Baiming Ren

Nathalie Lyczko

Yaqian Zhao
email: yaqian.zhao@ucd.ie

Ange Nzihou

Simultaneous hydrogen sulfide removal and wastewater purification in a novel alum sludge-based odor-gas aerated biofilter

Keywords:

Simultaneous hydrogen sulfide removal and wastewater purification in a novel alum sludge-based odor-gas aerated biofilter

Baiming Ren, Nathalie Lyczko, Yaqian Zhao, Ange

Introduction

Traditionally, the mission of wastewater treatment is to remove various pollutants (i.e. organic and nutrients) from wastewater and reduce the environmental impact on receiving waters. However, wastewater treatment processes also release unpleasant odors even virus (i.e. SARS-CoV-2) via aerosol [START_REF] Ji | Where do we stand to oversee the coronaviruses in aqueous and aerosol environment?[END_REF] and pose a threat to public health [START_REF] Ren | Current status and outlook of odor removal technologies in wastewater treatment plant[END_REF]. Generally, odors emanating from wastewater treatment processes are composed of a mixture of various chemical compounds, including nitrogenous compounds (e.g., ammonia (NH 3 )), inorganic sulfur compounds (e.g., hydrogen sulfide (H 2 S)), volatile organic compounds (e.g., butanone) etc. [START_REF] Gruchlik | Use of alum for odor reduction in sludge and biosolids from different wastewater treatment processes[END_REF]. Seventy-eight kinds of main odor-producing compounds have been reported [START_REF] Talaiekhozani | An overview of principles of odor production, emission, and control methods in wastewater collection and treatment systems[END_REF]. In particular, majority of the odor emissions from sewerage facilities contain H 2 S, and H 2 S could also cause the corrosion of wastewater networks, environmental pollution and human diseases [START_REF] Omri | Performance study of biofilter developed to treat H2S from wastewater odour[END_REF].

Due to the complex nature of odorants released from various treatment process in wastewater treatment plants (WWTPs), the efficient and low-cost odor removal strategy seems a "bottleneck issue" worldwide. Literature has highlighted physical/chemical and biological methods being used for odor abatement [START_REF] Barbusinski | Biological methods for odor treatment -A review[END_REF][START_REF] Lebrero | Odor assessment and management in wastewater treatment plants: a review[END_REF]. However, the physical/chemical odor control methods, such as adsorption approach and chemical scrubbing, normally have techno-economical demerits of high operation and management (O&M) costs to regularly change the sorbents and generate hazardous byproducts etc. [START_REF] Lin | Biological sulfur oxidation in wastewater treatment: a review of emerging opportunities[END_REF][START_REF] Vikrant | Biofiltration of hydrogen sulfide: Trends and challenges[END_REF]. It seems the "end-of-thepipes" technologies are relatively impractical for odor control both in environmental and "circular economy" terms. Recent reviews [START_REF] Alfonsín | Selection of odour removal technologies in wastewater treatment plants: a guideline based on Life Cycle Assessment[END_REF][START_REF] Barbusinski | Biological methods for odor treatment -A review[END_REF][START_REF] Ren | Current status and outlook of odor removal technologies in wastewater treatment plant[END_REF] have revealed that the investigation on odor removal has centered on developing the environmentally-friendly benign approaches that can remove odor via simple, efficient, and effective processes. For example, biofiltration (BF) is the most favorite applied biological methods for odor abatement, which had been intensively studied over the last several decades [START_REF] Barbusinski | Biological methods for odor treatment -A review[END_REF]. In BF, the humified odorants are forced through a packed bed where microbes and water are attached as a biofilm. Odorants are biodegraded by the microorganisms and sorbed by filter material [START_REF] Omri | Performance study of biofilter developed to treat H2S from wastewater odour[END_REF]. BF aims at maximize odor removal efficiency by utilizing various microbes and optimized design parameters such as moisture and pH of the media. However, there is one concern that BF presents large land requirements and likely further improvements to reduce its physical footprint [START_REF] Alfonsín | Selection of odour removal technologies in wastewater treatment plants: a guideline based on Life Cycle Assessment[END_REF].

As a biofilter-like facility, constructed wetlands (CWs) have been widely used as an efficient, cost-effective and sustainable wastewater treatment technology [START_REF] Parde | A review of constructed wetland on type, technology and treatment of wastewater[END_REF]. In particular, vertical flow constructed wetlands (VFCWs) have been developed due to their low footprint and high efficiency for pollutant removal [6,[START_REF] Ilyas | The performance of the intensified constructed wetlands for organic matter and nitrogen removal: a review[END_REF]. VFCWs only require 1-3 m 2 /PE area [START_REF] Liu | Intensified nitrogen transformation in intermittently aerated constructed wetlands: removal pathways and microbial response mechanism[END_REF]. Unfortunately, nitrogen (N) removal efficiency remains around 50% in most operated VFCWs under a nitrogen loading rate of 0.6-2.0 g m -2 d -1 [START_REF] Hu | Appropriate technologies for upgrading wastewater treatment plants: methods review and case studies in China[END_REF], which is still far from satisfactory and fails to meet increasingly stringent N discharge standards [START_REF] Liu | Intensified nitrogen transformation in intermittently aerated constructed wetlands: removal pathways and microbial response mechanism[END_REF]. Thus, improving the internal aerobic concentration is vital for enhancing the nitrification process and increasing the efficiency of biological N removal to improve the overall removal efficiency in VFCWs. Recent research has focused on enhanced N removal performance via intensified strategies [START_REF] Hou | Pathway governing nitrogen removal in artificially aerated constructed wetlands: Impact of aeration mode and influent chemical oxygen demand to nitrogen ratios[END_REF], such as effluent recirculation, artificial aeration, tidal flow and flow direction reciprocation [START_REF] Ilyas | The performance of the intensified constructed wetlands for organic matter and nitrogen removal: a review[END_REF][START_REF] Wu | Development of constructed wetlands in performance intensifications for wastewater treatment: a nitrogen and organic matter targeted review[END_REF]. In particular, artificial aeration has proven to be a good solution for creating aerobic conditions favorable for nitrification. Earlier, continuous artificial aeration strategies were employed in most studies [START_REF] Maltais-Landry | Nitrogen transformations and retention in planted and artificially aerated constructed wetlands[END_REF]. However, it inhibits denitrification process due to the lacking of anoxic zone which could decrease the removal of total nitrogen (TN). Thereafter, intermittent aeration was developed by creating alternating aerobic and anaerobic conditions. It benefits both nitrification and denitrification processes and enhances the N removal efficiency [START_REF] Stefanakis | Presence of bacteria and bacteriophages in full-scale trickling filters and an aerated constructed wetland[END_REF][START_REF] Zhou | Nitrogen removal responses to biochar addition in intermittent-aerated subsurface flow constructed wetland microcosms: enhancing role and mechanism[END_REF]. However, the high operation costs of the aeration strategy remains questionable [START_REF] Wu | Development of constructed wetlands in performance intensifications for wastewater treatment: a nitrogen and organic matter targeted review[END_REF][START_REF] Ilyas | The performance of the intensified constructed wetlands for organic matter and nitrogen removal: a review[END_REF].

Alum sludge is waterworks residue. It is an easily, largely available by-product produced in drinking water treatment plant when aluminum sulfate was used as coagulant for raw water purification. It has been intensively discussed as a recycle product for various environmental applications in line with "circular economy" [START_REF] Ren | Alum sludge as an efficient sorbent for hydrogen sulfide removal: experimental, mechanisms and modeling studies[END_REF]. Especially for the great adsorption capacity of phosphorus (P) [START_REF] Zhao | Pilot field-scale demonstration of a novel alum sludge-based constructed wetland system for enhanced wastewater treatment[END_REF], many studies have demonstrated that alum sludge could adsorb a number of heavy metals and semimetals from wastewaters and polluted soils, including Cd, Cr, Co, Cu, Pb, Hg, Ni, Zn, Mo, V, Ga, As, Se, and B [START_REF] Shen | Global profile of heavy metals and semimetals adsorption using drinking water treatment residual[END_REF]. Moreover, considerable research has been carried out to reuse dewatered alum sludge as substrate in CWs due to its large specific surface area (60-300 m 2 /g) and porous structure [START_REF] Zhao | Pilot field-scale demonstration of a novel alum sludge-based constructed wetland system for enhanced wastewater treatment[END_REF]. In particular, alum sludge has been successfully reused at first time for H 2 S removal in a fixed-bed reactor, with the capacity of 374.2 mg H 2 S g -1 [START_REF] Ren | Alum sludge as an efficient sorbent for hydrogen sulfide removal: experimental, mechanisms and modeling studies[END_REF]. Herein, it is highly promising and reasonable to employ alum sludge as the substrate in novel "odor-gas aerated" CW or biofilter for odor (H 2 S) eliminating.

As such, it could be a "win-win" solution to integrate BF concept with aerated biofilter by using the existing available resource "waste gas" for biofilter's aeration. It could counteract the shortage of each technology (e.g., partially remedy the high operation costs of the aeration process and reducing the extra land requirement) to achieve dualgoals of simultaneous waste gas purification and the enhanced wastewater treatment. Moreover, efficient resources recycle/reuse and odor abatement remains the challenges for WWTPs worldwide. In particular, majority of WWTPs in China have been required to meet third-grade (GB3838-2002) surface water standard [START_REF] Zhang | Current status of urban wastewater treatment plants in China[END_REF]. However, currently they are unable to meet the new guidelines. This proposed approach of using odor generated and collected from the WWTPs for biofilter aeration could be used for upgrading the existing WWTPs, by adding a series of novel "odor aerated" biofilters for ensuing effluent treatment as polishing stage in the existing plants to achieve enhanced wastewater efficiency and meanwhile for odor abatement. Hence, this study developed the alum sludge-based odor-gas aerated biofilter. It aims to remove the pollutants in wastewater and simultaneously to eliminate H 2 S generated from wastewater treatment facility.

Material and methods

Three columns setup and operation strategy

Dewatered alum sludge cakes were obtained from the Carmaux water treatment plant, located in Tarn, France. The plant supplies 3014 m 3 /day drinking water to 7,000 consumers, by treating reservoir water using aluminum chloride hydroxide sulfate (HYDREX 3531 Veolia, France) as a coagulant. The characterization of the alum sludge was found that the Al was 33%, with the details are reported in Ren et al., [START_REF] Ren | Alum sludge as an efficient sorbent for hydrogen sulfide removal: experimental, mechanisms and modeling studies[END_REF].

The experiment (as illustrated in Fig. 1) was composed of three parallel columns constructed with identical plexiglass columns of diameter 15 cm and height 100 cm. In each column, gravel (15 cm depth) was filled into the bottom as the support medium, while 80 cm depth air-dried alum sludge cake (moisture content 63%, particle size 3-6 cm) was filled as the main wetland medium layer, which gives a total volume of 16.8 L with initial porosity of 42% (7 L liquid contained). Aeration of column 1 and 2 was supplied with a diffuser placed on the support (gravel) layer (10 cm high from the bottom) and controlled with an air flow meter. Before the trial was formally operated, three columns were seeded with activated sludge collected from a local municipal WWTP for two weeks.

Synthetic wastewater (prepared from tap water, containing the compounds of NaAC 480 mg/L, NH 4 Cl 120 mg/L, KH 2 PO 4 40 mg/L, CaCl 2 15 mg/L, MgSO 4 12 mg/L, with the COD, TN, TP approximately of 300, 30, 10 mg/L, respectively), was introduced by peristaltic pumps into the column from the top while effluent was drained from the bottom. Prior to the trial, determination of the optimal aeration strategies was carried out by applying minimum aeration times and aeration rates in column in order to achieve the highest TN removal efficiency. Preliminary results suggest that the best TN removal efficiency (85%) was reached at the aeration time and rate of 4 h/d and 20 L/h, respectively. Thus, the system was operated in batch mode and intermittently aerated four hours (2 h + 2 h) per day at an airflow rate of 20 L/h (i.e. filling influent at 8:00 am, first aeration period was from 8:30 to 10:30 and second aeration period was from 16:30 to 18:30). To simulate the waste gas, column 1 (Al-OAF) was intermittently "aerated" with steel gas cylinder containing 200 ppm H 2 S based on air (a mixture of air and 200 ppm H 2 S). Column 2 (Al-AF) was intermittently aerated with compressed air from a compressor in the lab. Column 3 (Al-F) was un-aerated and served as a control. The experiment was then kept operation for six months under room temperature.

Analysis and data interpretation

Water samples were taken from the influent and effluent of the three columns at the same time (8:00 am) every 3 days. Samples were analyzed immediately for organic matter (as COD), TP, TN and ammonia-nitrogen (NH 4 -N). A Merck Nova 60 spectrophotometer was used to analyze COD; A Shimadzu VCPH 5050A was used for TN analysis; NH 4 + -N was analyzed by ion chromatography (Dionex ICS 3000); TP were determined by inductively coupled plasma-atomic emission spectroscopy (ICP-AES) on a HORIBA Jobin Yvon Ultima 2. Scanning electron microscopy (SEM) (Philips XL30 ESEM apparatus, FEI Company) coupled with an energy-dispersive X-ray spectroscopy (EDX analysis) was employed for alum sludge/substrate analysis. Fig. 2. Pollutants removal performance in three columns.

To collect effluent H 2 S, a plexiglass cap (as shown in Fig. 1) with rubber seal ring inside to prevent the H 2 S leakage was tightly covered on the column 1 (Al-OAF) during the aeration period. The output H 2 S concentration was monitored and record every two minutes by using a gas analyzer from BW Technologies (Gas Alert QUATTRO). After the aeration, the cap was immediately removed to keep the surface water in contact with atmosphere. Pollutant removal efficiency (RE) for the trials was calculated as a cumulative percent removal between the influent and the effluent. H 2 S loading rate (LR, g⋅(m 3 ⋅h) -1 ) and elimination capacity (EC, g⋅(m 3 ⋅h) -1 ) was calculated by Eq. ( 1) and [START_REF] Barbusinski | Biological methods for odor treatment -A review[END_REF].

LR = Q V × C in (1) EC = Q V × (C in -C out ) (2)
where Q is the gas flow rate (m 3 ⋅h -1 ), V is the net bed volume (m 3 ), C in and C out are the inlet and outlet H 2 S concentrations, respectively (mg⋅m -3 ).

Results and discussion

Pollutants removal performance in three columns

The COD removal performance of the three columns is shown in Fig. 2 (a). After one month's acclimatization, three parallel columns showed stable COD removal performance. The average effluent COD concentrations of column 1 (Al-OAF), 2 (Al-AF) and 3 (Al-F), which corresponds to the columns aerated with H 2 S, air and the blank column, were 17.27 ± 8.51, 15.99 ± 5.19 and 217.47 ± 31.66 mg/L, respectively. The COD removal performance in column 3 (Al-F) was not satisfied during the six-month trials. Previous studies clearly revealed that in un-aerated VFCWs the level of DO was low (0.12-1.3 mg/L) [START_REF] Ilyas | The performance of the intensified constructed wetlands for organic matter and nitrogen removal: a review[END_REF], and thus organic matter cannot be well oxidized and removed. Contrarily, it can be observed that intermittent aeration (column 1 and 2) significantly enhanced the COD removal by 92.1% and 92.6%, respectively. This is due to the intermittent air supply creating a better aerobic environment, as organic compounds are degraded by bacteria [START_REF] Tang | Microbial coupling mechanisms of nitrogen removal in constructed wetlands: a review[END_REF][START_REF] Li | Efficiency and kinetics of conventional pollutants and tetracyclines removal in integrated vertical-flow constructed wetlands enhanced by aeration[END_REF]. It is worth to note that the COD removal performance of column 1 (Al-OAF) presents almost the same level with column 2 (Al-AF). It indicated that H 2 S has limited influence on the COD effluent concentration as it can be observed that six-month average effluent COD concentration of column 1 (Al-OAF) was slightly higher (7%) than that in column 2 (Al-AF).

NH 4 -N removal performance is presented in Fig. 2(b). The average NH 4 -N effluent concentrations of column1 (Al-OAF), 2 (Al-AF) and 3 (Al-F) were 1.45 ± 2.98, 0.60 ± 1.66 and 27.34 ± 1.58 mg/L, respectively. Nitrification converts NH 4 -N into NO 3 -N for eventual TN elimination. Nitrification often requires a relatively high DO concentration, thus the NH 4 -N removal performance in column 1 (Al-OAF) and 2 (Al-AF) were greater than column 3 (Al-F). It also reveals that the aeration strategy is successful, and the nitrification process was sufficient to convert NH 4 -N to NO 2 -N and NO 3 -N. The TN removal performance of the three parallel columns is shown in Fig. 2(c). The average effluent TN concentrations of column 1 (Al-OAF), 2 (Al-AF) and 3 (Al-F) were 3.79 ± 2.44, 2.75 ± 1.55 and 30.46 ± 0.84 mg/L, respectively. Lu et al. [START_REF] Lu | Recent advances in the enhanced nitrogen removal by oxygen-increasing technology in constructed wetlands[END_REF] showed that N transformations in the biofilter mainly include volatilization, ammonification, nitrification, denitrification and matrix adsorption. Liu et al. [START_REF] Liu | Intensified nitrogen transformation in intermittently aerated constructed wetlands: removal pathways and microbial response mechanism[END_REF] states that different designs could achieve various N removal processing in VFCWs, and intermittent aeration could change N pathways when treating synthetic domestic wastewater.

In the un-aerated Al-F (column 3), the DO level was always low, resulted in negligible nitrification process. As a result, the TN removal performance was limited. Compared with the column 3 (Al-F), the intermitted aeration in column 1 (Al-OAF) and 2 (Al-AF) increased the TN removal of 87.6% and 91.0%, respectively. The turbulence of data between day 27 and day 45 in Fig. 2 was likely due to the room temperature decreasing before the heat supply season. The trial started from October and the temperature was about 6-8 • C in the early November, which affects mass transfer and biological reactions. Furthermore, the relatively unstable organic loading (during day 27 and day 45) may have an important effect on the denitrification process, which also leads to a turbulence removal rate of TN. After this period, TN removal performance of three columns remains well and stable.

TP removal is illustrated in Fig. 2 (d). The average TP effluent concentrations in column 1 (Al-OAF), 2 (Al-AF) and 3 (Al-F) were 0.18 ± 0.20, 0.15 ± 0.14 and 0.13 ± 0.16 mg/L, respectively. The major processes responsible for P removal in CWs and biofilters are adsorption by the substrate, chemical precipitation, and assimilation into microbial and plant biomass [START_REF] Ilyas | The effects of different aeration strategies on the performance of constructed wetlands for phosphorus removal[END_REF]. Furthermore, precipitation can refer to the reaction of phosphate ions with metallic cations such as Al, Ca or Fe, forming amorphous or poorly crystalline solids. Some important mineral precipitates in the wetland environment are: apatite, hydroxyl apatite, variscite, wavellite. In addition to direct chemical reaction, P can coprecipitate with other minerals, such as carbonate minerals, such as calcite (calcium carbonate).

From Fig. 2(d), all the columns performed equally well at removing P attributed to the P removing ability of the alum sludge, most likely P adsorption and precipitation reactions with the alum sludge. In most biofilters, it is often a challenge to achieve concurrent high removal efficiencies for P and organic matter. Therefore, the wetland substrate to own potential P adsorption ability is vital for concurrent P removal and high organic matter, i.e. COD, removal. That is why the dewatered alum sludge was used as main wetland substrate [START_REF] Zhao | Pilot field-scale demonstration of a novel alum sludge-based constructed wetland system for enhanced wastewater treatment[END_REF].

Overall removal efficiency in three columns

The six-month average removal efficiencies (RE, %) of COD, TN, NH 4 -N, TP in the three columns are shown in Fig. 3. The RE of COD in column 1 (Al-OAF), 2 (Al-AF) and 3 (Al-F) was 94.3 ± 3.0, 94.8 ± 1.9 and 30.2 ± 11.0%, respectively. Compared with the un-aerated Al-F (column 3), the introduction of H 2 S and air into column 1 (Al-OAF) and 2 (Al-AF) greatly improved COD removal efficiency. Generally, the unaerated VFCWs and biofilters could achieve 61-80% of COD removal [START_REF] Fan | Nitrogen removal in intermittently aerated vertical flow constructed wetlands: Impact of influent COD/N ratios[END_REF][START_REF] Ilyas | The performance of the intensified constructed wetlands for organic matter and nitrogen removal: a review[END_REF][START_REF] Wu | Decentralized domestic wastewater treatment using intermittently aerated vertical flow constructed wetlands: impact of influent strengths[END_REF]. However, in this study the column 3 (Al-F) presented a relatively low RE of COD. This is possibly due to the single stage CW as the wastewater cannot have sufficient time to contact with the biofilm. Intermittently aerated with H 2 S and air could achieve 64.1 and 64.6% additional RE of COD. In particular, no significant difference in COD removal was observed between column 1 (Al-OAF) and 2 (Al-AF). This demonstrates that H 2 S aeration has no negative influence on the COD removal. From the literature, the average value of RE of COD in intensified CWs is estimated at 84 ± 12% [START_REF] Ilyas | The performance of the intensified constructed wetlands for organic matter and nitrogen removal: a review[END_REF]. Obviously, H 2 S aeration is a kind of intensification of CWs. The TN RE of column 1 (Al-OAF), 2 (Al-AF) and 3 (Al-F) was 86.2 ± 14.2, 91.6 ± 5.4 and 8.8 ± 3.8%, respectively. This result indicates that the introduction of H 2 S and air could effectively improve TN removal. Literature survey showed a large variation in N removal of un-aerated VFCWs (26-65%) [START_REF] Ilyas | The performance of the intensified constructed wetlands for organic matter and nitrogen removal: a review[END_REF]. In this study, column 3 (Al-F) only presented a quite low average RE of 8.8 ± 3.8%, mainly due to the low DO level of batch operation with a negligible nitrification process. In column 1 (Al-OAF) and 2 (Al-AF), the additional TN RE was 77.4 and 82.8%, respectively was achieved. Compared with the study reported by Fan et al. [START_REF] Fan | Nitrogen removal in intermittently aerated vertical flow constructed wetlands: Impact of influent COD/N ratios[END_REF], additional removal of 64% of TN in an traditional intermittent aeration VFCW was obtained. Wu et al. [START_REF] Wu | Decentralized domestic wastewater treatment using intermittently aerated vertical flow constructed wetlands: impact of influent strengths[END_REF] also reported that a 63% additional removal of TN was achieved in a VFCW with intermittent aeration fed with domestic wastewater. Thus, it indicated that the H 2 S aeration in column 1 (Al-OAF) could achieved a high TN RE, even higher than the traditional intermittent aeration VFCWs. However, the six months average RE of TN in column 1 (Al-OAF) was 5.4% lower than the column 2 (Al-AF). This may be due to the long term H 2 S feeding, and in the presents of microorganisms which could oxidize sulfides such as thiobacillus bacteria. Some of the reaction products (e.g., sulfates) could be converted into sulfuric acid in the wastewater, while sulfuric acid may inhibit microbial activity, resulted in a reducing TN removal [START_REF] Vikrant | Biofiltration of hydrogen sulfide: Trends and challenges[END_REF]. More importantly, the current result was contrary with Zhang et al. [START_REF] Zhang | Simultaneous improvement of waste gas purification and nitrogen removal using a novel aerated vertical flow constructed wetland[END_REF] who conducted the trial of using waste gas for VFCW aeration. They pointed out that the best TN RE (51.88 ± 3.42%) was observed and the worse TN removal performance (23.14 ± 2.12%) was in the VFCW intermittently aerated with air. The reason may be from the different aeration strategies [START_REF] Tunçsiper | Nitrogen removal in an aerobic gravel filtration-sedimentation pondconstructed wetland-overland flow system treating polluted stream waters: effects of operation parameters[END_REF]. In this study intermittent aeration at an airflow rate of 20L/h for two cycles in a day was applied, while each cycle of 120 mins could successfully cerate the alternative nitrification-denitrification environment and thus facilitated the TN removal.

The NH 4 -N RE of the three columns was 92.1 ± 18.2, 97.8 ± 6.5 and 6.6 ± 5.9%, respectively. In column 1 (Al-OAF) and 2 (Al-AF), the RE of NH 4 -N was over 90%, indicating the functional nitrification process. However, in the un-aerated Al-F (column 3), the DO level was always low, causing an anaerobic environment, and thus may have negligible nitrification. Moreover, the six-month average RE of NH 4 -N in column 1 (Al-OAF) was slightly (5.7%) lower than in column 2 (Al-AF). This may be due to the long-term biological H 2 S degrading processes generating sulfuric acid to inhibit microbial activity. This is contrary to Zhang et al. [START_REF] Zhang | Simultaneous improvement of waste gas purification and nitrogen removal using a novel aerated vertical flow constructed wetland[END_REF], who demonstrated a better nitrification performance in a waste gas aerated system. However, Zhang et al. [START_REF] Zhang | Simultaneous improvement of waste gas purification and nitrogen removal using a novel aerated vertical flow constructed wetland[END_REF] used a lower H 2 S concentration (0.0045 ± 0.0006 mg/m 3 ), which comes from a sequencing batch reactor (SBR). The low concentration could be used by microorganisms for reproductive growth and enhanced nitrification.

The average TP RE of the three columns was 98.1 ± 1.7, 98.1 ± 1.7 and 98.3 ± 1.9%, respectively. No significant differences were found among the three columns. Likewise, Ilyas and Masih [START_REF] Ilyas | The effects of different aeration strategies on the performance of constructed wetlands for phosphorus removal[END_REF] found that artificial aeration did not have a significant influence on P removal as P removal in alum sludge based-CWs depends mainly on adsorption of the reactive substrate, i.e., alum sludge [START_REF] Zhao | Pilot field-scale demonstration of a novel alum sludge-based constructed wetland system for enhanced wastewater treatment[END_REF]. Previous studies suggest that in all types of CWs, TP removal varied between 40 and 60% with removed load ranging between 45 and 75 g P m 2 /year depending on CW types and flow loading [START_REF] Nuamah | Constructed wetlands, status, progress, and challenges. The need for critical operational reassessment for a cleaner productive ecosystem[END_REF]. In the current study, the superior TP removal performance was attributed by the alum sludge, which has been extensively investigated as a media for TP removal from various CWs, the P removal mechanisms lies in the various chemical composition of alum sludge, specifically its content of Ca, Al or Fe, three elements that can react with P under different environmental conditions [START_REF] Yang | Global development of various emerged substrates utilized in constructed wetlands[END_REF].

H 2 S removal efficiency

Previous studies have shown that acclimatization could shorten or even eliminate the biofilter start-up phase [START_REF] Yuan | Use of mature compost as filter media and the effect of packing depth on hydrogen sulfide removal from composting exhaust gases by biofiltration[END_REF]. Thus, before the H 2 S was introduced to column 1 (Al-OAF), air was supplied at a flow rate of 20 L/h and 4 h per day for two weeks to acclimatize and develop the biofilm (The same manner was applied for column 2).

After two weeks' acclimatization, 200 ppm H 2 S based on air was introduced into column 1 (Al-OAF) at the same flow rate of 20 L/h. The H 2 S concentrations in the outlet of column 1 (Al-OAF) were continuously recorded during the 4 h aeration period every day. As illustrated in Fig. 4, H2S was completely removed during the "2h + 2 h" aeration periods each day. In particular, this complete H 2 S removal efficiency was keeping stable on each day over the six-month trial period. The loading rate (100% removal) as well as elimination capacity of column 1 (Al-OAF) was 867.3 g⋅(m 3 ⋅h) -1 . This capacity was significantly higher than that of Zhang et al., [START_REF] Zhang | Simultaneous improvement of waste gas purification and nitrogen removal using a novel aerated vertical flow constructed wetland[END_REF]. This is likely due to an optimized intermediate aeration strategy (2 h + 2 h), which could create a proper aerobic environment for the H 2 S biodegradation process.

H 2 S removal mechanisms

In this study, the Al-OAF (column 1) acted as a "biofilter" to remove H 2 S. Indeed, microorganisms are the "engines" of CW as well as "biofilter" since the pollutant removal was highly depending on their activity. As shown in Fig. 5, the microbial degradation was occurred when H 2 S reaches the wastewater and specially biofilm. In fact, biofilm is a mixture of bacteria, fungi, yeasts, ciliated protozoa, amoebae, nematodes, algae and water [START_REF]Biotechnology for Odor and Air Pollution Control[END_REF]. In the present study, various bacteria have been contributed in bioprocesses to decompose H 2 S to elemental sulfur based on their functions as biocatalysts, such as phototrophic bacteria (also known as green sulfur bacteria), chemotrophs and Xanthomonas [START_REF] Yuan | Use of mature compost as filter media and the effect of packing depth on hydrogen sulfide removal from composting exhaust gases by biofiltration[END_REF]. In the literature [START_REF] Vikrant | Biofiltration of hydrogen sulfide: Trends and challenges[END_REF][START_REF] Zak | Sulphate in freshwater ecosystems: a review of sources, biogeochemical cycles, ecotoxicological effects and bioremediation[END_REF], bioconversion of H 2 S in presence of oxygen as a main electron acceptor can be described by Eqs. ( 3)- [START_REF] Fan | Nitrogen removal in intermittently aerated vertical flow constructed wetlands: Impact of influent COD/N ratios[END_REF]. In these reactions, H 2 S was oxidized by microbes and generating the odorless compounds such as elemental sulfur and sulfate. In particular, Fig. 5 (a) and (b) illustrate alum sludge before and after the trial, respectively. Large amount of S was observed on the surface of alum sludge after six-month trial. It indicated that the biodegradation process occurs on the surface (biofilm) of the alum sludge.

H 2 S + 0.5O 2 → S 0 + H 2 O (3)
S 0 + 1.5O 2 + H 2 O → SO 4 2-+ 82H + (4)
H 2 S + 2O 2 → SO 4 2-+ 2H + (5)
The microbes could also use the intermediate biodegradation products of H 2 S (e.g., HS -and S 2-) as energy sources. Moreover, chemotrophs could use organic and inorganic compounds (CO 2 ) as the carbon source to oxidize H 2 S and produce metabolic energy and new cell materials. In addition, the removal of sulfide could also be contributed by phototrophic bacteria. The biocatalytic reaction was descripted in Eq. (6). In strictly anaerobic conditions, in the presence of light, CO 2 and inorganic nutrients, sulfide was converted to elemental sulfur by green sulfur bacteria [START_REF]Biotechnology for Odor and Air Pollution Control[END_REF].

2nH 2 S + nCO 2 + hv→2nS + nCH 2 O (6)
Furthermore, heterotrophic (e.g., Thiobacillus sp.) bacterial biodegradation of H 2 S could also exist. The end product (sulfates) was generated through the intermittent polysulfides via bacterial degradation [START_REF] Jiang | Removal mechanisms of H2S using exhausted carbon in biofiltration[END_REF].

The biodegradation of H 2 S occurs under aerobic conditions, which means O 2 could acted as an electron acceptor. However, in anoxic conditions the biodegradation of H 2 S could also occur when nitrate (NO 3 -) was used as an electron acceptor instead of oxygen [START_REF] Vikrant | Biofiltration of hydrogen sulfide: Trends and challenges[END_REF]. This process (as described in Eq. ( 7)) results in generations of sulfate and elemental sulfur, while nitrate is reduced to nitrogen and nitrite. However, the anoxic biodegradation of H 2 S and reduction of NO 3-are a complex process. It highly depends on the operational parameters, such as N/S ratio.

12H 2 S + 15NO - 3 →6S 0 + 6SO 2- 4 + 5N 2 + 5NO - 2 + 9H 2 O + 2OH -+ 4H + (7)
In general, sulfate (SO 4 2-) is the main sulfur species that is formed during the bioconversion of H 2 S [START_REF] Das | Performance of a compost and biochar packed biofilter for gas-phase hydrogen sulfide removal[END_REF]. It has also been reported by Vikrant et al. [START_REF] Vikrant | Biofiltration of hydrogen sulfide: Trends and challenges[END_REF] that the accumulation of sulfate may inhibit microbial activity and decrease the mass transfer rate of H 2 S into the biofilm (due to reduced pH), reducing removal efficiency significantly. However, the effluent from the Al-OAF (column 1) can mitigate SO 4 2-accumulation problems and also be effective to avoid the inhibitory effect caused by SO 4 2-since the SO 4 2-is washed out by the effluent. Moreover, the end products of H 2 S biodegraded processes (sulfur or sulfate) are nonhazardous and ecology safe [START_REF] Lin | Biological sulfur oxidation in wastewater treatment: a review of emerging opportunities[END_REF].

Besides the biodegradation process, H 2 S sorption by alum sludge substrate is also a vital route for H 2 S removal. In a recent study [START_REF] Ren | Alum sludge as an efficient sorbent for hydrogen sulfide removal: experimental, mechanisms and modeling studies[END_REF], alum sludge have been demonstrated as an efficient sorbent for H 2 S removal in a fixed-bed column. The various flow rates, bed depth (mass of sorbent), breakthrough behavior simulation and modeling were investigated. In particular, the complex mechanisms behind H 2 S sorption onto alum sludge of a three-step (sorption-dissociation-oxidation) process were proposed. Firstly, alum sludge has a microporous structure and relatively great surface area (S p (BET) = 238 m 2 /g). Furthermore, due to the presence of various reactive sites (as shown in Fig. 5), e.g., carbon and metal oxides, H 2 S molecules are dissociated by these active sites into HS -and H + when H 2 S reaches the alum sludge surface and micropores. On the other hand, basic pH of alum sludge (pH = 10) could facilitated the dissociation of H 2 S and neutralize the acidification of alum sludge surface. Thereafter, the oxidation process with oxygenated organic and inorganic species leads to the formation of sulphate (Fig. 5).

Herein, the proposed H 2 S removal mechanisms of column 1 (Al-OAF) lies in a complex reaction of biodegradation as well as a sorption process by the substrate (alum sludge). Biofilm and the suspended microorganisms in the wastewater act as powerful "engine" for H 2 S conversion. Alum sludge also provided a proper media for H 2 S sorption process.

Perspective

Upgrading the existing WWTPs (e.g., improving the effluent quality to meet the updated standard and abating the unpleasant odor) is more difficult than building a new plant. This is due to the lack of practical experience worldwide as well as the variation of the wastewater qualities, treatment process and local circumstances [START_REF] Hu | Appropriate technologies for upgrading wastewater treatment plants: methods review and case studies in China[END_REF]. Building the effluent polishing or post-treatment (tertiary treatment) unit is one of the main upgrading options. It could minimize the interruptions in the daily operation of WWTPs and easy to implement. In especially, using biofilters or CWs as the polishing unit for WWTPs upgrading has become more favorable due to the low cost, simple operation & maintenance, natural landscape and biodiversity merits. Zhu et al. [START_REF] Zhu | Comparison of performance of two large-scale vertical-flow constructed wetlands treating wastewater treatment plant tail-water: contaminants removal and associated microbial community[END_REF] reported the use of a large-scale vertical-flow CWs as the polishing unit at the end of WWTP in Henan, China. Results of 14 months study showed that average removals of 40.05% COD, 45.47% NH 4 -N, 62.55% TP, 55.53% TN and 57.20% TSS were achieved. Cao et al. [START_REF] Cao | Comprehensive benefits assessment of using recycled concrete aggregates as the substrate in constructed wetland polishing effluent from wastewater treatment plant[END_REF] also states that a WWTP in China has been upgraded with the recycled concrete aggregate-based large-scale horizontal subsurface flow CWs. Indeed, China has faced urgent demand in large scale of existing WWTPs upgrading. To control the water environment in rivers, lakes and other surface water bodies, the current Chinese "National Standards for Environmental Quality of Surface Water (GB3838-2002)" requires the effluent COD ≤ 30 mg/L, NH 3 -N ≤ 1.5 mg/L, TP ≤ 0.1 mg/L. By June 2018, there are 5300 WWTPs across China, the upgrading of such large number of WWTPs is a great challenge while large scale CWs systems have been proposed as principal process [START_REF] Wu | Upgrading the Chinese biggest petrochemical wastewater treatment plant: Technologies research and full scale application[END_REF]. Wei et al. [START_REF] Wei | Development status of constructed wetland (CWs) for treatment of terminal effluent of wastewater treatment plants (WWTPs) in China[END_REF] reported five projects of WWTPs upgrading with CWs in China (i.e. by adding various CWs for terminal effluent treatment). However, few studies concern about odor treatment by using CWs for WWTPs upgrading.

Although alum sludge based-CWs have been intensively studied in recent years, to the best of our knowledge, this is the first time to use the alum sludge-based biofilter for dual goals of H 2 S or odor gas abatement and wastewater treatment. The novel technical strategy can be illustrated in Fig. 6. It has demonstrated that the alum sludge owns features of high P adsorption (over 98%), ability of H 2 S adsorption while biofilm developed on the alum sludge as wetland substrate owns capacity to have bio-reaction with H 2 S. In particular, the outstanding TP removal performance (over 98%) of alum sludge could ensure that the effluent concentration of the upgraded WWTP complies with the strict discharge standard (TP < 0.1 mg/L in GB3838-2002). Thus, the single biological P removal process cannot fulfill this requirement. A combination of biological P removal with tertiary treatment should be applied under this circumstance. The current study opens a technical option for the WWTPs upgrading of using a "waste" by-product (alum sludge) from water industry as a raw material in biofilters for effluent polishing while for odor gas (generated from the WWTPs) purification. No doubt, the importance of this study will help the WWTPs not only for water quality improvement, but also for waste gas purification in the ensuing large-scale upgrading engineering practice.

However, this is an inaugurating study to demonstrate the "dual-goals" of upgrading approach, while only single odorant (H 2 S) was tested. The real waste gas generated from wastewater treatment process and sludge treatment facilities in WWTPs is far more complicated than H 2 S. Even if H 2 S was completely removed in the Al-OAF in this study, WWTPs could release various volatile organic sulfur compounds (VOSCs) and inorganic sulfur compounds. Nitrogenous compounds and volatile fatty acids (VFAs) are also potential sources of odor. Odorous volatile aromatic compounds (OVACs), such as toluene, ethylbenzene, styrene, p-cresol, indole and skatole, have been identified from stored biosolids while the treatment of biosolids/sludge could form different kind of odor-causing compounds [START_REF] Gruchlik | Use of alum for odor reduction in sludge and biosolids from different wastewater treatment processes[END_REF]. The concentrations of those odorants account in a wide range depending on the nature of WWTPs. Obviously, those kinds of odorants with different concentrations need further investigation in the ensuing study.

Conclusions

This study developed a novel alum sludge-based odor-gas aerated biofilter, aimed to provide an effective approach to simultaneously enhance overall pollutants removal in wastewater and purify H 2 S. Through the six-month trial, the high removal efficiency of TP (>98%) was achieved due to sorption onto the alum sludge, while H 2 S was completed removed (100%). The introduction of H 2 S and air for CW aeration could significantly enhance the removal efficiency of COD and TN compared with an unaerated Al-F. More interestingly, there was no difference regarding the COD and TN removal performances between the "H 2 S aerated" and air aerated Al-F. The data suggest that intermittently aerated Al-OAF using H 2 S would be a potential "wiser choice" to intensify pollutant removal performance for wastewater and odor gas purification. This has great potential in China for large scale WWTPs upgrading. However, further studies are suggested to determine: (a) the metabolic process and microbial community involved in the elimination of H 2 S; (b) their potential to remove other pollutants such as NH 3 and VOCs from the WWTP odor stream; and (c) the application of this novel Al-OAF in full-scale and long-term system.

Fig. 1 .

 1 Fig. 1. Testing system of the three parallel columns.

Fig. 3 .

 3 Fig. 3. Overall removal efficiency of COD, TN, NH 4 -N and TP in three columns.

Fig. 4 .

 4 Fig. 4. Online analysis of H 2 S effluent and removal efficiency: (a) daily H 2 S data, (b) overall H 2 S removal performance.

Fig. 5 .

 5 Fig. 5. SEM-EDX data and proposed H 2 S removal mechanisms, (a) raw alum sludge, (b) alum sludge substrate from column 1 after six-month's trial.

Acknowledgements

The first author greatly acknowledges RAPSODEE Research Centre (CNRS UMR 5302), France for the financial and technological support. This study also jointly supported by the Fundamental Research Funds for the Central Universities, CHD 300102291302.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Raphael Soubeyran
email: raphael.soubeyran@inrae.fr

Pro-social Motivations, Externalities and Incentives *

Keywords: JEL classification: D91, D62, D63, D86 incentives, externality, principal, agents, coordination, pro-social preferences

This paper analyzes how pro-social motivations shape the relationship between incentives and inequality. I consider a principal who offers individual rewards to a group of agents to induce them to exert effort and to coordinate at least-cost. The agents value the payoffs of the other agents, and they are averse to inequality. My analysis highlights that pro-social motivations have an a priori ambiguous effect on inequality in the reward distribution. Despite this initial ambiguity, I show that the rewards are more unequal and lower when the agents have pro-social preferences. The model delivers empirical implications for intervention programs supporting the adoption of new health or agricultural technologies.

Introduction

Pro-social motivations are at the center of many important economic problems. For example, people adopt protection technologies for the Covid-19 virus because they care about the welfare of others (Kar Keung Cheng, 2020; [START_REF] Barzilay | Resilience, covid-19-related stress, anxiety and depression during the pandemic in a large population enriched for healthcare providers[END_REF]. While these motivations often determine how individuals respond to incentives in practice [START_REF] Bandiera | Social preferences and the response to incentives: Evidence from personnel data[END_REF], little is known on how they should theoretically affect optimal incentives and the resulting distribution of wealth.

In this paper, I examine how pro-social preferences shape the relationship between incentives and inequality. My analysis considers the optimal incentives that will induce a group of agents with pro-social preferences to make complementary efforts (i.e., positive peer effects). Pro-social motivations are conceptually different from peer effects because pro-social motivations depend on both the actions and the rewards of others. In contrast, peer effects depend only on the actions of others. Here I consider a principal who offers individual rewards, or bilateral contracts, to a group of agents to induce them to coordinate and to exert effort at least cost. The optimal reward scheme is called a least-cost unique implementation scheme. An agent's social utility is a linear combination of their private utility, which depends only on their own material payoff, and a function of the other agents' individual material payoffs. The pro-social function is assumed to increase in each of the other agents' material payoffs. It is also assumed to be concave and supermodular, meaning the agents are averse to inequality. These assumptions encompass, among others, the constant elasticity of substitution (CES) function.

I perform my analysis in three steps. First, I study specific incentive schemes (global ranking schemes) wherein agents are ranked and each agent is indifferent between exerting and not exerting effort while the preceding agents exert effort and the remaining agents do not. I show that within these schemes, agents with pro-social preferences get lower and more unequal rewards. Second, I show that any least-cost unique implementation scheme is necessarily a global ranking scheme. This implies that a least-cost unique implementation scheme can never lead to an increase in the rewards of an agent nor to less inequality in the reward distribution. Third, I show that any global ranking scheme is a unique implementation scheme if the agents are not too averse to inequality, and thus that the solution exists in this case.

The fact that pro-social preferences lead to more inequality in the reward distribution is not straightforward. If the agents have no pro-social preferences, as more agents exert effort, fewer rewards are needed to induce the remaining agents to exert effort. In this case of the leastcost implementation scheme, identical agents get different rewards. Pro-social motivations may lead to more or less inequality in the reward distribution. Indeed, pro-social motivations have two opposing effects on inequality: an extensive margin effect and an intensive margin effect. With the extensive marginal effect, each additional agent who decides to make effort generates a positive externality for the previous agent. Since the additional agent values this externality, the principal can give them fewer rewards. Therefore, the extensive margin effect tends to increase inequality in the reward distribution. However, with the intensive margin effect, each additional agent who decides to make effort generates positive externalities for all the previous agents, who already benefit from the positive externalities they generate for each other. Since the agents are averse to inequality, the additional agent's marginal contribution to the pro-social motivation is lower when more agents make effort. Thus, when more agents exert effort, the principal must increase the rewards given to the remaining agents to induce them to also exert effort. This shows how the intensive margin effect tends to decrease inequality in the reward distribution. Moreover, the extensive margin effect applies to the agent who obtains the lowest material payoff. The intensive margin effect applies to the other agents who exert effort and who receive higher rewards. Since the agents are averse to inequality, the extensive margin effect is always stronger than the intensive margin effect. Finally, inequality in the reward distribution is higher when the agents have pro-social preferences.

This paper introduces the analysis of pro-social preferences in the literature on contracts with externalities [START_REF] Segal | Contracting with externalities[END_REF][START_REF] Segal | Coordination and discrimination in contracting with externalities: divide and conquer?[END_REF][START_REF] Bernstein | Contracting with heterogeneous externalities[END_REF]. Here I consider a situation where these agents make binary decisions, and their actions are strategic complements (as in [START_REF] Winter | Incentives and discrimination[END_REF]. The literature typically characterizes optimal incentive schemes as generating heterogeneity among symmetric agents. My analysis reveals that pro-social preferences, which play an important role in current major issues, lead to an increase in inequality, while this was a priori unclear. 1The remainder of the paper is organized as follows: the model is introduced in Section 2, the main results are delivered in Section 3, a discussion on heterogeneity is provided in Section 4 and conclusions are summarized in Section 5. All proofs are provided in the appendix.

The Model

A principal offers individual bilateral contracts to several agents in an environment characterized by positive externalities between the agents. First, the principal proposes a publicly observable incentive scheme to a set of agents. Second, the agents observe the principal's proposition and simultaneously decide whether to exert effort at their individual price.

An agent who decides to exert effort generates a positive externality w ≥ 0 to the other agents that exert effort and no externality to the agents who do not exert effort. An agent who decides not to exert effort receives an outside option c. The principal aims to induce effort from all agents at the lowest possible cost. To reach this goal, the principal proposes an incentive scheme v = (v 1 , v 2 , ..., v n ) ∈ R n to the agents in the set of agents N , with i = 1, 2, ..., n, to incentivize them to exert effort. The reward is conditional on the agent exerting effort: agent i receives v i from the principal if he exerts effort and 0 otherwise. The incentive scheme v is designed such that each agent receives a unique offer υ i , i ∈ N , meaning the principal is able to use individualized rewards. The vector of agents' decisions is x = (x 1 , ..., x n ) ∈ {0, 1} n , where x i = 1 means that agent i chooses to exert effort and x i = 0 means that that agent decides not to exert effort. I will denote y y i =a vector y = (y 1 , ..., y n ) ∈ R n when its ith component is set to a. I will denote y -i vector y when the ith component is removed. y k = (y, ..., y) denotes the vector of dimension k with each component set to y ∈ R. Finally, t = (z, y) denotes the vector whose first elements are those of vector z followed by those of vector y. Agent i's material payoff is:

π i (x, v i ) = x i   v i + w j =i x j   + (1 -x i )c.
(1) I assume that the agents have social preferences and that they give weight to their own payoff (a "selfish" motive) and the payoffs of all other agents (a pro-social motive). Formally, the social utility of agent i is:

U i (x, v) = u i (π i (x, v i )) + θW i (π -i (x, v -i )), (2)
where π -i (x, v -i ) is the vector of the agents' payoffs but i, u i is the selfish part of the utility function and W i is the agent's pro-social function. Parameter θ ≥ 0 indicates the strength of the pro-social motivations for the agents. When θ = 0, the agents are purely selfish and when θ > 0 the agents have pro-social preferences. The individual utility function u i is strictly increasing and the pro-social function W i is increasing in the payoff of each of the other agents.

In the rest of the paper, I make the two following assumptions:

Assumption C-SupM: The pro-social functions W i are concave and supermodular.

Assumption S: The agents are symmetric, that is U i ≡ U and then u i ≡ u and W i ≡ W for all i.

Notice that Assumption S implies that the level of the pro-social component W i is the same for any permutation of the material payoff of the other agents (Anonymity).

The two assumptions hold for the following example of pro-social function:

Example 1: If u is concave, then the constant elasticity of subsitution (CES) pro-social func-

tion W i (π -i ) = k =i u(π k ) s-1 s s s-1
, where s ∈]0, +∞[ and s = 1, which is increasing in its arguments, satisfies assumptions S and C-SupM.

The main results of this paper are that pro-social motivations lead to a decrease in the rewards of the agents and cannot lead to a decrease in inequality. These results hold for this CES specification of the pro-social function.

In the following, I study the least-cost scheme that implements effort by all agents as a unique Nash equilibrium (i.e., the least-cost unique implementation scheme). I require schemes to induce a unique equilibrium only when the individual rewards are increased by a positive amount.2 More precisely, I characterize the reward vector v * that solves the following optimization problem:

M in v∈ n j∈N v j (3) s.t. for all > 0, full effort (x = 1 n ) is a Nash equilibrium: U i (1 n , v + n ) > U i (1 n x i =0 , v + n ), (NE)
for all i ∈ N , and, there is no other Nash equilibrium, that is, for all x = 1 n , ∃i ∈ N such that x i = a, a ∈ {0, 1} and:

U i (x x i =1-a , v + n ) > U i (x, v + n ). (UC)
The set of constraints (NE) ensures that each agent has an incentive to exert effort when all other agents exert effort, and the set of constraints (UC) ensures that for each for each of the other outcomes, at least one agent has an incentive to deviate.

Optimal Incentive Scheme

In this section, I present the results of the analysis of the problem of the principal.

Ranking Agents

It will be useful for the analysis to define when I consider a class of incentive schemes to be a ranking scheme for an agent: Definition 1: An incentive scheme v is a ranking scheme for an agent if the agents are ranked in a given order and agent i is indifferent to exerting or not exerting effort when the previous agents also exert effort and the subsequent agents do not.

Formally, if the agents are ranked from 1 to n (without loss of generality), and v is a ranking scheme for agent i, we must have:

u i (c) -u i (v i + (i -1)w) = θ W i (π -i ((1 i , 0 n-i ), v -i )) -W i (π -i ((1 i-1 , 0 n-i+1 ), v -i )) (4)
' Proposition 1: If v is a ranking scheme for agent i and all previous agents exert effort while the subsequent agents do not, then the material payoff of this agent is lower than the opportunity cost,

v i + (i -1)w ≤ c.
The intuition of this result is clear since when an agent exerts effort, he generates positive externalities for other agents who also exert effort, which increases the level of his pro-social function. It is thus not necessary to compensate the full opportunity cost to induce this agent to exert effort.

I now show a less intuitive result:

Proposition 2: If assumptions S and C-SupM hold and v is a ranking scheme for the i first agents according to a common ranking (1 to n without loss of generality), then

v k-1 ≥ v k + w for all 2 ≤ k ≤ i.
Therefore, agents ranked earlier obtain a higher material payoff. This inequality is binding when the agents have no pro-social motivations and strict when they have pro-social motivations. I explain the intuition in both cases below.

If the agents have no pro-social preferences (θ = 0), the first agent must be indifferent between exerting and not exerting effort when no other agents exert effort. In this case, the agent does not benefit from positive externalities. The second agent has to be indifferent between exerting and not exerting effort when the first agent also exerts effort. The second agent thus benefits from a positive externality and has a greater incentive to exert effort. The second agent thus obtains a lower reward than the first agent. This reasoning holds for any two subsequent agents. Now, consider the case where the agents have pro-social preferences (θ > 0). The difference between the utility levels agent i + 1 and i + 2 obtain in the situation where the agents ranked before them exert effort while the agents that are ranked after them do not is such that:

1 θ [u(v i+1 + iw) -u(v i+2 + (i + 1)w)] = W (π -(i+2) ((1 i+2 , 0 n-i-2 ), v -(i+2) )) -W (π -(i+2) ((1 i+1 , 0 n-i-1 ), v -(i+2) )) -W (π -(i+1) ((1 i+1 , 0 n-i-1 ), v -(i+1) )) -W (π -(i+1) ((1 i , 0 n-i ), v -(i+1) )) . (5)
The right side of condition ( 5) captures the difference between the increase of the pro-social component that agent i + 2 and i + 1 obtain when they exert effort. This difference can be written more explicitly as follows:

W (v 1 +(i+1)w, v 2 +(i+1)w, ..., v i+1 +(i+1)w, c n-i-2 )-W (v 1 +iw, v 2 +iw, ..., v i+1 +iw, c n-i-2 ) -W (v 1 + iw, ..., v i + iw, c n-i-1 ) -W (v 1 + (i -1)w, ..., v i + (i -1)w, c n-i-1 ) . (6)
The sign of this difference is a priori ambiguous. There are two diverging effects: a positive effect at the extensive margin and a negative effect at the intensive margin. First, when agent i + 2 decides to exert effort, he generates a positive externality for an additional agent (agent i + 1) compared to when agent i + 1 decides to exert effort. This extensive margin effect suggests that the difference should be positive. But when i + 1 agents exert effort, the material payoff of the i first agents is greater than when the i first agents only exert effort. Since W is concave, the increase in utility for the i first agents is greater when both the i first agents and agent i + 1 decide to exert effort than when the i + 1 first agents and agent i + 2 decide to exert effort. This second effect is an intensive margin effect, and it suggests the difference should be negative.

Let me illustrate this intuition using the utilitarian example, where W i (π -i ) ≡ k =i u(π k ). In this case, the two effects can be easily separated. Notice that since W must be concave, u must also be concave. The proof works by induction. First notice that v 1 = c, and by Proposition 1, the result holds for i = 1 (v 1 ≥ v 2 + w). Assume that the result of Proposition 2 holds for all k ≤ i + 1. Using the utilitarian specification, condition (6) can be written as follows:

u(v i+1 + (i + 1)w) -u(v i+1 + iw) Extensive margin (+) + 1≤k≤i ([u(v k + (i + 1)w) -u(v k + iw)] -[u(v k + iw) -u(v k + (i -1)w)])
Intensive margin (-) (7) Using ( 7) and concavity of u I can conclude that the extensive margin is larger than the intensive margin. This can be easily shown by rewriting condition (7) as follows:

u(v 1 + (i + 1)w) -u(v 1 + iw) + 2≤k≤i+1 ([u(v k + (i + 1)w) -u(v k + iw)] -[u(v k-1 + iw) -u(v k-1 + (i -1)w)]) ≥ 0 (8)
The condition above proves that the result of Proposition 2 holds in the utilitarian case with concave utility u. The extensive margin applies to the agent with the lowest material payoff, while the intensive margin applies to the agents who also exert effort and whose material payoffs are higher. The concavity of W implies that the extensive margin is larger than the intensive margin.

Proposition 2 can be applied to the CES function of Example 1 and more general functions. As in the utilitarian case, the general proof uses the symmetry assumption (where the payoff of two agents can be permuted without affecting the level of W ) and the concavity of W . Supermodularity enables one to prove the result in the general case.

Optimal Incentive Scheme and Inequality

I first prove a preliminary result that will be used to analyze the optimal scheme: Lemma 1: If assumption S holds and v is such that each agent k ≤ i prefers to exert effort when the previous agents (according to a common ranking, 1 to n without loss of generality) also exert effort and the remaining agents do not, then each agent k also prefers to make an effort when j agents, with k ≤ j ≤ i, exert effort.

The intuition of this result is simple if the agents have no pro-social preferences. As more agents exert effort, the agents who make effort benefit from more positive externalities and thus they are more likely to exert effort.

If the agents have pro-social preferences, the result is less straightforward. The monotonicity result of Proposition 2 is needed to prove the result. The proof works as follows: if v is a ranking scheme for the i first agents according to a common ranking, then their rewards are characterized by condition (4). Consider the situation where the i first agents exert effort. If i or k decide not to exert effort, the material payoff of each agent ranked before i is reduced by w. Moreover, if i does not exert effort, the payoff of k is reduced by w and if k does not exert effort, the payoff of i is reduced by w. Given that v k > v i and W is concave, the pro-social component of agent k's utility decreases more when k decides not to exert effort than the pro-social component of agent i's utility when i decides not to exert effort. Thus, agent k has fewer incentives to deviate than agent i. Since agent i is indifferent between exerting and not exerting effort, agent k strictly prefers to exert effort.

It is helpful to define a class of incentive schemes, or global ranking schemes, to characterize the least-cost unique implementation scheme: Definition 2: An incentive scheme v is a global ranking scheme if it is a ranking scheme for all the agents according to a common ranking.

Notice that this scheme is unique for a given ranking. Each agent is indifferent between exerting and not exerting effort when the agents ranked before him also exert effort, and the remaining agents do not. This definition is used to provide the necessary conditions for an incentive scheme to be optimal:

Theorem 1: Assuming that assumptions S and C-SupM hold, any least-cost unique implementation scheme is a global ranking scheme.

This Theorem proves that the optimal scheme is necessarily a global ranking scheme. Notice that the global ranking scheme is unique, up to a reordering of the (ex-ante identical) agents. Using the results from Propositions 1 and 2, I conclude that individual rewards decrease and inequality in the reward distribution increases when the agents have pro-social motivations.

To prove sufficiency, I need an additional assumption:

Assumption BIA: An utility function U ≡ u + θW exhibits bounded inequality aversion over I if, for all i ∈ N , π ∈ I ⊂ R n and δ > 0

θ W (π -i + w n-1 ) -W (π -i ) -W (π -i + w n-1 + δ n-1 ) -W (π -i + δ n-1 ) < u(π i + w + δ) -u(π i + w)
Let me illustrate this assumption with the two examples:

Example 2: The quasi-maximin utility function, which is such that u(π) ≡ π and W (π) ≡ γmin{{π j } j =i } + (1 -γ) j =i π j , γ ∈ [0, 1] satisfies assumption BIA.

This utility function introduced in [START_REF] Charness | Understanding social preferences with simple tests[END_REF] specifies that pro-social motivations such as altruism -captured by the sum of all the agents' payoffs-and inequality aversion -captured by the Rawlsian component-are additively separable. Assumption BIA holds for this example.

Example 3: The quadratic utilitarian utility function, which is such that u

(π) = aπ i -b(π i ) 2 , where π i ∈ [0, a/2b] and W (π) ≡ j =i u(π j ), γ ∈ [0, 1] satisfies assumption BIA if and only if b < a 2θ(n-1)w .
This second example illustrates that, for assumption BIA to hold, the agents cannot be too averse to inequality, meaning the utility function cannot be too concave.

Theorem 2: Assuming that assumptions S, C-SupM, and BIA hold, any global ranking scheme is a least-cost unique implementation scheme.

Theorem 2 proves sufficiency under an additional assumption. Namely, if the agents are not too averse to inequality, then full effort is a unique Nash equilibrium under any global ranking scheme. Suppose assumption BIA does not hold, and the agents are extremely inequality averse. An agent may prefer not to exert effort when the preceding agents exert effort and the remaining agents do not if the rewards of each agent are increased by a positive amount. Indeed, an increase in all the agents' rewards increases the marginal effect of exerting effort on the selfish part of the utility function and decreases the marginal effect of exerting effort on the pro-social part of the utility function. Thus, full effort is a unique Nash Equilibrium under a global ranking scheme if the latter effect is larger than the former for each agent when the preceding agents exert effort while the remaining agents do not.

Discussion

Up to now, I have focused on the case identical agents and homogenous externalities. One may wonder whether the main results of the paper hold in a situation where the agents have heterogeneous preferences and generate heterogeneous externalities.

Let me analyze the role of heterogeneity using a simple form of pro-social preference. Assume that the agents have heterogeneous preferences for the payoffs of the other agents and that the pro-social function of an agent is a weighted form of the other agents' payoffs:

W i (x) = j =i γ ij π j , (9)
where γ ij ≥ 0 represents the intensity of the preference of agent i for the payoff of agent j. Also assume that externalities are heterogeneous: when agents i and j make effort, agent i benefits from a positive externality from agent j denoted w ij ≥ 0.

To keep things simple, also assume that the selfish part of the utility is such that u i (π i ) ≡ π i . Using all the assumptions above, the utility of agent i can be written as:

U i (x, v) = π i + θ j =i γ ij π j , (10)
where

π i = x i   v i + j =i x j w ij   + (1 -x i )c. (11)
Thanks to the linearity of the utility function, I can derive the optimal reward scheme using similar proofs as in [START_REF] Bernstein | Contracting with heterogeneous externalities[END_REF]. Indeed, if each agent i j with j < k make effort, then agent k also prefers to make effort if:

v i k + j<k w i k i j + θ j<k γ i k i j w i j i k > c. (12)
Thus, the least cost unique implementation scheme is such that the agents are ranked from i 1 to i n and the optimal reward of agent i k is given by:

v * i k = c - j<k (w i k i j + θγ i k i j w i j i k ), (13)
Hence, the agents still obtain lower rewards when they have pro-social preferences (θ > 0) than when they do not (θ = 0).

The optimal ranking depends on the virtual popularity tournament described in section A in [START_REF] Bernstein | Contracting with heterogeneous externalities[END_REF]. In the context of the present model, agent j beats agent k if:

(1 -θγ i j i k )w i k i j < (1 -θγ i k i j )w i j i k (14)
This result is interesting since it implies that pro-social preferences affect the optimal ranking of the agents. Indeed, one may have w i k i j > w i j i k and (1 -θγ i j i k )w i k i j < (1 -θγ i k i j )w i j i k . This occurs for instance when agent k receives large externalities from agent j and agent j gives a strong weight to the payoff of agent k.

How the introduction of pro-social preferences (θ > 0) affect inequality is not straightforward. Let me first consider the spread of the rewards distribution:

max k∈N v i k -min k∈N v i k = j<l (w i l i j + θγ i l i j w i j i l ), ( 15
)
where l is such that min k∈N v i k = v l . Notice that we have l > 1.

The spread is clearly larger when the agents have pro-social preferences. Thus, in this sense, pro-social preferences also lead to more inequality when one introduces heterogeneity in the model.

It is however important to notice that the results of this section are limited to the case where the utility is linear in the actions of the players. They cannot be easily extended to the more general case of non linear preferences (that encompasses inequality aversion) considered in the rest of the paper.

Conclusion

In this paper, I examine a situation where a principal designs a least-cost unique implementation scheme by offering a group of agents with pro-social preferences rewards to exert effort. I show that the least-cost unique implementation scheme is necessarily a global ranking scheme. Additionally, I show that the agents get lower and more unequal rewards when they have pro-social preferences. The least-cost unique implementation scheme exists if the agents are not too averse to inequality. These results have practical implications for implementing schemes to induce agents to adopt new technology. When the agents have pro-social preferences, the optimal scheme must be more unequal and its cost is reduced. This is especially relevant when relatives or peers have to decide whether to adopt new technologies such as health [START_REF] Kremer | The illusion of sustainability[END_REF] 3 or agricultural technologies [START_REF] Foster | Learning by doing and learning from others: Human capital and technical change in agriculture[END_REF]. 4Proof of Lemma 1: Assume that Assumption S holds. Assume that agent i prefers to make effort when the previous agents also exert effort while the remaining do not (assuming they are ranked from 1 to n, without loss of generality). Hence:

u(v i + (i -1)w) + θW (π -i ((1 i , 0 n-i ), v -i )) ≥ u(c) + θW (π -i ((1 i-1 , 0 n-i+1 ), v -i )) (22)
Assume that agent k < i weakly prefers not to exert effort when each agent j ≤ i, j = k exerts effort while the remaining agents do not. Hence, we must have

u(c) + θW (π -k ((1 i x k =0 , 0 n-i+1 ), v -k )) ≥ u(v k + (i -1)w) + θW (π -k ((1 i , 0 n-i ), v -k )) (23)
Using ( 22), I obtain:

u(v i + (i -1)w) + θ W (π -i ((1 i , 0 n-i ), v -i )) -W (π -i ((1 i-1 , 0 n-i+1 ), v -i )) ≥ u(v k + (i -1)w) + θ W (π -k ((1 i , 0 n-i ), v -k )) -W (π -k ((1 i x k =0 , 0 n-i+1 ), v -k )) (24) which is equivalent to θ W (π -i ((1 i , 0 n-i ), v -i )) -W (π -i ((1 i-1 , 0 n-i+1 ), v -i )) -θ W (π -k ((1 i , 0 n-i ), v -k )) -W (π -k ((1 i x k =0 , 0 n-i+1 ), v -k )) ≥ u(v k + (i -1)w) -u(v i + (i -1)w) (25)
The second term in brackets in the left hand side of ( 25) is identical to the first term in brackets except that v k is replaced by v i , which is smaller. Thus, concavity of W implies that the left hand side in (25) is negative. Hence:

u(v k + (i -1)w) -u(v i + (i -1)w) ≤ 0 (26)
Thus, we must have v k ≤ v i , where k < i. This contradicts Proposition 2.

Proof of Theorem 1: Assume that v is a least cost unique implementation scheme. Hence, if no agent exert effort, one agent, say 1, must prefer to exert effort (otherwise 0 n is a Nash equilibrium). Assume that each agent k ≤ i prefers to exert effort when the agents ranked before this agent (according to a common ranking, 1 to n without loss of generality) exert effort while the remaining agents do not. When the i first agents exert effort, none of these agents has an incentive to deviate. Hence, another agent, say i + 1 has an incentive to exert effort, otherwise (1 i , 0 n-i ) is a Nash equilibrium. Hence, the set of constraints (UC) can be reduced to:

u(v i+1 + iw + ) + θW (π -(i+1) ((1 i+1 , 0 n-i-1 ), v -(i+1) + n-1 ))

> u(c) + θW (π -(i+1) ((1 i , 0 n-i ), v -(i+1) + n-1 )), ( 27) for all i + 1.

Given that the objective of the principal is linear, the optimal level v i+1 is either characterized by the following condition:

u(v i+1 +iw)-u(c) = θ W (π -(i+1) ((1 i , 0 n-i ), v -(i+1) )) -W (π -(i+1) ((1 i+1 , 0 n-i-1 ), v -(i+1) )) , (28) or by U (1 n , v) = U (1 n x i+1 =0 , v), which is equivalent to

u(v i+1 +(n-1)w)-u(c) = θ W (π -(i+1) (1 n x i+1 =0 , v -(i+1) )) -W (π -(i+1) (1 n , v -(i+1) )) . ( 29
)
Let me denote v * i+1 the solution to (28) and v i+1 the solution to (29). We must have v i+1 ≤ v * i+1 , which means that ( 28) is more constraining than (29). Hence v is a global ranking scheme.

Proof of Theorem 2: Let v be a global ranking scheme. Theorem 1 ensures that the solution cannot be a scheme that has is not a global ranking scheme. Lemma 1 (see the proof) ensures that constraints (NE) are not violated for i = 1, ..., n -1. It is thus sufficient to check that each agent strictly prefers to exert effort under scheme v + where > 0 and infinitesimal when all the previous agents (according to the ranking 1,...,n without loss of generality) also exert effort while the remaining agents do not. This is immediate from Assumption BIA.

CEE-M Working Papers

See Halac et al. (2020) for an analysis of discriminatory incentives with heterogeneous agents and their implications in terms of inequality. Another related paper is[START_REF] Gueye | Inequality Aversion and the Distribution of Rewards in Organizations[END_REF]. They consider inequality aversion à la[START_REF] Fehr | A theory of fairness, competition, and cooperation[END_REF] and show that inequality can increase or decrease when the degree of inequality aversion increases. The present paper differs substantially since, by definition of pro-social preferences, the utility of an agent cannot decrease when the payoff of another agent increases, even if the agents dislike inequality.

This solution concept corresponds to that of[START_REF] Winter | Incentives and discrimination[END_REF] and[START_REF] Halac | Raising capital from heterogeneous investors[END_REF].[START_REF] Halac | Rank uncertainty in organizations[END_REF] use a similar concept in a Bayesian game.

See also[START_REF] Oster | Determinants of technology adoption: Peer effects in menstrual cup take-up[END_REF];[START_REF] Dupas | Short-run subsidies and long-run adoption of new health products: Evidence from a field experiment[END_REF];[START_REF] Tarozzi | Micro-loans, insecticide-treated bednets, and malaria: evidence from a randomized controlled trial in Orissa, India[END_REF];[START_REF] Adhvaryu | Learning, misallocation, and technology adoption: Evidence from new malaria therapy in Tanzania[END_REF]

See also[START_REF] Munshi | Social learning in a heterogeneous population: technology diffusion in the indian green revolution[END_REF];[START_REF] Bandiera | Social networks and technology adoption in northern Mozambique[END_REF];[START_REF] Conley | Learning about a new technology: pineapple in Ghana[END_REF];[START_REF] Carter | Subsidies and the african green revolution: Direct effects and social network spillovers of randomized input subsidies in mozambique[END_REF].

Appendix

Proof of Proposition 2:

Assume that v is a ranking scheme for the first i agents. Using (4) for i = 1, we have

and then v 1 = c. Now I use induction to show that v k + w ≤ v k-1 for all k ≤ i. Assume that this inequality holds for all k ≤ j + 1, where j + 2 ≤ i. Using condition (4) for agents j + 1 and j + 2, we have:

It is sufficient to show that the right hand side of condition ( 17 [START_REF] Rothschild | Increasing risk: I. a definition[END_REF]. Letting

for 1 ≤ k ≤ j and ∆ k = c for j + 1 ≤ k ≤ n -1, we must have:

≥ W (v 1 + jw, ..., v j + jw, c n-j-1 ) + W (v 2 + jw, ..., v j+1 + jw, c n-j-1 ). ( 18)

Since the agents are symmetric, we can permute the material payoffs without affecting the level of W . Hence, I obtain:

Moreover, using supermodularity of W and v 1 = c, we have

Combining ( 19) and (20), we find:

≥ W (v 1 + jw, ..., v j + jw, c n-j-1 ) -W (v 1 + (j -1)w, ..., v j + (j -1)w, c n-j-1 ). ( 21)

Since W is an increasing function, condition (21) implies that the right hand side in ( 17) is indeed positive.