Gisèle Krysztofiak

Yao Veng Té

Valéry Catoire
email: valery.catoire@cnrs-orleans.fr

Gwenaël Berthet

Geoffrey C Toon

Fabrice Jégou

Pascal Jeseck

Claude Robert

Carbonyl sulfide (OCS) variability with latitude in the atmosphere

Carbonyl sulfide (OCS) is an important precursor of sulfate aerosols and consequently a key species in stratospheric ozone depletion. SPIRALE and SWIR balloon-borne instruments have flown in the tropical region and in the Arctic polar region, and ground-based measurements have been performed by QualAir FTS instrument in Paris. Partial and total columns and vertical profiles have been obtained to study the OCS variability with altitude, latitude and season. The annual total column variation in Paris reveals a seasonal variation with a maximum in April-June and a minimum in November-January. Total column measurements above Paris and from SWIR balloon-borne instrument are compared with several MkIV measurements, several NDACC stations, aircraft, ship and balloon measurements to highlight the OCS total column decrease from tropical to polar latitudes. OCS high resolution in situ vertical profiles have been measured for the first time in the altitude range between 14 and 30 km at tropical and polar latitudes. OCS profiles are compared with ACE satellite measurements and show good agreement. Using OCS:N 2 O correlation from SPIRALE, OCS stratospheric lifetime has been accurately determined. We find a stratospheric lifetime of 70±20 years at polar latitude and 58±14 years at tropical latitude leading to a global stratospheric sink of 54±14 Gg S yr -1 .

Introduction

Carbonyl sulfide (OCS) is the most abundant sulfur-containing compound in the atmosphere. OCS is mainly emitted into the troposphere by biogenic processes from oceans, directly and indirectly by oxidation of DMS and CS 2 . OCS is also emitted directly by biomass burning (10 to 20% of the overall sources; [START_REF] Notholt | Enhanced Upper Tropical Tropospheric COS: Impact on the Stratospheric Aerosol Layer[END_REF] and by soil and wetlands (Kettle et al., 2002a). In addition, OCS can be released by anthropogenic sources, particularly from CS 2 oxidation, itself released by industry and use of natural gas. Uptake by vegetation, soils and reaction with the hydroxyl radicals are the main tropospheric sinks in the lower troposphere (Kettle et al., 2002a). Due to its relatively low reactivity in the troposphere, OCS can reach the stratosphere and can react with OH or O radicals by the following reactions: OCS + OH  CO 2 + HS [START_REF] Toon | The JPL MkIV Interferometer[END_REF] OCS + O  CO + SO (2)

But the main OCS sink in the stratosphere is the photolysis reaction [START_REF] Chin | A reanalysis of carbonyl sulfide as a source of stratospheric background sulfur aerosol[END_REF][START_REF] Sander | Chemical Kinetics and Photochemical Data for Use in Atmospheric Studies Evaluation Number 17[END_REF]:

OCS + hν  CO + S (λ ≤ 300 nm)

Produced sulfur compounds S, HS and SO are rapidly oxidized by O 2 , O, OH, O 3 or HO 2 to form the product SO 2 . Finally, SO 2 is converted into H 2 SO 4 by reaction with OH or heterogeneously. H 2 SO 4 , is the principal gaseous precursor for stratospheric sulfate aerosol (Chin andDavis et al., 1993, 1995), which catalyzes ozone depletion.

However, the OCS mass budget and its contribution to the stratospheric aerosol layer are difficult to estimate [START_REF] Wilson | Steady-state aerosol distributions in the extra-tropical, lower stratosphere and the processes that maintain them[END_REF], mainly due to recurrent moderate volcanic eruptions disrupting the sulfur burden in the stratosphere by direct SO 2 injection into the stratosphere [START_REF] Vernier | Major influence of tropical volcanic eruptions on the stratospheric aerosol layer during the last decade[END_REF]. More OCS measurements at different latitudes during non-volcanic periods could reduce the uncertainties on the OCS mass budget.

The international Network for the Detection of Atmospheric Composition Change (NDACC) (http://www.ndsc.ncep.noaa.gov/) is composed of several stations providing OCS measurements, at ground level principally, of total column measurements or surface concentration. Additionally, OCS measurements have been obtained from several aircraft campaigns from ground to 14 km altitude maximum, principally by whole air sampling [START_REF] Simpson | Characterization of trace gases measured over Alberta oil sands mining operations: 76 speciated C 2 -C 10 volatile organic compounds (VOCs)[END_REF] during ARCTAS [START_REF] Harrigan | Anthropogenic emissions during Arctas-A: mean transport characteristics and regional case studies[END_REF], INTEX [START_REF] Blake | Carbonyl sulfide (OCS): Large-scale distributions over North America during INTEX-NA and relationship to CO 2[END_REF][START_REF] Barletta | Characterization of volatile organic compounds (VOCs) in Asian and north American pollution plumes during INTEX-B: identification of specific Chinese air mass tracers[END_REF] and TRACE-P [START_REF] Blake | Carbonyl sulfide (OCS) and carbon disulfide (CS2): Large-scale distributions over the western Pacific and emissions from Asia during TRACE-P[END_REF]. Only stratospheric balloons provide an altitude range from 10 km to 40 km altitude. However only few balloon campaigns have led to OCS measurements, and mainly by remote sensing instruments using solar occultation spectrometry (e.g. MkIV, [START_REF] Toon | The JPL MkIV Interferometer[END_REF] during for example ACE validation [START_REF] Velazco | Validation of the Atmospheric Chemistry Experiment by noncoincident MkIV balloon profiles[END_REF], STRAT, POLARIS, SOLVE I and II campaigns [START_REF] Leung | Isotopic fractionation of carbonyl sulfide in the atmosphere: Implications for the source of background stratospheric sulfate aerosol[END_REF], or by whole air sampling [START_REF] Engel | vertical profile measurements of carbonyl sulfide in the stratosphere[END_REF]. There are no OCS measurements by an in situ high resolution technique. et al., 2005) and SWIR [START_REF] Té | Balloonborne calibrated spectroradiometer for atmospheric nadir sounding[END_REF] balloon instruments participating in these campaigns. In addition, recent OCS total columns from ground-based measurements in Paris by the QualAir FTS instrument provide new reference information at mid-latitude (48.9°N, 2.4°E).

The paper is organized as follows. The QualAir FTS, SPIRALE and SWIR instruments, the Teresina and Esrange balloons flights and the ground based measurements over Paris are described in Section 2. Section 3 presents the observations performed during the two balloon campaigns and their comparisons with other results.

Then, we discuss about the role of OCS in the stratospheric sulfur budget.

Instrument description

a QualAir Fourier transform Spectrometer (QualAir FTS)

The Fourier transform spectrometer of the QualAir platform (QualAir FTS) is a When locked on the solar disk centre with an accuracy of 4 arcmin by using a photodiode system, the sun-tracker collects the solar radiation after transmission through the atmosphere and transfers it to the lower level experimental room before injecting it into the interferometer. The solar spectra recorded in presence of clouds are excluded. In order to optimize the signal-to-noise ratio and to focus on the species of interest, the appropriate optical filters and the detectors combination have to be chosen.

In this study, we have used an optical filter from 3.8 to 5.1 µm to retrieve especially OCS.

Spectra analyzed for this study were recorded from March 2011 to March 2012 in order to cover a whole year. We have used the PROFFIT algorithm developed by [START_REF] Hase | Intercomparison of retrieval codes used for the analysis of high-resolution: ground-based FTIR measurements[END_REF] to retrieve the atmospheric species abundance [START_REF] Té | Carbon monoxide urban emission monitoring: a ground-based FTIR case study[END_REF].

PROFFIT is a radiative transfer code based on the Beer-Lambert law for the analysis of solar absorption spectra. -windows (2038.8-2039.1, 2047.8-2048.7, 2051.2-2051.6 and 2052.5-2057.4 cm -1 ) are selected to retrieve OCS. The interfering species are H 2 O, CO 2 , O 3 and CO. In this case, solar lines were also taken into account and fitted. The degree of freedom (DOF) is about 1.5 (most of the information are from the troposphere) allowing only retrieval of the total column of OCS. Retrievals can be affected by two kinds of errors: random and systematic.

Random errors are the uncertainties in temperature profiles, solar zenith angles (SZA), effects of the instrument noise, and errors in the calculation of the interfering solar lines absorption. Systematic errors concern the spectroscopic parameters uncertainty, the a priori profile, and the errors in the instrument line shape function (ILS). Table 1 shows the retrieval errors on the OCS values. The assumptions used in generating the retrieval errors have been more detailed in the paper by [START_REF] Té | Carbon monoxide urban emission monitoring: a ground-based FTIR case study[END_REF].

Table 1. OCS retrieved total column uncertainties.

Error source OCS error (%) Temperature <0.5 Instrument noise <0.5 Solar zenith angle 0.9 Interfering solar lines <0.5

A priori profile <0.5 Instrument line shape 1 Spectroscopic parameters 4.5 to 9.5 Total error 4.7 to 9.7

In conclusion, we have estimated the total error of the OCS retrieval through the PROFFIT algorithm between 4.7% and 9.7%. The error bars in Figure 1 and in [START_REF] Té | Balloonborne calibrated spectroradiometer for atmospheric nadir sounding[END_REF]. This instrument is an infrared remote sensing instrument based on a Fourier Transform interferometer. The interferometer uses two InSb detectors to cover both thermal infrared (3-5 μm) and SWIR (1.8-2.4 µm)

domains. In the nadir-looking configuration, the maximum optical path difference (OPD) is fixed at 10 cm. The SWIR-balloon observes the upwelling radiation which is composed of several contributions from the Earth surface emission, the emission and absorption of atmospheric constituents, the emission of the clouds and the reflected solar radiation. Two reference sources on the optical head are used to perform an absolute calibration [START_REF] Revercomb | Radiometric calibration of IR Fourier transform spectrometers: solution to a problem with the High-resolution Interferometer Sounder[END_REF][START_REF] Té | A method to retrieve blackbody temperature errors in the two points radiometric calibration Infrared[END_REF]. The LPMAA Atmospheric Retrieval Algorithm LARA [START_REF] Té | Balloonborne calibrated spectroradiometer for atmospheric nadir sounding[END_REF] is a radiative transfer model coupled with an inversion code to retrieve the total columns (from ground to 34 km, the gondola height) of CO, O 3 , OCS, CO 2 , N 2 O and H 2 O. For the OCS retrieval, the spectral window from 2065 to 2145 cm -1 was selected for SWIR01, using the HITRAN 2012 database [START_REF] Rothman | The HITRAN 2012 molecular spectroscopic database[END_REF]. In addition to OCS, the contribution of six other atmospheric species H 2 O, CO 2 , O 3 , N 2 O, CO and CH 4 , has been taken into account. The main interfering species in these spectral windows are H 2 O, CO 2 , O 3 and CO. For the other two constituents, the absorption signals are

comparable to the noise level. OCS is retrieved from 0.32 km to 34 km for SWIR01, using UPMC 2D model [START_REF] Bekki | 2-D assessment of the impact of aircraft sulphur emissions on the stratospheric sulphate aerosol layer[END_REF][START_REF] Weisenstein | Modeling of Stratospheric aerosols[END_REF] profiles as a vmr profiles. The error bars in Figure 2 are the standard deviations of all retrieved columns during the flight.

 SPIRALE INSTRUMENT

SPIRALE is a spectrometer with six tunable laser diodes for in situ measurements of trace gas species from the upper troposphere to the middle stratosphere (~34 km height). A detailed description of the instrument can be found in a previous paper [START_REF] Moreau | A multispecies in situ balloon-borne experiment with six tunable diode laser spectrometers[END_REF]. In brief, six mid-infrared region (3 -8 μm) laser beams are passed between two mirrors spaced 3.50 m apart in a Herriott configuration multipass cell located at the extremities of a deployable mast below the gondola, leading to an average path length of 430.5 m. Several species, such as O 3 , N 2 O, CH 4 , HCl, NO 2 , HNO 3 and CO were measured with high frequency sampling (~1 Hz), which leads to a vertical resolution of a few meters (3 to 5 m), depending on the vertical velocity of the balloon. The overall uncertainties take into account the random and systematic errors, combined as the square root of their quadratic sum. The two main sources of random errors are the fluctuations of the laser background emission signal and the signal-tonoise ratio. At low altitudes (< 20 km), the overall uncertainties have the same contribution. Systematic errors originate essentially from the laser line width (an intrinsic characteristic of the laser diode), which contributes more at lower pressure than at higher pressures (lower altitude). For OCS, the overall uncertainty is 3.31.0% (1 confidence level) below 18 km, increasing from 10.73.0% at 18 km to 30% at 26 km and higher than 30% above 26 km. Absorption micro-windows for the three flights were 2056.1-2056.5 cm -1 , unperturbed by ozone or water absorption. The concentration of OCS was retrieved with the new HITRAN 2012 database [START_REF] Rothman | The HITRAN 2012 molecular spectroscopic database[END_REF].

The following time sequences characterized the SPIRALE flights: During the night of 9 to 10 June 2008 in Brazil, the measurements started at 23:30 UT (i.e. 20:30 local time) and the maximum altitude of 33.6 km (6.9 hPa) was reached at 01:14 UT.

Results and discussion

a Column variability 2 presents the values and the errors corresponding to the standard deviation used to make Figure 1. Seasonal patterns appear in the OCS data with a maximum in April-June (10.9×10 15 molecules cm -2 and 544 pptv) and a minimum in November-January (9.8×10 15 molecules cm -2 and 507 pptv). OCS variability with season is equal to 10% (peak to peak variation). The annual variation and the period of maximum observed are consistent with the study of [START_REF] Lejeune | Trend evolution and seasonal variation of tropospheric and stratospheric carbonyl sulfide (OCS) above Jungfraujoch[END_REF] at Jungfraujoch (NDACC station, 46.6°N). However, these authors observed a minimum during October-December with a peak to peak variation of 7.9% whereas we found a minimum shifted to November-January. Similar periods are observed at Izana (28.30°N, NDACC station) as reported by [START_REF] Notholt | Precursor gas measurements[END_REF] with a minimum in October-December and an annual variation of 10%. However a minimum during November-January period, as in our work, is observed at Kitt Peak (32°N) [START_REF] Rinsland | Ground-based infrared spectroscopic measurements of carbonyl sulfide: Free tropospheric trends from a 24-year time series of solar absorption measurements[END_REF]. To sum up, a strong seasonal variation exists for many ground-based sites in the Northern hemisphere, with a minimum between October and January and a maximum between April and June [START_REF] Notholt | Precursor gas measurements[END_REF]. The seasonal variations are directly related to the natural sinks and sources of OCS (Kettle et al., 2002a). The amplitude of the variation depends on surface processes such as the imbalance between vegetation and soil uptakes and oceanic fluxes. The maximum in the total column occurs during the highest production of OCS (direct flux), DMS and CS 2 (indirect flux) by biogenic processes in the ocean during late spring and the beginning of summer (Kettle et al., 2002a) and just before the maximum of OCS uptake by land plants during July-September [START_REF] Kettle | Comparing forward and inverse models to estimate the seasonal variation of hemisphere-integrated fluxes of carbonyl sulphide[END_REF]. The minimum period in October-January occurs during the lowest production by oceans and during less stronger but still significant uptake by plants, soils and OH reaction (Kettle et al., 2002a).

 SEASONAL VARIABILITY ABOVE PARIS

 LATITUDINAL VARIABILITY

An update of the long-term OCS total column shows an annual increase lower than 0.79% (Montzka and Reimann, WMO 2011). Consequently, for the rest of this study, we assume a negligible inter-annual variation and can compare measurements from different years. OCS total columns are extracted from SWIR-balloon measurements at Esrange and from QualAir FTS Paris station measurements. These total columns are compared with other weighted average measurements (the weight of each measurement is relative to its standard deviation) OCS total column from NDACC stations (http://www.ndsc.ncep.noaa.gov/), from Polarstern cruise campaigns well referenced [START_REF] Notholt | Latitudinal variations of trace gas concentrations in the free troposphere measured by solar absorption spectroscopy during a ship cruise[END_REF][START_REF] Xu | Measurements of carbonyl sulfide (COS) in surface seawater and marine air, and estimates of the air-sea flux from observations during two Atlantic cruises[END_REF] and from other campaigns of the MkIV instrument at different latitudes.

Most of the measurements used for this study come from instruments using FTIR to detect OCS, and mostly in the same domain of wavelengths (2040 cm -1 and 2080 cm -1 ).

A total of 19 stations of measurements are used with 11 datasets coming from the MkIV instrument [START_REF] Toon | The JPL MkIV Interferometer[END_REF]. MkIV uses the bands between 2041 and 2077 cm -1 to detect OCS in the atmosphere. The other stations measurements use the same approach to derive OCS. All the dataset coming from MkIV use the new HITRAN database updated in 2012 [START_REF] Rothman | The HITRAN 2012 molecular spectroscopic database[END_REF] [START_REF] Notholt | Latitudinal variations of trace gas concentrations in the free troposphere measured by solar absorption spectroscopy during a ship cruise[END_REF][START_REF] Xu | Measurements of carbonyl sulfide (COS) in surface seawater and marine air, and estimates of the air-sea flux from observations during two Atlantic cruises[END_REF] and from the SWIR-balloon instrument from the Esrange balloon campaign (in red, [START_REF] Té | Balloonborne calibrated spectroradiometer for atmospheric nadir sounding[END_REF].

A latitudinal trend seems to appear in the OCS total columns whatever the season. OCS total column decreases from tropical to polar latitudes, from (10-11)×10 15 molecules cm -2 to (6.5-9)×10 15 molecules cm -2 . SWIR measurements from Esrange and QualAir FTS measurement (in red symbols in Figure 2) are in accordance with the latitudinal trend observed within the uncertainties. Polarstern cruise campaigns highlight a maximum in the values near 20°N of latitude in the Atlantic Ocean (17°N-21°W for JFM months and 23°N-30°W for OND). [START_REF] Bates | Sulfur Emissions to the Atmosphere from Natural Sources[END_REF] show maximum oceanic sulfur emissions in latitude between 5°N-20°N. In addition, over the Atlantic Ocean, the atmosphere is clearly affected by the African and South American biomass burning leading to OCS emission [START_REF] Junkermann | On the budget of photooxidants in the marine boundary layer of the tropical South Atlantic[END_REF].

The high latitude part seems to be more affected by seasonal variations. While a seasonal variation of 14% is observed at Thule (76.53°N), a mean value of 10.5×10 15 molecules cm -2 without any significant variation for the six months of measurements (JFM and OND) is measured for latitudes between 20°S and 20°N. A minimum of 6.61×10 15 molecules cm -2 is observed at Thule during OND compared to the maximum value during AMJ of 7.58×10 15 molecules cm -2 . [START_REF] Barkley | Global distributions of carbonyl sulfide in the upper troposphere and stratosphere[END_REF] observed the same phenomena in the OCS vertical profile from ACE for different seasons with the tropical latitude (20°S-20°N) with OCS values remaining rather constant over the year.

OCS has a long lifetime, resulting in a quasi-uniformity of its vmr in the troposphere. However, is the tendency in the total column also observed in the partial column above 13 km? Figure 3 shows the OCS partial column in the upper troposphere and lower stratosphere (UTLS) (above 13 km) for the balloon campaigns of SWIR- 3 and the POLARIS, STRAT, StraPolEte campaigns in June and JAS months and are represented in blue. As for total column, a trend of the OCS column with the latitude appears for altitudes above 13 km, from 0.5×10 15 molecules cm -2 at the polar latitudes to 2.0×10 15 molecules cm -2 . This trend observed also in the stratosphere shows that the tropical UTLS is composed of tropospheric young air transported from local source compared to the polar stratosphere composed of aged air coming mainly from tropical stratosphere.

As previously for total column in the tropics, a maximum value is found at 17°N of latitude in the Atlantic Ocean (17°N-66°W, Near Dominican Republic). Again, this is both due to the high oceanic emissions found at these latitudes (5°N-20°N) and transported by convection and to the African biomass burning transported over the Atlantic Ocean [START_REF] Ricaud | Impact of land convection on troposphere-stratosphere exchange in the tropics[END_REF]. 1957) were located at 16.9 km, 9.5 km and 9.3 km height on 9 June 2008, 24 August latitude contains values in the upper troposphere, and in particular in the tropical tropopause layer (TTL) [START_REF] Fueglistaler | Tropical tropopause layer[END_REF]. The altitude of the TTL for the SPIRALE flight identified by blue line in Figure 4 is 14.4 km -18.0 km [START_REF] Marécal | Impact of deep convection on the tropical tropopause layer composition in Equatorial Brazil[END_REF]. The transition from the TTL to the stratosphere is clearly seen by a decrease of the OCS vmr from more than 550 ppt to less than 500 ppt. The TTL is characterized by the transition from a pure tropospheric regime to a pure stratospheric regime. The bottom of the TTL is directly affected by the main convective outflow and consequently the OCS vmr is almost constant, with values comparable to those found in the troposphere. The upper TTL is not affected by convection and consequently OCS sinks (photolysis, OH reaction) are predominant. The OCS vertical distribution is well shown by the profiles. OCS mixing ratios at the tropical latitude are approximately constant between 14 and 17 km altitude with values of 540-570 pptv and then start to decrease above the thermal tropopause (Tp)

from 550 pptv at 17 km height to 34 pptv at 28.5 km height. Indeed, OCS is destroyed in the stratosphere mainly by photolysis [START_REF] Notholt | Precursor gas measurements[END_REF] with a maximum absorption for λ < 250 nm [START_REF] Molina | Temperature dependent UV absorption cross sections for carbonyl sulfide[END_REF]. According to [START_REF] Demore | Chemical kinetics and photochemical data for use in stratospheric modelling[END_REF], for λ < 250 nm, the actinic flux is maximum for altitudes higher than 20 km. The OCS mixing ratio behavior at polar latitudes is similar to that at tropical latitude. Due to instrumental problems, the polar profile is more scattered than the tropical one. We observe for polar latitude a decrease of OCS vmr with altitude, from 420±100 pptv for altitudes lower than 17 km and then a decrease from 460 pptv to 150 pptv at 22 km.

During the STRAT campaign in July 1996, the MkIV instrument observed similar behaviors with OCS constant vmr values of 440 pptv below 14 km [START_REF] Leung | Isotopic fractionation of carbonyl sulfide in the atmosphere: Implications for the source of background stratospheric sulfate aerosol[END_REF] and then a decrease to 120 pptv at 22 km.

The latitudinal variation of OCS in the stratosphere is significant, with a large vmr difference increasing from about 150 pptv to about 300 pptv between the tropopause and 22 km. [START_REF] Barkley | Global distributions of carbonyl sulfide in the upper troposphere and stratosphere[END_REF] with the OCS ACE satellite data observed a similar difference in OCS vmr, of about 250 pptv, between the tropical and polar latitudes. This phenomenon is also observed in the OCS partial column explained in the Section 3a (Latitudinal variability). This is due to the global atmospheric transport regulating the OCS polar stratospheric amount. OCS tropospheric air reaches the stratosphere at tropical latitude. Subsequently, OCS is transported poleward by the Brewer-Dobson circulation while being photodissociated. Tropical and midlatitude poleward intrusions of tropospheric air into the extratropical stratosphere are also possible stratospheric sources of OCS [START_REF] Roiger | In-situ observation of Asian pollution transported into the Arctic lowermost stratosphere[END_REF][START_REF] Krysztofiak | Detection in the summer polar stratosphere of air plume pollution from East Asia and North America by balloon-borne in situ CO measurements[END_REF] which possibly explain the various OCS vmr enhancements on the polar profiles.

 COMPARISONS WITH ACE SATELLITE DATA

The Atmospheric Chemistry Experiment (ACE) on board the Canadian SCISAT satellite [START_REF] Bernath | Atmospheric Chemistry Experiment (ACE): Mission overview[END_REF]  STRATOSPHERIC LIFETIME [START_REF] Plumb | Interrelationships between mixing ratios of long-lived stratospheric constituents[END_REF] highlighted the use of tracer-tracer correlations to determine the atmospheric lifetime of species. Simultaneous N 2 O measurements were performed by SPIRALE in the tropical and polar regions. As OCS, N 2 O has purely tropospheric sources [START_REF] Seinfeld | Atmospheric chemistry and physics: from Air Pollution to Climate Change, chapter 7[END_REF] and possess a long lifetime compared to horizontal and vertical air motions. The lower stratosphere (below 22 km) correlation between OCS and N 2 O is linear and the relationship between their atmospheric lifetimes is represented by this equation:

O N OCS O N OCS vmr vmr A 2 2 / /     (9)
Where τ represents the stratospheric lifetime, vmr the tropospheric volume mixing ratio, A the slope of the correlation [START_REF] Plumb | Interrelationships between mixing ratios of long-lived stratospheric constituents[END_REF].

Due to the absence of data below 18 km in the OCS profile at the polar latitude on 21 April 2011, the lower stratosphere correlation between OCS and N 2 O cannot be plotted and no lifetime has been calculated for this date. mean tropospheric mixing ratio of 328±8 ppbv N 2 O and 550±40 pptv OCS from SPIRALE measurements to calculate the OCS stratospheric lifetime at tropical and polar latitudes. We find a lifetime of 70±20 years for the polar latitude case and 58±14 years for the tropical latitude case, which leads to a mean global stratospheric lifetime of 64±17 years. Actinic fluxes for λ < 250 nm (OCS photodissociation) and for altitudes higher than 20 km show a difference with latitude [START_REF] Finlayson-Pitts | Chemistry of the upper and lower atmospheretheory, experiments and applications[END_REF].

The correlations between N 2 O and OCS from the data of ACE-FTS used in the section 3.b (Comparisons with ace satellite data) in polar region and from the MkIV measurements during several campaigns (SOLVE I and II) in the Kiruna region are used to extract the OCS lifetime in polar region. These results are compared with our lifetime from SPIRALE measurements in polar region and the other lifetime found by the studies of [START_REF] Engel | vertical profile measurements of carbonyl sulfide in the stratosphere[END_REF] and by [START_REF] Barkley | Global distributions of carbonyl sulfide in the upper troposphere and stratosphere[END_REF] only at polar latitude.

Table 3 shows these comparisons. All the lifetimes found in the polar region are consistent each other and allows us to establish an OCS lifetime by taking into account the overall uncertainties in polar region of 71±10 years (1σ). MkIV measurements (G. Toon and http://mark4sun.jpl.nasa.gov/m4data.html) compared with the study of [START_REF] Engel | vertical profile measurements of carbonyl sulfide in the stratosphere[END_REF] and [START_REF] Barkley | Global distributions of carbonyl sulfide in the upper troposphere and stratosphere[END_REF] We can calculate the stratospheric OCS sink by dividing the total OCS mass in the atmosphere, 5.86×10 12 g (inferred from the total atmospheric mass of 5.148×10 21 g [START_REF] Trenberth | The mass of the atmosphere : A constraint on global analyses[END_REF] and our tropospheric OCS vmr) by our global OCS lifetime. We find a global stratospheric sink of 99±26 Gg OCS yr -1 , corresponding to 54±14 Gg S yr -1 . Table 4 shows a comparison between our values of OCS stratospheric sink and several values from literatures. Our range values are in general accordance with the average 50±15 (1) Gg S yr -1 of the OCS stratospheric sinks given by previous studies.

Conclusions and perspectives

Carbonyl sulfide (OCS) is the most abundant sulfur-containing compound in the Due to its long lifetime, OCS can easily reach the stratosphere and is, with SO 2 , the precursor of sulfate aerosols. Consequently, it is a key species for understanding heterogeneous chemistry of stratospheric ozone depletion.

A seasonal variation is extracted from the ground-based FTS measurements of the

  The first flight of SWIR-balloon (SWIR01 flight) took place in polar atmosphere and the instrument was flown around Esrange-Kiruna region (Sweden, 67.9°N, 21.1°E) on 14 August 2009 at 9:26 UT for 5h58min. The first position of the gondola at float was [68.1°N, 21.0°E] and the last position before the balloon cut-off was [68.4°N, 20.6°E]. The altitude at float varied around 34 km.

Figure 1

 1 Figure 1 presents the monthly mean OCS total column in Paris and the tropospheric OCS vmr averaged from March 2011 to March 2012 measured by QualAir FTS. Table2presents the values and the errors corresponding to the standard deviation

Figure 1 .

 1 Figure 1. Average OCS total column and volume mixing ratio (vmr) with error bars (1 standard deviation σ) above Paris measured by FTIR from March 2011 to March 2012. The grey dashed lines represent the annual weighted averages (the weight of each measurement is relative to its standard deviation).

  . For the other stations utilizing the HITRAN spectroscopic parameters before 2008, the columns were multiplied by 0.86, corresponding to the increase in the line strengths of OCS. All comparisons are shown in Figure 2. The Figure is divided into four plots according to the seasonal variation found in Section 3a (Seasonal variability): January/February/March (JFM)/December (OND). The total columns of OCS plotted in Figure 2 are scaled to sea level by multiplying the total column by the ratio between the standard pressure (1013 hPa) and the surface pressure observed in each station.

Figure 2 .

 2 Figure 2. Pressure corrected total column of OCS with error bars (standard deviation) versus latitude (°N). The ground based (NDACC and other) and balloon-borne measurements from the Bruker IFS125HR interferometer, from the MkIV instrument (in orange, Toon et al. 1991) and from the QualAir FTS instrument (in red) are weighted average of several years with minimum and maximum of each range shown by crosses and dashed lines. The other measurements are from Polarstern cruise campaigns measured by Bruker 120M interferometer(Notholt et al.,

Figure 3 .

 3 Figure 3. OCS partial column above 13 km with error bars (representing the 1- measurement precisions) versus latitude (N°) from several MkIV campaign (STRAT, POLARIS, SOLVE I and II and AASE II)[START_REF] Toon | The JPL MkIV Interferometer[END_REF][START_REF] Toon | Airborne observation of the composition of the 1992 tropical stratosphere by the FTIR solar absorption spectrometry[END_REF], aircraft P-AVE campaigns measured by FTIR instrument[START_REF] Coffey | The temporal trend of stratospheric carbonyl sulphide[END_REF] and from StraPolEte balloon campaign measured by the SWIR instrument (in dark cyan)[START_REF] Té | Balloonborne calibrated spectroradiometer for atmospheric nadir sounding[END_REF]. The values in blue (blue and dark cyan) are for campaigns occurring during June, July and August and in black for campaigns occurring during January, February and March.

Figure 4 .

 4 Figure 4. SPIRALE measurements of OCS volume mixing ratios (in ppt) with error bars on 9 June 2008 (blue full circles) above Teresina (5.1°S-42.9°W, Brazil), on 24 August 2009 (red open circles) and 20 April 2011 (black full circles) above Esrange (67.9°N-21.1°E, Sweden). The Thermal Tropopause (Tp) and the tropical tropopause layer (TTL) top and bottom are represented by blue line and blue dashed lines respectively for the flight at tropical latitude on June 2008.

  was launched on 12 August 2003. Using the solar occultation technique, two instruments measure the vertical profiles of atmospheric constituents from the troposphere to the lower mesosphere. For this study, we use the OCS version 3.0 (v.3.0) of the vmr profiles from the instrument ACE Fourier Transform Spectrometer (ACE-FTS). A total of thirteen microwindows are used for the OCS ACE-FTS v. 3.0 vmr retrievals, positioned at 1950.10 cm -1 for the first one, and between 2039.01 cm -1 and 2057.52 cm -1 for the other ones. In order to compare ACE-FTS and SPIRALE we must take into account the difference in the vertical resolution of these two instruments. Indeed, ACE-FTS has a vertical resolution of 2-6 km and that of SPIRALE is of a few meters. For comparisons, we use the ACE-FTS retrieved results which are interpolated onto a 1 km grid from 0.5 km to 149.5 km. A weighting triangular function of 3 km at the base has been applied to SPIRALE data interpolated first onto a 5 m grid. Consequently, we have truncated the bottom and the top of SPIRALE profile from 1.5 km.As explained in Section 3a (Latitudinal variability), we assume a negligible OCS variation between 2003 (start date of ACE measurements) and the year of our SPIRALE balloon flights(i.e. 2008 and 2009). Moreover, as seen in Section 3a (Seasonal variability), there is a seasonal trend in the OCS vmr. Consequently, for ACE-FTS comparisons with SPIRALE at Teresina on June 2008, we use OCS profiles from April to June and for comparisons at Esrange in August 2009, we use OCS profiles from July to September and only the profiles above land (balloon flights occurred above land).There are not a lot of ACE profiles available at tropical latitudes thus for the comparison at Teresina, we used 6 profiles (between 2004 and 2007) acquired at 5±6°S and 43[+20,-10]°W. For the comparison at Kiruna, we used 56 profiles (between 2004 and 2011) acquired at 67.9±6°N and 21.1±10°E.

Figure 5 (

 5 Figure 5 (left) shows the comparisons between SPIRALE OCS measurements and the weighted average (weight of each measurement relative to its standard deviation) of the ACE-FTS OCS profiles. Figure 5 (right) shows the difference between SPIRALE and ACE average (SPIRALE -ACE %) compared to the mean value and the error bars. At tropical latitudes, for altitudes lower than 22.5 km, ACE-FTS underestimates OCS values compared to SPIRALE. The differences between SPIRALE and ACE indicate a bias between 15 % and 20%. For altitudes higher than 22.5 km at Teresina, SPIRALE and ACE-FTS are in very good agreement (within 10%).For polar latitudes, at Esrange, SPIRALE and ACE-FTS are also in very good agreement (within 11%) by taking into account the uncertainties of both instruments except at the altitude of 16.5 km where ACE-FTS underestimates OCS values compared to SPIRALE with a bias of 20% .Velazco et al. (2011, Figure 18) show similar bias of 15% between ACE-FTS and MkIV OCS profiles for altitudes between 12 km and 23 km (350-550 K) at 35°N of latitude.

Figure 5 .

 5 Figure 5. Left panels: SPIRALE measurements of OCS volume mixing ratios (in pptv) with error bars (red triangles) compared to the weighted average (weight of each measurement relative to its 1- error) of the ACE-FTS OCS profiles with error bars (1-) (black squares). The grey dashed lines represent the spread (minimum and maximum values found in all the ACE profiles). Right panels: percentage differences between SPIRALE and ACE-FTS with error bars. Top panels: Comparisons for SPIRALE flight on 9 June 2008 at Teresina (Brazil) and Bottom panels: Comparisons for SPIRALE flight on 24 August 2009 at Esrange (Sweden).

Figure 6 .

 6 Figure 6.A represents the N 2 O:OCS correlation in the lower stratosphere at the tropical latitude in 2008 and Figure 6.B at the polar latitude in 2009 from SPIRALE measurements. Both linear correlations lead to an r² correlation coefficient higher than 0.75. We use N 2 O lifetime from Montzka and Fraser et al. (2003) of 117±20 years and a

  QualAir (Qualité de l'Air) platform at Jussieu campus in Paris. A maximum value of the total column and the tropospheric mean vmr are found in April-June and a minimum value in November-January with a peak to peak variability of 10%.Total column measurements from QualAir FTS measurements above Paris and SWIR balloon-borne instruments are compared with several MkIV measurements, NDACC stations measurements, ship measurements and balloon-borne measurements, to highlight the OCS variability with latitude. OCS total column decreases with increasing latitude. The variability is also found for the partial column above 13 km showing the atmospheric OCS transport from the tropical troposphere into the tropical stratosphere and then into the polar stratosphere. The OCS total columns show a seasonal variability more pronounced at polar latitudes compared to tropical latitudes.The first high resolution vertical profile of OCS at two different latitudes is presented in this paper. The SPIRALE balloon-borne instrument has flown over the tropical region (5.1°S, Teresina, Brazil) on 9 June 2008 and over the Arctic polar region

  The inverse code of PROFFIT supports both Optimal

	Estimation and Twomey-Tikhonov constraints. The atmosphere is modeled by 49
	altitude levels where the a priori volume mixing ratios (vmr) profiles are sampled in
	parts per million by volume (ppmv) unit (see http://waccm.acd.ucar.edu, Whole
	Atmosphere Community Climate Model). Pressure and temperature vertical profiles are
	from	NCEP	(National	Centers	Environmental	Prediction,	see
	http://www.ncep.noaa.gov) and H 2 O continuum from Clough et al. (2005). HITRAN
	2008 database is used for the spectroscopic parameters (position, intensity, pressure line
	shift and broadening; see Rothman et al. 2009). No change has been noticed for OCS in
	our micro-windows between the HITRAN 2008 database and HITRAN 2012 (which
	was released after the submission of the present paper). Four micro

Table 2

 2

	represent the standard deviations of all retrieved columns during one month, without
	taking into account the retrieval errors. This allows a better assessment of the daily
	fluctuation during one month.
	b Balloon-borne instruments
	This study use data from balloon-borne measurements from three large balloon
	campaigns. The first was conducted from Teresina (5.1°S, 42.9°W) in North-East Brazil
	in June 2008 with a SPIRALE flight on 9 June 2008. The second campaign was
	conducted from Esrange (Swedish Space Corporation) close to Kiruna (67.9°N-21.1°E,
	Sweden). The SWIR-balloon instrument flew on 14 August and SPIRALE on 24
	August 2009. The last campaign was also conducted from Esrange, with a SPIRALE
	instrument flight on 21 April 2011.

Table 2 .

 2 Monthly averaged OCS total column and volume mixing ratio, with their errors, above Paris measured by QualAir FTS (described in Section 2a) from March 2011 to March 2012. For September, only one day measurements is available, which is not representative of monthly averaging.

		Months	Total column (×10 15 molecules cm -2 )	Error (1σ) on column (×10 15 molecules cm -2 )	Tropospheric vmr (pptv)	Error on tropospheric vmr (pptv)
		January	9.74	0.29	500.7	13.0
		February	10.05	0.30	517.3	13.4
		March	10.03	0.30	511.9	13.3
		April	10.46	0.31	541.2	14.0
		May	10.75	0.32	553.6	14.3
		June	10.62	0.32	552.7	14.3
		July	10.36	0.31	538.1	13.9
		August September	10.02 --1	0.30 --	520.7 --	13.5 --
		October	10.08	0.30	519.9	13.5
		November	10.00	0.30	517.5	13.4
	1	December	9.93	0.30	512.7	13.3

  campaigns where OCS is measured by FTIR (P-AVE, Coffey and Hannigan, 2010) AASE II; http://espoarchive.nasa.gov/). All these datasets use the HITRAN 2008 or 2012 databases. The SOLVE I and II, P-AVE and AASE II occurred in December and JFM months and are represented in black in Figure

balloon instrument at Esrange compared with balloon campaigns of the MkIV instrument (SOLVE I and II, STRAT and POLARIS, http://espoarchive.nasa.gov/) and several aircraft

Table 3 .

 3 OCS lifetimes (years) in polar region found from SPIRALE measurements, from ACE-FTS measurements (data from the section3.b (Comparisons with ace satellite data)) and from

  .

		Engel and Schmidt (1994)	Barkley et al. (2008)	ACE-FTS	MkIV	SPIRALE
	Lifetime (Years)	69±28 1	76±32 1	70±16 2	74±24 2	70±20 2
	1 CFC-12/OCS correlation 2 N 2 O/OCS correlation

Table 4 .

 4 OCS stratospheric sink (Gg S yr-1) comparison with previous studies

		Crutzen (1976)	Turco et al. (1980)	Crutzen et Schmailzl (1983)	Engel et Schmidt (1994)	Chin et Davis (1995)	Weisenstein et al. (1997)	Barkley et al. (2008)	This work
	OCS
	Stratospheric sink	50	80	43	49±20	30	49	50±16	54±14
	(Gg S an -1 )

Acknowledgements

The authors thank the LPC2E technical team (L. Pomathiod, B. Gaubicher, G. Chalumeau, B. Coûté, T. Vincent, F. Savoie and G. Jannet, S. Chevrier) for the SPIRALE instrument preparation, the LPMAA technical team (I. Pépin, C. Rouillé and P. Marie-Jeanne) the SWIR-balloon instruments preparation, the CNES balloon launching team and the Swedish Space Corporation at Esrange for successful operations. The instrumental funding of the QualAir FTS was supported by the university Paris VI (Université Pierre et Marie Curie). The ETHER database (Pôle thématique du CNES-INSU-CNRS) and the "CNES sous-direction Ballon" are partners of the project. This work was supported by the European integrated project SCOUT-O3 (GOCE-CT-2004-505390), by the European Space Agency (ESA) as part of the Envisat validation program, by the French national LEFE/INSU project "UTLS Tropicale", by the French national Labex "VOLTAIRE" (ANR-10-LABX-100-01), by the "StraPolÉté" project funded by the "Agence Nationale de la Recherche (ANR-BLAN08-1_31627), the "Centre National d'Etudes Spatiales (CNES), and the Institut Polaire Paul-Emile Victor (IPEV), and by the "ENRICHED" project funded by the CNES, INSU-CNRS and IPEV. The Atmospheric Chemistry Experiment (ACE), also known as SCISAT, is a Canadian-led mission mainly supported by the Canadian Space Agency. The data used in this publication were obtained as part of the Network for the Detection of Atmospheric Composition Change (NDACC) and are publicly available (see http://www.ndacc.org). Part of this work was performed at the Jet Propulsion Laboratory, California Institute of Technology, under contract with NASA. The authors thank also J. Notholt and O. Schrems for their data from the Polarstern campaign. We thank M. Coffey and J. Hannigan for their results from the P-AVE campaign.

Indeed, for the same solar time and months (here June and August), the solar zenith angle is smaller at tropical latitudes compared to polar latitudes and for λ < 250 nm the actinic fluxes increase with decreasing solar zenith angles. Consequently, OCS undergoes stronger photolysis at tropical latitudes and has a lower stratospheric lifetime. Using the correlation with CFCl 3 (lifetime of 55 years), [START_REF] Engel | vertical profile measurements of carbonyl sulfide in the stratosphere[END_REF] found an OCS stratospheric lifetime of 69±28 years for latitudes higher than 43°N from measurements at Kiruna (68°N) and Aire sur l'Adour (43°N). Using correlations with CF 2 Cl 2 and CFCl 3 , [START_REF] Barkley | Global distributions of carbonyl sulfide in the upper troposphere and stratosphere[END_REF] found a global OCS stratospheric lifetime of 64±21 years. These values are consistent with our global value.

The value found by [START_REF] Engel | vertical profile measurements of carbonyl sulfide in the stratosphere[END_REF] is consistent with our polar value.

Unfortunately, [START_REF] Barkley | Global distributions of carbonyl sulfide in the upper troposphere and stratosphere[END_REF] did not estimate OCS lifetime at latitudes between 20°N and 20°S, and so no comparison is possible with our OCS lifetime found at tropical latitude (Teresina).

(67.9°N, Esrange, Sweden) on 24 August 2009. The OCS vertical dependence is similar for both latitudes with a significant decrease in OCS vmr above 20 km. OCS profiles are also compared with non-coincident ACE satellite measurements and show consistency for the polar latitude case and a bias of 15-20% for the tropical latitude case for altitudes lower than 22 km.

Using OCS:N 2 O correlation from SPIRALE, OCS stratospheric lifetime is accurately determined. We found a lifetime of 70±20 years at the polar latitude and 58±14 years at the tropical latitude which leads to a mean global stratospheric lifetime of 64±17 years. The global lifetimes is used to determine the stratospheric sink of OCS.

In the lower stratosphere, the OCS stratospheric degradation produces 54±14 Gg S yr -1 .

To go further, the contribution of OCS to the stratospheric aerosol burden should be more investigated by modeling study.


Guy Meunier
email: guy.meunier@ivry.inra.fr

Risk aversion and technology mix in an electricity market

Keywords: Electricity market, Technology mix, Risk aversion

This article analyzes the effect of risk and risk-aversion on the long-term equilibrium technology mix in an electricity market. It develops a model where firms can invest in baseload plants with a fixed variable cost and peak plants with a random variable cost, and demand for electricity varies over time but is perfectly predictable. At equilibrium the electricity price is partly determined by the random variable cost and the returns from the two kinds of plants are negatively correlated. When the variable cost of the peak technology is high the return of peak plants is low but the return to baseload plants is high. Risk-averse firms reduce the capacity of the riskiest technology and develop the capacity of the other, compared to risk-neutral firms.

In the particular case where a risk-neutral firm invests heavily in baseload technology and only sparely in peak capacity, a risk-averse firm would invest less in baseload, increase peak capacity, and increase total installed capacity.

introduction

The purpose of this article is to analyze the influence of firms' risk aversion on the technology mix in an electricity market if the variable cost of a technology is random. In the electricity industry, the fundamental explanation for the existence of a technology mix is the great variability of the demand for electricity over the course of a year and the limited potential to utilize intertemporal arbitrage to smooth these variations. To address the time-variable nature of electricity demand, firms invest in several types of technology; baseload technologies are more efficient for frequent production, whereas peak technologies are employed for more sporadic production.

In addition to this well-known variability of the electricity demand curve, electricity producers face numerous uncertainties with respect to costs of production. In particular, the prices of fossil fuels and CO 2 emissions are uncertain. These uncertainties are likely to influence the technology mix that is chosen to meet the variable demand, particularly if firms are risk-averse.

A canonical model of an electricity market is used to consider a situation in which the demand curve is variable and two different technologies are available to produce electricity: a baseload technology and a peak technology. The variable cost of the peak technology is random. It is assumed that electricity producers anticipate the variability of the demand curve; moreover, these producers are regarded as risk-averse. In the risk-neutral benchmark situation, these firms invest in the technology mix that minimizes the expected cost of servicing the electricity demand. We consider how risk aversion modifies the equilibrium technology mix and investigate how this modification is related to the cost structure and the variability of electricity demand.

2

This study demonstrates that risk-aversion modifies the technology mix in either one of two directions: (i) the baseload capacity is increased, although the peak and total capacity are reduced; or (ii) the baseload capacity is reduced, but the peak and the total capacity are increased. In the latter situation, compared with the risk-neutral benchmark, risk-averse firms overinvest in both the risky technology and their total capacity.

The uncertainty with respect to the variable cost of the peak technology translates into an uncertainties in the electricity prices during the fraction of the year in which the peak technology establishes these prices. Both technologies have risky returns. The risk faced by peak units is related to the fraction of the year in which the peak technology is a sub-marginal source of electricity, whereas the risk faced by baseload units is related to the fraction of the year during which the peak technology establishes the price. Thus, the two technologies face negatively correlated risks. In certain circumstances, the baseload technology involves higher risk than the peak technology, and risk-averse firms will therefore tend to decrease their quantity of baseload capacities and increase their capacities of the risky peak technology to hedge against the risk incurred by baseload capacities.

The influence of the cost structure and the variability of electricity demand is also investigated. Among other conclusions, this study demonstrates that over-investment in the peak technology is more likely to arise when demand is less variable. This result is of particular interest if the framework of this study is applied to considerations of environmental policy because in these considerations, the peak technology may represent a polluting method of generating electricity. Thus, the development of renewable intermittent electricity production might induce an increase in peak capacity through increasing the variability of the (residual) load [START_REF] Lamont | Assessing the long-term system value of intermittent electric generation technologies[END_REF]. Our results reveal that this effect could be softened if firms are risk-averse and if the future price of CO 2 emissions is uncertain. However, the environmental benefit from real-time-pricing [START_REF] Holland | Is real-time pricing green? the environmental impacts of electricity demand variance[END_REF]) might be reduced if producers are risk-averse.

The questions of whether firms are risk-averse and what the consequences of firms' potential risk aversion may be are issues that have received a considerable attention both in the general literature and in studies of energy and resource economics. With a complete sets of markets [START_REF] Arrow | Existence of an equilibrium for a competitive economy[END_REF], firms' profits are not random and whether they are risk-averse or not does not matter. If markets are incomplete, [START_REF] Diamond | The role of a stock market in a general equilibrium model with technological uncertainty[END_REF] suggests that firms should be risk-neutral, and shareholders should use capital-asset markets to hedge their risks. However, there are several theoretical arguments (for a review, see [START_REF] Banal-Estanol | Mergers with Product Market Risk[END_REF] and certain empirical evidence that suggests that firms are risk-averse or at least behave as if they are risk-averse.2

Numerous authors have considered how risk aversion influence production choices in situations that feature random demand and perfect competition [START_REF] Dhrymes | On the theory of the monopolistic multiproduct firm under uncertainty[END_REF][START_REF] Mccall | Competitive production for constant risk utility functions[END_REF][START_REF] Baron | Price uncertainty, utility, and industry equilibrium in pure competition[END_REF][START_REF] Sandmo | On the theory of the competitive firm under price uncertainty[END_REF][START_REF] Appelbaum | Measures of risk aversion and comparative statics of industry equilibrium[END_REF] or scenarios that involve monopolies [START_REF] Baron | Demand uncertainty in imperfect competition[END_REF][START_REF] Leland | Theory of the firm facing uncertain demand[END_REF] or oligopolies [START_REF] Tessitore | Market segmentation and oligopoly under uncertainty[END_REF][START_REF] Wambach | Bertrand competition under cost uncertainty[END_REF][START_REF] Asplund | Risk-averse firms in oligopoly[END_REF][START_REF] Banal-Estanol | Mergers with Product Market Risk[END_REF]. The assumption regarding the risk aversion of firms has also been used to analyze the firms' use of not only financial futures and option contracts [START_REF] Mckinnon | Futures markets, buffer stocks, and income stability for primary producers[END_REF][START_REF] Newbery | The theory of Commodity Price Stabilization[END_REF][START_REF] Moschini | The hedging role of options and futures under joint price, basis, and production risk[END_REF] but also vertical integration [START_REF] Hirshleifer | Risk, futures pricing, and the organization of production in commodity markets[END_REF][START_REF] Aïd | Hedging and vertical integration in electricity markets[END_REF].

In a situation featuring random input prices, [START_REF] Stewart | Factor-price uncertainty with variable proportions[END_REF] demonstrates that a risk-averse firm over-invests in riskless factors. Input price risks have also been analyzed by [START_REF] Blair | Random input prices and the theory of the firm[END_REF] and by [START_REF] Okuguchi | Input Price Uncertainty and the Theory of the Firm[END_REF], but none of these analyses have addressed the issue of technological diversification. Recently, [START_REF] Meunier | Risk aversion and technology portfolio[END_REF] considers the effect of risk aversion on choices between two risky technologies; however, he does not consider the variability of the demand but instead regards risk-aversion as the sole explanation for technological diversification. Furthermore, he assumes the existence of an exogenous correlation that arises endogenously in the current study.

In the electricity industry, the influence of producers' risk-aversion on investment decisions is a central issue because electricity prices demonstrate a high variability and because of the uncertainty that surround the development of fossil fuel markets and environmental regulations. [START_REF] Neuhoff | Insufficient incentives for investment in electricity generations[END_REF] provide a formal analysis of the influences of risk and of electricity producers' risk-aversions on the producers' investment choice with respect to a single technology. [START_REF] Willems | Market completeness: How options affect hedging and investments in the electricity sector[END_REF], who build on the framework of [START_REF] Bessembinder | Equilibrium pricing and optimal hedging in electricity forward markets[END_REF], consider the effect of financial options on both welfare and investment decisions. They do not consider the choice of technology mix but consider the incentive of a risk-averse producer to invest in one plant. They show that the initial development of financial markets can reduce the incentive to invest in peak plants. They do not consider the fact that investment in one type of plant can be used to hedge risks associated with other types of plants.

In the presence of several risky technologies, a financial portfolio framework (à la [START_REF] Markowitz | Portfolio selection[END_REF] has been utilized by [START_REF] Bar-Lev | A portfolio approach to fossil fuel procurement in the electric utility industry[END_REF] to analyze the "technology portfolio". In particular, [START_REF] Bar-Lev | A portfolio approach to fossil fuel procurement in the electric utility industry[END_REF] evaluate the mix of fossil fuels of regulated electricity utilities. 3 Roques et al. (2008) adopt a more positive perspective and determine the efficiency frontier (the expected return versus variance) of portfolios that are composed of combined cycle gas turbine (CCGT) plants, coal plants and nuclear plants for a price-taking firm that faces random electricity and gas prices. These researchers consider exogenous electricity prices and do not analyze the market equilibrium. In particular, they stress the role of the correlation between electricity and gas prices; however they had to assume this correlation, whereas this relationship arises endogenously in our framework.

Two recent works possess similarities to the present study. [START_REF] Ehrenmann | Generation capacity expansion in a risky environment: a stochastic equilibrium analysis[END_REF] use numerical simulation of an electricity industry equilibrium to assess the influence of electricity producers' risk-aversion on the total capacity that is built and on the producers' technology mix. They consider several sources of uncertainty, including the design of the EU CO 2 permit trading scheme and the evolution of fossil fuel markets. [START_REF] Fan | Risk aversion and co 2 regulatory uncertainty in power generation investment: Policy and mod-eling implications[END_REF][START_REF] Fan | Electricity capacity investment under risk aversion: A case study of coal, gas, and concentrated solar power[END_REF]) also perform numerical simulations of the equilibrium of an electricity industry; they focus on the uncertainty surrounding CO 2 prices and the effect of rules for allocating free allowances. They show that risk-aversion tends to favor investment in peak units (using gas) if permits are auctioned and not grandfathered. Compared with these two articles, we provide a tractable model; this model allows us to perform a formal analysis that contributes to a better understanding of the effects that risk produces on the technological mix and capacity of electricity producers.

The rest of the article is organized as follows: Section 2 introduces the model for this investigation. Subsequently, Section 3 describes the equilibrium in the risk-neutral benchmark situation and in the case of a risk-averse firm. Section 4 provides a generalized version of the model that incorporates more than two technologies. Section 5 considers the normative aspects of the model. Section 6 concludes.

The model

We consider a simple electricity system. The demand side is represented by a variable inelastic demand with a year's duration normalized to a value of 1. Electricity demand x is assumed to be distributed in [0, X] with a cumulative distribution function F (x). The function F is positive, increasing and differentiable, and its derivative F ′ = f is the distribution of the load; in other words, f (x) represents the duration of the year during which the demand for electricity is x. The "load duration curve" is the curve F -1 (1-t).4

The typical shape of a load duration curve is depicted in Figure 1(b).

The real time demand for electricity is x if the price is below v and 0 otherwise, v is the "value of the lost load" (VoLL). The instantaneous surplus when the demand is x and a quantity y < x (resp. y > x) of electricity is served is vy (resp. vx) in $ per time. The total surplus is the sum over the year of the flow of instantaneous surplus.

There are two technologies to produce electricity; these are labeled t = b and t = p to represent the baseload and peak technologies. As a concrete illustration we have in mind nuclear for the baseload technology and CCGT for peak units. Each technology t is characterized by a variable cost c t ($ per W yr) and a capacity cost I t ($ per W ). The variable cost of the peak technology is assumed to be random at the time of investment. We assume that the variations of c p are sufficiently small to ensure that v > c p > c b for all realizations. Finally, the expected value of c p is cp and its standard deviation σ.

The baseload technology b provides less costly production per unit of electricity throughout the year than the peak technology p, but the baseload technology is more costly than the peak technology for production over a short period: the aggregate expected cost from each technology is lower than the consumer surplus. The duration r p is assumed to be less than one, i.e. v > cp + I p , and lower than r, r p < r; this last assumption ensures that technology p is used at equilibrium; it is equivalent to the statements that r p < r b and r b < r.

c b + I b < cp + I p ,
These features are illustrated in figure 1(a).

It is assumed that there is a set I of n firms. Individual capacities of firm i ∈ I for each technology are denoted by k i b and k i p and its aggregate capacity

is k i = k i b + k i p .
Thus, the aggregate quantities of the capacities of all firms for each technology are:

k t = i k i t for t = b, p and k = k b + k p .
Once capacities are fixed, in the short term, the price is set at the marginal cost of the last unit that is utilized when all demand is satisfied, and at v in case of "rationing", i.e., when x > k. If demand is less than k b , the wholesale price is c b , and baseload capacities produce. When demand is greater than k b and smaller than k the price is c p , baseload capacities are fully utilized and the quantity xk b that remains is produced by peak capacities. For higher levels of the demand the price is v; because only k units of electricity are consumed, a portion xk of the demand is not satisfied. The profit of a firm i ∈ I may be expressed as follows:

π i = k k b (c p -c b )k i b f (x) dx + X k (v -c p )k i p + (v -c b )k i b f (x)dx (1) -I b k i b -I p k i p .
The portion of the year during which rationing occurs is T = 1-F (k) and the portion of the year during which the baseload capacity is fully utilized is

T b = 1 -F (k b )
. With these notations, the profit of a firm may be rewritten as a function of its aggregate capacity and baseload capacity:

π i (k i b , k i , c p ) = [T (v -c p ) -I p ]k i + [T b (c p -c b ) -(I b -I p )]k i b .
(2)

For i ∈ I, the risk-aversion of firm i is denoted by λ i , a positive real number. The objective function of firm i is

U i = E[π i ] - λ i 2 var(π i ). ( 3
)
Firms are assumed to be price takers, they consider the distribution of electricity prices to be given and they choose their capacity in a manner that maximizes their objective. The risk-aversion of firms is related to the uncertainty of the variable cost c p but not to the variability in the demand for electricity which is perfectly anticipated.

The essential feature of the framework is that the variable cost c p determines the price of electricity during a fraction of the year. Consequently, with endogenous electricity prices, the revenues from peak plants and baseload plants are negatively correlated: when c p is large, the former is low and the latter is large. Risk-averse firms adjust their technology mix in order to reduce the variation of their profit. The influence of the variable cost c p on the profit of a firm determines in which type of plant the firm invest to smooth its profit's variation.

The effect on a firm's profit of an increase of the variable cost c p depends on the quantities of each type of plants and on the number of hours the peak technology is marginal and inframarginal. The overall effect of an increase of

c p is dπ/dc p = -T k i + T b k i b ,
which is either positive or negative; depending on this sign, a risk-averse firm will tend to distort its mix toward the peak or the baseload technology to hedge its risk.

Risk aversion and equilibrium.

Before we examine the effect of risk aversion, we consider the risk-neutral benchmark situation.

The risk-neutral case

If a firm is risk-neutral, i.e. λ i = 0, then the equilibrium technology mix will generate a null expected profit for the firm (because of the linearity of the framework). From equation (2), it implies the following relationships for the risk-neutral duration T * and T * b :

(v -cp )T * = I p ⇔ T * = r p ; (4) (c p -c b )T * b = I b -I p ⇔ T * b = r. (5)
The first of these two relationships states the typical association that relates the value of lost load v and the loss of load probability T to the variable and capacity costs of the peak technology. The second of these equations indicates the technology mix that minimizes the expected production cost.

From these equations, an expression for the risk-free technology mix is:

k * = F -1 (1 -r p ), k * b = F -1 (1 -r) (6)
In Figure 1, the risk-neutral benchmark scenario is depicted. The optimal mix is obtained from the load duration curve, F -1 (1 -t), and the durations r and r p .

The quantities k * and k * b correspond to the technology mix chosen in the absence of risk aversion or in the absence of risk; in these cases, they correspond to the welfare-maximizing investment. In Section 4, we further elaborate on the normative properties of this technology mix.

Risk aversion

Consider the situation in which all firms are risk-averse. Each firm i ∈ I maximizes its objective function, expressed by equation (3). Given our choice of variables, the firm will choose its total capacity as if the entire capacity was based on the peak technology, and the firm will determine the share of baseload capacity by considering the effects of substituting a peak unit for a baseload unit. At an interior solution, the marginal increase in expected profit is equalized with the marginal increase in weighted variance for each technology.

Lemma 1. At equilibrium, both technologies are used and the following equations are satisfied

(v -cp )T -I p = Λσ 2 T (T k -T b k b ) (7) (c p -c b )T b -(I b -I p ) = Λσ 2 T b (T b k b -T k) (8) in which Λ = i∈I 1 λ i -1 . ( 9
)
This lemma allows for a representative firm with risk aversion Λ and capacities k and k b to be considered through the analysis of equations ( 7)

and ( 8).5

Compared to the risk-neutral benchmark, described by ( 4) and ( 5), a risk-averse firm has an incentive to soften the variations of its profit. This incentive is reflected in the right-hand sides of equations ( 7) and ( 8) which are null in the risk-neutral case. When c p is large the return from peak plants is low and the return from baseload plants is large. The short-term revenue is either increasing or decreasing with respect to c p . Depending on this monotonicity investing in a peak plant will tend to either increase or reduce the variability of the firm's profit. Overall, the effect on the variance of a firm's profit of adding a peak plant or substituting a peak plant to a baseload plant have opposite sign. The former is

T (T k i -T i b k i b ) and the latter T b (T b k i b -T i k i ).
This feature implies that the comparison with the risk-neutral benchmark is clear-cut.

Proposition 1. There is at least one equilibrium in the case of risk aversion.

Each equilibrium with risk aversion is of one of the following two types:

Type B: the total capacity is lower than k * , the baseload capacity is larger than k * b , and the peak capacity is lower than k * p .

Type P: the total capacity is larger than k * , the baseload capacity is lower than k * b , and the peak capacity is larger than k * p .

The proof is presented in Appendix B. 6 There are only two possible distortions of the technology mix; the effect on the total capacity is similar to the effect on the quantity of peak plants because those plants are the marginal ones. It is possible to specify conditions for the parameters of a particular scenario that ensures that a Type B or a Type P equilibrium will exist.

Proposition 2. If T * k * > T * b k * b , then there is one equilibrium of Type B; if T * k * < T * b k * b
, then there is one equilibrium of Type P.

The proof is presented in Appendix C. The difference T * k * -T * b k * b corresponds to the risk exposure of the firm in the risk-neutral benchmark. It could be rewritten as the difference:

T * k * -T * b k * b = T * (k * -k * b ) -(T * b -T * )k * b The quantity T * (k * -k * b )
is the quantity of electricity that is sold at price v and costs c p , and (T * b -T * )k * b is the quantity of electricity from baseload capacity that is sold at price c p . The difference between the two determines whether the profit of a firm with the risk-neutral portfolio is positively correlated to the marginal cost c p . And this sign, in turn, determines in which direction the firm should modify its mix to smooth the variations of its profit.

If T * k * -T * b k * b is positive (resp. negative), the profit of the firm is increasing (resp. decreasing) with respect to c p and developing its baseload capacity (resp. peak capacity) smooths its profit variations.

In Figure 2, an equilibrium of Type P is depicted. The difference T * k * -T * b k * b is the difference between the two areas below the load duration curve, it is negative for the case depicted. From equations ( 7) and ( 8), the risk-averse equilibrium is similar to a risk-neutral equilibrium with an adjusted peak variable cost. The equilibrium "risk premium" added to the variable cost is equal to the difference Λσ 2 (T k -T b k b ). 7 The effect of risk aversion could be visualized by rotating the peak line I p + c p t; on Figure 2 In Figure 2, relative to the risk-neutral situation, risk-averse firms will tend to display lower shares of the baseload technology but greater total capacity. In this situation, the expected return from peak capacity is negative, whereas both the return from baseload capacity and the overall expected profit are strictly positive. Figure 2: A Type P equilibrium. This equilibrium involves a larger total capacity and a lower baseload capacity than the equilibrium in the risk-neutral benchmark scenario.

The influence of the cost structure

The answer to whether an equilibrium with risk-averse firms will be of Type B or Type P relates to the cost structure and the shape of the load duration curve. We begin this facet of the analysis with a discussion of the influence of the cost structure and subsequently consider the influence of the load duration curve. The Voll and the four cost parameters determine the two durations r and r p that will be used to conduct the discussion.

From Proposition 2, the existence of an equilibrium of Type B or Type P is related to the sign of the difference

T * k * -T * b k * b .
The risk that is associated with a technology is the product of two factors: the duration of the full use of this technology, which is either r or r p , and the corresponding capacity. The latter is decreasing with respect to the former so that the product of the two is not monotonic. For any r there is a value of the parameter r p such that the difference T * k * -T * b k * b is strictly negative; moreover, if r is sufficiently large there is another value such that the aforementioned difference is strictly positive. By constraining the shape of the distribution F , it is possible to obtain a stronger result.

Corollary 1. If tF -1 (1 -t) is strictly concave, then for any r there is a threshold rp such that there is a Type B (resp. Type P) equilibrium if r p < rp (resp. r p > rp ).

Proof. If tF -1 (1 -t) is strictly concave, there is a unique t max ∈ (0, 1) that maximizes tF -1 (1 -t) (because this function is null at 0 and 1). For r ≤ t max , r p F -1 (1 -r p ) < rF -1 (1 -r) for all r p ∈ [0, r), and rp = r.

For r > t max there is rp < r such that rp

F -1 (1 -rp ) = rF -1 (1 -r) and r p F -1 (1 -r p ) < rF -1 (1 -r) ⇔ r p < rp .
In Corollary 1, the duration r is maintained at a fixed level, and variations of the duration r p are considered. Figure 3(b) graphically illustrates this corollary. Three extreme situations are worth considering: the two technologies could be very different (large r and small r p ) or relatively similar with both technologies being used for baseload supply (large r and r p ) or to meet peak demand (small r and r p ).

If the two technologies are very different (r is large and r p is small), there is a large investment in the peak technology, and the variable cost of this technology establishes the price of electricity for a large portion of the year; however firms are not frequently at full capacity (r p is small). Consequently, the risk that is faced by technology p is small, while the risk that is faced by the baseload technology is large. As a result, relative to the benchmark case of risk-neutrality, the overall investment of these firms will be increased, and the technological mix will be adjusted towards the peak technology.

If both technologies are similar and peak-oriented, p is effectively a peak technology and b could be described as a midload one. There is a large investment in this technology and a small investment in the peak technology. The technology b faces a large risk and risk-averse firms increase their investment in the peak technology to hedge this risk. Compared to the previous case, the large risk associated with technology b has a different explanation. In the previous case, this risk was due to a large duration of full use with a price at c p ; by contrast, in the current case it comes from the large investment in this technology.

If both technologies have similar cost structure and both r and r p are close to 1, technology p is a midload technology and the two technological options are substitutable for production throughout the course of the year. In this situation, there is a small investment in technology p, it is frequently fully utilized and is rarely infra-marginal. In this scenario, technology p is highly risky and risk-averse firms will therefore attempt to reduce their risks by decreasing their quantities of this technology and increasing their quantities of the baseload technology.

The influence of the variability of the load

It has been shown that risk-aversion tends to modify the technology mix in either one of two directions. This is related to the variability of the load.

A linear load duration curve is used on Figure 3 In Figure 3(b), two load duration curves with different variability are depicted. With a more variable load, the quantity of total capacity increases, and the quantity of baseload capacity decreases. With a highly variable load, as represented by the thick line in Figure 3, the riskiest component of firms' profits are obtained from their total capacities rather than from their baseload capacities; in these cases, a Type B equilibrium exists, and firms will tend to increase their baseload capacities and reduce their total capacities. With a relatively stable load, as represented by the thin line in Figure 3, firms will possess small quantities of peak capacity and a Type P equilibrium will exist. In these situations, the return from peak units will be negatively correlated with the overall profit of the firm, and the firm will tend to increase its investment in peak units to hedge the risk from the baseload production, which constitutes its main source of profit.

Welfare

In the above analysis, we determined how risk and risk aversion modify the technology mix. The benchmark used was the risk-neutral situation that also described the absence of risk. However, it is also interesting to compare the long-run equilibrium with a welfare maximizing mix that would account for risk-aversion.

Welfare function and 1 st best technology mix

For a realization c p of the variable cost of the peak technology, the consumer surplus is the sum of the difference between the gross surplus and the price:

CS(k, k b , c p ) = k b 0 (v -c b )xf (x)dx + k k b (v -c p )xf (x)dx (10)
If consumers are risk-neutral, their expected surplus is CS(k, k b , cp ). Welfare is defined as the sum of the expected consumer surplus and the firms' utility:

W (k, k b ) = CS(k, k b , cp ) + i U (λ i , k i b , k i ) (11)
With this welfare function, it is implicitly assumed that transfers between consumers and firms are feasible ex-ante, when c p is unknown, and not expost, when c p is known.

If transfers could be implemented ex-post and made conditional on the realization of the marginal cost c p , a social planner would ensure that firms earns a constant profit to cancel out the cost due to risk-aversion.9 In such a case, the first-best optimal investment would be equal to the risk-neutral equilibrium investment, and it could be decentralized via a complete set of markets. Such a complete set of markets would associate a price for electricity for each realization of c p and each hour of the year (i.e., each demand level

x in our framework).

Constrained optimum

From the equation ( 11), the second best optimal investment schedule would satisfy:

(v -c)T -I p = λ i σ 2 T (T k i -T b k i b ) + j∈I λ j σ 2 (T k j -T b k j b )T ′ k j (12) (c -c b )T b -(I b -I p ) = λ i σ 2 T b (T b k i b -T k i ) + j∈I λ j σ 2 (T b k j b -T k j )T ′ b k j b (13)
Compared with the risk-neutral (and first best) investment, there are additional terms related to risk aversion. The investment of a firm does not only directly affect its own risk but also indirectly affects the risk of all firms (including itself) via the change of the durations T and T b .

Because of market incompleteness, there is an externality associated with each firm investment. The sign of the externality associated with the peak capacity of a firm is opposite to the sign of the externality associated with the baseload capacity of this firm. At the market equilibrium, these signs are related to the equilibrium type. For instance, if the equilibrium is of Type B, an increase of the total capacity of a firm and a decrease of its baseload capacity would reduce the risk faced by all firms. The reverse holds for an equilibrium of Type P.

In order to compare the equilibrium portfolio with the second best, a representative firm could be considered so that the second best technology mix satisfies the two equations:

(v -c)T -I p = Λσ 2 (T k -T b k b )(T + T ′ k) (14) (c -c b )T b -(I b -I p ) = Λσ 2 (T b k b -T k)(T b + T ′ b k b ). ( 15
)
The comparison between the market equilibrium and the second best is not straightforward because of the substitutability between the two types of capacity. Even though at an equilibrium of Type B an increase of k or a decrease of k b would increase welfare,10 it is not possible to conclude that there is an under-investment in total capacity and an over-investment in baseload capacity. This is so because an increase of the total capacity can increase the marginal benefit from a baseload capacity via its effect on the risk-exposure of firms. In such a case there might be an under-investment in both types of technology.

Formally this issue is related to the sign of the cross derivative of welfare with respect to k and k b , which is

∂ 2 W/∂k b ∂k = Λσ 2 (T + T ′ k)(T + T ′ k b ).
The sign of this expression varies with capacities; if both quantities are close (so that k b is a large share of k) it is positive which would ensure that the comparison is intuitive. However, if the two quantities are sufficiently different it could be negative and the comparison cumbersome. This issue would require further work. However, if the risk-aversion of firms or the risk are sufficiently small, the cross effect would be dominated by the direct effect and the comparison between the market equilibrium and the second best mix would be similar than the comparison between the market equilibrium and the first best mix.11

A generalization to a situation with more than two technologies

In this section, a broader version of the model is developed to understand the generalizability of results that have previously been described.

In this broader version of the model, there are m technologies labeled j = 1..m, which are ordered in terms of increasing variable cost c j and decreasing capacity cost I j :

0 < c 1 < c 2 < ... < c m < v and 0 < I m < I m-1 < ... < I 1
The variable costs are assumed to be independently distributed random variables with mean cj and standard variation σ j . The risks are assumed to be sufficiently small such that the merit order is never modified. In accordance with the two-technology framework, the duration r j is defined as:

r m = I m v -cm and r j = I j -I j+1 cj+1 -cj for j ≤ m -1
Moreover, we assume that 0 < r m < r m-1 < ... < r 1 < 1 to ensure that all technologies are used.

We consider a representative firm with risk aversion Λ. This firm's capacity of technology j is k j , and we use K j to denote the sum of the capacities of technologies lower than j, that is

K j = j l=1
k l , and K 0 = 0

The price of electricity is the variable cost c j if K j-1 ≤ x < K j and the price is v in case of rationing. Therefore, the price of electricity is c j during a portion F (K j ) -F (K j-1 ) of the year and v during the remaining 1 -F (K n ) of the year. We use T j to denote the portion of the year during which the price is strictly above c j : T j = 1 -F (K j ). The profit of the representative firm may then be expressed as follows:

Π(k, c) = [T n (v -c n ) -I n ] K n + m-1 j=1 [T j (c j+1 -c j ) -(I j -I j+1 )] K j . (16)
In the risk-neutral benchmark situation, the values of T j are equal to the ratios r j for j = 1..m. The risk-averse firm has an objective function that is similar to (3). The variance of this firm's profit may be expressed as follows:

var(Π) = m j=2 σ 2 j (T j K j -T j-1 K j-1 ) 2 + σ 2 1 T 1 K 2 1 . (17)
At equilibrium, the following first-order conditions are satisfied:

T m (v -cm ) -I m = ΛT m σ 2 m (T m K m -T m-1 K m-1 ) (18) T j (c j+1 -cj ) -(I j -I j+1 ) = ΛT j σ 2 j+1 (T j K j -T j+1 K j+1 ) +σ 2 j (T j K j -T j-1 K j-1 ) (19) T 1 (c 2 -c1 ) -(I 1 -I 2 ) = ΛT 1 σ 2 2 (T 1 K 1 -T 2 K 2 ) + σ 2 1 T 1 K 1 . (20)

If only one technology is risky

If l > 1 is the sole risky technology, then the right-hand sides of the above first-order conditions are null for all j except for l and l -1, and for these two technologies the first-order conditions are:

T l (c l+1 -cl ) -(I l -I l+1 ) = ΛT l σ 2 l (T l K l -T l-1 K l-1 ) T l-1 (c l -c l-1 ) -(I l-1 -I l ) = ΛT l-1 σ 2 l (T l-1 K l-1 -T l K l )
These two equations are similar to the first-order conditions ( 7) and (8) in the two technology framework with the VoLL v replaced by the variable cost of technology l + 1. Thus, the two aggregate capacities K l and K l-1 are the only capacities modified by the introduction of risk aversion. The two technology case can then be applied to understand how the mix is distorted.

If we further assume that the two technologies l and l -1 are sufficiently similar to ensure that r l and r l-1 are close to each other, then we can deduce from the previous analysis that there will be an over-investment in the risky technology if it is a peak technology and that there will otherwise be an under-investment in this technology.

If several technologies are risky

If several technologies are risky, the effect of risk aversion is more difficult to assess. Indeed, if the risky technologies are "isolated", that is, if technology l is risky but technologies l -1 and l + 1 are not risky, then there will be local distortions of the technology mix that could be analyzed in isolation.

If all the available technologies except for the baseload technology are risky, then the sum of the right-hand sides of equations ( 18), ( 19) and (20) will be null; thus, except for a single specific case, the introduction of risk aversion into the situation causes at least one of the K j to increase and at least one K j to decrease.

Furthermore, if the technologies, i.e. the r j , are sufficiently close to each other, then it is possible to develop a notion of the effect of risk aversion on the two extreme technologies. In this situation, it is likely that the righthand side of equation ( 18) will be negative and that the right hand side of equation (20) will be positive; this result would imply that the introduction of risk aversion into the scenario would produce an increase in the total capacity and a reduction in the baseload capacity.

Conclusion

In an electricity market, the technology mix is fundamentally driven by the variability of the load. In this article, we have investigated the effect of the risk aversion of firms on these firms' technology mixes when the cost of a technology is random.

In the model that was developed for this investigation, two technologies, a baseload technology and a peak technology, were assumed to be available to satisfy a variable demand for electricity. The variable cost of the peak technology was assumed to be random. In the presence of risk aversion, the total capacity may be either smaller or larger than the total capacity in the risk-neutral benchmark case. In the latter situation, the quantity of baseload capacity is lower and the quantity of peak capacity is larger with risk aversion than without. In this case firms tend to over-invest in the risky peak technology to hedge the risks associated with the baseload technology. These risks originate from the peak variable costs because those costs determine the electricity prices and the baseload short-term revenue.

This work clarifieq the effects of risk aversion on firms' investment decisions and hels to elucidate the mechanisms underlying these decisions. This study contributes to a greater understanding of how uncertainties surrounding fossil fuel prices impact the investment decisions of electricity producers.

For instance the uncertainty surrounding gas prices has consequences for gas plants but also for other technologies with lower variable costs (e.g. nuclear, coal, renewable). Because the variable cost of a technology determines the short-term rent of technologies with lower variable costs, uncertainty only propagates down the merit order.

Moreover, this investigation might also be applicable to an analysis of the ways in which electricity producers may be affected by certain environmental regulations, such as the uncertainty surrounding the future regulation of greenhouse gas emissions. For instance, if the baseload technology is a CO 2free process, the analysis of this study could demonstrate how the uncertainty surrounding the CO 2 price could lead to a reduction of investment in the clean baseload technology because of the risks that are associated with the CO 2 -and k i b that maximize firm i objective for each i ∈ I and such that i k i t = k t , for instance k i = Λ/λ i k and k i b = Λ/λ i k b (but they are not unique). From equations ( 7) and ( 8), we will use the fact that the risk-averse equilibrium is identical to a risk-neutral equilibrium with a peak variable Consequently, there is at least one solution to ρ = Ψ(ρ). For any such solution, the corresponding quantities k(ρ) and k b (ρ) describe an equilibrium.

Type of equilibrium

At an equilibrium the difference T k -T b k b is either positive or negative.

Let us show that in the former case the equilibrium is of Type B, and that in the latter case it is of Type P.

• If at the equilibrium T k -T b k b > 0:

  and I p < I b $/W.yr × 1 yr + $/W The ratio r = (I b -I p )/(c p -c b ) is the portion of a year such that technology b is more efficient than p for production over a longer period than r. The ratios r p = I p /(vcp ) and r b = I b /(vc b ) are the minimal duration of production with the peak and baseload technologies, respectively, such that

  Figure 1: Cost structure, load curve (F -1 (1 -t)) and equilibrium investment in the riskneutral case

  (a), the rotation is clockwise. The corresponding shift of capacity is indicated in Figure 2(b).

  The effect of risk is similar to a clockwise rotation of the peak plant cost. The risk-averse equilibrium (dashed lines) compared to the risk-neutral benchmark (dotted lines).

Figure 3 :

 3 Figure 3: The effect of the load duration curve on the marginal risk from investment, with a linear load duration curve X[(1 + u)/2 -u.t].

  cost cp + Λσ 2 (T k -T b k b ). Graphically, on Figure 2, we will rotate the line of the cost of the peak technology until the premium is equal to Λσ 2 (T k -T b k b ).Let us denote by k(ρ), k b (ρ), T (ρ) and T b (ρ) the risk-neutral equilibrium quantities and durations for a variable cost cp + ρ. These are the solutions ofT (ρ) = Ip v-(cp+ρ) , T b (ρ) = I b -Ip cp+ρ-cb , k = F -1 (1 -T ) and k b = F -1 (1 -T b ) for ρ between ρ min = (I b + c b ) -(I p + cp ) and ρ max = (vcp ) + I p /r b , at ρ min (resp. ρ max ) the peak (resp. base) technology is the sole technology used. Finally, let us define Ψ(ρ) = T (ρ)k(ρ) -T b (ρ)k b (ρ). (B.1) This function is continuous. For ρ = ρ min it is strictly positive because k b (ρ min ) = 0, and for ρ = ρ max it is strictly negative because k p (ρ max ) = 0.

For instance risk aversion can explain corporate hedging activity[START_REF] Amihud | Risk reduction as a managerial motive for conglomerate mergers[END_REF][START_REF] Nance | On the determinants of corporate hedging[END_REF][START_REF] May | Do managerial motives influence firm risk reduction strategies?[END_REF], and[START_REF] Wolak | A model of homogeneous input demand under price uncertainty[END_REF] have empirically investigated how risk aversion can explain Japanese firms' choices of coal suppliers.

This framework has been used by several authors to evaluate national portfolios[START_REF] Humphreys | Reducing the impacts of energy price volatility through dynamic portfolio selection[END_REF][START_REF] Awerbuch | Investing in photovoltaics: risk, accounting and the value of new technology[END_REF][START_REF] Awerbuch | Energy Security and Diversity in the EU: A Mean-Variance Portfolio Approach[END_REF][START_REF] Jansen | Application of portfolio analysis to the Dutch generating mix[END_REF].

The load duration curve is obtained by ranking hourly demands in decreasing order, so that for each date t the corresponding quantity is such that the demand is larger than this quantity during t.

At equilibrium, each firm has a degree of freedom with respect to its choices. Each firm is indifferent with respect to its total capacity but not to its technological mix.

Because of market incompleteness there might be several equilibria, and some restrictions on the load duration curve would ensure uniqueness of the equilibrium.

This risk premium is not common to all firms, firm i premium isλ i σ 2 (T k i -T b k i b ).

The demand x is homogeneously distributed over[ X(1 -u)/2, X(1 + u)/2], the corresponding load duration curve is X[(1 + u)/2 -u.t].The quantity X represents the total quantity of electricity that is demanded over the course of one year, and u represents the variability of the load.

If a transfers scheme t i (c p ) could be implemented, then, for a variable cost c p , the profit of firm i would be π i + t i (c p ) and consumers surplus CSi t i (c p ). The regulator could set t i (c p ) = πiπ i to stabilize the firm revenue and strictly increase the welfare.

At an equilibrium of Type B, the derivative of welfare w.r.t. k (resp. k b ) is -(T k -T b k b )T ′ k (resp. (T k -T b k b )T ′ b k b )which is positive (resp.negative).

Note that the comparison between the first best (i.e. risk-neutral) mix and the 2nd best mix is ambiguous because the signs of T + T ′ k and T b + T ′ b k b vary with k and k b .

Consider that an equilibrium exists.

• If technology b is not used, then, T b = 1 and the derivative of the objective of firm i, with respect to k i b , would be (c p -c b ) -(I b -I p ) + λ i T k i which is strictly positive; therefore, firm i would invest in technology b.

• If technology p is not used, then k = k b and T = T b , the profits of the firms would not be random and the quantity k b would be such that (vc b )T b = I b , and the effect of a peak unit on the objective function would be (vcp ) I b v-c b -I p which is strictly positive, because r p > r. Hence, firm i would invest in peak units.

Therefore, both technologies are used at equilibrium.

Then, with the expression of profit (2), the objective function (3) of a firm is

This function is maximized for positive and finite quantities if and only if

Therefore, this equality is satisfied and firms choose their capacities so that k i > k i b ≥ 0 and the following two first order conditions are satisfied:

Then, dividing all sides by λ i and summing over i ∈ I one gets that the aggregate quantities of capacity are the solution of the two equations ( 7) and ( 8) with the expression of Λ given by ( 9).

Appendix B. Proof of Proposition 1

Existence of an equilibrium

We show that there is k and k b with k > k b > 0 such that (7) and (8) are satisfied. Then, if these quantities exist, there are individual quantities k i -(vcp ) -I p /T > 0 by ( 7), so T > T * p and the total quantity is lower than the risk-neutral total quantity.

b and the quantity of baseload capacity is lower than in the risk-neutral benchmark.

-The quantity of peak capacity is kk b , from the two previous comparisons it is lower than the risk-neutral quantity. 7), so T < T * p and the total quantity is larger than the risk-neutral total quantity.

b and the quantity of baseload capacity is larger than in the risk-neutral benchmark.

-The quantity of peak capacity is kk b , from the two previous comparisons it is larger than the risk-neutral quantity.

Appendix C. Proof of proposition 2

We use the function Ψ defined by (B.1) to establish the result. The risk-neutral situation corresponds to ρ = 0, and

is negative, there is a solution to the equation ρ = Ψ(ρ) in (0, ρ max ). Such a solution corresponds to an equilibrium of Type B because T k -T b k b = Ψ(ρ) = ρ > 0. A similar procedure could be performed for r p F -1 (1 -r p ) -rF -1 (1 -r) < 0.