Elsa Fristot

Guillaume Cambray
email: cambray.guillaume@gmail.com

Jerome Bonnet
email: jerome.bonnet@inserm.fr

LactoSpanks: a collection of IPTG inducible promoters for the commensal lactic acid bacteria Lactobacillus gasseri

Lactic acid bacteria (LAB) are important for many biotechnological applications, such as bioproduction and engineered probiotics for therapy. Inducible promoters are key gene expression control elements, yet those available in LAB are mainly based on bacteriocin systems and have many drawbacks, including large gene clusters, costly inducer peptides and little portability to in vivo settings. Using Lactobacillus gasseri, a model commensal bacteria from the human gut, we report the engineering of Lactospanks promoters (Pls), a collection of variable strength inducible promoters controlled by the LacI repressor from B. subtilis and induced by isopropyl β-D-1-thiogalactopyranoside (IPTG). We first show that the Phyper-spank promoter from Bacillus subtilis is functional in L. gasseri, albeit with substantial leakage. We then construct and screen a semi-rational library of Phyper-spank variants to select a set of four IPTG-inducible promoters that span a range of expression levels and exhibit reduced leakages and operational dynamic ranges (from ca. 9 to 28 fold-change). With their low genetic footprint and simplicity of use, Lactospanks will support many applications in L. gasseri, and potentially other lactic acid and gram-positive bacteria.

MAIN TEXT

Lactic acid bacteria (LAB) encompass a group of disparate microorganisms present in various ecological niches from fermented food to mammalian microbiota [START_REF] George | Occurrence and Dynamism of Lactic Acid Bacteria in Distinct Ecological Niches: A Multifaceted Functional Health Perspective[END_REF] . LAB are gram-positive, rod-shaped, non-spore forming, acido-tolerant and facultative anaerobic bacteria that share the ability to convert carbohydrates to lactic acid [START_REF] Kandler | Carbohydrate Metabolism in Lactic Acid Bacteria[END_REF] . Due to their GRAS (generally recognized as safe) qualification, they are of great interest for biotechnological applications, including fermentation [START_REF] Ayivi | Lactic Acid Bacteria: Food Safety and Human Health Applications[END_REF] , production of food preservatives [START_REF] Parada | Bacteriocins from Lactic Acid Bacteria: Purification, Properties and Use as Biopreservatives[END_REF] and, more recently, engineered probiotics for therapy [START_REF] Plavec | Engineering of Lactic Acid Bacteria for Delivery of Therapeutic Proteins and Peptides[END_REF] . Many of these applications primarily rely on the precise control of gene expression.

Several systems have been deployed to control gene expression in LAB. While initial approaches leveraged constitutive promoters from housekeeping genes (e.g. PslpA and Ppgm from L. acidophilus) [START_REF] Nguyen | Constitutive Expression and Cell-Surface Display of a Bacterial β-Mannanase in Lactobacillus Plantarum[END_REF] , inducible gene expression systems provide greater versatility for rapid prototyping, controllable production of physiologically costly proteins or sensitive in vivo applications. The main induction systems used in lactic acid bacteria are based on bacteriocin- [START_REF] Kleerebezem | Controlled Gene Expression Systems for Lactic Acid Bacteria: Transferable Nisin-Inducible Expression Cassettes for Lactococcus, Leuconostoc, and Lactobacillus Spp[END_REF][START_REF] Karlskås | Heterologous Protein Secretion in Lactobacilli with Modified pSIP Vectors[END_REF] , stress- [START_REF] Benbouziane | Development of a Stress-Inducible Controlled Expression (SICE) System in Lactococcus Lactis for the Production and Delivery of Therapeutic Molecules at Mucosal Surfaces[END_REF] or carbohydrates-inducible systems [START_REF] Miyoshi | A Xylose-Inducible Expression System for Lactococcus Lactis[END_REF] . The Nisin inducible pNICE system is the reference, but it involves a large gene cluster that limits plasmid real estate and reduces transformability. Furthermore, the system can impose a burden on the host cell, as it requires expression of a membrane receptor that can alter bacterial physiology and the induction with Nisin peptide, a bacteriocin which can be toxic in some expression strains [START_REF] Neu | New Thermosensitive Delivery Vector and Its Use to Enable Nisin-Controlled Gene Expression in Lactobacillus Gasseri[END_REF] .

In widely used and studied model organisms, such as Escherichia coli, a wide variety of inducible systems with low genetic burden have been developed. Examples include the Ptet system induced by anhydrotetracycline, the Pbad system controlled by arabinose, or the Plac system inducible with lactose or its non metabolisable analog isopropyl β-D-1-thiogalactopyranoside (IPTG) [START_REF] Lutz | Independent and Tight Regulation of Transcriptional Units in Escherichia Coli via the LacR/O, the TetR/O and AraC/I1-I2 Regulatory Elements[END_REF] . These inducible systems are extensively applied to control the levels and timing of gene expression in liquid culture and in vivo applications. Contrasting with those currently used in LAB, these classical systems rely on the expression of a single repressor gene, which involves lower genetic footprint, complexity and burden, and consequently improved portability. They also respond to cheap inducers that are non toxic and, for some of them, non-metabolizable (e.g. aTc or IPTG). Although some Plac systems have been reported in lactic acid bacteria, their use has remained anecdotal due to limited dynamic ranges (at most four-fold) and induction restricted to lactose instead of non-metabolizable analogs [START_REF] Zhang | Development of Strong Lactose/galactose-Inducible Expression System for Lactobacillus Plantarum by Optimizing Promoter[END_REF][START_REF] Heiss | Evaluation of Novel Inducible Promoter/repressor Systems for Recombinant Protein Expression in Lactobacillus Plantarum[END_REF][START_REF] Pérez-Arellano | Optimization of the Green Fluorescent Protein (GFP) Expression from a Lactose-Inducible Promoter in Lactobacillus Casei[END_REF] . Optimization of these systems to reduce leakage while maintaining strong expression upon induction is thus essential to improve their utility. Such systems (based on Tet, LacI, or T7) have been successfully developed in model gram-positive bacteria, such as B. subtilis, and several have proven functional in various lactobacilli (e.g. PSfrA) [START_REF] Guan | Development of a Novel Expression System in Lactic Acid Bacteria Controlled by a Broad-Host-Range Promoter PsrfA[END_REF] .

We are interested in developing inducible systems for Lactobacillus gasseri ATCC33233 [START_REF] Azcarate-Peril | Analysis of the Genome Sequence of Lactobacillus Gasseri ATCC 33323 Reveals the Molecular Basis of an Autochthonous Intestinal Organism[END_REF] , an easily manipulated strain that is naturally present in the vaginal and colorectal microbiota and has the potential to be engineered as a boosted probiotic, with existing proof of principles for efficient mucosal vaccination 18 . To develop reliable inducible promoters for the precise engineering of protein delivery by this bacteria, we chose to use the Phyper-spank promoter, an inducible system developed for B. subtilis IPTG induction derived from the spac system carrying the bacteriophage promoter SP01 modified with lac operator and an E. coli lac repressor [START_REF] Vavrová | Comparison of Different Bacillus Subtilis Expression Systems[END_REF] .

As Phyper-spank was obtained from a B. subtilis shuttle vector, we first transferred the Phyper-spank to drive sfGFP expression along with its LacI repressor on the broad range lactobacilli shuttle vector pTRKH2 carrying the pAMB1 high-copy replication origin for gram-positive bacteria 20 , and transformed it into L. gasseri for characterization assays (Figure 1A). This construct showed first to be efficiently induced in the presence of IPTG but shows a sizable basal activity -about 5 times the background fluorescence of cells devoid of plasmid grown in the same conditions-thus indicating the B. subtilis promoter to be functional but substantially leaky in this strain (Figure 1B and Table 1). When induced with 100µM IPTG, Phyper-spank produced a strong response with a 16-fold increase in measured fluorescence levels. Considering the leakiness of the repressed Phyper-spank (referred as Phyp) and its otherwise strong intrinsic activity, we reasoned that improved dynamic ranges could be obtained by reducing the strength of the promoter. To achieve this, we separately introduced random nucleotide substitutions in three distinct regions known to modulate the activity of σ 70 promoters to various extent : the -35 motif, the -10 motif, and the spacer between those two regions. To further investigate the potential impact of translation on the response, we also generated a fourth library targeting the ribosome binding site (RBS). (Figure 1C). Each sub-library was generated using degenerate oligonucleotides, cloned in E. coli, transformed in L. gasseri and screened by randomly picking 12-40 clones and coarsely evaluating their response to induction in a single replicate (Figure 1D).

All isolated clones exhibited reduced leakage due to decreased gene expression caused by the mutated promoter regions (Supplementary Figure S3). Broadly speaking, mutations in the -10 motif or RBS resulted in strong reductions in reporter expression, while mutation in the -35 and between the -35 and the -10 regions leads to a more moderate decrease. Based on improved dynamic range (ratio between the fluorescence intensities of the fully activated state vs the non-induced state).. Based on these initial results, we chose 22 promising clones (10, 9, 1 and 2 for the -35, spacer, -10 and RBS regions, respectively) to perform three additional replicates induction experiments (Supplementary Figure S4).

We categorized these 22 clones into four levels of induced expression, then chose the clone with maximal dynamic range from each bin, which yielded four final variants Pls1 to 4. We compared their expression profiles with the original Phyper-spank construct under saturating induction with 100 μM IPTG (Figure 2A and2B). Depending on the mutation, these variants showed a 2 to 10 fold reduction in basal leakage while the maximal induced expression is comparatively less affected. For example, Pls4's leakage is reduced by 50%, but its induced expression is only reduced by 25%. Across all variants, the reduction in leakage tends to be twice as marked as the reduction in induced expression, which results in better dynamic ranges, with a fold change up to twice as high as that of Phyper-spank.

We next proceeded to measure the variants' response to increasing concentration of IPTG, ranging from 10nM to 1mM (Figure 2C, supplementary figure S5). The four variants cover a wide range of maximum expression levels, ranging from 25 to 75% that of Phyper-spank (Figure 2C andtable 1). The half maximal effective concentration (EC50) of IPTG derived from these titration curves tends to increase with reduced promoter strength (Table 1). We thus improved promoter fold changes compared to wt Phyper-spank, with Fold changes from 16 for the wild type version up to ~28 for Pls3. Promoters activation kinetics were then assessed (Figure 2D). The number of cells in the ON state increased with time, with the fraction of switched cells ranging from 21% for Pls1 up to 67% for Pls3 as early as two hours post induction, and a maximum reached after 6 post induction, with 73~77% of switched cells for the Pls variants. Cells harboring Phyp show a switch from 62% 2 hours post-induction to 86% at 6 hours (Table 1, supplementary table 1, supplementary figure S6). A sizable fraction (from 14% for pHyp to 29% for Pls3) of the cell population maintains background fluorescence level, indicating the absence of reporter signal. We observed this behavior with all promoters we tested, including constitutive promoters of varying strength (e.g. reference promoter pSlpA on figure 1B).

The Pls1-4 variants were sequenced (Figure 2D, and supplementary data). The lowest expression mutants Pls1 and Pls2 showed distinct substitutions of two nucleotides at the same location of the -35 region while Pls3 showed a single mutation in the same region, supporting that the -35 motif is a strategic target to fine-tune gene expression. The strongest variant, Pls4 showed a 6 nucleotides substitution in the spacer region. Despite it harboring the largest number of mutations, the spacer variant had the smallest reduction in expression (supplementary Figure S3D). During the course of our project, we identified limitations with the use of the sfGFP reporter in lactic acid bacteria. In the time-course experiment shown in figure 2D, the Phyper-spank construct did not reach saturation after 8h. In contrast, the fluorescence signal all of the weaker Pls variants saturated in 6h and showed a slight decrease at 8 hours post-induction, despite the stability of the sfGFP reporter protein. As with the fraction of non-expressing cells, the mechanism underlying this observation is unknown but may be linked to the acidification of the cytosol over time. In this perspective, the decrease in fluorescence may reflect the pH effect on the protonation of the sfGFP chromophore [START_REF] Shaner | Advances in Fluorescent Protein Technology[END_REF] , which would have a more prominent impact as the protein is expressed at lower levels. A similar observation was previously made in L. casei [START_REF] Pérez-Arellano | Optimization of the Green Fluorescent Protein (GFP) Expression from a Lactose-Inducible Promoter in Lactobacillus Casei[END_REF] .

The Lactospank collection provides a suite of four promoters (Pls1-4) with characteristics that can fit different user requirements. For instance, Pls3 demonstrates reduced leakage while reaching two-third the maximum expression level of the original Phyper-spank (Table 1), which should be useful for the majority of applications. Phyp and Pls2-4 Pls3 and Pls4 as well as the Phyp showed higher expression than the widely used strong constitutive promoter PslpA from Lactobacillus acidophilus, which placed them well for strong protein expression applications. In contrast, Pls1 provides lower expression levels, with no observable leakage, and will be suitable for expressing more toxic and complex compounds that require tight repression. In Pls2-4, higher dynamic range was achieved through reducing basal leakage at the expanse of maximal induced expression with respect to the original Phyper-spank. Other approaches could be investigated to maintain or increase maximal expression, including tuning the expression of the LacI repressor, modulating the position and number of Lac operators, or using more advanced circuit topologies [START_REF] Castillo-Hair | Subtilis Inducible Promoter System with over 10 000-Fold Dynamic Range[END_REF][START_REF] Ho | A Suppressor tRNA-Mediated Feedforward Loop Eliminates Leaky Gene Expression in Bacteria[END_REF] .

A variety of orthogonal tools enabling fine tuning of gene expression are critically needed for lactic acid bacteria. Following this work, other low footprint inducible systems could be developed for these species. Some existing systems could be transposed from other species as we did. Alternatively, new inducible systems could be discovered in the wide realm of LAB and repurposed. Improved control will support the implementation of complex genetic circuits such as logic gates, oscillators, and memory systems, and allow synthetic biologists to unleash the potential of lactic acid bacteria as programmable agents for industrial, environmental, and medical applications.

METHODS

Plasmids

The pTRKH2 plasmid was akind gift from Rodolphe Barrangou & Todd Klaenhammer (Addgene plasmid #71312). pTRKH2 is a theta replicative plasmid carrying a pAMβ1 origin (high copy number), a p15A E. coli replication origin (low copy number) and an erythromycin resistance gene Erm(B). The Phyper-spank-sfGFP sequence used in this study originated from our lab [START_REF] Guiziou | A Part Toolbox to Tune Genetic Expression in Bacillus Subtilis[END_REF] . The Phyper-spank device and the pTRKH2 were PCR-amplified using the Q5 polymerase (NEB) and cloned by Gibson assembly [START_REF] Gibson | Enzymatic Assembly of DNA Molecules up to Several Hundred Kilobases[END_REF] in E. coli NEB 10 beta chemocompetent cells. The resulting plasmid pEF1-Phyperspank-sfGFP was sequenced and transformed in electrocompetent L. gasseri cells. The pSlpAs-sfGFP cassette was synthesized as a linear DNA fragment(Twist Bioscience) and cloned by Gibson assembly in pTRKH2. All constructs are flanked by standard spacers to facilitate downstream cloning via Gibson assembly. All plasmids are available from Addgene (pEF1-Phyper-spank-sfGFP: #205747, pEF1-Pls1-sfGFP: #205748, pEF1-Pls2-sfGFP: #205749, pEF1-Pls3-sfGFP: #205750, pEF1-Pls4-sfGFP: #205751). Primers are listed in Supplementary Table 2.

Library generation

Phyper-spank variant libraries were constructed by amplifying the vector pEF1-Phyperspank-sfGFP with two set of primers to insert mutation in promoter region and reassemble the vector back using Gibson assembly in E. coli NEB 10 beta chemocompetent cells. Primer sequences are listed in Supplementary Table 2. Transformed cells were plated on LB agar plate with erythromycin (150µg/mL; E6376 Sigma-Aldrich) for selection. After overnight growth, all resulting colonies were scrapped from the plates and resuspended in 2mL LB with antibiotics. The four resulting cultures were grown overnight and subjected to plasmid extraction by miniprep. The plasmid libraries were then transformed into L. gasseri electrocompetent cells. After screening, plasmids clones were purified from Lactobacillus gasseri using the QIAprep spin Miniprep kit (Qiagen) with addition of 100µg/mL of Lysozyme and 5U/mL of Mutanolysin to buffer P1 with an incubation time of 30 minutes prior to lysis buffer addition, and sequence-verified by Sanger sequencing (Eurofins Genomics, EU).

Strains and cell culture E. coli NEB10 beta was obtained from NEB. Cells are grown in LB media supplemented with Erythromycin (E6376 Sigma-Aldrich) at 150 μg/mL, as needed.Lactobacillus gasseri ATCC 33323 was obtained from LGC STANDARDS. ells are grown in liquid MRS medium (De Man, Rogosa and Sharpe, DIFCO) supplemented with erythromycin at 7 μg/mL, as needed. Growth is performed without agitation at 37°C in anaerobic conditions using CO 2 incubator Eppendorf (10% CO 2 ). MRS composition is as follow: peptone proteose: 10 g/L, beef extract: 10 g/L, yeast extract: 5.0 g/L, dextrose: 20 g/L, polysorbate 80: 1.0 g/L, ammonium citrate: 2.0 g/L, sodium acetate: 5.0 g/L, magnesium sulfate: 0.1 g/L, manganese sulfate: 0.05 g/L, dipotassium phosphate: 2.0 g/L. Agar plates are obtained by adding 1% agar to the appropriate liquid medium.

Electrocompetent cells preparation

A 50 mL culture of L. gasseri cells was grown overnight. 98 mL of fresh MRS medium was then inoculated with 2 mL of the overnight culture and incubated to an OD of 0.5-0.6 (ca. 4-5 hours). Cells were harvested by centrifugation at 3,500 rpm for 5 min and the supernatant was discarded. From this step on, cells were always maintained on ice or at 4°C in refrigerated centrifuges. Pelleted cells were then resuspended in 15 mL ice-cold 3X SMEB electroporation buffer (1X Sucrose Magnesium Electroporation Buffer: 298 mM sucrose, 1 mM MgCl2 in cold sterile water), centrifuged at 5,000 rpm for 10 min and the supernatant was discarded. This procedure was repeated twice. Cell pellets were then concentrated in 1.0 mL of ice-cold 3X SMEB buffer and 200 μL aliquots were used directly for electroporation up to one hour after preparation or placed at -80°C for long term storage.

Electroporation procedure

A GenePulser X Cell apparatus (Bio-rad Laboratories, Richmond, CA) was used for electroporation of L. gasseri. 200 μL competent cells were thawed on ice, transferred in a 0.2 cm electroporation cuvette (Bio-rad Laboratories, Richmond, CA) after addition and gentle mixing of 1 μg of DNA (up to 5uL), and incubated on ice for 5 minutes. After drying the electrodes with a clean paper wipe, cells were electroporated with a tension of 1.8 V, a resistance of 600 Ω and a capacitance of 25 µF. Successful electroporations usually showed a time constant of 8.0-11 ms. Directly after, cells were gently resuspended in 800 μL MRS medium pre-heated at 37°C and transferred to culture tubes for a 2 hours recovery at 37°C without shaking in anaerobic incubator. 100 μL of the resulting culture were then spread on MRS agar plate (1% agar) supplemented with 7μg/mL erythromycin using glass beads (0.4 cm diameter).

Induction assays

Overnight pre-cultures were started from single colonies on agar plates. The next day, saturated cell cultures (OD600~2) were diluted 1:100 in 400 µL MRS in 96 deepwell plates and incubated up to eight hours at 37°without agitation. IPTG was added at the dilution stop as needed. 10 uL from these culture were resuspended in 96 well plates in 200 µL focusing fluid for flow cytometry analyses.

Cytometry analysis

Flow cytometry was performed on an Attune NxT flow cytometer (Thermo Fisher) equipped with an autosampler and Attune NxT Version 2.7 Software. Experiments on Attune NxT were performed in 96-well plates with the following setting: FSC: 200 V, SSC: 380 V, and green intensity BL1: 460 V (488 nm laser and 510/10 nm filter). All events were collected with a cutoff at 20,000 single cell events. Every experiment included a negative control consisting in cells without plasmid grown in the same conditions to generate the gates. SeeSupplementary Figure S2 for the gating strategy. Data were analyzed using Flowjo (Treestar, Inc) and Prism (GraphPad).

Figure 1 .

 1 Figure 1. Generation and characterization of IPTG responsive promoter in L. gasseri. A. Transfer of Phyper-spank (Phyp) from B. subtilis to L. gasseri. Phyp was cloned in the shuttle vector pTRKH2 and transformed into L. gasseri. The plasmid map is shown in Supplementary Figure S1. B. Representative distribution of fluorescence intensities from Phyp operating in L. gasseri after 8 hours of growth in the absence (blue) or presence (green) of IPTG (100µM) as measured by flow cytometry. NC: negative control (grey), the WT L. gasseri without plasmid. PslpA (yellow): widely constitutive promoter used as a reference C. Schematic overview of four Phyper-spank regions independently targeted for random substitutions. Mutations are

Figure 2 :

 2 Figure 2: Characterization of the Lactospank promoter collection. A. Representative distribution offluorescence intensities from Phyper-spank and the four Pls variants after 8 hours of growth in the absence or presence of IPTG (100µM ), as measured by flow cytometry. B. Mean of the median sfGFP fluorescence intensities for the different variants after 8 hours of growth in the absence or presence ofIPTG (100µM). Means are calculated over three replicate cultures grown on different days. Error bars: +\-standard deviation. Resulting dynamic range for each variant is shown above the corresp[onding bars. . NC: negative control (grey), the WT L. gasseri without plasmid. C. Response of the different variants to increasing concentrations of IPTG after 8 hours of growth.. Data points show the means of the median fluorescence intensities for three replicates cultures grown on different days. Error bars: +\-SD. NC: negative control, the WT L. gasseri without plasmid. Data is color-coded for the different strains, as shown. D. Kinetic response of the different variants after induction by IPTG (100µM). Shown are the medians of sfGFP fluorescence intensities measured by flow cytometry. NC: negative control, the WT L. gasseri without plasmid. Data is color-coded for the different strains, as shown. E. Sequences of the four Pls variants and comparison with Phyper-spank original sequence. Mutations are shown in red.

Table 1 :

 1 Phyper-spank and Pls variants characteristics measured after 8 hours of growth in the absence and presence of IPTG (100µM). Leakage is the median fluorescence intensity in the non-induced state minus median autofluorescence of negative control cells (without plasmid). The fold change (or dynamic range) is the ratio of median fluorescence intensities measured in induced over uninducedconditions. The swing isthe median fluorescence intensities measured in induced minus that measured in uninduced conditions. The EC50 is the half-maximal effective concentration derived from the fit shown in Figure2C.

Acknowledgements

We thank members of our groups and of the CBS for fruitful discussions and feedback. We thank M. Jules and Olivier Delumeau (INRA MICALIS, Jouy-en-Josas) for raising our interest in using B. subtilis parts in lactic acid bacteria. This work was supported by grants from INSERM, the Bettencourt-Schueller Foundation, the Canceropole Grand Sud-Ouest (JB), and a CNRS-ATIP (GC). EF is a recipient of Ph.D fellowships from the French Ministry of Research and the French Ligue

Data processing

Each experiment was replicated three times on three different days. For each strain, data are summarized as the mean and standard deviation of the median of the sfGFP fluorescence intensity across these three replicates. Fold change and swing of each variant were calculated from the mean of the fluorescence median. The goodness of fit and the EC50 for each data set were calculated by applying nonlinear regression using the agonist versus response variable slope function in GraphPad Prism.

Authors contribution

EF, JB, and GC designed the study. EF performed the experiments. EF, JB, and GC analyzed the results and wrote the paper.

Associated content

Supporting Information is available.

-Supplementary materials -Raw data -DNA sequences

Competing interests:

The authors declare that no competing interests exist.


Xavier Pautrel
email: xavier.pautrel@univ-nantes.fr

Health-enhancing activities and the environment: How competition for resources make the environmental policy beneficial

Keywords: Growth, Environment, Health, Overlapping generations

In a two-period overlapping generations model, this paper demonstrates that the relationship between environmental taxation and economic activity (output level and output growth) becomes inverted-U shaped when the detrimental impact of pollution on health and the private decision of each working-age agent to improve her health are taken into account. In particular, a tighter environmental tax is more likely to promote (rather than to harm) output-level and -growth when health is very sensitive to pollution, the weight of health in preferences is high, the polluting capacity of the production technology is high and the rate of natural purification of pollutants is low.

The inverted-U shaped relationship between environmental tax and economic activity is due to a positive effect arising from the competition for resources between the final output sector and the health-enhancing activities. This offsets the conventional detrimental "drag-down effect" for low values of the environmental tax.

We also demonstrate that the link between environmental tax and lifetime welfare is inverted-U shaped as well. Finally, we investigate the social optimum and the determinants of the optimal environmental tax.

Introduction

Is environmental policy harmful to economic activity, both in terms of output level and growth? Does the reduction of pollution imply such a heavy cost for the economy that the gains from a better environment quality are not able to offset it? At the theoretical level, the answers are not clear-cut.

The aim of this article is to contribute to the debate focusing on one of the more striking features of pollution: its detrimental impact on health. In contrast to previous works in the field that take into account the impact of pollution on life expectancy (i.e. mortality), we focus our attention on the influence of pollution on illness and disability (morbidity). Indeed, a growing set of empirical evidence finds a link between pollution and chronic diseases such as cancer, diabetes, hypertension, stroke, heart disease, pulmonary conditions and mental disorders, amongst others. Even if pollution is not the main source of these diseases, it contributes as a factor favouring their emergence or their worsening.

In contrast to mortality which affects mainly the oldest old, illness and disability due to chronic diseases primarily impacts the working-age population, leading to important losses of productivity and rising health-expenditure for the 30-50 old age people. [START_REF] Devol | An unhealthy america: The economic burden of chronic disease -charting a new course to save lives and increase productivity and economic growth[END_REF], from the Milken Institute, estimate that the seven common chronic diseases represent $277 billion spent annually on treatment and a lost productivity equal to $1.1 trillion per year for the United-States.

Rising health-expenditure for working-age people and the time they have to devote in order to manage to their chronic disease creates competition for resources that could be used in alternative ways, for example growth-led activities or final production activities. 1 The main contribution of this paper is to demonstrate that such resource competition is a channel of transmission between the environmental policy, economic activity and welfare, when the detrimental impact of pollution on the health of the working-age population is taken into account.

To demonstrate this point and analyze its implications, we use a two-period overlapping generations model, in the line with previous articles addressing intergenerational environmental issues, 2 introducing an explicit link between the environment and health.

1 Competition for resources in the relation between health and growth has already been studied in several articles (see Dormont et al., 2007, for details and references). Nevertheless, most of these contributions view better health as an increase in life expectancy. Empirically, [START_REF] Dormont | Health expenditure growth: Reassessing the threat of ageing[END_REF] find changes in morbidity that induce savings which more than offset the increase in spending due to population ageing.

2 For example, [START_REF] John | An overlapping generations model of growth and the environment[END_REF] who analyze the potential conflict between economic growth and the maintenance of environmental quality when consumption degrades environmental quality while Following empirical evidence, we assume that health is negatively influenced by pollution but is improved by the investment in health-enhancing activities made by each agent during her working-age. 3 Pollution is a by-product of final output production and in the competitive economy the government taxes final output to limit pollution emissions.

Our first contribution is to demonstrate that if the detrimental effect of pollution on health-status and an endogenous investment in health by the working-age are taken into account, the link between environmental taxation and economic activity (final output level and growth) is inverted-U shaped. Indeed, a tighter environmental policy has two opposite effects. First, because the environmental tax is imposed on final output, it reduces the rewards to production factor: the conventional "drag-down effect". Second, it reduces pollution and therefore improves health-status of the working-age population. Agents reduce their investment in health-enhancing activities and the freed resources are used to increase consumption and production. This second effect, called the "resource competition effect" is positive. As a result, for low values of the environmental tax, the second effect offsets the first one, and the environmental policy promotes the level and the growth rate of output, as well as overall welfare.

Our second contribution is to show that the greater the room for improvement in the health status, the more likely the environmental policy promotes economic activity and growth. This occurs when the rate of natural health decay is low, the efficiency of health care spending is low, the weight of health in preferences is high, the share of labor in final output is high, the rate of natural purification of pollutants is low, the polluting capacity of production technology is high, and the detrimental impact of pollution on health and the elasticity of pollution stock with respect to the net flow of pollution are high. Most of these criteria exist in the most developed countries, and because the detrimental impact of pollution on health is well-documented worldwide, our results show that an active environmental policy in these countries is highly probable to promote growth and output levels: ie the positive gains in terms of health and growth should be higher than the losses from factor rewards.

Finally, we investigate the social optimum and the optimal environmental tax. We investment in environmental maintenance promotes it. See also [START_REF] John | Short-lived agents and the long-lived environment[END_REF] who investigate the effects of environmental taxation distinguishing the horizon of the agents and the economy. For models with non-renewable resources, see [START_REF] Agnani | Growth in overlapping generation economies with non-renewable resources[END_REF], [START_REF] Kemp | The under-exploitation of natural resources: A model with overlapping generations[END_REF], [START_REF] Mourmouras | Competitive equilibria and sustainable growth in a life-cycle model with natural resources[END_REF][START_REF] Mourmouras | Conservationist government policies and intergenerational equity in an overlapping generations model with renewable resources[END_REF].

3 Because we consider that agents are suffering from chronic diseases that required medical care when they are young we do not assume that the poor health agents expend more on medical care when they are elderly like [START_REF] Gutiérrez | Dynamic inefficiency in an overlapping generation economy with pollution and health costs[END_REF]. And, conversely to [START_REF] Williams | Environmental tax interactions when pollution affects health or productivity[END_REF][START_REF] Williams | Health effects and optimal environmental taxes[END_REF], we do not assume that agents who are ill do not work.

demonstrate that the higher the weight of health in preferences, the elasticity of pollution stock with respect to the net flow of pollution, the detrimental impact of pollution on health and/or the part of labor in production; the higher the optimal environmental tax.

The paper is set out as follows. Section 2 gives empirical evidence on the link between pollution related illness and disability. Section 3 presents the model. Section 4 studies the competitive equilibrium and the impact of environmental taxation on the steady-state.

Section 5 investigates the social optimum and the optimal environmental tax. Section 6 examines two extensions: AK endogenous growth and the impact of health on labor productivity. Section 7 concludes.

Pollution, illness and disability

The majority of environmental economic contributions investigating the detrimental impact of pollution on health focus on life expectancy.4 Nevertheless, pollution also affects morbidity by favoring the development of certain chronic diseases that do not always lead to death but have durable detrimental impacts in terms of illness and disability.

These chronic diseases, that include cancer, diabetes, hypertension, stroke, heart disease, asthma and obesity are a growing burden because they represent 60% of all deaths worldwide and they are a major source of disability (see WHO, 2004WHO, , 2005)). While they greatly impact developing countries, they also represent a significant burden for developed economies, such as the United-States, England, Canada, Israel, and Australia, as reported by [START_REF] Suhrcke | Chronic disease: An economic perspective[END_REF] and [START_REF] Zhang | Chronic diseases and labour force participation in australia[END_REF] among others. [START_REF] Devol | An unhealthy america: The economic burden of chronic disease -charting a new course to save lives and increase productivity and economic growth[END_REF] find that more than half of all Americans (55.8%) suffered from one or more chronic diseases in 2003.

The causes of increased risk of developing chronic diseases are well-established: mainly unhealthy diet, physical inactivity and tobacco use. Nevertheless, recent studies emphasize the importance of environmental factors in the development and the worsening of some of these chronic diseases, especially in developed countries. According to [START_REF] Briggs | Environmental pollution and the global burden of disease[END_REF] about 8-9% of the total disease burden may be attributed to pollution in developed countries.5

In the case of air pollution for example, it is well-documented that particulate matter pollution, nitrogen dioxide, sulfur dioxide and ozone favour the onset of asthma crises and aggravate respiratory diseases, and that carbon monoxide affects mental function... (for a study on European countries see [START_REF] Katsouyanni | Ambient air pollution and health[END_REF]. Particulate matter air pollution also plays a role in the development (pathogenesis in medical terms) of cardiovascular disease [START_REF] Brook | Air pollution and cardiovascular disease: a statement for healthcare professionials from the expert panel on population and prevention science of the american heart association[END_REF] and lung cancer [START_REF] Pope | Lung cancer, cardiopulmonary mortality, and long-term exposure to fine particulate air pollution[END_REF], and it may be particularly harmful to people with diabetes and people with hypertension [START_REF] Pope | cardiovascular mortality and long-term exposure to particulate air pollution: epidemiological evidence of general pathological pathways of disease[END_REF]. Moreover, air pollution could be deleterious to vascular health especially for the people with diabetes [START_REF] Rajagopalan | Particulate pollution and endothelial function: Déjà vu all over again in the air[END_REF][START_REF] O'neill | Diabetes enhances vulnerability to particulate air pollutionassociated impairment in vascular reactivity and endothelial function[END_REF]. Water pollution [START_REF] Paulu | Tetrachloroethylene-contaminated water in Massachusetts and the risk of colon-rectum, lung, and other cancers[END_REF], [START_REF] Valent | Burden of disease attributable to selected environmental factors and injury among children and adolescents in Europe[END_REF]] and industrial pollution [START_REF] Nadal | Metal pollution of soils and vegetation in an area with petrochemical industry[END_REF], [START_REF] Chen | Health risk assessment on humans exposed to environmental polycyclic aromatic hydrocarbons pollution sources[END_REF], [START_REF] Schuhmacher | Long-term study of environmental levels of dioxins and furans in the vicinity of a municipal solid waste incinerator[END_REF]] are also reported as detrimental to health. [START_REF] Lang | Association of urinary bisphenol a concentration with medical disorders and laboratory abnormalities in adults[END_REF] recently found that higher unary BPA concentration (chemical pollution) is associated with cardiovascular diagnoses and with diabetes for adults. [START_REF] Smink | Exposure ti hexachlorobenzene during pregnancy increases the risk of overweight in children aged 6 years[END_REF] show that children in the higher exposure group of HCB (a pesticide extensively used before being banned in the United-States) have an increased risk of being overweight and obese, even if the mother is normal weight. The epidemiological association between persistent organic pollutants and diabetes has been also demonstrated by [START_REF] Rylander | A cross-sectional study of the association between persistent organochlorine pollutants and diabetes[END_REF] and [START_REF] Porta | Persistent organic pollutants and te burden of diabetes[END_REF]. Even if such an association does not necessarily prove a causal relation, as noted by [START_REF] Jones | Environmental pllution and diabetes: a neglected association[END_REF], such a relation could be biologically explained (see [START_REF] Remillard | Linking dioxins to diabetes: epidemiology and biological plausibility[END_REF].

Furthermore, the link between obesity and diabetes seems to be dependant on pollution. [START_REF] Lee | Association between serum concentrations of persistent organic pollutants and insulin resistance among nondiabetic adults: results from the national health and nutrition examination survey[END_REF] find that the expected association between obesity and diabetes is absent in people with low concentrations of persistent organic pollutants in their blood. Furthermore, they find that the association between obesity and diabetes become stronger as the concentrations of such pollutants in the blood increase. [START_REF] Lockwood | Diabetes and air pollution[END_REF] give further references on the association between exposure to dioxins and the development of diabetes or altered insulin metabolism. He notes that obesity by itself may increase the risk of diabetes: because insulin is concentrated in body fat, obese individuals are likely to have an increased dioxin body burden that could explain the link between the rise in the prevalence of diabetes and the "epidemic" of obesity (see also Ando et al., 2002, for empirical evidence).

One important feature of chronic diseases is that they do not only affect the really old. They also impact working-age people through illness and disability: according to the World Health Organization (WHO, 2004) 56% of those suffering from some form of disease are people aged 15-59 in high-income countries. [START_REF] Lakdawalla | Are the young becoming mre disabled[END_REF] demonstrate that disability increases more rapidly among the young while it decreases for the elderly. They find that one factor is the growth in asthma which appears to be enough to explain the change in disability rate. This is confirmed by [START_REF] Bhattacharya | Chonic disease and severe disability among working-age populations[END_REF]: "Recent work has shown that rates of severe disability, measured by the inability to perform basic activities of daily living, have been rising in working age populations. At the same time, the prevalence of important chronic diseases has been rising, while others falling, among working age populations. Chronically ill individuals are more likely than others to have activity of daily living limitations. " [START_REF] Perlkowski | The impact of health on employment, wages, and hours worked over life-cycle[END_REF] study the influence of health on working conditions (wages and hours worked) making a distinction between short-term and long-term illness and the age at which each illness appears (because it implies different adjustments for young people at the beginning of careers and for old people closed to retirement). They distinguish between temporary and permanent illness and find that the adverse effects of permanent health problems peak at an age of onset in the 40s for men and in the 30s for women. The biggest decline in wages and hours worked are observed for individuals whose problems started at those ages. This leads to a major economic impact in terms of labor productivity, labor supply, education or savings, as shown by Suhrcke et al. (2006a). Chronic diseases mainly lead to a reduction of the productivity of the labor-force (even if agents are not sick enough to stop working or even if they are not sick but it is the member of the family who is sick) and the increase in disability of working people. For the US in 2003, [START_REF] Davis | Health and productivity amoung u.s. workers[END_REF] estimated that 55 million workers out of 148 million, aged 19 to 64 reported an inability to concentrate at work due to their own illness or that of a family member and 69 million workers reported missing days due to illness. Regarding the effects of chronic diseases on workers' productivity, [START_REF] Blanc | The work impact of asthma and rhinitis: Findings from a population-based survey[END_REF] demonstrate, with a sample of 125 adults in Northern-California that "Both asthma and rhinitis negatively affect work productivity.

Those with asthma are less likely to be employed at all, while among those remaining on the job, rhinitis is a more potent cause of decreased work effectiveness. The economic impact of asthma and rhinitis and related conditions may be under-appreciated". For Australia, van [START_REF] Van Leeuwen | Chronic pain and reduced work effectiveness: The hidden cost to australian employers[END_REF] estimate that "while the impact of reduced work effectiveness on days worked with pain on productivity is uncertain, it has the potential to account for the majority of lost productivity costs associated with chronic pain.". [START_REF] Devol | An unhealthy america: The economic burden of chronic disease -charting a new course to save lives and increase productivity and economic growth[END_REF] from the Milken Institute estimate that the seven common chronic diseases represent for the United-States, lost productivity equal to $1.1 trillion per year.

The second important feature of chronic illness is that it places a burden on health care and welfare systems. For the United States in 2003, [START_REF] Devol | An unhealthy america: The economic burden of chronic disease -charting a new course to save lives and increase productivity and economic growth[END_REF] estimate that $277 billion is spent annually on treatment. All these resources could be used in alternative activities promoting final (non health) consumption and growth. 6Furthermore, chronic illness has major implications in terms of occupational choices, that can not be supported (or funded) by the public health-care system or insurance contracts. Since Grossman's seminal work of 1972, in addition to goods, time appears as an important input in the health production function by influencing the next period's health capital level. More recently, time and time costs in health production have become important in the economic analysis of obesity, for example (see [START_REF] Cutler | Why have americans become more obese?[END_REF]Philipson and Posner, 2008, among others). As emphasized by [START_REF] Mullahy | No time to lose? time constraints and physical activity[END_REF], increasing levels of physical activity is now viewed as a means to improve health outcomes. In their study based on the Bureau of Labor Satistics' American Time Use Survey, [START_REF] Russell | Health-related activities in the american time use survey[END_REF] noted that 11.3% of American adults (in 2003-2004) reported spending time (mean, 108 minutes) on activities related to health on their designed day and 5.6% (86 minutes) reported making medicine, giving themselves a shot, exercising or therapy for medical reasons. Physical activity has been shown to reduce the risk of developing or dying from heart diseases, diabetes, colon cancer and high blood-pressure. The US Department of Health and Human services advises to be in good health, to do 150 minutes of physical activity at moderate level or 75 minutes at vigourous level each week.

As a result, chronic illness forces agents to allocate more time to health-enhancing activities, time they could use to home or market production. Thus, both the increase in health-care expenditure and the rising investment in health-enhancing activities lead to a competition for resources that could be detrimental for economic activity, growth and/or welfare. And because pollution favours chronic illness and disability, this influence could be a new channel of transmission between the environmental public policy and the economy.

We investigate this point in the following sections.

The model

Let's consider an overlapping generations model. A new generation is born at each date t = 1, 2, ..., and lives for two periods. The number of individuals born at time t is L.

Population is constant. Individuals are non-altruistic: the old do not care for the young and the young do not care for the old. The preferences of an agent born in period t are represented by the following utility (from van Zon and Muysken ( 2001)):

log c φ 1t h 1-φ t + θ log c φ 2t+1 h 1-φ t+1
where c 1t and c 2t+1 are respectively consumption in young and old age, h t and h t+1 are respectively private health-status in young and in old age. Parameter θ = (1 + ι) -1 where ι > 0 is the subjective discount rate of the agent. Parameter φ > 0 (respectively 1φ)

captures the relative importance of consumption (respectively health), in utility.7

Each young agent is endowed with one unit of time. She supplies ν t ∈ [0, 1[ of this unit of time in final production and uses her remaining time 1ν t as an investment in health care activities to improve her health status.8 She earns a wage income ν t w t where w t is the wage rate.

The private health status of an agent born at period t evolves from period t and period t+1 according to two opposing forces [START_REF] Aisa | Endogenous longevity, health and economic growth: a slow growth for a longer life[END_REF]). On the one hand, biological processes involve a natural decay in health simply as time passes. Following [START_REF] Grossman | On the concept of health capital and the demand for health[END_REF] and [START_REF] Cropper | Measuring the benefits from reduced morbidity[END_REF] we further assume that health depreciates over time with the stock of pollution (denoted S t ). On the other hand, the time invested in health-enhancing activities (1ν t ) mitigates against this deterioration. Therefore, for an agent born at t, private health status evolves from period t to period t + 1 as:

h t+1 -h t = η(1 -ν t ) -ξS γ t h t (1)
with η > 0 is a productivity scalar of health-enhancing activities. 9 . Parameter γ ≥ 0 measures the influence of pollution stock on the natural decay ξ ∈]0, 1[.10

A consumer, born at t, works during the first period of her life, consumes an amount c 1t and saves the remainder of her revenue. The budget constraint of a young agent is

c 1t + s t = ν t w t
where s t denotes saving in youth. The budget constraint of an old person is

c 2t+1 = (1 + r t+1 )s t
where r t+1 is the interes rate paid on saving held from period t to t + 1.

Firms operate through perfect competition using physical capital and labor to produce a final good with a constant return Cobb-Douglas technology:

Y t = Ãt K α t N 1-α t , α ∈]0, 1[
where Y t is the aggregate output, K t is the aggregate productive capital and N t is labor. Ãt is a productive scalar, assumed as constant for the moment: Ãt ≡ A. Capital depreciates fully in the production process.11

The stock of pollution S from period t to period t + 1 increases because of the net flow pollution in the current period t and decreases according to the rate of natural purification of pollutants µ ∈]0, 1[. The net flow of pollution in period t is the ratio between pollution emissions in period t, denoted E t , and the abatement activities funded by the government, denoted D t . We assume, as conventional, that polluting emissions arise from final production such that

E t = zY t , z ∈]0, 1[
Parameter z measures the polluting capacity of the technology. Consequently the stock of pollution in period t + 1 is defined as:

S t+1 = zY t D t χ + (1 -µ)S t (2)
where χ > 0 is the exogenous elasticity of pollution stock with respect to the net flow of pollution E/D.

Competitive equilibrium

A representative agent born in period t maximizes her utility function taking wages, the interest rate and the stock of pollution as given. She chooses consumption at both ages (c 1t , c 2t+1 ) and the proportion of time ν t she uses in production:

max {c 1t ,c 2t+1 ,νt} log c φ 1t h 1-φ t + θ log c φ 2t+1 h 1-φ t+1 s.t.      c 1t + s t = ν t w t c 2t+1 = (1 + r t+1 )s t h t+1 = η(1 -ν t ) + (1 -ξS γ t ) h t
The first-order condition gives saving:

s t = θ 1 + θ ν t w t (3)
and the allocation of time into production:

ν t = φ(1 + θ) η(1 -φ)θ h t+1 ∈]0, 1[ (4)
Because ν t < 1, the health status of the old h t+1 is bounded to

η(1 -φ)θ φ(1 + θ) . 12
We assume that abatement D t is provided by the government as a public good and financed by an environmental tax τ on the source of pollution Y t such that the public budget is balanced at each date:

D t = τ Y t .
The law of motion of the stock of pollution,

given by equation (2) becomes:

S t+1 = z τ χ + (1 -µ)S t
Firms maximize their profit π t = (1τ )Y t -(1 + r t )K tw t N t and the demand for capital and labor is

(1 -τ )αY t /K t = 1 + r t (1 -τ )(1 -α)Y t /N t = w t
The good market clearing yields:

K t+1 = s t L
and the labor market clearing equates labor demand N t to labor supply ν t L N t = ν t L

12 See van Zon and Muysken (1997, p.5) for a justification of the health status boundary. .

The competitive equilibrium may be summarized by the following relations:

K t+1 = (1 -α) θ 1 + θ (1 -τ )AK α t (ν t L) 1-α (5) ν t = φ(1 + θ) η(1 -φ)θ h t+1 ⇒ ν t = (1 -φ)θ φ(1 + θ) + 1 -1 1 + (1 -ξS γ t ) h t η (6) h t+1 = η(1 -ν t ) + (1 -ξS γ t )h t (7) S t+1 = z τ χ + (1 -µ) S t (8)
The steady-state is defined here as an equilibrium where physical capital, private health-status, pollution stock and the allocation of labor in production are constant at K ⋆ , h ⋆ , S ⋆ and ν ⋆ respectively, defined as:

S ⋆ = S(τ ) ≡ (z/τ ) χ µ (9) h ⋆ = H(τ ) ≡ η φ(1 + θ) (1 -φ)θ + ξ (z/τ ) χ µ γ -1 (10)
ν ⋆ = V(τ ) ≡ φ φ + ξ(1 -φ)θ (1 + θ) (z/τ ) χ µ γ -1 (11)
Consequently, the health status and the allocation of time in production are positively affected by the environmental tax τ .

From equations ( 5), ( 10) and ( 11), the steady-state value of the physical capital stock is:

K ⋆ = A (1 -τ ) 1 1-α V(τ ) with A ≡ (1 -α) θ 1 + θ A 1/(1-α) L. Because Y ⋆ = 1 + θ θ(1 -α) (1 -τ ) -1 K ⋆ , 13
we obtain the steady-state value of final output as a function of the environmental taxation τ :

Y ⋆ = A 1 (1 -τ ) α 1-α B + ξ z χ µ γ τ -χγ -1 with A 1 ≡ B 1 + θ θ(1 -α) A and B ≡ φ(1 + θ) (1 -φ)θ . 13 Note that Kt+1 = stL = " θ 1 + θ « νt(1 -τ )(1 -α)Yt/νt = " θ 1 + θ « (1 -τ )(1 -α)Yt.
Proposition 1. When endogenous investment in private health-status and the detrimental impact of pollution on health are taken into account, the relationship between the steady-state output and environmental taxation has an inverted-U shape.

Below (respectively above) an environmental tax-level denoted τ and defined as

χγξ z χ µ γ τ -1 - α 1 -α Bτ χγ - α 1 -α + χγ ξ z χ µ γ = 0. ( 12
)
a tighter environmental taxation raises (respectively lowers) the steady-state level of output Y ⋆ .

Proof. See Appendix B

To understand the basic mechanism of this result, let us remember that

Y ⋆ = A 1 (1 -τ ) α 1-α Ia V(τ ) Ib (13)
Environmental tax influences the steady-state level of output through two channels: the direct impact of the environmental taxation on the rewards to labor (see overbrace Ia in equation 13) and the (indirect) impact on the allocation of labor in the manufacturing sector (see overbrace Ib in equation 13).

The former (negative) is the conventional "drag-down" effect of the environmental tax that reduces factor rewards -captured by (1-τ ) α 1-α . The latter (positive) is a new channel of transmission due to the "competition for resources" between health enhancing activities and production activities that affects the supply of labor into the final production sector ν ⋆ = V(τ ). Because pollution has a detrimental impact on the evolution of private healthstatus, by reducing the net flow of pollution and therefore the stock of pollution, the environmental policy improves the private health-status of the agents. Consequently, each agent decides to reduce her investment in health enhancing activities (1-ν ⋆ decreases) and to raise her labor supply to productive activities (ν ⋆ increases). In this way, the tighter environmental tax frees ressources that were allocated to health enhancing activities and that are now reallocated to production, leading to a higher level of steady-state output and a steady-state physical capital. Consequently the competition for resources between output production and health enhancing activities associated with the negative impact of pollution to health is a source of a new channel trough which the environmental policy may promote economic activity.

When γ = 0, the evolution of health-status is independent from pollution and therefore the investment of each agent in health-enhancing activities is not affected by the environmental tax: ν ⋆ is independent from τ . In such a case, the competition for resources is not affected by the environmental policy and only the "drag-down"effect remains: the environmental policy is detrimental for growth. In the same way, the "competition for resources effect" no longer holds when there is no endogenous investment in health-enhancing activities.

Proposition 2. Endogenous investment in health and the detrimental impact of pollution on health are two necessary conditions to obtain Proposition 1.

Proof. See above and Appendix B.

Considering the influences of parameters on the environmental tax-level τ , enables us to understand why these two opposite effects of the environmental policy leads to an inversed-U shaped relationship between the environmental tax and the steady-state output level. These influences are summarized in the following table (see equation ( 12) in Proposition 1 and Appendix B for the demonstration):

ξ η φ θ α µ z ---+ --+ Table 1.

Parameter Changes and Responses of τ

Because it is cumbersome to obtain analytically the influence of γ and χ on the tax-level τ , we use a numerical application. We first calibrate the model assuming that the length of each period is 30 years, as usual in the literature. The first period covers ages 20 to 50, and the second period covers ages 50 to 80. We use the U.S. economy as a benchmark.

From De La Croix and Michel ( 2002), we choose α and θ following the standard choice in the RBC literature, that is α = 0.36 and a quarterly psychological discount factor equal to 0.99. This implies that θ = 0.99 (4×30) = 0.3. We use the calibration by [START_REF] Van Zon | Health, education and endogenous growth[END_REF] for the values of ξ and φ. Finally, parameter η is chosen to obtain a private health-status higher than unity to enable the welfare to be positive.

Benchmark value of parameters are summarized in Table 2:

α θ φ ξ A η
L χ µ γ 0.36 0.3 1/2 0.2 50 0.8 1 1 0.5 1.5 Proof. See Appendix B and Table 3.

When the detrimental effects of a dirty environment on health are great, the gains in terms of health status of reducing the pollutant emissions are very significant and the "competition for resources effect" that leads to an increase in labor supply has a greater influence than the "drag-down effect" that reduces factor rewards and as a consequence saving and physical capital accumulation. Neverthless, these positive gains diminish with the increase in the tax rate because the possible improvements in health-status due to the tax are reduced. At the same time, the losses from the reduction of factor rewards increase in the tax rate such that for the environmental tax-level τ , they offset the gains, and a further increase in τ leads to a decrease in the steady-state output level.

Consequently, the greater the room for improving the environment and private healthstatus through environmental policy, the more beneficial the environmental policy is likely to be for the economy.

The numerical application also enables us to investigate the impact of environmental taxation on the steady-state lifetime welfare, as well. As shown by Figure 1, there is a relationship that is inverted-U shaped for similar reasons as the relationship between the steady-state output and the environmental tax is inverted-U shaped. The purpose of this section is to investigate the determinants of the optimal environmental taxation in the presence of endogenous investment in private health and the detrimental impact of pollution on private health.

In the centralized economy, the central planner aims at maximizing the welfare of agents:

max {c 1 ,c 2 ,ν,K,D} log c φ 1 h 1-φ + θ log c φ 2 h 1-φ s.t.            Y = AK α (νL) 1-α = Lc 1 + Lc 2 + D + K h = η(1 -ν)/(ξS γ ) S = (E/D) χ /µ E = zY
As demonstrated in Appendix C, consumption at a young and old age are related 14 c1 = θc 2 14 A bar ¯denotes optimal value.

with c1 = (1 -α)φ α(1 + θ) Aαφ (1 -α)χγ(1 -φ) + φ 1/(1-α)
The optimal allocation of time to production is:

ν = φ
and the optimal stock of physical capital is

K = Aαφ (1 -α)χγ(1 -φ) + φ 1/(1-α) φL
Consequently, the optimal final output is

Ȳ = A 1/(1-α) αφ (1 -α)χγ(1 -φ) + φ α/(1-α) φL Abatement activity is given by D = (1 -α)χγ(1 -φ) (1 -α)χγ(1 -φ) + φ Ȳ
and the optimal stock of pollution in the steady-state is

S = z (1 -α)χγ(1 -φ) + φ (1 -α)χγ(1 -φ) χ /µ
Consequently, the optimal value of the environmental tax that enables the decentralized economy to attain the optimal stock of pollution in the steady-state is

τ = (1 -α)χγ(1 -φ) (1 -α)χγ(1 -φ) + φ (14)
The following proposition arises from this expression.

Proposition 4. The higher the weight of health in preferences (1φ), the elasticity of pollution stock with respect to the net flow of pollution (χ), the detrimental impact of pollution on health (captured by γ) and/or the share of labor in production (1α); the higher the optimal environmental tax is.

Proof. From equation ( 14), it is straightforward that ∂ τ /∂φ < 0, ∂ τ /∂χ > 0, ∂ τ /∂γ > 0 and ∂ τ /∂α < 0.

Nevertheless, an optimal environmental tax is not sufficient to enable the steadystate equilibrium to be optimal because in the decentralized economy the agents do not internalize the impact of their labor supply decisions on final output and the net flow of pollution. Consequently, agents do not supply enough time to output production. Hence, to obtain the optimal individual labor supply ν, the government have to subsidise healthenhancing activities at a rate (see Appendix D):

τ ν = 1 - ξθ (1 + θ) z χ µ γ (1 -α)χγ(1 -φ) (1 -α)χγ(1 -φ) + φ -γχ
The environmental tax τ associated with the subsidy τ ν make the steady-state decentralized equilibrium optimal.

Extensions

AK endogenous growth

In this section, we consider that external learning by doing à la Romer (1986) exists , 15 such that the productivity scalar Ãt evolves as physical capital:

Ãt = AK α t
to obtain an interest rate independent from physical capital.

Consequently, the final output becomes

Y t = AK t (ν t L) 1-α
and the law of motion of physical capital is given by

K t+1 = (1 -α) θ 1 + θ (1 -τ )AK t (ν t L) 1-α
At the steady-state, physical capital and output evolves at a constant positive rate of growth g ⋆ ≡ K t+1 /K t -1, that is, using equation ( 11)

g ⋆ = A ′ (1 -τ ) B + ξ z χ µ γ τ -χγ α-1 with A ′ ≡ (1 -α) θ 1 + θ AL 1-α B 1-α .
The influence of the environmental taxation on the growth rate at the steady-state is given by

∂g ⋆ /∂τ = -A ′ B + ξ (z/τ ) χ µ γ α-1-1 τ -χγ × B + ξ z µ γ 1 + (1 -α)χγ(1 -τ -1 )
Consequently ∂g ⋆ /∂τ > 0 if the last term in the right-hand side of the previous expression is negative:

Bτ χγ + ξ z µ γ 1 + (1 -α)χγ(1 -τ -1 ) < 0
Because the left-hand side of the inequality is a monotonic increasing function of τ with lim τ →0

= -∞ and lim τ →1

= Bτ χγ + ξ z µ γ > 0, there exists a unique τg defined as

Bτ χγ g + ξ z µ γ 1 + (1 -α)χγ(1 -τ -1 g ) = 0
such that for τ < τg (respectively τ > τg ) we have ∂g ⋆ /∂τ > 0 (resp. ∂g ⋆ /∂τ < 0).

Proposition 5. Under the assumption of a learning-by-doing source of growth à la [START_REF] Romer | Increasing returns and long run growth[END_REF], the introduction of an endogenous private health care expenditure and a detrimental impact of pollution on health makes the environmental taxation policy good for growth when the level of taxation is not too high.

Proof. See above.

Health affects labor productivity

As emphasized in the introduction, chronic diseases affect the economy through the huge losses of productivity they create. As a result, it is expected that a tighter environmental taxation will reduce this productivity loss by reducing pollution and increasing the health-status of workers. To investigate how the "productivity effect" associated with the "competition for resources effect" could further improve the beneficial impact of an environmental policy, we introduce the impact of health on the productivity of labor.

We continue to consider that poor health agents expend more on medical care when they are young and not elderly as does [START_REF] Gutiérrez | Dynamic inefficiency in an overlapping generation economy with pollution and health costs[END_REF]. We do not assume that ill agents do not work [START_REF] Williams | Environmental tax interactions when pollution affects health or productivity[END_REF][START_REF] Williams | Health effects and optimal environmental taxes[END_REF]). Instead we consider that a better health-status makes workers more productive and that absenteeism due to illness does not occur.16

The technology to produce final output becomes:

Y t = A t K α t (h ε t N t ) 1-α (15)
where ε ≥ 0 measures the effect of health on labor productivity. The introduction of health-dependent labor productivity leaves the model unchanged except for the law of motion of physical capital (equation 5):

K t+1 = (1 -α) θ 1 + θ (1 -τ )AK α t (h ε t ν t L) 1-α (16)
As a result, the steady-state value of physical capital becomes:

K ⋆⋆ = A (1 -τ ) 1 1-α V(τ )H(τ ) ε
and the steady-state expression of final output is now:

Y ⋆⋆ = A 2 (1 -τ ) α 1-α B + ξ z χ µ γ τ -χγ -(1+ε)
with A 2 ≡ A 1 η ε . Proposition 6. When the effect of health on labor productivity is taken into account, positive effect of an environmental tax on ouput-level is enhanced and the tax level under which a thigher environmental tax increases output level is higher. The environmental policy is more likely to increase final output.

Proof. The tax level, denoted τ , for which ∂Y /∂τ = 0 is

(1 + ε)χγξ z χ µ γ τ -1 - α 1 -α B τ χγ - α 1 -α + (1 + ε)χγ ξ z χ µ γ = 0.
It is straightforward that for ε = 0, τ = τ . Furthermore the LHS of the equation is increasing in ε because τ ∈]0, 1[. Therefore from the theorem of implicit function we find that ∂ τ /∂ε > 0. Therefore τ > τ when ε > 0.

Conclusion

This paper investigates how environmental tax affects the economy (output-level andgrowth, welfare) when the detrimental impact of pollution on health is taken into account and working-age individuals have to invest in health-care activities to limit the damaging influence of pollution.

In a two-period overlapping generations model, this paper demonstrates that the relationship between environmental taxation and economic activity ( output level and output growth) becomes inverted-U shaped, when the detrimental impact of pollution on health and the private decision of each working-age agent to improve her health are taken into account. In particular, a tighter environmental tax is more likely to promote (rather than to harm) output-level and -growth when health is very sensitive to pollution, the weight of health in preferences is high, the polluting capacity of the production technology is high and the rate of natural purification of pollutants is low.

The inverted-U shaped relationship between environmental tax and economic activity is due to a positive effect arising from the competition for resources between the final output sector and the health-care sector that offsets the conventional detrimental "dragdown effect" for low values of the environmental tax.

We also demonstrate that the link between environmental tax and lifetime welfare is inverted-U shaped as well. Finally, we investigate the social optimum and the determinants of the optimal environmental tax.

This contribution shows that, besides life expectancy, there are other ways in which pollution may affect health and health affect economic activity. Those ways may be new channels of transmission of the environmental pollution to economic activity.

Firms in the health sector operate under perfect competition and maximize their profit

m t H t = w t (1 -ν t ) such that: m t A H = w t (17)
The final output sector is defined as in section 2.

In the competitive equilibrium, besides her consumption of final good, the consumer buys when she is young H t units of health care service for an amount of health-care expenditure equal to m t H t . The private health status of the agent evolves between period t and t + 1 as:

h t+1 -h t = ηH t -ξS γ t h t
The program of the consumer consists in choosing consumption when young and old and health-care services H t in order to maximize her lifetime utility subject to her budget constraint when young and old and the evolution of her health-status:

max {c 1t ,c 2t+1 ,Ht} log c φ 1t h 1-φ t + θ log c φ 2t+1 h 1-φ t+1 s.t.      c 1t + m t H t + s t = w t c 2t+1 = (1 + r t+1 )s t h t+1 = ηH t + (1 -ξS γ t ) h t
The first-order condition gives saving (using equation 17):

s t = θ 1 + θ (w t -m t H t ) = θ 1 + θ ν t w t
Furthermore, ν t is given by

h t+1 = ηA H (1 -φ)θ φ(1 + θ) ν t
that is ν t increases in h t+1 . These two expressions are similar to those found when healthcare investment is in the form of time (see equations ( 3) and ( 4)) when A H = 1.

B Environmental taxation in competitive equilibrium

The influence of environmental tax on the steady-state level of output is given by:

dY ⋆ /dτ = Y ⋆ τ -γχ (1 -τ ) -1 B + ξ z χ µ γ τ -χγ -1 × χγξ z χ µ γ τ -1 - α 1 -α Bτ χγ - α 1 -α + χγ ξ z χ µ γ .
First-order conditions are φc -1 1 = λL (18)

θφc -1 2 = λL that is c 2 = θc 1 λ(F ′ K (•) -1) = (1 + θ)χγ(1 -φ)F ′ K (•)/F (•) (19) (1 -φ)(1 + θ) γχF ′ ν (•)/F (•) + (1 -ν) -1 = λF ′ ν (•) (20) λ = (1 + θ)χγ(1 -φ)D -1 (21)
Equations ( 19) and ( 21) give:

D = Y (1 -1/F ′ K (•)) = Y -α -1 K
Furthermore, from (18), we obtain λ = φ c 1 L , consequently

D = φ -1 (1 + θ)χγ(1 -φ)c 1 L
and consequently the market equilibrium gives

Y = c 1 L(1 + θ) 1 + φ -1 χγ(1 -φ) + K that is c 1 L = Y -K (1 + θ) [1 + φ -1 χγ(1 -φ)]
In the same way, the market equilibrium may be written as

Y = (1 + θ)c 1 L + K + Y -α -1 K that is (1 + θ)c 1 L = (α -1 -1)K Consequently Y -K [1 + φ -1 χγ(1 -φ)] = (α -1 -1)K that is Y = α -1 φ -1 [(1 -α)χγ(1 -φ) + φ] K (22)
Finally, equation ( 20) gives

(1 -φ)(1 + θ) (1 -α)γχ + ν 1 -ν = φ(1 + θ) (α -1 -1) (1 -α)Y /K that is ν 1 -ν = φ 1 -φ ⇒ ν = φ
From ( 22), we can obtain the express of the steady-state physical capital in the centralized economy, denoted K c :

K = Aαφ (1 -α)χγ(1 -φ) + φ 1/(1-α)

D Subsidy to health-enhancing activities

We re-write the competitive equilibrium assuming that the government subsidises the health-enhancing activities of agents by paying a subsidy τ ν to the opportunity cost of health-enhancing activities (that is the foregone wage (1ν t )w t ) that is funded by a lump-sum tax denoted a t .

The budget-constraint for the young agent born at period t becomes: The expression of savings s t is the same than equation (3) and the percentage of time into output production ν t is still given by equation ( 4). But now, the percentage of time in health care expenditure:

c 1 +
u t = 1 - φ(1 + θ) η(1 -φ)θ h t+1 - φ 1 η(1 -φ)θ h t+1 = 1 - φ(1 + θ) + φ 1 η(1 -φ)θ h t+1
and the percentage of time to leisure is

1 -u t -ν t = φ 1 η(1 -φ)θ h t+1
The competitive equilibrium may be summarized by the following relations:

K t+1 = (1 -α) θ 1 + θ (1 -τ )AK α t (ν t L) 1-α ν t = φ(1 + θ) η(1 -φ)θ h t+1 h t+1 = ηu t + (1 -ξS γ t )h t u t = 1 - φ(1 + θ) + φ 1 η(1 -φ)θ h t+1 S t+1 = z τ χ + (1 -µ) S t
In the steady-state, ν, u, h, S, K, Y remain constant. Consequently, the stock of pollution in the steady-state S ⋆ is always given by ( 9), and the private health-status is

h ⋆ = η B ′ + ξ z χ µ γ τ -χγ -1 with B ′ ≡ φ(1 + θ) + φ 1 (1 -φ)θ
. When leisure is taken into account, health care expenditure is lowered and the steady-state health status too, but the influence of the environmental tax is not modified.

The percentage of time allocated to the output production is:

ν ⋆ = B B ′ + ξ z χ µ γ τ -χγ -1

Figure 1 :

 1 Figure 1: Steady-state lifetime welfare as a function of τ

  α)χγ(1φ) (1α)χγ(1φ) + φ Ȳ

Table 2 .

 2 Benchmark value of parametersand the results of the numerical application is reported in Table3

		τ	Ŷ	ν⋆	ĥ⋆	Ŝ⋆	Ŵ ⋆
	Benchmark 21.6%	9.73 0.896 2.068 1.84	1.28
	γ = 0.5	7.77% 10.77 0.90 2.089 5.15	1.46
	γ = 1	15.73% 10.12 0.894 2.065 2.54	1.35
	γ = 2	25.97% 9.48 0.901 2.080 1.54	1.23
	γ = 2.5	29.13% 9.31 0.907 2.094 1.37	1.20
	χ = 1.25	23.14% 9.769 0.909 2.099 1.667 1.28
	χ = 1.5	24.07% 9.821 0.921 2.125 1.515 1.29
	χ = 1.75	24.65% 9.875 0.930 2.146 1.387 1.293
	χ = 2	25.01% 9.928 0.937 2.163 1.279 1.298

Table 3 .

 3 Steady-state τ for different values of γ and χThe third proposition stems from the Table1 and Table 3.

Proposition 3. When the negative impact of the environment on health and the endogenous decision of each agent to invest her resources in health enhancing activities are taken into account, we demonstrate that the environmental taxation will be more likely to improve the steady-state level of output if the rate of natural health decay (ξ) is low, the efficiency of health care spending (η) is low, the weight of health in preferences (1φ) is high, the share of labor in final output (1α) is high, the rate of natural purification of pollutants (µ) is low, the polluting capacity of production technology (z) is high, and the detrimental impact of pollution on health (γ) and the elasticity of pollution stock with respect to the net flow of pollution χ are high.

  s t + a t = ν t w t + τ ν (1ν t )wand in the competitive equilibrium, the program of the agent ismax {c 1t ,c 2t+1 ,νt,ut} 1t + s t = ν t w t c 2t+1 = (1 + r t+1 )s t h t+1 = ηu t + (1 -ξS γ t ) h t

			log c φ 1t h 1-φ t	(1 -u t -ν t ) φ 1 + θ log c φ 2t+1 h 1-φ t+1
				  	c
				s.t.
				 
	s t =	θ 1 + θ	ν t w t

t

The maximization of lifetime utility gives

s t = θ 1 + θ (ν t w t + τ ν (1ν t )w ta t ) and

because government budget constraints require a t = τ ν (1ν t )w t , we obtain

Among others, see[START_REF] Jouvet | Longevity and environmental quality in an olg model[END_REF],[START_REF] Mariani | Life expectancy and the environment[END_REF],[START_REF] Varvarigos | Envvironmental quality, life expectancy and sustainable economic growth[END_REF] or[START_REF] Pautrel | Reconsidering the impact of pollution on long-run growth when pollution influences health and agents have a finite-lifetime[END_REF][START_REF] Pautrel | Pollution and life expectancy: How environmental policy can promote growth[END_REF]. The aforementioned articles by[START_REF] Gutiérrez | Dynamic inefficiency in an overlapping generation economy with pollution and health costs[END_REF] and[START_REF] Williams | Environmental tax interactions when pollution affects health or productivity[END_REF][START_REF] Williams | Health effects and optimal environmental taxes[END_REF] are exceptions.

See the 2003 special issue of the British Medical Bulletin (issue 1, volume

68) for related references.

Here, we are reasoning for a given life expectancy and we do not consider that life expectancy may rise or decrease. As a result, we do not integrate the fact that additional years of life increases health-care expenditure (seeSuhrcke et al., 2008, p.15).

We do not integrate green preferences because we will assume in the following that health status is affected by pollution.

We could assume that there exists a sector that produces health care services with labor and therefore a part ν of labor is allocated to manufacturing production and a part 1 -ν is allocated to health-care production. We would find the same qualitative results (see Appendix A). Consequently, what we call investment in health-enhancing activities could be viewed as health-care expenditure. Our modelling has the advantage of leading to a simpler exposition of the model and the results. In Appendix E, we demonstrate that results are not modified when leisure time is introduced.

Note that here, we model a linear relationship between the health-enhancing activities and the evolution of health-status which could be not empirically relevant. As demonstrated by[START_REF] Skinner | The efficiency of medicare[END_REF]: "nearly 20 percent of total Medicare expenditure appears to provide no benefit in terms of survival, nor is it likely that this extra spending improves the quality of life". Our assumption is made for simplicity.

We impose γ ≥ 0 to investigate the absence of a detrimental impact of pollution on health, that is γ = 0. Nevertheless, it is expected that γ > 1, that is the higher the stock of pollution, the higher the detrimental effect of pollution, even if there is no empirical evidence on such a linear relationship.

The production process is over the course of a generation. If the annual depreciation rate is 10% (which is empirically relevant), 96% of the capital stock is depreciated over the course of a 30 year generation. Therefore, we assume that capital is fully used up in the production process. Considering a positive depreciation rate would not change the qualitative results.

Following Romer (1986), production factors remain paid at their marginal after environmental tax cost.

We take into account presenteeism, i.e. a worker present but with reduced productivity rather than absenteeism, i.e. a worker absent, because it accounts for not only worker health but also health of his family. For the US in 2003,[START_REF] Davis | Health and productivity amoung u.s. workers[END_REF] estimated that 55 million workers over 148 million ages 19 to 64 reported the inability to concentrate at work because of their own illness or that of their family.

Appendix

A The basic model with a health sector

Let us consider in this section, that there exists in the economy a health sector that produces an amount H t of health-care services at period t, using labour and the following technology

where A H is a productivity scalar and 1ν t is the part of labour (normalized to unity) allocated to the health sector.

The influence of environmental tax on the steady-state level of output is positive if and only if

Because the left-hand side of the inequality is a decreasing monotonic function of τ with lim τ →0

= +∞ and lim

, there exists a unique τ ∈]0, 1] under which the inequality is verified. This τ is denoted τ and is defined as:

When γ = 0, the left-hand side of the inequality is independent from τ and negative.

Consquently, when γ = 0, we have dY ⋆ /dτ < 0.

To find how parameters affect the tax level τ , let rewrite the expression of τ as:

From the theorem of implicit function, we obtain

C The optimum

In the centralized economy, the central planner aims at maximizing the welfare of the agents:

The Lagrangian may be written as:

and the individual labor supply is given by

In the steady-state equilibrium, the private health-status remains constant at:

The higher the subsidy to health-enhancing activities, the higher the health-status in the steady-state and the lower the individual supply of labor ν ⋆ .

The subsidy to health-enhancing activities that enables replication of the optimal allocation of time between health-enhancing activities and production (denoted

E The competitive equilibrium with leisure

Up to this point, we investigated the competition for resources assuming that the time unit at the disposal of the agent is divided between two different "occupational" activities: production which enables an income to be earned and investment in health care which enables the agent to stay in good health.

In this section, we consider that each agent also values leisure-time and that she may adjust her leisure-time according to the level of her health-status: the healthier is the agent, the greater the utility of one minute of leisure.

To do so, we continue to denote ν ∈]0, 1[, the percentage of time the agent chooses in output production and we denote u ∈]0, 1[ the percentage of time spent as investment in health status. Consequently 1uν ∈]0, 1[ represents the percentage of time To keep things simple, preferences are written as follows

with φ 1 > 0 captures the weight of leisure in utility.

The law of motion of the private health-status (equation 1) is modified as follows

Finally, we obtain

A.

Because A ′ 1 < A 1 , the steady-state level of output is lower when leisure is taken into account, but the effect of the environmental taxation in the steady-state level of output and the expression of τ , the environmental taxation under which a higher tax promotes economic activities, are not modified.