Sergey Stepanyan

Jun Hayashi

Arthur Salmon

Gabi D Stancu

Christophe O Laux

Large-volume excitation of air, argon, nitrogen and combustible mixtures by thermal jets produced by nanosecond spark discharges

This work presents experimental observations of strong expanding thermal jets following the application of nanosecond spark discharges. These jets propagate in a toroidal shape perpendicular to the interelectrode axis, with high velocities of up to 30 m/s and over distances of centimeter scale. Their propagation length is much larger than the thermal expansion region produced by the conventional millisecond sparks used in car engine ignition, thus greatly improving the volumetric excitation of gas mixtures. The shape and velocity of the jets is found to be fairly insensitive to the shape of the electrodes. In addition, their spatial extent is found to increase with the number of nanosecond sparks and with the discharge voltage, and to decrease slightly with the pressure between 1 and 7 atm at constant applied voltage. Finally, this thermal jet phenomenon is observed in experiments conducted with many types of gas mixtures, including air, nitrogen, argon, and combustible CH 4 /air mixtures. This makes Nanosecond Repetitively Pulsed (NRP) discharges particularly attractive for aerodynamic flow control or plasma-assisted combustion because of their ability to excite very large volumes of gas.

Introduction

Nanosecond discharges have shown strong potential for various applications such as Plasma-Assisted Ignition (PAI) [1][2][3][4][5], lean flame stabilization [6][7][8], aerodynamic flow control [9][10][11][12] and nanomaterial synthesis [13][14][15].

In air-containing mixtures, the phenomenon of ultrafast gas heating by nanosecond high voltage (HV) pulses was first experimentally observed in [16]. This phenomenon was shown to be caused by dissociative quenching reactions between molecular oxygen and electronically excited nitrogen molecules produced by electron impact during the discharge [17,18]. This effect was also predicted numerically [19][20][21]. The ultrafast gas heating in atmospheric pressure air plasmas occurs within a few tens of nanoseconds following a nanosecond spark pulse, and it also leads to the nearly full dissociation of molecular oxygen [18,[22][23][24] thus producing large amounts of reactive chemical species and making NRP discharges attractive for the applications mentioned above.

An expected consequence of the ultrafast gas heating following nanosecond sparks is the formation of shock waves and hydrodynamic effects. Some of the earliest experimental observations and numerical modeling of these effects for a short duration spark of a few mJ were presented in [25]. The authors observed that the hot gas produced by a single spark propagated over a few millimeters within tens of microseconds. In further developments, Leonov et al [26] investigated high current (~1.5 kA) sparks of 50-ns duration across long interelectrode gaps (50-60 mm) in air, and observed the onset of strong turbulence attributed to Rayleigh-Taylor instabilities, with a radial expansion of the hot gas core to distances up to 50 times the radius of the initial spark channel. More recently, this effect was also observed in nanosecond sparks with lower current (~100 A) [27][28][29][30]. Xu et al [31,32] further showed that the hydrodynamic effects induced by a burst of nanosecond pulses in high pressure, lean propane-air mixtures can cause wrinkling of the flame front and accelerate flame propagation. This was also observed by Bane et al [33].

Insights into the mechanism responsible for the onset of these wrinkled combustion front were recently proposed by Castela et al [35] who demonstrated, through a comparison of experimental results and numerical modeling, that the hot gas core produced by the spark develops into a torus of advected reactive hot gas, and that fresh gas recirculates inside the interelectrode region. This was confirmed by Schlieren diagnostics in [27] and by PIV (particle image velocimetry) measurements in [START_REF] Seydou | 18th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics[END_REF].

In this paper, we present a detailed investigation of the expanding hot gas core produced by single nanosecond discharges and by bursts of nanosecond discharges. The motion of the hot gas core is followed up to the millisecond time scale. We present a parametric study of the effects of various parameters, namely the number of nanosecond sparks, applied voltage amplitude, shape of the electrodes, and gas composition, on the hydrodynamic effects induced by nanosecond sparks.

Experimental

Nanosecond Repetitively Pulsed (NRP) discharges were produced between two metallic pin electrodes separated by a distance of 1 mm. Most experiments reported here were performed with paraboloid pins with tapered 150-µm tip radius, but additional experiments were conducted with conical or cylindrical pins. The NRP discharges were generated with an FID FPG 30-100MC4K pulser producing pulses of 10-ns duration, amplitudes up to 42 kV and Pulse Repetition Frequency (PRF) up to 100 kHz. The energy deposited into the plasma was measured with a high voltage probe (Lecroy PPE 20 kV), current probe (Pearson model 6585) and recorded on a LeCroy Wavepro 7100 oscilloscope. A description of technique used for energy measurements can be found elsewhere [16,18].

To characterize the hydrodynamic effects induced by the spark up to 1 ms after the discharge, phase-locked Schlieren measurements were performed with the experimental setup, similar to the one used in [28], shown in Figure 1. The gate of the ICCD PIMAX camera used for the Schlieren measurements was set to 0.1 µs. Fast imaging of spontaneous emission of OH*(308 nm) was recorded with a Photron SA-X2 camera, using a camera gate of 5 µs.

To ensure a complete characterization of the gas density gradients induced by the discharge, the knife-edge of the Schlieren setup was placed either horizontally or vertically. In the horizontal position (i.e. parallel to the interelectrode axis), the knife-edge cuts the light deflected toward the bottom side of the image; in the vertical position (i.e. perpendicular to the interelectrode axis), it cuts the light deflected toward the right side of the image. For most Schlieren images presented here, the knife-edge was set to cut about 90% of the light (perpendicular to the interelectrode axis) in order to increase contrast.

Results

Gas recirculation and transport of active speciesfollowing a single nanosecond spark

Figure 2 presents a set of Schlieren images obtained from 10 to 300 µs after a single nanosecond spark. The discharge was initiated in ambient air by a high voltage (HV) pulse of 36-kV amplitude and 10-ns duration. Figure 2 shows that the hydrodynamic disturbance starts within a few microseconds after the discharge. Two main phenomena can be observed: shock wave propagation during the first ~10 µs, and expansion of the hot gas core up to 300 µs. The shock waves induced by nanosecond discharges have been observed before [27,[START_REF] Nagaraja | [END_REF]38]. The subsequent expansion of the hot gas shows how the energy initially deposited into the interlectrode discharge channel redistributes spatially. The Schlieren results presented in Figure 2 show that the shape of the heated gas region is symmetrical around the pin axis, and that the hot gas expands by about 6 mm within 300 µs. So, the energy initially deposited into a thin nanosecond spark channel (1mm length and 100-µm diameter) redistributes over a large volume of 6-mm diameter within about 100 µs.

Figure 3 presents Schlieren images for the same conditions as in Figure 2 but with the knife-edge placed vertically in order to visualize the gas density gradients in the direction parallel to the pin axis. The combination of the data presented in Figures 2 and3 provides a full map of the hot gas region and evidences the gas recirculation pattern between the pin electrodes: after about 30 µs the hot gas and active species created by the discharge between the electrodes are transported outwards in a toroidal jet and the interelectrode gap is refilled by fresh gas. This recirculation effect was already reported by Kono [25] and Maly [START_REF] Maly | Fuel Economy in Road Vehicles Powered by Spark Ignition Engines[END_REF]. Two-dimensional modeling by Thiele, Warnatz et al [START_REF] Thiele | [END_REF] describes the mechanism of gas recirculation. Recently, three-dimensional Direct Numerical Simulations (DNS) of the hydrodynamic effects induced by a nanosecond spark for the conditions considered in the present work were performed by Castela et al [35]. These DNS simulations confirm the creation of a torus of ejected hot gas and the recirculation of fresh gases along the interelectrode axis.

Figure 4 compares a Schlieren image and the distribution of OH* (308 nm) spontaneous emission intensity at the same instant (30 µs). The discharge conditions for Figure 4 are the same as for Figures 2 and3. The distributions of gas density perturbation and of OH* emission intensity are found to have the same toroidal shape and radius. This indicates that reactive species are transported in the hot torus.

The blue curve in Figure 4c presents the evolution of the diameter d, defined in Figure 4a, of the hot gas torus produced by a single nanosecond spark. One can see that the hot gas torus first expands rapidly (dd/dt~60 m/s) and then slows down after ~100 µs. The energy deposited in the plasma is thus redistributed into a volume of characteristic scale d max in approximately 100 µs.

The red curve in Figure 4c presents the expansion of the hot gas after ignition by a spark produced by a conventional automotive ignition coil with energy of about 54 mJ and charging duration of 4 ms. The hot gas torus induced by the nanosecond spark expands significantly faster and further than that produced by the conventional millisecond spark, even though the energy deposited by the conventional spark (~54 mJ) is much higher than that of the nanosecond spark (~6 mJ). The faster gas motion in the case of nanosecond spark is explained by the higher temperature gradients following the short (10 ns) energy deposition.

Parametric studies

The following sections examine the dependence of the speed and spatial extent of the hot gas torus expansion as a function of shape and material of the electrodes, gas composition, applied pulse voltage amplitude, and number of pulses in the case of multiple nanosecond sparks.

i) Shape of the electrodes and their material

Figure 5 presents Schlieren images obtained after a single nanosecond spark in air (U=36 kV, P=1 atm, 10 ns pulse) with different electrode systems: a) conic stainless steel pins with 50 µm tip; b) parabolic tungsten pins tapered with 150 µm tip radius; c) cylindrical tungsten electrodes of 2 mm diameter. All images were taken 300 µs after the discharge. It can be seen that the scale and the shape of the expanding hot gas core are nearly identical in all three cases, thus the observed phenomenon does not depend on the electrode shape or the material used here.

The effects of the electrode shape were investigated by Bane et al [33] who presented Schlieren images taken after a nanosecond spark in an H 2 /air mixture at P=1 atm. These authors demonstrated that factors such as the heat exchange between the electrodes and the hot gas or viscous effects may play a significant role in the energy balance of the system and affect the initiation of the ignition kernel. However, as shown above, the spatial scale of the energy redistribution is practically the same for very different electrode geometries investigated here. It should be also noted that the results of calculations for nanosecond spark presented in [35] demonstrate that the effect of gas recirculation that governs the spatial energy redistribution is not induced or affected significantly by presence of electrodes (or their shape) near the plasma.

ii) Single nanosecond pulse vs burst of nanosecond pulses

For applications of nanosecond discharges in repetitive mode (NRP) in plasma assisted combustion and flow control, it is important to understand how the hot gas core expansion and fresh gas recirculation are affected by multiple sparks. The hot core expansion process depends on the value of the energy density deposited by the discharge, and this parameter strongly depends on the initial gas number density in the space between the electrodes. When a burst of pulses is applied, it is often considered that the gas density before each new pulse is lower than before the previous one because the gas heated by previous pulses diffuses slowly. If this were the case, then the highest gas temperature would always be between the electrodes and each new discharge would be initiated in the hot gas with lower number density. However, we have shown that the energy deposited into plasma does not remain confined between the electrodes, but spatially redistributes owing to the motion of the hot gas torus. If the period between nanosecond pulses is longer than the recirculation time of the fresh gas, then each new discharge will be triggered at the same conditions as the first one, i.e. in the fresh gas. If the period between the nanosecond pulses is shorter than the flow recirculation time, then all discharges after the first one will be initiated at higher temperature, i.e. at lower gas number density. This means that the energy deposited in each new pulse will significantly change at some critical frequency.

Figure 6a,b presents the comparison of a single pulse discharge in air at P=1 atm, U=28 kV with the sequence of discharges initiated by a burst of 30 pulses at f=70 kHz, U=28 kV. Both images were obtained 500 µs after the first pulse. The shape of the expanded hot gas in the case of multiple nanosecond pulses is seen to be a superposition of the toruses produced by individual pulses. However, we note a strong synergy between pulses, as the torus produced by a given pulse appears to be pushed outwards by the previous pulses. One can see that the size of the hot gas core is much larger for the burst of sparks. Furthermore, Figure 6c shows that the spatial scale of the hot gas torus depends linearly on the number of the pulses in a burst, i.e. linear to the energy deposited in plasma. This means that the volume of the hot gas core grows disproportionally large with energy deposited in plasma. The red dashed line in Figure 6c corresponds to the dimension of the hot gas at 500 µs after ignition of a standard automotive spark. As can be seen from Figure 6c, the spatial extent of the gas region heated by a single pulse of energy 2.5 mJ is already much larger than that produced by the standard automotive spark, and it extends even further when multiple nanosecond pulses are applied.

iii) Dependence on gas composition, pressure and applied voltage

We now present results on the expansion dynamics of the hot gas core in various gases, namely air, nitrogen, argon, and a stoichiometric CH 4 /air mixture. A single 10-ns discharge was initiated for all cases at P=1 atm, U=36 kV at 1-mm gap between the pin-to-pin electrodes. As can be clearly seen in Figure 7a, the dynamics of the expansion as well as the maximal propagation length are similar for all gases considered here. We note that the maximal radius is slightly larger for the two mixtures containing air molecules, which may be attributed to the stronger effect of ultrafast heating via dissociative quenching of excited N 2 with O 2 .

Figure 7b presents the dependence of d max on pressure in the range 1-7 atm for a single nanosecond spark and for a burst of 10 sparks (10 ns each) applied at a frequency of 70 kHz. The applied voltage is U=36 kV for all pulses, and the interelectrode gap is fixed at 1 mm. The increase of pressure slightly decreases the maximal propagation length d max for both the single spark and the burst of sparks. Interestingly, at 7 atm there is practically no difference in maximum propagation length between the single spark and the burst. This suggests that the multiple toruses merge into a single torus of high cumulative energy.

Finally, Figure 7c presents the dependence of d max vs applied voltage for a single spark and for multiple sparks. For both single and multiple sparks, the maximal propagation length of the hot gas core increases up to 50 percent over the voltage range considered here.

Based on the foregoing observations, it is clear that d max can be significantly increased at sub millisecond scale by increasing the number of multiple nanosecond sparks and by increasing their voltage amplitude. This provides a convenient and effective way to control the spatial distribution of the energy deposited in the plasma, which is promising for ignition of lean combustible mixtures in volume.

Discussion and conclusions

We have shown that the energy deposited by single nanosecond spark is initially concentrated in a thin hot channel about 100 µm in diameter (see Figure 2). Then this hot gas expands and forms a torus whose diameter increases up to about 5 mm within about 100 µs and then slows down. This expansion of the hot gas is more than twice higher than for the conventional spark even though the energy deposited in a single ns spark is almost 10 times lower than in the conventional spark. The advected hot gas also contains the active species produced by the spark discharge. Fresh gas is drawn into the interlectrode gap, thus creating a strong recirculation zone. This mechanism has an interesting effect of redistributing the energy deposited by the discharge into a very large volume of gas within a very short time. This phenomenon is shown to occur in different gases (Air, CH 4 /Air, N 2 , Ar) over a wide range of pressures (1-7) atm, voltage amplitudes kV and to be almost independent on the shape and material of the pin electrodes. The temporal dynamics of the gas expansion after a single nanosecond spark were found to be practically the same for all considered conditions and for all gases investigated. Furthermore, the experiments conducted with multiple nanosecond sparks demonstrated that the dimension of the expanded hot gas torus increases linearly with the number of multiple sparks. It is therefore possible to affect a large gas volume (torus of radius >10 mm), thus making NRP discharges a promising tool for stabilization or ignition of lean combustible mixtures and aerodynamic flow control.

Figure 1 .

 1 Figure 1. Experimental setup. The focal length of the lens is 1 m. Both spherical mirrors have focal distance of 2 m.

Figure 2 .Figure 3 .

 23 Figure 2. Schlieren images obtained for a single nanosecond spark in air at P=1 atm, U=36 kV. The distance between the pins is 1 mm. The knife edge is horizontal and cuts 90 % of the light

Figure 4 .Figure 5 .Figure 6 .

 456 Figure 4. Comparison of a) Schlieren image and b) distribution of spontaneous OH* emission (308 nm) at 30 µs after a single nanosecond spark in air at P=1 atm, with a 10 ns pulse of amplitude U=36 kV, ε=6 mJ; c) Evolution of the hot core diameter d; Nanosecond spark (blue curve): U=36 kV, 10 ns pulse, deposited energy: ε=6 mJ. Conventional automotive spark (red curve): ε=55 mJ. Distance between pins: 1 mm.

Acknowledgments

This work was supported by the FAMAC (Grant no. ANR-12-VPTT-0002) and PLASMAFLAME (Grant no.ANR-11-BS09-0025) projects. The authors thank Dr. Denis Packan(ONERA) for providing the Schlieren light source.


Saad Labyad
email: saad.labyad@student.ecp.fr

Mehdi Senouci
email: mehdi.senouci@centralesupelec.fr

Deriving multiple-input production and utility functions from elasticities of substitution functions *

Keywords: Production functions, utility functions, elasticity of substitution, marginal productivity, marginal utility, factor shares. JEL codes: C60, D11, D20, D24, D33, E23, E25

For each production or utility function, we can define the corresponding elasticities of substitution functions; but is the reverse true? This paper shows that yes, and that this link is fruitful. By inverting the system of partial differential equations defining the elasticities of substitution functions, we uncover an analytical formula which encompasses all production and utility functions that are admissible in Arrow-Debreu equilibria. We highlight the "Constant Elasticities of Substitution Matrix" (CESM) class of functions which, unlike the CES functions, does not assume uniform substitutability among all pairs of goods. A shortcoming of our method is that it permits only to control for local concavity while it is difficult to control for global concavity.

Introduction

Production and utility functions are the tools that neoclassical economics use to model aggregation, respectively from the technical and subjective points of view. They constitute the most important building blocks of theory and empirics in several subfields of economics: economic growth, macroeconomics, international trade, industrial organization, etc.. In general equilibrium theory, the set of admissible intensive-form production functions over n ≥ 1 inputs is the set of functions of * + n → * + of class 2 that are strictly increasing with respect to each variable and strictly concave. The same set describes admissible utility functions. 1 We call these functions 'aggregation functions' (AFs).

Despite their importance in serving theoretical and empirical representations, only a handful of functional forms of AFs are actually used by researchers and practitioners. Cobb-Douglas and CES 2 functions vastly dominate, while the use of other specifications has remained marginal. 3 As a matter of fact, the modeler's toolbox becomes more restrictive as the number of inputs increase. 4 For instance CES production and utility functions rest on the assumption that all pairs of goods have the same substitutability parameter, which can be a costly assumption even at moderate disaggregation levels. 5 AFs are most often classified according to the associated elasticity of substitution functions (ESFs). ESFs are defined through the second-order derivatives of the AF and govern the evolution of first-order quantities as inputs vary. 6 As such, ES functions carry a central information on the associated AF.

In this paper, we show that the link between AFs and the associated ESFs is so tight that it permits to derive an analytical formula encompassing all AFs over a finite number of arguments (n ≥ 1). The method that we use consists in inverting the system of partial differential equations defining the ESFs. 7 We hinge on the method developed by [START_REF] Moysan | A Note on 2-Input Neoclassical Production Functions[END_REF] for oneinput AFs and extend it to any number of inputs. Doing so, we uncover a new -to the best of our knowledge -duality result between AFs and ESFs, in the form of an analytical expression. 8 All functions derived through our formula, however, are not AFs. Our formula generates functions that are strictly increasing w.r.t each variable, strictly concave w.r.t. each variable, but which might not be strictly concave. Still, we show how to control for local strict concavity around some reference input vector, which proves enough for local uses of our formula.

The rest of the paper is organized as follows. In section 1, we recall the theorem stated in Moysan and Senouci (2016) on one-input AFs, in section 2 we treat the case of two-input AFs before turning to the general n -input AFs in section 3. We investigate concavity issues in section 4 and conclude in section 5.

One-input aggregation functions

Let's call 1 the set of on-input AFs :

1 = f ∈ 2 : * + → * + f > 0; f < 0 All functions f ∈ 1 verify: ∀x > 0, f (x ) > x f (x )
, we prove this in appendix A.1. [START_REF] Kadiyala | Production Functions and Elasticity of Substitution[END_REF] defined some two-input, Variable Elasticity of Substitution (VES) production functions. Other existing functions include the translog class [START_REF] Christensen | Transcendental Logarithmic Production Frontiers[END_REF][START_REF] Christensen | Transcendental Logarithmic Utility Functions[END_REF]), the LINEX class [START_REF] Kümmel | Technical Progress and Energy Dependent Production Functions[END_REF], Ayres and Warr (2005)) and the Isoelastic Elasticity of Substitution class (Growiec and Mućk (2016)). See [START_REF] Mishra | A Brief History of Production Functions[END_REF] on the history of production functions. 4 The usual procedure of nesting CES production functions has the drawbacks that the parameters lose their meaning of elasticities of substitution and that the analytical expression depends on the nesting order. 5 preferences. 6 Quantities governed by the ES include relative marginal rates of substitution, relative marginal costs, competitive factor shares, etc.. See [START_REF] Hicks | The Theory of Wages[END_REF]Hicks ( [1963]]), Robinson (1933Robinson ( [1969]]), [START_REF] Allen | Mathematical Analysis for Economists[END_REF] and [START_REF] Samuelson | Two Generalizations of the Elasticity of Substitution[END_REF]. 7 Lau (1976) attempted to link ESFs to cost functions. 8 Our results can be seen as an extension of [START_REF] Arrow | Capital-Labor Substitution and Economic Efficiency[END_REF] result to any number of inputs and ES functions that are not necessarily constant nor equal across pairs of inputs.

For any f ∈ 1 , we can define functions α, Π and σ:

α(x ) = x f (x ) f (x ) ∈ (0, 1) (1)
Π(x ) = α(x ) 1-α(x ) ∈ (0, +∞) (2) σ(x ) = - f (x ) ( f (x )-x f (x ) ) x f (x )f (x ) ∈ (0, +∞) (3)
For a production function, α(x ) is the competitive x share, Π(x ) is the ratio of the competitive x share over the other factor competitive share, and σ is the ESF associated to f . σ is linked to Π through the equation:

Π (x ) Π(x ) = σ(x ) -1 σ(x ) 1 x (4)
By inverting equation ( 3), [START_REF] Moysan | A Note on 2-Input Neoclassical Production Functions[END_REF] have shown the following result:

Theorem 1. -Let f ∈ 1 and let σ ∈ σ ∈ 0 : * + → * + be the associated ESF. Let x > 0.
Then, there exists two constants Ā > 0 and Π > 0 such that:

∀x > 0, f (x ) = Ā. exp     x x dx x 1 + exp - x x σ-1 σ (x ) dx x Π     (5) Besides, Ā = f ( x ), Π = Π ( x ).
-Conversely, let σ ∈ σ ∈ 0 : * + → * + and let Ā > 0 and Π > 0. Then, function f : * + → * + defined by equation ( 5) is in 1 .

Proof. See Moysan and Senouci (2016, p. 81).

Theorem 1 shows that to design a function f ∈ 1 amounts to (a) choose an ESF σ from the admissible set which is non other than σ ∈ 0 : * + → * + , and (b) choose the value taken by f (= Ā > 0) and the value taken by Π (= Π > 0) at some specific input level x . Consequently, [START_REF] Growiec | Isoelastic Elasticity of Substitution Production Functions[END_REF] constitute a 'recipe' for all one-input AFs.

Two-input aggregation functions

For the ease of understanding of the n -input case in the next section, we first treat the two-input case in this section. We call 2 the set of two-input AFs:

2 = f ∈ 2 : * + 2 → * + f 1 > 0, f 2 > 0, f strictly concave
where f i denotes the partial derivative of function f w.r.t. the i th variable, for i ∈ {1, 2}.

We denote by 2 the set of functions of class 2 that are strictly increasing and strictly concave w.r.t. each variable:

2 = f ∈ 2 : * + 2 → * + f 1 > 0, f 2 > 0, f 11 < 0, f 22 < 0
Clearly, 2 ⊂ 2 and 2 = 2 . We show now how theorem 1 can be extended to write functions belonging to 2 in terms of the associated ESFs as well as some constants.

Let f ∈ 2 . Then for any (x 1 ,

x 2 ) ∈ * + 2 , functions f (•, x 2 ) and f (x 1 , •) belong to 1 . Conse- quently, for any (x 1 , x 2 ) ∈ * + 2 , f (x 1 , x 2 ) > x 1 f 1 (x 1 , x 2 ) and f (x 1 , x 2 ) > x 2 f 2 (x 1 , x 2 ). f (x 1 , x 2 ) = f (x 1 , x2 ). exp       x 2 x2 dx 2 x 2 1 + exp - x 2 x2 σ 2 -1 σ 2 ( x 1 ,x 2 ) dx 2 x 2 Π 2 (x 1 , x2 )       (6)
with:

∀(x 1 , x 2 ) ∈ * + 2 , σ 2 (x 1 , x 2 ) = - f 2 f -x 2 f 2 x 2 f f 22 (x 1 ,x 2 )
and:

∀(x 1 , x 2 ) ∈ * + 2 , Π 2 (x 1 , x 2 ) = x 2 f 2 f -x 2 f 2 (x 1 ,x 2 )
Remark that, by the definition of σ 2 and like in equation ( 4):

Π 1 (x 1 , x 2 ) = Π 1 ( x1 , x 2 ) . exp x 1 x1 σ 1 -1 σ 1 x 1 , x 2 dx 1 x 1 (7)
Furthermore, f (•, x2 ) ∈ 1 , so we can apply theorem 1 to function f (•, x2 ) in equation ( 6), which yields:

f (x 1 , x 2 ) = f ( x1 , x2 ). exp       x 1 x1 dx 1 x 1 1 + exp - x 1 x1 σ 1 -1 σ 1 ( x 1 , x2) dx 1 x 1 Π 1 ( x1 , x2 )       . exp       x 2 x2 dx 2 x 2 1 + exp - x 2 x2 σ 2 -1 σ 2 ( x 1 ,x 2 ) dx 2 x 2 Π 2 (x 1 , x2 )       (8)
Equation ( 8) contains the quantity Π 2 (x 1 , x2 ) which remains unknown.

Lemma 1. Let f ∈ 2 and let functions α 1 and α 2 be defined by:

∀(x 1 , x 2 ) ∈ * + 2 , α i (x 1 , x 2 ) = x i f i f (x 1 , x 2 ) = Π i 1+Π i (x 1 , x 2 ). Then, ∀(x 1 , x 2 ) ∈ * + 2 : x 2 α 1 ∂ α 1 ∂ x 2 = α 2 (x 1 , x 2 ) (σ 12 (x 1 , x 2 ) -1) (9) x 1 α 2 ∂ α 2 ∂ x 1 = α 1 (x 1 , x 2 ) (σ 12 (x 1 , x 2 ) -1) ( 10
)
where

σ 12 (x 1 , x 2 ) = f f 12 f 1 f 2 (x 1 , x 2 ).
Proof. Immediate from log-differentiation of α 1 and α 2 .

For convenience, let's call β (x 1 ) = α 2 (x 1 , x2 ) . From equations ( 7) at (x 1 , x2 ) and ( 10), function

β (•)
is a solution to the differential equation:

β (x 1 ) = β (x 1 ) σ 12 (x 1 , x2 ) -1 x 1 1 + exp - x 1 x1 σ 1 -1 σ 1 ( x 1 , x2) dx 1 x 1 Π 1 ( x1 , x2 )
which easily integrates into:

β (x 1 ) = β ( x1 ) exp     x 1 x1 σ 12 x 1 , x2 -1 1 + exp - x 1 x1 σ 1 -1 σ 1 ( x 1 , x2) dx 1 x 1 Π 1 ( x1 , x2 ) dx 1 x 1     (11)
We can now state the central result of this section:

Theorem 2. Let f ∈ 2 and let σ 1 , σ 2 and σ 12 be the elasticity of substitution functions associated to f , defined by:

∀(x 1 , x 2 ) ∈ * + 2 ,          σ 1 (x 1 , x 2 ) = - f 1 ( f -x 1 f 1 ) x 1 f f 11 (x 1 ,x 2 ) ∈ * + σ 2 (x 1 , x 2 ) = - f 2 (f -x 2 f 2 ) x 2 f f 22 (x 1 ,x 2 ) ∈ * + σ 12 (x 1 , x 2 ) = - f f 12 f 1 f 2 (x 1 ,x 2 ) ∈ Let ( x1 , x2 ) ∈ * + 2 , and let Ā = f ( x1 , x2 ) ∈ * + , Π1 = x 1 f 1 f -x 1 f 1 ( x1 , x2 ) ∈ * + , Π2 = x 2 f 2 f -x 2 f 2 ( x1 , x2 ) ∈ * + , ᾱ1 = Π1 1+ Π1
∈ (0, 1), and ᾱ2 = Π2 1+ Π2

∈ (0, 1). Then, for all (x 1 ,

x 2 ) ∈ * + 2 : f (x 1 , x 2 ) = Ā. exp       x 1 x1 dx 1 x 1 1 + exp - x 1 x1 σ 1 -1 σ 1 (x 1 , x2 ) dx 1 x 1 Π1 + x 2 x2 dx 2 x 2 1 + exp - x 2 x2 σ 2 -1 σ 2 (x 1 ,x 2 ) dx 2 x 2 Π 2 (x 1 , x2 )       (12)
with:

Π 2 (x 1 , x2 ) = 1            exp    - x 1 x1 σ 12 (x 1 , x2 )-1 1+ exp - x 1 x1 σ 1 -1 σ 1 (x 1 , x2 ) dx 1 x 1 Π1 dx 1 x 1    ᾱ2 -1            (13)
It also holds that, for all (x 1 ,

x 2 ) ∈ * + 2 : f (x 1 , x 2 ) = Ā. exp       x 1 x1 dx 1 x 1 1 + exp - x 1 x1 σ 1 -1 σ 1 (x 1 ,x 2 ) dx 1 x 1 Π 1 ( x1 ,x 2 ) + x 2 x2 dx 2 x 2 1 + exp - x 2 x2 σ 2 -1 σ 2 ( x1 ,x 2 ) dx 2 x 2 Π2       (14)
with:

Π 1 ( x1 , x 2 ) = 1            exp    - x 2 x2 σ 12 ( x1 ,x 2 )-1 1+ exp - x 2 x2 σ 2 -1 σ 2 ( x1 ,x 2 ) dx 2 x 2 Π2 dx 2 x 2    ᾱ1 -1            (15)
Proof. We get equations ( 12)-( 13) directly from equations ( 8) and [START_REF] Griliches | Capital-Skill Complementarity[END_REF]. Equations ( 14)-( 15) are obtained by symmetry in the arguments x 1 and x 2 -equivalently, equations ( 14)-( 15) are obtained by applying the same method to f (•, x 2 ) and then to f ( x1 , •).

So all two-input AFs admit a representation of the form of equations ( 12)-( 13) or, equivalently, of the form of equations ( 14)-( 15) which, despite appearances, are equivalent.

However, the reciprocal is wrong: not all functions of the form of ( 12)-( 13) or ( 14)-( 15) are AFs.

For instance, differentiating equation ( 12) with respect to x 2 we get:

f 2 (x 1 , x 2 ) = f (x 1 ,x 2 ) x 2   1+ exp - x 2 x2 σ 2 -1 σ 2 (x 1 ,x 2 ) dx 2 x 2 Π 2 (x 1 , x2 )    f 22 (x 1 , x 2 ) = -1 σ 2 (x 1 ,x 2 ) f 2 (x 1 ,x 2 ) exp - x 2 x2 σ 2 -1 σ 2 (x 1 ,x 2 ) dx 2 x 2 Π 2 (x 1 , x2 ) x 2   1+ exp - x 2 x2 σ 2 -1 σ 2 (x 1 ,x 2 ) dx 2 x 2 Π 2 (x 1 , x2 )   
with Π 2 (x 1 , x2 ) like defined in equation ( 13). While f (x 1 , x 2 ) > 0 by construction, it appears that

f 2 (x 1 ,
x 2 ) might be negative since nothing ensures that 1 + exp -

x 2 x2 σ 2 -1 σ 2 (x 1 , x 2 ) dx 2 x 2 Π 2 (x 1 , x2 ) is strictly positive, since Π 2 (x 1 , x2
) in equation ( 13) might be negative, or even undefined. The sign of f 22 is also undefined in this expression.

All in all, this means that equations ( 12)-( 13) or ( 14)-( 15) do a poor job in providing a 'general recipe' for all two-input AFs.

For instance, for constant σ 1 = 1, σ 2 = 1, σ 12 ∈ , our method yields:

       f (x 1 , x 2 ) = Ā. 1 -ᾱ1 + ᾱ1 x 1 x1 σ 1 -1 σ 1 σ 1 σ 1 -1 . 1 -α 2 (x 1 , x2 ) + α 2 (x 1 , x2 ) x 2 x2 σ 2 -1 σ 2 σ 2 σ 2 -1 with: α 2 (x 1 , x2 ) = ᾱ2 1 -ᾱ1 + ᾱ1 x 1 x1 σ 1 -1 σ 1 (σ 12 -1) σ 1 σ 1 -1 (16)
or, equivalently:

       f (x 1 , x 2 ) = Ā. 1 -α 1 ( x1 , x 2 ) + α 1 ( x1 , x 2 ) x 1 x1 σ 1 -1 σ 1 σ 1 σ 1 -1 . 1 -ᾱ2 + ᾱ2 x 2 x2 σ 2 -1 σ 2 σ 2 σ 2 -1 with: α 1 ( x1 , x 2 ) = ᾱ1 1 -ᾱ2 + ᾱ2 x 2 x2 σ 2 -1 σ 2 (σ 12 -1) σ 2 σ 2 -1 (17)
It is apparent from ( 16) or ( 17) that α 1 (x 1 , x 2 ) and α 2 (x 1 , x 2 ) might not be confined to the (0, 1) interval, implying that functions f defined through [START_REF] Revankar | A Class of Variable Elasticity of Substitution Production Functions[END_REF] or (17) might not be an AF over * + 2 .

However, as we will see in section 4, it is possible to use formulas [START_REF] Revankar | A Class of Variable Elasticity of Substitution Production Functions[END_REF] or (17) together with some criterion over (σ 1 , σ 2 , σ 12 , ᾱ1 , and ᾱ2 ) to characterize functions that are 'local AFs', i.e. functions which satisfy the definition of AFs in the neighborhood of ( x1 , x2 ).

But first, we replicate in the next section the result of theorem 2 for n -input AFs.

n -input aggregation functions

Let n ≥ 2. We call n the set of two-input AFs:

n = f ∈ 2 : * + n → * + ∀i ∈ {1, . . . , n} , f i > 0, f strictly concave
We denote by n the set of functions of class 2 that are strictly increasing and strictly concave w.r.t. each variable:

n = f ∈ 2 : * + n → * + ∀i ∈ {1, . . . , n} , f i > 0, f i i < 0 Again, n ⊂ n and n = n .
For any function f ∈ n , we define the elasticities of substitution functions:

∀i ∈ {1, . . . , n}, ∀(x 1 , . . . , x n ), σ i (x 1 , . . . , x n ) = f i (f -x i f i ) -x i f f i i (x 1 ,...,x n ) ∈ * + (18) ∀i , j ∈ {1, . . . , n }, i = j , ∀(x 1 , . . . , x n ), σ i j (x 1 , . . . , x n ) = f f i j f i f j (x 1 ,...,x n ) ∈ (19)
as well as each factor's competitive share function α i and the corresponding ratios:

∀i ∈ {1, . . . , n},

α i = x i f i f ∈ (0, 1
)
Π i = α i 1-α i ∈ (20)
By definition, ESFs σ i and σ i j verify:

∀i ∈ {1, . . . , n }, ∀(x 1 , . . . ,

x n ) ∈ * + n , ∂ Π i ∂ x i (x 1 , . . . , x n ) = σ i -1 σ i (x 1 , . . . , x n ) Π i (x 1 , . . . , x n ) x i (21) ∀i , j ∈ {1, . . . , n }, i = j , ∀(x 1 , . . . , x n ) ∈ * + n , ∂ α i ∂ x j (x 1 , . . . , x n ) = σ i j (x 1 , . . . , x n ) -1 α j (x 1 , . . . , x n ) x j (22)
Finally, let ( x1 , . . . , xn ) ∈ * + n . We first present a simple lemma that shall be useful in this section.

Lemma 2. Let p ∈ * and let h be a function of * + p into * + of class 1 . For any i ∈ 1, . . . , p , define function m i by m i = h i /h , where h i is the first derivative of h w.r.t. its i th variable. Then:

∀(x 1 , . . . , x p ) ∈ * + p , h (x 1 , . . . , x p ) = h x1 , . . . , xp . exp p i =1 x i xi m i x 1 , . . . , x i -1 , x i , xi+1 , . . . , xp Proof. See appendix A.2.
First, let's apply lemma 2 to f . Let (x 1 , . . . , x n ) ∈ * + n . By the definition of α i , for all i ∈ {1, . . . , n },

it holds that (f i / f )(x 1 , . . . , x n ) = α i (x 1 , . . . , x n )/x i .
Then, for all :

f (x 1 , . . . , x n ) = f ( x1 , . . . , xn ) . exp n i =1 x i xi α i x 1 , . . . , x i -1 , x i , xi+1 , . . . , xn dx i x i (23)
The next step is then to find a formula for α i (x 1 , . . . , x i -1 , x i , xi+1 , . . . , xn ). To do so, we use equations ( 21)-( 22). Lemma 3. For all i ∈ {2, . . . , n }:

α i (x 1 , x i -1 , xi , . . . , xn ) = α i ( x1 , . . . , xn ) . exp i -1 j =1 x j
x j σ i j -1 α j x 1 , . . . , x j -1 , x j , x j +1 , . . . , xn dx j x j

Proof. Let i ∈ {2, . . . , n }. Then, for all j ∈ {1, . . . , i -1}, equation ( 22) together with lemma 2 leads to the stated formula.

Lemma 4.

For all i ∈ {2, . . . , n }:

α i (x 1 , . . . , x i -1 , x i , xi+1 , . . . , xn ) = 1 1+     exp -i -1 j =1
x j x j (σ i j -1) α j x 1 ,...,x j -1 ,x j , x j +1 ,..., xn dx j

x j

α i ( x1 ,..., xn ) -1     exp - x i xi σ i -1 σ i ( x 1 ,...,x i -1 ,x i , xi +1 ,..., xn ) dx i x i
Proof. See appendix A.3.

Lemma 4 links α i to the α j 's, with j < i . Again, a reasoning by induction allows us to derive all α i 's.

For the ease of notation, we call ᾱi = α i ( x1 , . . . , xn ).

We also define for all i ∈ {1, . . . , n} the operator Γ i :

Γ i : 1 * + i , * + × 1 * + i , * + × 1 * + i -1 , * + × * + × * + → 1 * + i , *
+ by:

Γ i (ϕ, γ, η, x 0 , δ) :      * + i → * + (x 1 , . . . , x i ) → exp - x i x 0 γ(x 1 ,...,x i -1 ,x i )-1 1+exp - x i x 0 ϕ(x 1 ,...,x i -1 ,x i ) x i dx i . η(x 1 ,...,x i -1 ) δ -1 . dx i x i ( 24
)
with the convention * + 0 = ∅ and that all functions of ∅ into * + are constant equal to 1.

We also define, for i ∈ {1, . . . , n } functions ψ(i ) ∈ 1 * + i -1 , * + by induction:

ψ(1) = 1 ∀i ∈ {2, . . . , n }, ∀(x 1 , . . . , x i -1 ) ∈ * + i -1 , ψ(2)(x 1 , . . . , x i -1 ) = i -1 j =1 Γ j σ j -1 σ j , σ i j , ψ( j ), x j , ᾱj (25)
Lemma 5. For all i ∈ {1, . . . , n}:

α i (x 1 , . . . , x i , xi+1 , . . . , xn ) = 1 1 + exp - x i xi σ i -1 σ i (x 1 , . . . , x i -1 , x i , xi+1 , . . . , xn ) dx i x i ψ(i )(x 1 ,...,x i -1 ) ᾱi - 1
Proof. See appendix A.4.

We can now state the most important result of this section:

Theorem 3. Let f ∈ n . Then: f (x 1 , . . . , x n ) = Ā. exp n i =1 x i xi 1 1+exp - x i xi σ i -1 σ i (x 1 ,...,x i -1 ,x i , xi +1 ,..., xn ) dx i x i ψ(i )(x 1 ,...,x i -1 ) ᾱi -1 dx i x i (26)
where Ā = f ( x1 , . . . , xn ).

Proof. Immediate from equation (23) and lemma 5.

Theorem 3 states that for any AF f , f is representable in terms of its n (n + 1)/2 ESFs σ i and σ i j as well as n + 1 constants, out of which n are linked to the first derivatives of f -i.e. ( ᾱ1 , . . . , ᾱn )and one is linked to the level of f -i.e. Ā = f ( x1 , . . . , xn ). Like in theorem 2, the i 's can be permuted to yield other formula -so there exists n ! formulas equivalent to (26).

However, like for theorem 2, the reciprocal of theorem 3 does not hold. It is not enough to take n (n + 1)/2 functions plus n + 1 constants and apply formula (26) to get a function f ∈ n . The resulting function might not belong to n neither, and formula (26) might even be undefined.

In the next section, we show how formula (26) can be used to design functions that are AFs near ( x1 , . . . , xn ).

A local characterization of n -input aggregation functions

As we already highlighted in the n = 2 case, formula (26) does not yield a 'general recipe' for all AFs.

More precisely: if one takes some ESFs (σ i ) i and (σ i j ) i = j and some constants Ā, ( ᾱi ) i , and consider function f defined in (26); then function f is not necessarily an AF. However, although it is difficult to use equation ( 26) to design functions that are global AFs, we now show that it is straightforward to use equation ( 26) to get functions that are AFs around ( x1 , . . . , xn ).

Theorem 4. Let n ∈ , n ≥ 2. Let x = ( x1 , . . . , xn ) ∈ * + n .
For all i ∈ {1, . . . , n }, let

σ i ∈ 0 * + n → * + . For all i = j ∈ {1, . . . , n }, let σ i j ∈ 0 * + n →
with σ i j = σ j i . Let Ā > 0 and for all i ∈ {1, . . . , n }, let ᾱi ∈ (0, 1),

with n i =1 ᾱi < 1. Define for all i , Πi = ᾱi /(1 -ᾱi ).
We call ( x ) the following symmetric matrix:

( x ) =       -1 σ 1 ( x ) Π1 σ 12 ( x ) • • • σ 1n ( x ) σ 12 ( x ) -1 σ 2 ( x ) Π2 • • • σ 2n ( x ) . . . . . . . . . σ 1n ( x ) σ 2n ( x ) • • • -1 σ n ( x ) Πn      
Let f be the function Proof. See appendix A.5.

The practical use of theorem 4 is straightforward. Imagine a modeler willing to model an AF f ∈ n . Suppose that the modeler is interested in the AF only around some vector of reference x = ( x1 , . . . , xn ) -maybe because theory or real data points to this vector of reference. Assume that the modeler has some a priori assumptions on the ESFs (σ i ) i and (σ i j i = j as well as on the values of the constants Ā and ( ᾱi ) i . Finally, assume that the matrix ( x ) is negative definite for such assumptions on ESFs and constants.

Then, theorem 4 states that the modeler should consider f as being perfectly described by equation ( 26). As long as the modeler makes only a local use of this formula -meaning that the modeler abstains from applying formula (26) to vectors x ∈ * + n that do not lie in the neighborhood of x -function f yielded by formula ( 26) is an AF. Also, by this method, all 'local AFs' can be generated with the proper choice of ESFs and constants.

To illustrate the method we have in mind, we lay down a simple model of production with three inputs: physical capital (X 1 ), human capital (X 2 ), raw labor (L ): Y = F (X 1 , X 2 , L ). We assume that F has constant-returns-to-scale and we normalize by L :

y = Y L = f (x 1 , x 2 ) with x 1 = X 1 L , x 2 = X 2 L and f (•, •) = F (•, • , 1)
The competitive input shares are α 1 for physical capital, α 2 for human capital and 1-α 1 -α 2 for raw labor. For simplicity, let's assume that α 1 does not depend on x 1 and that α 2 does not depend on x 2 , which amounts to assume that σ 1 ≡ 1 and σ 2 ≡ 1. For the crossed ESFs, the traditional assumption in this three-input framework is that physical capital is more complementary with human capital than with raw labor. We take σ 12 > 1, so that when x 1 (resp. x 2 ) increases, α 2 (resp. α 1 ) increases and 1 -α 1 -α 2 increases (idem). If in real-world data y = Ā, x i = xi and α i = ᾱi for i = 1, 2; then from equation ( 16) the production function near ( x1 , x2 ) has to be:

f (x 1 , x 2 ) = Ā x 1 x1 ᾱ1 x 2 x2 ᾱ2 x 1 x1 ᾱ1 (σ 12 -1) (27)
Function f is a local AF around ( x1 , x2 ) if and only if the eigenvalues of the corresponding matrix ( x ) are strictly negative:

( x ) = -1-ᾱ1 ᾱ1 σ 12 σ 12 -1-ᾱ2 ᾱ2
The trace of ( x ) is strictly negative, and so ( x ) is negative definite if and only if:

σ 2 12 < 1 -ᾱ1 ᾱ1 1 -ᾱ2 ᾱ2 (28)
To sum up, for σ 12 , ᾱ1 and ᾱ2 that satisfy constraint (28), function f defined by equation ( 27) is strictly increasing in each argument and strictly concave near ( x1 , x2 ).

Conclusion

This paper has uncovered the link between production and utility functions on the one hand, and the elasticities of substitution functions on the other. We brought new duality results which lead us to a formula encompassing all production and utility functions that are admissible in the literature.

This approach also provides a general recipe for all local production and utility functions. We hope that modeling, especially over quite large numbers of inputs, will benefit from the results presented here. We also keep in mind that other definitions could be taken for the elasticities of substitution functions, which we keep in mind for future work.

A Appendix

A.1 Proof that f ∈ 1 =⇒ ∀x > 0, f (x ) > x f (x ) Let f ∈ 1 and let x > x > 0. Then: f (x ) = f ( x ) + x x f (x ) dx . Since f ∈ 1 ,
f is strictly decreasing, and so:

f (x ) = f ( x ) + x x f x dx > f ( x ) + (x -x ) f (x ) (29)
Since f is strictly increasing and strictly positive, it admits a limit near 0 + . Let's call this limit l 0 ≥ 0. As x tends to 0 + , inequality (29) yields:

f (x ) ≥ l 0 + x f (x ) ≥ x f (x )
The passage to the limit makes the inequality wide rather than strict. We now show that it is, indeed, strict.

Let x > 0 and assume f (x ) = x f (x ). Then, for all x < x , inequality (29) implies: 0 > f ( x ) -x f (x ). But we also

know that f ( x ) ≥ x f ( x ). So f ( x ) < f (x ), which constitutes a contradiction. Consequently, ∀x > 0, f (x ) > x f (x ).

A.2 Proof of lemma 2

We prove lemma 2 by induction on p .

• If p = 1, then h (x ) = m(x )h (x ) is integrable into h (x ) = h ( x ) .
x

x m (x ) dx . Since α i = Π i /(1 + Π i ), the above equation leads to:

α i (x 1 , . . . , x i -1 , x i , xi+1 , . . . , xn ) = 1 1 + exp - x i xi σ i -1 σ i (x1,...,xi-1,x i , xi+1 ,..., xn ) dx i x i Π i (x 1 ,...,x i -1 , xi , xi+1 ,..., xn )
From lemma 3, and since 1/Π i = 1/α i -1, it holds that:

α i (x 1 , . . . , x i -1 , x i , xi+1 , . . . , xn ) = 1 1+ 1 α i (x1,...,xi-1, xi , xi +1 ,..., xn ) -1 exp - x i xi σ i -1 σ i (x1,...,xi-1,x i , xi+1 ,..., xn ) dx i x i = 1 1+     exp -i -1 j =1
x j x j (σ i j -1)α j x 1 ,...,x j -1 ,x j , x j +1 ,..., xn dx j

x j

α i ( x1 ,..., xn ) -1     exp - x i xi σ i -1 σ i (x1,...,xi-1,x i , xi+1 ,..., xn ) dx i x i
which completes the proof.

A.4 Proof of lemma 5

For simplicity, let's call A i (x 1 , . . . , x i ) = α i (x 1 , . . . , x i , xi+1 , . . . , xn ).

We show by induction the following proposition H (i ), for i ∈ {1, . . . , n }:

H (i ) : A i (x 1 , . . . , x i ) = 1 1 + exp - x i xi σ i -1 σ i (x 1 , . . . , x i -1 , x i , xi , . . . , xn ) dx i x i ψ(i )(x 1 ,...,x i -1 ) ᾱi -1 • Equation (21) for i = 1 implies Π 1 (x 1 , x2 , . . . , xn ) = Π 1 ( x1 , x2 , . . . , xn ) . exp x 1 x1 σ 1 -1 σ 1 x 1 , x2 , . . . , xn dx 1 x 1
, which leads to:

A 1 (x 1 ) = α 1 (x 1 , x2 , . . . , xn ) = 1 1 + exp - x 1 x1 σ 1 -1 σ 1 x 1 , x2 , . . . , xn dx 1 x 1 1 ᾱ1 -1 =⇒ H (1) is true.
• Let i ∈ {1, . . . , n -1} such that for all j ≤ i , H ( j ) is true:

∀ j ≤ i , A j (x 1 , . . . , x j ) = 1 1 + exp - x j x j σ j -1 σ j (x 1 , . . . , x j -1 , x j , x j +1 , . . . , xn ) dx j x j ψ( j )(x 1 ,...,x j -1 ) ᾱj -1 (30)
Using lemma 3:

A i +1 (x 1 , . . . , x i +1 ) = 1 1+ 1 ᾱi +1
expi j =1 ((σi j -1)A j ) x 1 ,...,x j -1 ,x j , x j +1 ,..., xn dx j

x j -1 . exp -

x i +1 xi +1 σ i +1 -1 σ i +1 (x1,...,xi ,x i +1 , xi+2 ,..., xn ) dx i +1 x i +1
Using equation (30), the above equation leads to:

A i +1 (x 1 , . . . , x i +1 ) = 1 1+exp x i +1 xi +1 - σ i +1 -1 σ i +1 (x1,...,xi ,x i +1 , xi+2 ,..., xn ) dx i +1 x i +1 .     1 ᾱi +1 i j =1 exp     - x j x j σ i j x 1 ,...,x j -1 ,x j , x j +1 ,..., xn -1 1+exp   x j x j - σ i -1 σ i x 1 ,...,x j -1 ,x j , x j +1 ,..., xn dx j x j   ψ( j )(x 1 ,...,x j -1 ) ᾱ j -1 dx j x j     -1    
Therefore:

A i +1 (x 1 , . . . , x i +1 ) = 1 1 + exp x i +1 xi+1 σ i +1 -1 σ i +1 x 1 , . . . , x i , x i +1 , xi+1 , . . . , xn dx i +1 x i +1 ψ(i +1)(x 1 ,...,x i ) ᾱi - 1
which means that H (n + 1) is true.

A.5 Proof of theorem 4

-By construction, f is well-defined around x , strictly positive and of class 2 around x .

-It is also obvious by the construction of f that:

∀i ∈ {1, . . . , n },

f i ( x ) = f ( x ) xi ᾱi > 0
By continuity of f i we conclude that f i (x ) > 0 for all x in some neighborhood of x .

-The Hessian matrix of f around x is ( x ) = f i j ( x ) i , j ∈{1,. In the spirit of lemma 6, for each i , j ∈ {1, . . . , n } we multiply the i th row of ( x ) by 1/ f i ( x ) and the j th column of ( x ) by 1/ f j ( x ). We also multiply all rows and columns by f ( x ). This proves that: ( x ) is negative definite ⇐⇒ ( x ) = f .f i j f i . f j ( x ) i , j ∈{1,...,n} is negative definite.

The non-diagonal elements of ( x ) are σ i j ( x ) i = j . The diagonal elements of ( x ) are

f f i i f 2 i ( x ) = -1 σ i f -xi f i xi f i ( x ) = -1 σ i Π i ( x ).
If ( x ) is definite negative, then by continuity f is strictly concave over some neighborhood of x .

  [START_REF] Uzawa | Production Functions with Constant Elasticities of Substitution[END_REF] section VI) extended the multiple-input CES formula to allow for different elasticities of substitution among pairs of goods. Zhelobodko et al. (2012) and Parenti et al. (2017) have extended the Dixit-Stiglitz monopolistic competition framework to allow for more flexible

  , define m i = h i /h where h i is the first derivative of h w.r.t. the i th variable. In particular, m p +1 = h p +1 /h . Then, according to the bullet point above, for all (x 1 , . . . , x p +1 ) ∈ * +1 x 1 , . . . , x p , x p +1 dx p +1 Function g : (x 1 , . . . , x p ) → h x 1 , . . . , x p , xp+1 is a function of * + p → * + of class1 and such that ∀i ∈ {1, . . . , p }, g i = m i h . Then, by the assumption of induction, for all (x 1 , . . . , x p ) ∈ * (x 1 , . . . , x p ) = h x 1 , . . . , x p , xp+1 = h x1 , . . . , xp , xp+1 . exp 1 , . . . , x i -1 , x i , xi+1 , . . . , xp+1 dx i Hence, for all (x 1 , . . . , x p +1 ) ∈ * Π i (x 1 , . . . , x i -1 , x i , xi+1 , . . . , xn ) = Π i (x 1 , . . . , x i -1 , xi , xi+1 , . . . , xn ) . exp

	• Assume that lemma 2 is valid for some p ∈ * . Let x1 , . . . , xp+1 ∈ * +	+ p +1 . Let then h be a function of *	p +1
	into * + and forall i +	p +1 :
				x p +1
	h (x 1 , . . . , x p +1 ) = h x 1 , . . . , x p , xp+1 . exp		xp+1	m p +	p ,
	x i xi m i x + g p i =1 p +1 :
	p +1	x i
	h (x 1 , . . . , x p +1 ) = h x1 , . . . , xp , xp+1 . exp			m i x 1 , . . . , x i -1 , x i , xi+1 , . . . , xp+1 dx p +1
	i =1	xi
	A.3 Proof of lemma 4
	By equation (21):
	x i xi	σ i -1 σ i	x 1 , . . . , x i -1 , x i , xi+1 , . . . , xn	dx i x i

  Let S be a symmetric n × n matrix and let λ > 0. Let S be the square matrix obtained by multiplying one column of S by λ, and let S be the square matrix obtained by multiplying one line of S by λ.Then, S is negative definite if and only of S is negative definite, and if and only if S is negative definite.Proof. Let S be a symmetric n ×n matrix and let λ > 0. Without loss of generality, let's assume that S is obtained by multiplying the first column of S by λ, while S is obtained by multiplying the first row of S by λ:According to Sylvester's criterion, S is negative definite if and only if its leading principal minors alternate in signs:S is negative definite ⇐⇒ µ 1 = a 11 < 0, µ 2 =a 11 a 12 a 21 a 22 > 0 , . . . For i ∈ {1, . . . , n } Let's call respectively µ i and µ i the i th leading principal minor of S and S . It is obvious that: ∀i ∈ {1, . . . , n }, µ i = µ i = λµ i µ i is the same sign as µ i as well as µ i . Consequently, S is negative definite if and only if S is negative definite, and if and only if S is negative definite.

			a 11 a 12 • • • a 1n				λa 11 a 12 • • • a 1n				λa 11 λa 12 • • • λa 1n	
	S =	    	a 21 a 22 • • • a 2n . . . . . . . . .	    	, S =	    	λa 21 a 22 • • • a 2n . . . . . . . . .	    	, S =	    	a 21 . . .	a 22 • • • a 2n . . . . . .	    
			a n 1 a n2 • • • a n n				λa n 1 a n 2 • • • a nn				a n 1	a n 2 • • • a nn

..,n } . We now show that is negative definite if and only if ( x ) is negative definite.

Lemma 6.