved two main results. First, we exhibited an example in which an equilibrium in underaccumulation is not Pareto-e¢ cient. Under-accumulation is no longer a suf-…cient condition ensuring e¢ ciency. Secondly, we gave a general condition ensuring Pareto-e¢ ciency: the limit value of the wage does not make it possible to …nance a fertility level higher than the limit value of the interest factor.

In this article, we focus on a particular case: an economy with CES production and utility functions. In this framework, we are able to obtain additional results. We …rst give a condition that ensures the existence and uniqueness of the intertemporal equilibrium. This condition is that the sum of the two elasticities of substitution must not be smaller than one. Secondly, we provide a set of parameters such that the stationary state of the economy is both in under-accumulation and ine¢ cient. This result generalizes the example given in Michel and Wigniolle (2006), and provides a better understanding of the causes of ine¢ ciency. Thirdly, we prove a su¢ cient condition for the Pareto-e¢ ciency of an equilibrium converging in under-accumulation: the sum of the two elasticities of substitution must not be greater than two. This result shows that a trade-o¤ between the two substitution elasticities in production and utility exists, in order to ensure the e¢ ciency of a competitive equilibrium converging in under-accumulation.

The intuition behind this last result is that an allocation can Paretoimprove a competitive equilibrium converging in under-accumulation, if it leads, in the long run, to a fertility level higher than the interest factor. Our result gives two arguments capable of precluding this increase in the fertility level. The …rst argument is the complementarity of production factors. Indeed, the increase in fertility is linked to a fall in capital accumulation. If capital and labor are substitutable, this change is not too detrimental to production; if capital and labor are complementary, however, it can lead to a signi…cant drop in resources. The second argument is the complementarity of consumption and fertility in the utility function. If consumption and fertility levels are highly substitutable, an increase in fertility allows for a substantial reduction in consumption, which is compatible with a constant level of utility. When they are complementary, however, the admissible fall in consumption is low.

Our results can be compared with existing literature. Two recent studies are concerned with the de…nition of Pareto-e¢ ciency criteria in an endogenous fertility framework. Conde-Ruiz, Giménez and Perez-Nievas ( 2004) use a slightly more general framework, where technical progress can a¤ect both the production technology and the children cost function. They use the same criterion of e¢ ciency which they call "Millian e¢ ciency". They undertake a detailed analysis of e¢ ciency criteria and provide several conditions for Millian-e¢ ciency. With respect to their work, this article focuses on a less general framework, which allows us to provide a suf-…cient condition for the e¢ ciency of an equilibrium in under-accumulation that directly involves the parameters of the production and utility functions. Golosov, Jones and Tertilt (2004) provide a general analysis of Pareto-e¢ ciency criteria in a framework in which fertility level is a discrete variable and in the presence of intragenerational heterogeneity. This article appears as the more general and thorough on the question of Pareto-e¢ ciency. But its results cannot be applied to a large range of simple macroeconomic models with endogenous fertility, particularly to models in which fertility is a continuous variable.

Section Two presents the model. Section Three studies the intertemporal equilibrium. Section Four introduces the main de…nitions concerning Pareto-e¢ ciency. Section Five gives a su¢ cient condition ensuring the Pareto-e¢ ciency of a competitive equilibrium converging in under-accumulation. Section Six concludes.

The model 2.1 Consumers

We make use of an overlapping generations model. Each individual lives for two periods. He or she inelastically supplies one unit of labor during his/her youth and retires in old age. During their …rst period of life, agents work, consume, save and choose their number of children. Children entail a cost for their parents. In their second period of life, agents consume the proceeds of theirs savings.

At each date t; the new generation is populated with a number N t of identical agents. Each agent is endowed with an intertemporal CES utility function

U t = c 1 t + d 1 t+1 + m 1 t 1 ;
(1)

where ; the elasticity of substitution, is such that > 0 and 6 = 1: c t and d t+1 are respectively the consumptions of good during youth and old age, and m t is the number of children1 . For the limit case = 1; we consider the log-linear function

U t = 1 ln c t + 2 ln d t+1 + 3 ln m t ;
(2)

with 3 P i=1 i = 1:
The number of agents of generation t + 1 is equal to

N t+1 = m t N t : (3)
The …rst period budget constraint faced by a generation t agent is

c t + m t + s t = w t ; (4)
where w t is the wage earned during this period and s t the savings. Each child entails a constant educational cost . Savings are invested in capital and yield a gross return in t + 1; equal to R t+1 . The second period budget constraint is

d t+1 = R t+1 s t : (5)
From ( 4) and ( 5), the lifetime budget constraint can be stated as

c t + d t+1 R t+1 + m t = w t : (6)
Maximizing the utility functions (1 or 2) under the constraint (6) leads to the following demand functions

c t = w t 1 (R t+1 ); (7) m t = w t 3 (R t+1 ); (8) s t = d t =R t+1 = w t 2 (R t+1 ): (9)
When the utility function is given by: (1), the functions i (R) are

1 (R) = 1 1 + R 1 + 1 ;
(10)

2 (R) = R 1 1 + R 1 + 1 ;
(11)

3 (R) = 1 1 + R 1 + 1 : (12)
When the utility function is given by: (2), the functions i (R) are constant and equal to i : At date t = 0; the N 1 old agents consume the proceeds of their savings: d 0 = R 0 s 1 : N 1 ; s 1 and N 0 come from past choices and are given.

Firms

We assume that at each period t one competitive …rm exists, which produces Y t quantity of good, according to a CES production function with two inputs (capital: K t ; labor: L t )

F (K; L) = A K 1 + bL 1 1 ; (13)
with > 0 and 6 = 1: When the elasticity of substitution is equal to 1; the production function is Cobb-Douglas

F (K; L) = AK L 1 : (14)
The capital fully depreciates within the production period.

The pro…t maximization of the …rm gives

w t = F 0 L (K t ; L t ) w(K t =L t ); (15) R t = F 0 K (K t ; L t ) R(K t =L t ): (16)
When the production function is given by ( 13), w and R have the following expression

w(x) = Ab x 1 + b 1 1 ; (17) R(x) = A 1 + bx ( 1) 1 1 : (18)
When it is given by ( 14), we obtain

w(x) = A(1 )x ; (19) R(x) = A x 1 : (20)
3 The intertemporal equilibrium

Equilibrium characterization

The initial conditions are the following. At period t = 0; N 1 old agents and N 0 young agents are living. The old agents hold equal shares of the capital stock K 0 : This stock comes from their past savings behavior:

K 0 = N 1 s 1 .
The consumption of the old agents is

d 0 = R 0 K 0 =N 1 : (21)
Along an intertemporal equilibrium, all markets are balanced. The labor market is balanced at each period:

L t = N t ; with N t following the relation N t+1 = m t N t :
Prices result from the …rms'behavior at equilibrium

R t = R(k t ) and w t = w(k t ); with: k t K t =N t : (22)
The commodity market is balanced, or

F (K t ; N t ) = N t c t + N t 1 d t + K t+1 + N t m t : (23)
Taking consumers' budget constraints into account, this last relation is equivalent to

K t+1 = N t s t , m t k t+1 = s t : (24)
Finally, using optimal behaviors and equilibrium conditions, it is possible to give the general de…nition of the intertemporal equilibrium:

De…nition 1 Starting from the positive initial conditions N 1 ; N 0 ; K 0 and

d 0 = R 0 K 0 =N 1 ; an intertemporal equilibrium is a sequence (K t ; N t ; c t ; d t ; m t ) t 0 ; which satis…es c t + d t+1 R t+1 + m t = w t ; (25)
U ct = R t+1 U d t+1 = U mt ; (26)
N t+1 = m t N t ; (27) F (K t ; N t ) = N t c t + N t 1 d t + K t+1 + N t m t ;
(28) with R t = R(k t ) and w t = w(k t ); where:

k t = K t =N t : (29)
This general de…nition allows us to encompass the di¤erent choices of utility and production functions. When it is possible, it will generally be more simple to reason with general functions.

Existence and convergence of intertemporal equilibria

In this part, we study the existence of the intertemporal equilibrium and its convergence toward a steady state.

Using equations ( 8), ( 9), ( 22) and ( 24), an intertemporal equilibrium is characterized by a sequence (k t ) t 0 starting from a given initial value k 0 = K 0 =N 0 ; and following the equation

k t+1 = s t m t = 2 [R(k t+1 )] 3 [R(k t+1 )] : (30)
This equation only depends on k t+1 and not on k t : This property is obtained as the utility function is homothetic. As s t and m t are proportional to w t ; k t+1 no longer depends on w t : The two adjustments in savings and fertility lead to a constant ratio K t+1 =N t+1 in one period.

There is existence and uniqueness of the intertemporal equilibrium if this equation has a unique solution k t+1 = k: In this case, the economy reaches its stationary state k in one period. k is de…ned by

k 3 R(k) = 2 R(k) : (31)
If 8t 1; k t = k; we obtain that the variables R t = R(k t ); w t = w(k t ); c t ; d t and m t are also equal to constant values R; w; c; d; and m. As in Diamond's model with exogenous fertility, we say that the steady state is in under-accumulation if R > m; and in over-accumulation if R < m:

It remains to study the conditions ensuring a unique solution for equation (31). We successively consider the case of a log-linear utility and the case of a CES utility.

When the utility is log-linear (given by ( 2)), the economy converges to the steady state in one period

k = 2 3 : (32)
When a CES utility function is assumed (given by ( 1)), the intertemporal equilibrium exists and is unique if the expression kR(k) 1 increases from 0 to +1 when k goes from 0 to +1. Indeed, k is such that

k 1 = R(k) 1 ; (33) or kR(k) 1 = : (34)
For the CES production function (13), k exists and is unique when + 1: Indeed, it is the unique solution of the equation

k 1 + bk 1 1 1 = A 1 : (35)
For the Cobb-Douglas production function ( 14), there is no condition on for the existence and uniqueness of the intertemporal equilibrium and k is equal to

k = ( A) 1 1 1+(1 )( 1) : (36)
Finally, we can summarize these results by the following proposition:

Proposition 1 If + 0; the intertemporal equilibrium exists and is unique. The economy reaches its stationary state in one period.

E¢ cient allocations and endogenous fertility

In the [START_REF] Diamond | National debt in a neoclassical growth model[END_REF] model with exogenous fertility, the Pareto-optimality properties of the competitive equilibrium are well-known. An intertemporal equilibrium converging to a stationary state in under-accumulation is Paretooptimal. On the other hand, an intertemporal equilibrium converging to a stationary state in over-accumulation is not Pareto-optimal. De la Croix and [START_REF] De La Croix | A Theory of Economic Growth: Dynamics and Policy in Overlapping Generations[END_REF] provide a careful presentation of these results.

In this section, we show that the concept of Pareto-optimality needs a new de…nition in a framework of endogenous fertility. We …rst introduce the basic notion of feasible allocations. This allows us to introduce e¢ ciency.

Feasible allocations with representative consumers

We de…ne the notion of a feasible allocation with representative consumers as an intertemporal allocation that satis…es the resource constraints of the economy.

De…nition 2 (RC-allocation)2 A feasible allocation with representative consumers (or RC-allocation) is a sequence (K t ; N t ; c t ; d t ; m t ) t 0 of positive variables that satisfy 8t 0

F (K t ; N t ) = K t+1 + N t c t + N t 1 d t + N t+1 ; N t+1 = m t N t : (37)
For a competitive equilibrium, the initial conditions N 1 ; N 0 ; K 0 and d 0 = R 0 K 0 =N 1 are given. Following de…nition 2, an intertemporal equilibrium is a particular case of feasible allocation, which moreover satis…es equations ( 7), ( 8) and ( 9), with (22). From assumption 1, if K 0 ; N 0 and N 1 are positive, all the variables (K t ; N t ; c t ; d t ; m t ) t 0 will be positive.

Remark 1 To simplify the presentation, we will always consider the four initial quantities (N 1 ; N 0 ; K 0 ; d 0 ) as given. It will be hereafter implicit that all feasible allocations or competitive equilibria considered will start from these same initial conditions, even if they are not mentioned.

A new characterization of feasible RC-allocations will be useful in what follows. In equation ( 37), c t and m t are chosen by generation t agents when d t is determined by generation t 1 agents. By a simple change in variables, we can introduce a new de…nition of feasible allocations, in which all variables are decided by the same generation.

Lemma 1 (Characterizations of feasible RC-allocations)

1. Any feasible RC-allocation can be characterized as a sequence (y t ; N t ; c t ; d t ; m t ) t 0 of positive variables starting from the initial condition y 0 = F (K 0 =N 0 ; 1) N 1 d 0 =N 0 ; and satisfying 8t 0;

m t y t+1 = F (y t c t m t ; m t ) d t+1 ; N t+1 = m t N t ; y t > c t + m t : (38)
Proof: This result is immediately obtained by a simple change in variables in the resource constraint of the economy

y t = K t+1 =N t + c t + m t : (39)
Remark 2 Let K t ; N t ; c t ; d t ; m t t 0 be an intertemporal equilibrium. Along such a trajectory, the new variable y t is equal to the wage

y t = w t = F 0 L K t ; N t : (40)

E¢ ciency of RC-allocations

Pareto-e¢ ciency is usually de…ned for a given set of agents. The model with endogenous fertility introduces a new di¢ culty, because the set of agents is the result of individual behaviors. We only consider the case of identical agents within each generation. In order to compare RC-feasible allocations, we use the main de…nition introduced in Michel and Wigniolle (2006).

De…nition 3 (RC-dominance)3 Let (K i t ; N i t ; c i t ; d i t ; m i t ) t 0 for i = 1; 2 be two feasible RC-allocations. Allocation 1 is said to RC-dominate allocation 2 if it leads to a higher level of utility for all generations, with a strict improvement for (at least) one generation. Formally,

8t 0; U (c 1 t ; d 1 t+1 ; m 1 t ) U (c 2 t ; d 2 t+1 ; m 2 t ); (41) 9t 0 0; such that: U (c 1 t 0 ; d 1 t 0 +1 ; m 1 t 0 ) > U (c 2 t 0 ; d 2 t 0 +1 ; m 2 t 0 ): (42)
From the notion of RC-dominance, a de…nition of e¢ ciency follows:

De…nition 4 An RC-allocation is said to be RC-e¢ cient, if there does not exist another RC-allocation that RC-dominates it.

Conde-Ruiz, Giménez and Perez-Nievas (2004) use the same concept of e¢ ciency which they call "Millian e¢ ciency", and they provide a general discussion of its relevance. Michel and Wigniolle (2006) discuss another notion, which is obtained in adding to the de…nition of RC-dominance the constraints

8t 0; m 1 t m 2 t : (43)
For this last notion, an allocation can dominate another one only if it includes at least an equal number of children in each period. On the other hand, RCdominance allows to improve the utility level of one (or many) generation(s) to the detriment of the size of one (or many) generation(s). In the current article, we will neglect this second notion, as it does not allow us to obtain more results. Indeed, it will appear that these two notions lead to the same classi…cation of competitive equilibria. The reason behind this result is that non-RC-e¢ cient equilibria are dominated by allocations that include a higher level of fertility.

E¢ ciency of the intertemporal equilibrium

In this section, we focus on the question of the Pareto-e¢ ciency of the competitive equilibrium with endogenous fertility, using our framework with CES utility and production functions. First, we provide a set of parameters such that the competitive equilibrium is both in under-accumulation and ine¢cient. This result enlarges the simple example given in Michel and Wigniolle (2006). Secondly, we give a su¢ cient condition ensuring that a competitive equilibrium converging in under-accumulation is Pareto-e¢ cient. This condition only involves the two elasticities of substitution of the production and utility functions.

Under-accumulation does not imply e¢ ciency

The following proposition gives a set of parameters such that the stationary state of our CES economy is both in under-accumulation and ine¢ cient. The following notations will be useful

~ ln(A ) ln A ; (44)
A 1 4 A 1 1 + 1 + A 1 1 + A 1 2 ; ( 45
)
A 2 1 + 1 + A 1 : (46)
Proposition 2 Let us assume that one of these 3 conditions holds

< A < 1; (47) or < 1 < A and > ~ ; (48)
or A < 1 < and < ~ ;

(49) then we have A 1 < A 2 ; and, for each value of b such that

A 1 < b < A 2 ; (50)
the competitive stationary equilibrium is both in under-accumulation and in-e¢ cient for su¢ ciently high .

Proof: see Appendix 1.

As shown in Appendix 1, the proof is obtained by building a RC-allocation that gives the same consumption levels, and a higher fertility level, with respect to the competitive equilibrium. The fertility level rises in such a way that it becomes higher than the interest factor. The result is obtained taking the limit case ! +1: At the limit, the interest factor becomes constant and independent of the capital-labor ratio k: In this case, it is possible to devote more resources to children and fewer resources to capital accumulation in such a way that the fertility level becomes higher than the interest factor.

To obtain this result, the initial stationary equilibrium must be such that the share of the wage devoted to savings 2 R(k) is higher than the share devoted to children 3 R(k) : This property can be obtained under various values of the di¤erent parameters. At the limit when ! +1; this property occurs if one among the three conditions ( 47), ( 48) or (49) holds. If ( 47) is satis…ed, no condition on the elasticity of substitution of the utility function is needed. Condition (48) requires a su¢ ciently high value of and ( 49) a su¢ ciently low value of .

A su¢ cient condition for the e¢ ciency of an intertemporal equilibrium

In standard microeconomics, for a …nite number of agents endowed with a utility function, a su¢ cient condition for an allocation to be Pareto-optimal is that this allocation maximizes a weighted sum (with positive weights) of each agent's utilities. In our model, in which we have considered an in…nite number of agents of di¤erent generations, this property always holds, as illustrated in the following lemma.

The consequence is that for each T t 0

T X t=0 t U (c t ; d t+1 ; m t ) U (c t ; d t+1 ; m t ) (54)
t 0 U (c t 0 ; d t 0 +1 ; m t 0 ) U (c t 0 ; d t 0 +1 ; m t 0 ) > 0: (55)
The allocation (y t ; N t ; c t ; d t ; m t ) t 0 does not maximize the objective function (51).

The preceding lemma provides a method for proving that a competitive equilibrium that converges in under-accumulation is RC-e¢ cient: it is suf-…cient to prove that this allocation maximizes the objective function (51) in the set of feasible RC-allocations with an appropriate choice of weights ( t ). This maximization problem is not relevant in the general case, however, because it is not concave. The lack of concavity is due to the presence of the term m t y t+1 in the resource constraint (38). Consequently, we only consider particular cases with CES utility and production functions, where the properties of ( 51) can be explicitly studied.

The optimization problem de…ned by lemma 2 can be solved in two stages: a static optimization problem that is concave in the general case; a dynamic optimization problem that is generally not concave. We begin by studying the static problem. This optimization problem is ( max

(ct;d t+1 ;mt) U (c t ; d t+1 ; m t ) s. t. m t y t+1 = F (y t c t m t ; m t ) d t+1 (56)
Our method consists of studying this problem in the particular case of CES utility and production functions before considering the dynamic problem.

Even with these particular functions, however, we cannot …nd an explicit solution to this static problem. Below we introduce another optimization problem, which will be useful. We de…ne the following functions:

v(
By de…nition, we have v(y t ; y t+1 ) z(y t ; y t+1 ): Indeed, the …rst program looks like the second program with an additional constraint: x t = m t :

Recalling that y t = w t along an intertemporal equilibrium, the objective function de…ned by ( 51) can now be set as

lim inf T !+1 T X t=0 t [v(y t ; y t+1 ) v(w t ; w t+1 )] ; ( 59
)
and it is also possible to de…ne a second objective as

lim inf T !+1 T X t=0 t [z(y t ; y t+1 ) z(w t ; w t+1 )] : (60)
The following lemma establishes two properties that are satis…ed by v and z. First, these two functions coincide when (y t ; y t+1 ) is taken along a competitive trajectory: (y t ; y t+1 ) = (w t ; w t+1 ): Secondly, for an appropriate choice of the sequence ( t ), the competitive equilibrium satis…es the …rstorder conditions of the two dynamic optimization problems de…ned by ( 59) and (60).

Lemma 3 Let y t ; N t ; c t ; d t ; m t t 0 be a competitive equilibrium with w t = y t as the competitive wage and R t+1 = F 0 K (y t c t m t ; m t ) as the competitive interest factor. We have:

1. v(w t ; w t+1 ) = z(w t ; w t+1 ) = U (c t ; d t+1 ; m t ):
2. Let ( t ) be the sequence de…ned by: 0 = 1 and 8t 0;

t+1 t = U 0 c c t ; d t+1 ; m t m t U 0 c c t+1 ; d t+2 ; m t+1 R t+1 ; (61)
we have 8t 0

t v 0 2 (w t ; w t+1 ) + t+1 v 0 1 (w t+1 ; w t+2 ) = 0; (62) t z 0 2 (w t ; w t+1 ) + t+1 z 0 1 (w t+1 ; w t+2 ) = 0: (63)
Proof: see Appendix 2. The …rst result is obtained by showing that the …rst order conditions of the two static optimization programs coincide when (y t ; y t+1 ) = (w t ; w t+1 ) :

The second result proves that the marginal conditions of the two dynamic optimization programs ( 59) and ( 60) are satis…ed by y t = w t 8t; when t is de…ned by (61).

We can now consider the trajectory of the competitive equilibrium ( y t ; N t ; c t ; d t ; m t ) t 0 obtained using the CES production technology given by ( 13), or by ( 14) in the limited case of an elasticity of substitution equal to one, and a CES utility function given by (1). We know that the steady state k is reached from period 1; and that it is unique under the assumption + 1: Denoting the stationary values of the interest factor and the fertility level by R = R(k) and m; respectively, we obtain the following result: Proposition 3 Assume that + 1 (the competitive equilibrium is determined), that 1 (production factors are complements in production) and that the competitive economy converges in under-accumulation (R > m). Then, if + 2; the competitive equilibrium is RC-e¢ cient.

Proof: see Appendix 3. The result is obtained by explicitly calculating the function z(y t ; y t+1 ) and proving that the objective function

lim inf T !+1 T X t=0 t [z(y t ; y t+1 ) z(w t ; w t+1 )] (64)
reaches a global maximum in y t = w t : As v(y t ; y t+1 ) z(y t ; y t+1 ) with v(w t ; w t+1 ) = z(w t ; w t+1 ); the objective function

lim inf T !+1 T X t=0 t [v(y t ; y t+1 ) v(w t ; w t+1 )] (65)
reaches a global maximum in y t = w t :

The intuition behind Proposition 3 is that an allocation can Paretoimprove a competitive equilibrium converging in under-accumulation, if it leads, in the long run, to a fertility level higher than the interest factor. Our result gives two arguments capable of precluding this increase in the fertility level. The …rst argument is the complementarity of production factors. Indeed, the increase in fertility is linked to a fall in capital accumulation. If capital and labor are substitutable, this change is not too detrimental to production; if capital and labor are complementary, however, it can lead to a signi…cant drop in resources. The second argument is the complementarity of consumption and fertility in the utility function. If c; d and m are highly substitutable, the increase in m allows for a substantial reduction in c and d; which is compatible with a constant level of utility. When they are complementary, however, the admissible fall of c and d is low.

Proposition 3 shows that a trade-o¤ between the two substitution elasticities in production and utility exists, such that the competitive equilibrium (converging in under-accumulation) is RC-e¢ cient. This condition is the same as the one found in [START_REF] Michel | Population growth and optimality: when does serendipity hold[END_REF], who are interested in the optimal fertility rate in a framework of exogenous fertility. In their model, the condition guarantees that the serendipity theorem holds. Michel and Wigniolle (2006) provide a general condition which ensures Pareto-e¢ ciency for the competitive equilibrium of an economy endowed with general utility and production functions: an intertemporal equilibrium converging toward a stationary state such that R > w is RC-e¢ cient. This condition is stronger than under-accumulation. Indeed, at the steady state of the competitive economy, we have: m w < R ) m < R:

With respect to this general condition, our results in Proposition 3 are more precise for an economy endowed with a CES utility and production functions. We have proved that if + 1; 1 and + 2 hold, underaccumulation is su¢ cient for RC-e¢ ciency of the competitive equilibrium. The condition R > w is stronger and not relevant in this case.

Proposition 3 gives su¢ cient conditions for RC-e¢ ciency of a competitive equilibrium in under-accumulation. But it is unlikely that these conditions are necessary. Proposition 2 allows us to conclude on the ine¢ ciency only for high values of : There exists a large range of the parameters ( ; ) for which no conclusion is reached. Indeed, even if and do not satisfy the conditions given in Proposition 3, building a RC-allocation that dominates the competitive equilibrium needs extra assumptions on other parameters. For instance, in Proposition 2, we also need some constraints on the parameters A and b of the production technology, on the child cost ; and on the coe¢ cients and of the utility function.

Our results give some insights on the role of the two parameters and ; but the understanding is not complete and some further results could be achieved. More precisely, the proofs of Propositions 2 and 3 give some intuitions that we were not able to prove. In Proposition 2, the assumption that is su¢ ciently high plays a crucial role. A value of greater than one implies that the marginal productivity of capital tends to a …nite value when the capital stock tends toward 0. It is also well known that this case allows the existence of unbounded realizable trajectories for the variable k t : These properties allow us to build an allocation that RC-dominates the competitive equilibrium in under-accumulation. Now, assuming 1 and high enough , such that the assumption + 2 in Proposition 3 is violated, or even such that ! +1, is it possible to construct such an allocation dominating the competitive equilibrium? We have not succeeded in answering this question. Conversely, it could be true that the condition 1 alone, without the extra condition + 2; is su¢ cient to ensure RC-e¢ ciency, but we have not succeeded in proving this property.

Conclusion

This article has studied the e¢ ciency of the competitive equilibrium for an overlapping generations economy with endogenous fertility. Utility and production functions were assumed to be CES. We have given a su¢ cient condition, which ensures the existence and uniqueness of a competitive equilibrium: the sum of the two elasticities must not be smaller than one. Then, we have exhibited a set of parameters such that a competitive equilibrium in under-accumulation is not e¢ cient. Finally, we have proved that a competitive equilibrium converging in under-accumulation is Pareto-e¢ cient if the sum of the two elasticities of substitution is not greater than two.

It would be interesting to study whether these results remain true in considering more general environments. Notably, a generalization could emerge concerning the cost of children. In our simple framework, this cost is constant for each child. Conde-Ruiz, Giménez and Perez-Nievas ( 2004) consider a more general formulation where the cost is a convex function of the number of children. Another interesting generalization could be to introduce a cost in time. Under this assumption, children have an opportunity cost that depends on the wage. According to [START_REF] Greenwood | The baby boom and baby bust[END_REF], the rise of this opportunity cost has played a crucial role in the historical decline in fertility observed in America over the last 200 years.

Using policy-oriented models with endogenous fertility and applying e¢ciency criteria to these models could be an important topic. Indeed, e¢ ciency criteria should in ‡uence the design of economic policies such as PAYG pension systems or child bene…ts, as appears from Conde-Ruiz, Giménez and Perez-Nievas'results (2004).

we obtain su¢ cient conditions that respectively ensure that k 1 > 0 and k t+1 > 0 8t 1;

Ab 1 1 w > 0; (75) Ab 1 1 w 2 > 0: (76)
The only condition that must hold is

Ab 1 > 1 w + 2 : (77)
Finally, we impose the inequality > m; which leads to

R=w > 2 3 = 2 : (78)
Condition (78) implies U (c; d; ) > U (c; d; m) as > m: All utilities are higher along the feasible trajectory with m t = :

The three conditions ( 68), ( 77) and ( 78) remain to be studied at the limit when ! +1: When ! +1; lim R = A and lim w = Ab:

Conditions ( 68) and ( 78) are equivalent to

=b > max 2 (A); 2 3 (A)= 2 (A) : (79)
Condition (77) can be written as

2 < Ab [1 1 (A)] = Ab [ 2 (A) + 3 (A)] : (80)
Taking the square of the preceding inequality, we obtain

4 2 2 = 4 2 (A)A 2 b < A 2 b 2 [ 2 (A) + 3 (A)] 2 ; (81) or =b < [ 2 (A) + 3 (A)] 2 4 2 (A) : (82)
By ( 79) and ( 82), we must have

2 3 (A) 2 (A) < [ 2 (A) + 3 (A)] 2 4 2 (A) ; (83) which implies 2 (A) > 3 (A): (84) And, if 2 (A) > 3 (A); then max [ 2 (A); 2 3 (A)= 2 (A)] = 2 (A): max (ct;d t+1 ;mt;xt) ln c 1 t + d 1 t+1 + m 1 t 1 s. t. x t y t+1 = A h (y t c t m t ) 1 + b (x t ) 1 i 1 d t+1 (99)
First, we independently consider the maximization with respect to x t (as x t only appears in the constraint and not in the objective)

max xt d t+1 = A h (y t c t m t ) 1 + b (x t ) 1 i 1 x t y t+1 (100)
Before solving this program, we note that, as < 1;

A h (y t c t m t ) 1 + b (x t ) 1 i 1 < Ab 1 x t : (101)
We obtain:

d t+1 < Ab 1 y t+1 x t : (102)
As d t+1 must be positive, however, the condition y t+1 < Ab 1 must be satis…ed along any feasible trajectory. When y t+1 < Ab 1 ; the program (100) has an interior solution. We …nd

b (x t ) 1 = (y t c t m t ) 1 y t+1 Ab 1 1 1 : (103)
Replacing b(x t ) 1 in the constraint, we obtain

d t+1 = A (y t c t m t ) " 1 y t+1 Ab 1 1 # 1 1 ; (104)
so that we can also write

c t + m t + h(y t+1 )d t+1 = y t ; (105) with h(y t+1 ) = b 1 Ab 1 1 (y t+1 ) 1 1 1 : (
We study this function on 0; Ab 1 : From the lemma 3, we know that the derivative of this function cancels out when y t = w t = w for t 1: This derivative is calculated in using (107)

Z 0 (y t ) = t 1 y t 8 > > > > < > > > > : t t 1 h(y t ) 1 " (yt) 1 Ab 1 1 (yt) 1 # 1 + 1 + h(y t ) 1 9 > > > > = > > > > ; : (112)
First, we prove that the function Z has a global maximum on 0; Ab 1 when y t = w t = w:

We establish this property in introducing the function

Y (y t ) = h(y t ) 1 " (yt) 1 Ab 1 1 (yt) 1 # 1 + 1 + h(y t ) 1 ; (113)
which is such that

Z 0 (y t ) = t 1 y t t t 1 Y (y t ) : (114)
If the function Y (y t ) is strictly increasing, as t = t 1 Y (y t ) cancels out when y t = w t = w; we can deduce that Z 0 (y t ) < 0 for y t < w and that Z 0 (y t ) > 0 for y t > w: y t = w is the global maximum of Z on 0; Ab 1 :

To prove that Y (y t ) is strictly increasing, we calculate

By assumption, as < 1; the term h(y) 1 (1 ) Ab 1 1 is positive. If < 1; as < 1 the other term is positive and Y 0 (y) > 0: If > 1; as y < Ab 1 and > 1; we obtain (1 + 1 ) : (117)

A su¢ cient condition for obtaining Y 0 (y)=Y (y) > 0 is 2 0: Finally, we have proved that if + 2;

(118)

the function Z has a global maximum on 0; Ab 1 when y t = w t = w:

Next, we must prove that the objective function is maximum at this point. This result is achieved by writing

Second case: = 1: In this case, the production function is a Cobb-Douglas given by ( 14). We use the same method as for < 1: It is easy to check that equation ( 109

  (y t ; y t+1 ) v(w t ; w t+1 )] = y) reaches a global maximum for y = w; the …rst term is negativeT X t=0 t [Z(y t ) Z(w)] 0:(120)As h(y) is an increasing function, the second term is a decreasing function of y T +1 : This term is smaller than:

  If the maximum of the objective function on the set of RC-feasible allocations is reached in (y t ; N t ; c t ; d t ; m t ) t 0 ; this allocation is RC-e¢ cient.Remark 3The assumption of the preceding lemma simply means that (y t ; N t ; c t ; d t ; m t ) t 0 is weakly maximal in the sense of[START_REF] Brock | On the existence of weakly-maximal programmes in a multi-sector economy[END_REF]. Indeed, if (y t ; N t ; c t ; d t ; m t ) t 0 maximizes (51), it cannot be overtaken by another feasible allocation.Proof: Reductio ad absurdum. First, we can note that the objective function (51) is equal to zero for the allocation (y t ; N t ; c t ; d t ; m t ) t 0 : Let us assume that (y t ; N t ; c t ; d t ; m t ) t 0 is a feasible RC-allocation that maximizes (51) in the set of feasible RC-allocations, and that this allocation is not RC-e¢ cient. In that case, it is RC-dominated by a feasible RC-allocation (y t ; N t ; c t ; d t ; m t ) t 0 , such that such that U (c t 0 ; d t 0 +1 ; m t 0 ) > U (c t 0 ; d t 0 +1 ; m t 0 ):

	we de…ne 0; U (c (52) the objective function lim inf t=0 T !+1 T X t U (c 8t 9t 0 0;

Lemma 2 Let (y t ; N t ; c t ; d t ; m t ) t 0 be a feasible RC-allocation. Let ( t ) be a sequence, such that 8t; t > 0: For this sequence and for some feasible allocation (y t ; N t ; c t ; d t ; m t ) t 0 with the same initial conditions, t ; d t+1 ; m t ) U (c t ; d t+1 ; m t ) : (51) t ; d t+1 ; m t ) U (c t ; d t+1 ; m t );

  y t ; y t+1 ) = U (c t ; d t+1 ; m t ) s. t. x t y t+1 = F (y t c t m t ; x t ) d t+1

	z(y t ; y t+1 ) =	( (	max (ct;d t+1 ;mt) s. t. m t y t+1 = F (y t c t U (c t ; d t+1 ; m t ) m t ; m t ) d t+1 max (ct;d t+1 ;mt;xt)	(57)

  which is additively separable as the sum of a function of y t and a function of y t+1 .Let us now consider the objective ) de…ned by (61). For a CES utility function, we know from Section 3.3 that w t is constant: 8t 1; w t = w: As the function z is additively separable, it is possible to match together the two terms that depend on y t (8 t 1)

	It is straightforward to check that the function 1=h(y) is the factor price
	frontier that gives the interest factor R with respect to the wage y when
	…rms'behavior is competitive.
	Secondly, we must consider the program
							1		1	1	1
		max (ct;d t+1 ;mt)	ln c	t	+ d	t+1 + m	t	(108)
		s. t. c t + m t + h(y t+1 )d t+1 = y t
	The indirect utility associated with this program is by de…nition the func-
	tion z
	z(y t ; y t+1 ) = ln y t +		1	1	ln 1 +		1	+ h(y t+1 ) 1	;	(109)
		lim inf T !+1
	Z (y t )	t 1 1	ln 1 +
									106)
	It will be useful to keep in mind the following property of the function h
		h 0 (y t+1 )y t+1 h(y t+1 )	=	Ab 1	(y t+1 ) 1 1	(y t+1 ) 1	:	(107)

T X t=0 t [z(y t ; y t+1 ) v(w t ; w t+1 )] ;

(110)

with the sequence ( t 1 + h(y t ) 1 + t ln y t :

As usual, we consider m t as a continuous variable. This assumption is consistent with a model which assumes one representative agent per generation.

An RC-allocation means a feasible allocation for Representative Consumers.

RC-dominance means dominance for Representative Consumers.

Appendix 1: proof of Proposition 2.

We introduce the following notation in what follows i R(k) = i : (66)

We assume that the competitive equilibrium is in under-accumulation. This property is expressed by the condition

We can then build a feasible RC-allocation starting from the stationary state k 0 = k and m 1 = m; such that: 8t 0; c t = c; d t = d; m t = m; and k t satis…es the resource constraint

In t = 0; as m 1 is given and equal to m; this condition is

As d = Rs = Rmk; the term f (k) d=m is equal to f (k) Rk = w: The preceding condition becomes

For t 1; (69) becomes

We set 2 d= = 2 Rw= : We can write

We choose m = ; which is the value that minimizes the last term of the equation. At each date t; the condition k t > 0 must hold. As

Finally, we have proved that the conditions that are to be satis…ed are

where the inequality

is a consequence of 3 (A) < 2 (A). Using ( 11) and ( 12), ( 85) gives

that can be written

As > 0; it is straightforward to prove that if one condition among (47) ( 48) or ( 49) is satis…ed, then 2 (A) > 3 (A): Moreover, we know that in this case

: Considering now (86), it can be interpreted as a condition on b that can be written

Using ( 11) and ( 12), this last condition becomes (50). Finally, when conditions given in Proposition 2 hold, conditions (68), ( 77) and (78) are satis…ed in the limit case when ! +1: By continuity, they are satis…ed when is large enough. As a last comment, we see that the inequality 3 (A) < 2 (A) implies

< 2 (A): When ( 85) and ( 86) hold, we have

which is equivalent to the inequality

when ! +1: Consequently, the allocation (c; d; ) that has been built is associated with over-accumulation.

Appendix 2: proof of lemma 3.

The …rst part of the lemma is obtained by considering the two static optimization programs, which are both concave.

The …rst order conditions for the …rst program are

For (y t ; y t+1 ) = (w t ; w t+1 ) and (c t ; d t+1 ; m t ) = c t ; d t+1 ; m t ; these conditions are satis…ed, as they correspond to the conditions of the consumer maximization program along the competitive equilibrium. The resource constraint is identical to the budget constraint of the consumer.

The …rst order conditions for the second program are

All these conditions are satis…ed when (y t ; y t+1 ) = (w t ; w t+1 ) and (c t ; d t+1 ; m t ) = c t ; d t+1 ; m t : These two optimization programs also make it possible to …nd the following properties

The second result is obtained by using the following properties v 0 1 (w t+1 ; w t+2 ) = U 0 c t+1 c t+1 ; d t+2 ; m t+1 and v 0 2 (w t ; w t+1 ) = (97) From the de…nition of the sequence t ; the result is proved.

Appendix 3: proof of Proposition 3.

We separately consider the two cases: < 1 and = 1: First case: < 1:

For CES production and utility functions, we can calculate the explicit form of the function z: We can consider the utility function

z is de…ned by If < 1; as h(y) is a (strictly) increasing function, Y (y) is (strictly) increasing, and the property is proved.

When = 1; Y (y) is constant. As we know that Z 0 (w) = 0; we obtain that Z 0 (y) = 0 8y and …nally that Z(y) is constant. Using the same reasoning as for < 1; lim inf